 
 
 
An Exploratory Study of High School Geometry Teachers? Aplied Content 
Knowledge 
by 
Anna Wan 
 
 
A disertation submited to the Graduate Faculty of 
Auburn University 
in partial fulfilment of the 
requirements for the Degree of 
Doctor of Philosophy 
 
Auburn, Alabama 
December 14, 2013 
 
 
Keywords: mathematics, education, geometry, teacher knowledge, TACK 
 
 
Approved by 
Marilyn E. Strutchens, Chair, Profesor of Mathematics Education 
W. Gary Martin, Profesor of Mathematics Education 
Daniel J. Henry, Asistant Profesor of Educational Foundations, Leadership and 
Technology  
Huajun Huang, Asociate Profesor of Mathematics 
 
 
  
 
 
 
 
ii 
 
 
 
 
 
Abstract 
 
Results from the 2007 TIMS showed the grim reality that the United States did 
not measure up to other industrialized countries in both fourth- and eighth-grade results 
in geometry (Mullis, Martin, & Foy, 2008). Furthermore, Clements (2003) concluded 
from literature on teaching and learning geometry in kindergarten through twelfth grade 
(K-12) that the U.S.?s curriculum and teaching are weak. Since teacher knowledge 
impacts student achievement, it is pertinent to study teacher knowledge as a subset of the 
solution to improving student achievement in geometry.  
Case studies were conducted to se what connections there are between two high 
school geometry teachers? specialized content knowledge, knowledge of content and 
students, and knowledge of content and teaching (Hil, Bal, & Schiling, 2008) shown 
during the planning of lesons and the teachers? actual execution of the lesons. The three 
types of knowledge were collectively caled Teachers? Applied Content Knowledge 
(TACK) for this study.  
The two teachers chosen for the case study were dubbed as exemplary geometry 
teachers by the researcher. Each teacher participated in interviews and observations 
involving two units of her choice. One teacher was observed for 7 lesons, and the other, 
6; al lesons were also of their choosing. Interviews and clasroom observations were 
video taped and audio recorded. Clasroom observations were recorded with a video 
camera and with a voice recorder that the teachers carried with them. Qualitative data 
 
 
 
 
iii 
analysis was done through case study and grounded theory. These exemplary teachers? 
ability to execute what was planned with additional geometry TACK shown during 
observations was based on knowledge acumulated from many sources, but the most 
commonly referenced were profesional development and reflections from previously 
taught lesons.  
  
 
 
 
 
iv 
 
 
 
 
 
Acknowledgements 
 
                
  
 (One bee makes no honey, one grain makes no rice soup.) 
 
 This disertation was the result of hard work and the contributions of many. First 
and foremost, I thank God. To my family and friends, mentors, colleagues, and 
commite members: I am truly blesed to have you in my life. I wish to thank my 
maternal grandmother      for encouraging my curious nature to discover and learn 
about al things unknown. I have the confidence to do whatever I put my mind to from 
the love and support of my amazing parents, Li Yeh Chen and Kang Lai Wan, and my 
beloved sister, Katy Wan.  To Michel Bowen, who has helped me navigate through 
countles academic decisions, I am indebted to you for my academic achievements since 
high school.  
 I had the privilege of working with Dr. Marilyn E. Strutchens and Dr. W. Gary 
Martin, two of the most pasionate and profesional mathematics educators that I have 
ever met. Your sympathetic ears and compasionate hearts during some dificult times in 
the program wil never be forgotten. A special thanks goes to Dr. Strutchens for not only 
being a remarkable advisor, but also an exemplary mentor to me. Thank you to Dr. 
Daniel J. Henry for always putting things in perspective for me, and Dr. Huajun Huang 
for being positive encouragement for al things math. Thank you al for working together 
so wel to make this disertation possible.  
 
 
 
 
v 
 There are also close friends and colleagues that without whose care and support 
throughout this proces, this disertation would not be possible. Lisa Ross, Elaine Prust, 
Denise Peppers, Dr. Laureta Garret, Dr. Luke Smith, and many others made Auburn 
University sem like home away from home. Leah Calote, Jennifer Holt, David Mayorga, 
and Thomas Mathew Watson: thank you al for being so supportive throughout this 
proces. To my colleagues at Columbus State University, Dr. Debbie Gober, Dr. Marlene 
Alen, Dr. Kimberly Shaw, Dr. Timothy Howard, and many more, thank you al for your 
encouragement and support through the final stages of this disertation. This disertation 
was acomplished with the love and support from everyone with whom God has blesed 
to be in my life.  
 
 
          	 
(To enjoy a grander sight, climb to a greater height) 
  
 
 
 
 
vi 
 
 
 
Table of Contents 
 
Abstract ........................................................................................................................... ii 
Acknowledgements ........................................................................................................ iv 
List of Tables ............................................................................................................... xii 
List of Figures ............................................................................................................... xv 
Chapter 1: Introduction ....................................................................................................... 1 
Teacher Knowledge ........................................................................................................ 4 
Conceptual Framework ................................................................................................... 7 
Purpose of Study ............................................................................................................. 9 
Chapter 2: Literature Review ............................................................................................ 11 
Learning ........................................................................................................................ 11 
Knowledge and Understanding. ................................................................................ 11 
Conceptual and Procedural Knowledge. ............................................................... 12 
Relational and Instrumental Understanding. ......................................................... 13 
Summary. .............................................................................................................. 14 
Zone of Proximal Development. ............................................................................... 15 
Teaching ........................................................................................................................ 16 
Knowledge in Teaching. ........................................................................................... 17 
Knowledge in Teaching Mathematics. ..................................................................... 20 
Profesional Standards for Teaching Mathematics. .................................................. 23 
 
 
 
 
vii 
Summary. .................................................................................................................. 25 
Learning and Teaching Geometry ................................................................................ 25 
Learning. ................................................................................................................... 26 
Van Hiele Levels. .................................................................................................. 26 
Geometric Habits of Mind. ................................................................................... 28 
Teaching. ................................................................................................................... 31 
Van Hiele Levels. .................................................................................................. 31 
Principles for Fostering Geometric Habits of Mind. ............................................ 33 
Studies on Teaching and Learning Geometry. .......................................................... 36 
Dynamic Geometry Software. .............................................................................. 37 
Non Dynamic Geometry Software studies related to the van Hiele Levels. ........ 44 
Teacher Knowledge, Clasroom Instruction, and Student Achievement ..................... 50 
Teacher Knowledge and Student Achievement. ....................................................... 50 
Summary. .............................................................................................................. 53 
Clasroom Instruction and Student Achievement. .................................................... 54 
Teacher Knowledge and Clasroom Instruction. ...................................................... 58 
Teacher Knowledge. ................................................................................................. 63 
Asesing Teacher Knowledge ..................................................................................... 70 
Observation Protocols. .............................................................................................. 70 
Concept Maps. .......................................................................................................... 73 
Interviews. ................................................................................................................. 76 
Summary of Literature Review ..................................................................................... 78 
TACK. ....................................................................................................................... 81 
 
 
 
 
viii 
Knowledge of Content and Teaching. .................................................................. 81 
Knowledge of Content and Students. .................................................................... 82 
Specialized Content Knowledge. .......................................................................... 82 
SCK, KCT, and KCS Relationships. .................................................................... 83 
Teaching and Learning Geometry. ........................................................................... 84 
Teacher Knowledge, Clasroom Instruction, and Student Achievement. ................ 85 
Asesing Teacher Knowledge. ................................................................................ 86 
Chapter 3: Theoretical Perspectives and Methodologies .................................................. 87 
Theoretical Perspective ................................................................................................. 88 
Philosophical Asumptions ........................................................................................... 90 
Research Design ............................................................................................................ 91 
Participants. ............................................................................................................... 92 
Sources of Data. ........................................................................................................ 94 
Quality of Research ....................................................................................................... 96 
Researcher Bias. ........................................................................................................ 98 
Procedures for Collection of Data. .......................................................................... 100 
Initial Meting. .................................................................................................... 100 
Interviews for Leson Plans. ............................................................................... 101 
Observations of Lesons. .................................................................................... 101 
Interview for Reflections of Lesons. ................................................................. 101 
Teacher Self-Reflections of Lesons. ................................................................. 102 
Interview for Reflections of Units. ..................................................................... 102 
Analysis of TACK ...................................................................................................... 103 
 
 
 
 
ix 
Analysis through Geometry Filter .............................................................................. 108 
Analysis During Data Collection. ........................................................................... 112 
Analysis After Data Collection for Each Observation/Unit. .................................. 114 
Level 1. ............................................................................................................... 115 
Level 2. ............................................................................................................... 117 
Level 3. ............................................................................................................... 120 
Summary of Order of Analysis. .......................................................................... 121 
Summary of Theoretical Perspectives and Methodologies ......................................... 121 
Chapter 4: Research Findings ......................................................................................... 124 
Mrs. Orchid ................................................................................................................. 126 
Unit 1. ..................................................................................................................... 127 
Concept Maps. .................................................................................................... 128 
Unit 1 Planned. .................................................................................................... 131 
Observed Lesons: Planned and Executed. ......................................................... 134 
Unit 1 TACK....................................................................................................... 139 
Unit 1 Geometry Filter. ....................................................................................... 142 
Van Hiele Levels. ............................................................................................ 143 
Phases of Learning Based on the van Hiele Model. ....................................... 145 
Findings from Mrs. Orchid Unit 1. ..................................................................... 150 
Unit 2. ..................................................................................................................... 150 
Concept Map. ...................................................................................................... 151 
Unit 2 Planned. .................................................................................................... 152 
Observed Lesons: Planned and Executed. ......................................................... 152 
 
 
 
 
x 
Unit 2 TACK....................................................................................................... 158 
Unit 2 Geometry Filter. ....................................................................................... 160 
Findings from Mrs. Orchid Unit 2. ..................................................................... 162 
Summary of Unit 1 and Unit 2. ............................................................................... 162 
Themes from Unit 1 and Unit 2. ............................................................................. 163 
Mrs. Lotus ................................................................................................................... 164 
Unit 1. ..................................................................................................................... 165 
Concept Map. ...................................................................................................... 166 
Unit 1 Planned. .................................................................................................... 166 
Observed Lesons: Planned and Executed. ......................................................... 167 
Unit 1 TACK....................................................................................................... 169 
Unit 1 Geometry Filter. ....................................................................................... 169 
Findings from Mrs. Lotus Unit 1. ....................................................................... 171 
Unit 2 Concept Map and Overview. ....................................................................... 172 
Observed lesons: Planned and executed. ............................................................... 172 
Unit 2 TACK....................................................................................................... 175 
Unit 2 Geometry Filter. ....................................................................................... 175 
Findings from Mrs. Lotus Unit 2. ....................................................................... 177 
Summary of Unit 1 and Unit 2. ............................................................................... 177 
Themes from Unit 1 and Unit 2. ............................................................................. 178 
Connections betwen Planned and Executed TACK .................................................. 179 
Knowledge of Content and Teaching. .................................................................... 179 
Specialized Content Knowledge. ............................................................................ 180 
 
 
 
 
xi 
Geometry Filter for Mrs. Orchid and Mrs. Lotus ....................................................... 181 
Connections of Geometry Filter to TACK .................................................................. 182 
Comparison of the Cases: Mrs. Lotus and Mrs. Orchid ............................................. 185 
Similarities. ............................................................................................................. 185 
Diferences. ............................................................................................................. 186 
Addresing the Research Question from Analysis ...................................................... 187 
Axial Coding ............................................................................................................... 189 
Specialized Content Knowledge. ............................................................................ 191 
Knowledge of Content and Teaching. .................................................................... 193 
Knowledge of Content and Students. ...................................................................... 194 
Summary of Themes from Mrs. Orchid and Mrs. Lotus ............................................ 195 
Summary of Research Findings .................................................................................. 198 
Chapter 5: Conclusions, Discussions, and Suggestions for Future Research ................. 199 
Limitations .................................................................................................................. 200 
Conclusions ................................................................................................................. 201 
How did exemplary teachers explain concepts to students? ................................... 202 
How did exemplary teachers modify lesons for students? .................................... 203 
How did exemplary teachers addres student dificulties? ..................................... 204 
How did exemplary teachers conduct clasroom procedures? ............................... 205 
How did exemplary teachers utilize technology? ................................................... 206 
How did exemplary teachers discuss planned and executed lesons? .................... 206 
Summary. ................................................................................................................ 207 
Findings compared to literature on teaching and learning geometry .......................... 207 
 
 
 
 
xii 
Implications for Teaching and Learning ..................................................................... 211 
Teaching Geometry. ................................................................................................ 211 
Preparing Preservice Teachers. ............................................................................... 212 
Providing Profesional Development. .................................................................... 213 
Summary. ................................................................................................................ 213 
Implications for Research ........................................................................................... 214 
Further Areas of Study ................................................................................................ 214 
References ....................................................................................................................... 216 
Appendix ......................................................................................................................... 235 
Appendix A ................................................................................................................. 235 
Appendix B ................................................................................................................. 236 
Appendix C ................................................................................................................. 237 
Appendix E ................................................................................................................. 238 
Appendix G ................................................................................................................. 240 
Appendix H ................................................................................................................. 242 
Appendix I .................................................................................................................. 250 
Appendix J .................................................................................................................. 252 
Appendix K ................................................................................................................. 279 
 
 
 
 
  
 
 
 
 
xii 
 
 
 
List of Tables 
 
Table 2-1 Elbaz (1983) and Shulman?s (1987) conceptions of teacher knowledge ......... 19!
Table 2-2 Definitions of select types of mathematical knowledge for teaching ............... 22!
Table 2-3 The van Hiele theory. (Breyfogle & Lynch, 2010, p. 234) .............................. 27!
Table 2-4 Geometric Habits of Mind and Their Indicators (Driscoll, 2007, pp.12-15) ... 30!
Table 2-5 Phases of Learning from the van Hiele Model (Mistreta, 2000, p. 367) ......... 31!
Table 2-6 The van Hiele model of geometric understanding. .......................................... 32!
Table 2-7 Driscoll?s (2007) framework for questioning (p. 102) ..................................... 34!
Table 2-8 Articles involving geometry ............................................................................. 48!
Table 2-9 Teacher Knowledge and Student Achievement ............................................... 54!
Table 2-10 Clasroom Instruction and Student Achievement .......................................... 57!
Table 2-11 Teacher Knowledge and Clasroom Instruction ............................................. 63!
Table 2-12 Teacher in the Teacher Knowledge Literature ............................................... 69!
Table 2-13 Summary of Literature Asesing Teacher Knowledge ................................. 78!
Table 2-14 Mathematical Tasks of Teaching .................................................................... 83!
Table 3-1 Teacher information ......................................................................................... 93!
Table 3-2 Sequence of Data Collection .......................................................................... 102!
Table 3-3 Planning and Execution areas of focus of TACK .......................................... 104!
Table 3-4 a priori codes .................................................................................................. 106!
Table 3-5 Abridged version of Breyfogle and Lynch?s (2000) van Hiele Levels .......... 109!
Table 3-6 Phases of Learning from the van Hiele Model (Mistreta, 2000, p. 367) ....... 110!
 
 
 
 
xiv 
Table 3-7 Geometric Habits of Mind and Their Indicators (Driscoll, 2007, pp.12-15) . 110!
Table 3-8 Geometry Filter .............................................................................................. 111!
Table 3-9 TACK with subdomains and listed codes ...................................................... 116!
Table 4-1 Data Collected ................................................................................................ 124!
Table 4-2 Van Hiele Levels abridged from Breyfogle and Lynch (2010, p. 234) .......... 143!
Table 4-3 Phases of Learning from the van Hiele Model (Mistreta, 2000, p. 367) ....... 145!
Table 4-4 Geometric Habits of Mind and Their Indicators (Driscoll, 2007, pp.12-15) . 147!
Table 4-5 Aspects of SCK and related codes .................................................................. 192!
 
  
 
 
 
 
xv 
 
 
List of Figures 
 
Figure 1-1 Main NAEP Average Scores of Topics in Mathematics by Years Tested 
(National Center for Education Statistics, 2010) ......................................................... 2!
Figure 1-2 Teacher Knowledge, Clasroom Instruction, and Student Achievement .......... 7!
Figure 1-3 Domain map for mathematical knowledge for teaching as it relates to 
Shulman?s (1987) pedagogical content knowledge. .................................................... 8!
Figure 2-1 Domain map for mathematical knowledge for teaching as it relates to 
Shulman?s (1987) pedagogical content knowledge. .................................................. 21!
Figure 2-2 Rectangle Maker Task. .................................................................................... 40!
Figure 2-3 Teacher and student verbal interactions for one of the cases. ......................... 45!
Figure 2-4 Two Sample Questions on the Teacher Questionnaire. .................................. 52!
Figure 2-5 Sample problem from the Mathematics Teaching Questionnaire (An, Klum, & 
Wu, 2004, p. 152). ..................................................................................................... 65!
Figure 2-6 Sample concept map (Novak & Ca?as, 2008, p. 2). ....................................... 74!
Figure 2-7 Domain map for mathematical knowledge for teaching as it relates to 
Shulman?s (1987) pedagogical content knowledge. .................................................. 79!
Figure 2-8 Sample SCK question for Multiplication. ....................................................... 80!
Figure 3-1 Sample node diagram .................................................................................... 120!
Figure 4-1 Mrs. Orchid?s concept map for congruent triangles ...................................... 128!
Figure 4-2 An example of a truss for the roof of a house ............................................... 129!
Figure 4-3 Mrs. Orchid?s concept map for quadrilaterals ............................................... 131!
Figure 4-4 First two triangles of Triangle in a Bag activity ........................................... 135!
Figure 4-5 Triangle and asociated student questions. ................................................... 135!
Figure 4-6 Belringer Observation Day 3 ....................................................................... 138!
 
 
 
 
xvi 
Figure 4-7 Mrs. Orchid Concept Map Unit 2 ................................................................. 151!
Figure 4-8 Orientations of the triangles in the belringer problems ................................ 153!
Figure 4-9 Triangle with angles and sides labeled for reference to the formula for area 155!
Figure 4-10 Measure of angle A is 60?, the supplement of 120?. Height is side a. ........ 155!
Figure 4-11 Figure of Triangle Labeled for students to set up Law of Sines ................. 157!
Figure 4-12 Content source of student dificulties and misconceptions ......................... 197!
 
 
 
 
 
1 
Chapter 1: Introduction 
In 2010, President Obama highlighted the need for education in science, 
technology, engineering, and mathematics (STEM) in school systems in the United States 
in order for future generations to compete in a global marketplace. In the United States 
alone, Lacey and Wright (2009) projected 785,700 new jobs in computer and 
mathematical occupations for 2008 to 2018. As a group, these jobs ?wil grow more than 
twice as fast as the average for occupations in the economy? (p. 85). The notion that math 
is a necesary occupation requirement is not new. Acording to the National Research 
Panel (1989), many jobs require basic knowledge in algebra and geometry. A closer look 
at the 46 careers mentioned on WeUseMath.org (2011) showed 24 careers, which 
specificaly named geometry as one of the types of mathematics required in careers like 
high school math teacher, urban planner, atorney, political scientist, and animator. 
However, results from the 2007 TIMS showed the grim reality that the United 
States did not measure up to other industrialized countries in both fourth- and eighth- 
grade results in geometry (Mullis, Martin, & Foy, 2008). On the national level, Main 
National Asesment of Education Progres (NAEP) results from 1990 to 2009, as 
presented in Figure 1-1, showed that geometry and measurement are stil areas of need. 
 
 
 
 
 
 
 
2 
 
Figure 1-1 Main NAEP Average Scores of Topics in Mathematics by Years Tested 
(National Center for Education Statistics, 2010) 
Michael Shaughnesy, in the October 2011 President?s Mesage in the twice 
monthly newsleter Summing Up of the National Council of Teachers of Mathematics 
(NCTM), highlighted a concern that algebra is geting more atention in standards than 
geometry. ?When states or national organizations develop sample asesment tasks, they 
usualy begin the proces with tasks that involve arithmetic operations or algebraic 
concepts and procedures. Geometry tasks are often lower on their priority list? 
(Shaughnesy, 2011, ? 1). The Common Core State Standards, a set of mathematics and 
English standards adopted by 45 states and four territories in the United States, 
emphasizes topics in mathematics such as operations on numbers, algebra, and functions, 
?putting geometry a tad off to the side? (Shaughnesy, 2011, ? 1). ?Geometry is a crucial 
part of the mathematical education of our students and our citizens? (Shaughnesy, 2011, 
? 6). Shaughnesy?s (2011) concerns for geometry?s place in school mathematics are not 
new. Nearly a decade prior, Glenda Lappan, in her (1999) President's Mesage for the 
National Council of Teachers of Mathematics, cited Trends in International Mathematics 
and Science Study (TIMS) results up to 1995 as wel as National Asesment of 
245!
250!
255!
260!
265!
270!
275!
280!
285!
290!
1990?! 1996! 2000! 2003! 2005! 2007! 2009!
Algebra!
Data!Analysis,!Sta<s<cs,!
and!Probability!
Geometry!
Measure!
Number!
 
 
 
 
3 
Education Progres (NAEP) results up to 1996 as reasons for the need to enrich the 
geometry strand across al grades. Smith et al. (2005) made the observation that 
?geometry and measurement are the topic areas on which U.S. students have exhibited 
their worst performance on national and international asesments during and since the 
1990s? (p. xi). 
Clements (2003) concluded from literature on teaching and learning geometry in 
kindergarten through twelfth grade (K-12) that ?U.S. curriculum and teaching in the 
domain of geometry is generaly weak, leading to unaceptably low levels of 
achievement? (p. 152).  Thus, one way of ensuring students get the education they need 
to ?compete in an age where knowledge is capital, and the marketplace is global? 
(President Obama in the document from Ofice of Science and Technology Policy, 2010, 
p. vii) is to make sure that teachers have the content knowledge necesary.  
Student achievement is linked to teacher knowledge (Hil, Rowan, & Bal, 2005; 
Monk, 1994). However, diferent aspects of teacher knowledge have been used to ases 
relationships betwen teacher knowledge and student achievement. For example, Hil, 
Rowan, and Bal, (2005) asesed teachers? knowledge through a multiple choice test 
based on common and specialized content knowledge and compared their results with 
student achievement; whereas Monk (1994) used the number of mathematics courses a 
high school teacher took as the level of subject mater knowledge and compared that to 
student achievement (further details of those studies wil be expounded upon in chapter 
two). 
 
 
 
 
4 
Teacher Knowledge 
Teacher knowledge has been a subject of concern in the past twenty years (Hil, 
Schiling, & Bal, 2004). In order to study teachers? knowledge, teachers? knowledge has 
been asesed; but why and how teachers are asesed have varied among the diferent 
asesments (Hil, Slep, Lewis, & Bal, 2007). The political environment, that need to 
show what makes teaching a profesion, and improving teacher input and output were 
some reasons for the necesity of building asesments of teacher knowledge (Hil, Slep, 
Lewis, & Bal, 2007). Teachers? knowledge of teaching a topic should not be identical to 
an average educated person?s knowledge of that topic since the teaching of a topic 
requires a deeper understanding of the topic (Bal, 2003; Hil, Slep, Lewis, & Bal, 
2007; Shulman, 1987). Improving teacher input and output refers to establishing 
?evidence on the efects of teacher education on teachers? capacity, and of teachers? 
knowledge and skil on their students? learning? (Hil, Slep, Lewis, & Bal, 2007, pp. 
111-112). It is now possible and necesary to develop a coherent approach to asesing 
teachers? knowledge, specificaly, knowledge of teaching mathematics (Hil, Slep, 
Lewis, & Bal, 2007). 
?The past several years have sen ambitious policy making in the area of teacher 
quality and qualifications? (Hil, Slep, Lewis, & Bal, 2007). For example, the No Child 
Left Behind (NCLB) Act of 2001 
require[d] states to set standards for designating al public school teachers as 
highly qualified and require[d] districts to notify parents of students in Title I 
programs if their child?s teacher d[id] not met these standards. The requirements 
appl[ied] to al teachers of core academic subjects?English, reading or language 
 
 
 
 
5 
arts, mathematics, science, foreign languages, civics and government, economics, 
arts, history, and geography?and to teachers who provide[d] instruction in these 
subjects to students with limited English proficiency (LEP) and students with 
disabilities (Birman, Boyle, Le Floch, et al. 2009, p.xix) 
Teachers? knowledge is asesed to determine whether or not teachers are ?highly 
qualified? (Hil, Slep, Lewis, & Bal, 2007). In September 2010, the Ofice of Science 
and Technology, from the President?s Council of Advisors on Science and Technology 
made public a report caled ?Prepare and Inspire: K-12 Education in Science, 
Technology, Engineering, and Math (STEM) for America?s Future?. One of the 
recommendations in this report was to ?Recruit and train 100,000 great STEM teachers 
over the next decade who are able to prepare and inspire students? (Ofice of Science and 
Technology, 2010, p. vii). Acording to the Ofice of Science and Technology (2010), 
?The most important factor in ensuring excelence is great STEM teachers, with both 
deep content knowledge in STEM subjects and mastery of the pedagogical skils required 
to teach these subjects wel? (p. vii). 
Knowledge of the skils and concepts of the mathematics being taught, although 
important, is not enough to teach it efectively (Bal, 2003; Suzuka, et al., 2009). Bal 
(2003) gave an example of teaching 0.3!?!0.7. 
Knowing how to multiply
 
0.3!?!0.7, and being able to produce eficiently the 
answer of 0.21, is not sufficient to explain and justify the algorithm to students. In 
teaching fifth graders, a student wil likely ask why, in multiplication, you count 
the number of decimal places in the numbers you are multiplying and ?count 
over? the same number of places in the product to place the decimal point 
 
 
 
 
6 
correctly. The student may point out that when you add two decimals, you simply 
line the numbers up: 
 
    0.3  +0.3 
? 0.7   +0.7 
 0.21    1.0
  (p. 2)
 
 
Being able to do the calculations oneself is insufficient for being able to respond wel. 
Even understanding the procedure in the formal terms that one might learn in a 
mathematics course may not equip one to explain it in ways that are both mathematicaly 
valid and acesible to fifth graders. The capacity to do this is a form of mathematical 
work that has been overlooked in the current discussions of improving teaching quality 
(Bal, 2003, p. 2). 
From this example, the teacher needed to: 1. ?Interpret and make mathematical 
and pedagogical judgments about students? questions, solutions, problems, and insights 
(both predictable and unusual)? (Bal, 2003, p. 6), and 2. ?Be able to respond 
productively to students? mathematical questions and curiosities? (Bal, 2003, p. 6). Thus, 
mathematical knowledge for teaching not only needs to be built upon a firm foundation 
of understanding of mathematics content, but also incorporates instructional aspects 
specific to the teaching of mathematics (Bal, 2003). 
Bal?s (2003) example of teaching  ilustrates how teacher knowledge can 
influence clasroom instruction. Studies like Swaford, Jones, and Thornton (1997) and 
Hil et al. (2008) looked at teacher knowledge at the same time as observing clasroom 
instruction, and they found evidence that teacher knowledge influenced clasroom 
0.30.7?
 
 
 
 
7 
instruction. With regards to clasroom instruction and student learning, acording to 
Hiebert and Grouws (2007), one can logicaly claim that student learning is linked to 
clasroom instruction. Jacobson and Lehrer (2000) and Pesek and Kirshner (2000) are 
just two examples of studies that explored clasroom instruction and its impact on student 
achievement. Data from these studies substantiated Heibert and Grouws (2007) logical 
asertion that student learning is linked to clasroom instruction. 
Conceptual Framework 
?A conceptual framework is an argument that the concepts chosen for 
investigation, and any anticipated relationships among them, wil be appropriate and 
useful given the research problem under investigation? (Lester, 2010, p. 73). Summing up 
what has been discussed so far: student achievement in the area of geometry is an area of 
concern (Lappan, 1999; Mullis, Martin, & Foy, 2008); teacher knowledge influences 
student achievement (Hil, Rowan, & Bal, 2005; Monk, 1994); teacher knowledge 
impacts clasroom instruction (Swaford, Jones, & Thornton, 1997; Hil et al., 2008); and 
clasroom instruction impacts student achievement (Jacobson & Lehrer, 2000; Pesek & 
Kirshner, 2000). Figure 1-2 shows the summary of implications that have been discussed 
that are betwen teacher knowledge, clasroom instruction, and student achievement. 
 
 
Figure 1-2 Teacher Knowledge, Clasroom Instruction, and Student Achievement 
Literature on student learning of geometry show that improving student 
achievement in geometry is necesary. This disertation addreses this necesity by 
adding to existing literature on teacher knowledge; specificaly by examining teachers? 
Teacher!Knowledge! Classroom!Instruc<on! Student!Achievement!
 
 
 
 
8 
mathematical knowledge for teaching geometry in planning for and executing clasroom 
instruction. Learning Mathematics for Teaching (LMT) Project defined common content 
knowledge, knowledge at the mathematical horizon, specialized content knowledge, 
knowledge of content and students, knowledge of content and teaching, and knowledge 
of curriculum as categories of mathematical knowledge for teaching. Shulman (1987) 
proposed many categories for types of knowledge needed to teach, and of those, subject 
mater knowledge and pedagogical content knowledge were two categories. Hil, Bal, 
and Schiling (2008) of Learning Mathematics for Teaching (LMT) Project published a 
domain map showing mathematical knowledge for teaching as it relates to Shulman?s 
(1987) subject mater knowledge and pedagogical content knowledge.  
 
Figure 1-3 Domain map for mathematical knowledge for teaching as it relates to 
Shulman?s (1987) pedagogical content knowledge.  From ?Unpacking pedagogical 
content knowledge: Conceptualizing and measuring teachers? topic-specific knowledge 
of students.? by H.C. Hil, D.L. Bal, and S.G. Schiling, 2008, Journal for Research in 
Mathematics Education, 39(4), p. 377. Copyright 2008 by Journal for Research in 
Mathematics Education. Reprinted with permision. 
 
In this disertation, I examined the middle slice of the domain map, that is ? 
specialized content knowledge, knowledge of content and students, and knowledge of 
content and teaching. Specialized content knowledge is teachers' knowledge of 
 
 
 
 
9 
mathematics used in teaching (Bal, Hil, & Bas, 2005) ?but not directly taught to 
students? (Hil, Slep, Lewis, & Bal, 2007). Knowledge of content and teaching is the 
knowledge of teaching the content with emphasis on teaching; e.g. how to correct student 
mistakes so that they wil understand and how to build on what the students already know 
(Hil, Bal, & Schiling, 2008). Separate from both knowledge of content and teaching 
and specialized content knowledge; knowledge of content and students is the subset of 
pedagogical content knowledge that enables teachers to identify common student errors, 
interpret students' understanding of content, identify students' developmental sequences, 
and identify common student computational strategies (Hil, Bal, & Schiling, 2008). 
Specialized content knowledge, knowledge of content and students, and 
knowledge of content and teaching are sen by the author of this disertation as 
applications of content knowledge. Teachers are applying their content knowledge for 
explanations in specialized content knowledge. In knowledge of content and students, 
teachers are applying their knowledge of content in relation to how students learn; and 
similarly in knowledge of content and teaching, teachers are applying their knowledge of 
content in relation to how to teach it. Therefore, for the purposes of this disertation; 
specialized content knowledge, knowledge of content and students, and knowledge of 
content and teaching, collectively wil be named Teachers? Applied Content Knowledge 
(TACK). 
Purpose of Study 
Students need to learn mathematics. Teachers? Applied Content Knowledge 
(TACK) is an important component of mathematical knowledge for teaching, and 
mathematical knowledge for teaching is a factor of students? opportunities to learn 
 
 
 
 
10 
mathematics. For a teacher to know the content does not imply that the teacher can teach 
the content; in fact, in Borko et al. (1992) and Thompson and Thompson?s (1994) studies, 
the teachers had sufficient knowledge of the content they were teaching, but could not 
explain it in a way for students to understand. However, from what a teacher knows about 
teaching, how does the knowledge of how to teach connect with what is actualy taught? 
An exhaustive search using databases such as Academic Search Premier, 
Education Research Complete, ERIC, Profesional Development Collection, 
PsycARTICLES, and PsycINFO; profesional orgainzations? websites for journal articles 
such as nctm.org and amte.net; and general search engines such as Google Scholar 
yielded many articles on mathematics teacher knowledge. However, in looking at studies 
on teacher knowledge and student achievement, studies with meaningful descriptions of 
what the teachers did in the clasroom to foster student achievement were few ? and the 
majority of those studies focused on elementary teachers, with some in middle school, 
and very few at the high school level. Existing research of high school teachers? 
knowledge for teaching mathematics nearly echoed Michael Shaughnesy?s (2011) 
concerns that algebra is geting more atention than geometry, as there are few studies 
that shed light on high school teachers? mathematical knowledge for teaching. 
Based on my interest in the area of high school geometry teachers? knowledge, 
the research question for this disertation was: What connections are there betwen the 
high school geometry Teachers? Applied Content Knowledge shown during the planning 
of the leson and the teachers? actual executions of the leson? 
 
 
 
 
 
11 
Chapter 2: Literature Review 
 An element of quality mathematics education research is ?building on the work of 
others? (Simon, 2004, p. 161). This chapter has three main parts: defining what it means 
to learn and teach with understanding; studies relating teacher knowledge, clasroom 
instruction, and student achievement; and asesing teacher knowledge. The first part of 
this chapter lays the foundation of ?learning and teaching? and ?learning and teaching 
geometry? with studies of student learning and suggestions for teaching. The second part 
focuses on studies that look at the relationships betwen teacher knowledge, clasroom 
instruction, and student achievement. Since teacher knowledge is the focus of this 
disertation, the last section focuses on how other studies asesed teacher knowledge. 
Learning 
The ?Learning Principle? in NCTM?s (2000) Principles and Standards for School 
Mathematics stated that ?students must learn mathematics with understanding, actively 
building new knowledge from experience and prior knowledge? (p. 20). NCTM (2000) 
suggested that learning with understanding ?makes subsequent learning easier? (p. 20). 
However, what does learning with understanding mean? What does it mean to 
understand? The following section discusses what it means to learn with understanding 
and students? zone of proximal development. 
Knowledge and Understanding. ?For much of this century, most psychologists 
acepted the traditional thesis that a newborn?s mind is a blank slate (tabula rasa) on 
 
 
 
 
12 
which the record of experience is gradualy impresed? (Bransford, Brown, & Cocking, 
2000, p. 79). As time progresed and more studies were done about learning, ?chalenges 
to this view arose? (Bransford, Brown, & Cocking, 2000, p. 79). Studies showed that 
?very young children are competent, active agents of their own conceptual development? 
(Bransford, Brown, & Cocking, 2000, p. 79). The move away from tabula rasa 
perspective of the infant mind was atributed to Jean Piaget (Bransford, Brown, & 
Cocking, 2000, p. 79), whose theory of development proposed that cognitive 
development happens in invariant stages (Cobb, 2007). The knowledge atained at the 
various cognitive stages can be categorized as diferent types of knowledge; conceptual 
and procedural (Hiebert & Lefevre, 1986) and relational and instrumental understanding 
(Skemp, 1967). 
Conceptual and Procedural Knowledge. Hiebert and Lefevre (1986) proposed 
two types of knowledge, conceptual and procedural. Procedural knowledge has two parts, 
formal language of mathematics and rules of sequences of actions to complete a 
mathematical task (Hiebert & Lefevre, 1986). Hiebert and Lefevre (1986) gave an 
example of procedural knowledge of multiplying 3.82 by .43, where ?one usualy applies 
three subprocedures: one to write the problem in appropriate vertical form, a second to 
calculate the numerical part of the answer, and a third to place the decimal point in the 
answer? (p. 7). This example highlighted the two parts of procedural knowledge; 
familiarity with the multiplication symbol and knowing the steps to solve the problem. 
 Conceptual knowledge difers from procedural knowledge in that it is a 
?connected web of knowledge (Hiebert & Lefevre, 1986, p. 5) where ?the new material 
becomes part of an existing network? (Hiebert & Lefevre, 1986, p. 6). Conceptual 
 
 
 
 
13 
knowledge is defined as the procedures to solve the problem as wel as the understanding 
of what the procedures mean and the principles behind them (Hiebert & Lefevre, 1986; 
Ritle-Johnson, Siegler, & Alibali, 2001). ?Relationships betwen items of knowledge 
cannot be constructed if the knowledge does not exist? (Hiebert & Lefevre, 1986, p. 17). 
Hiebert?s and Lefevre?s (1986) example of incorrectly adding 
!
!
+
!
!
=!
!
 showed lack of 
conceptual knowledge of fractions. Proceduraly in adding whole numbers, 1+1=2, and 
2+3=5, but lacking the conceptual knowledge of fractions caused the incorrect answer of 
!
!
 (Hiebert & Lefevre, 1986, p. 17). Conceptual knowledge of fraction equivalencies 
would lead one to se that 
!
!
 is les than 
!
!
+
!
!
!since 
!
!
 is already les than!
!
 . ?Deficiencies 
in concepts or procedures, although sometimes hidden, can be a source of weak or 
mising connections? (Hiebert & Lefevre, 1986, p. 17). 
Relational and Instrumental Understanding. ?Understanding? and ?meaningful 
learning? have been terms used to describe the phenomenon of new information properly 
being atached to existing knowledge (Hiebert & Lefevre, 1986). Skemp (1967) described 
two types of understanding?relational and instrumental. Students that learn mathematics 
relationaly understand the concepts and the connections that exist betwen them; while 
students that learn instrumentaly learn skils and ideas about mathematics, but may not 
grasp the connections that exist betwen concepts (Skemp, 1967). Learning mathematics 
relationaly means that students learn it with understanding, and the knowledge has 
permanence in their minds (Skemp, 1967). However, whether students understand 
relationaly or instrumentaly is connected to how the teacher teaches to support a 
 
 
 
 
14 
particular type of understanding (Pesek & Kirshner, 2000). Pesek and Kirshner (2000) 
caled this relational and instrumental instruction. 
Pesek and Kirshner (2000) studied the impact of instrumental and relational 
teaching on 12 fifth grade students in mathematics. The length of this study was ?three 1-
hour lesons over a 3-day period? (Pesek & Kirshner, 2000, p. 529). Relational 
instruction for area and perimeter of squares, rectangles, triangles, and paralelograms 
was done in such a way that: 
Area and perimeter for each shape were presented together to help students 
contrast and compare these constructs. The shapes were discussed in the 
following order: squares, rectangles, paralelograms, and triangles. Connections 
were developed through concrete materials, questioning, student communication, 
and problem solving (Pesek & Kirshner, 2000, p. 529). 
Instrumental instruction focused on ?memorization and routine application of the 
formulas? (Pesek & Kirsher, 2000, p. 529). Pesek and Kirshner (2000) created writen 
pre- and post-tests and results of those showed that students that learned relationaly 
scored beter than students that learned instrumentaly. Interviews of the students showed 
that those who learned concepts relationaly beter applied the mathematics concepts they 
learned. In contrast, some students who learned instrumentaly atempted to apply the 
rules that they learned but misused the appropriate one for the task (Pesek & Kirshner, 
2000).  
 Summary. Conceptual knowledge includes some procedural knowledge, but not 
al elements of procedural knowledge are in conceptual knowledge (Hiebert & Lefevre, 
1986); thus relational understanding is having both conceptual and procedural knowledge 
 
 
 
 
15 
and knowing the connections betwen the two. Instrumental understanding is just the rote 
memorization of procedures (Pesek & Kirshner, 2000). In Pesek and Kirshner?s (2000) 
study, the term for how students learned was how the teachers taught; students of 
teachers that taught relationaly learned relationaly. Teaching relationaly yielded 
students? relational understanding, and teaching instrumentaly yielded students? 
instrumental understanding (Pesek & Kirshner, 2000). Thus in order for students to have 
relational understanding of a topic, teachers need to teach that topic relationaly. What 
and how teachers teach impact what and how students learn (Darling-Hamond, 2006). 
Zone of Proximal Development. Students posses ?a range of prior knowledge, 
skils, beliefs and concepts? that impact learning in school (Bransford, Brown, & 
Cocking, 2000, p. 10), and new knowledge is created from interaction with prior 
knowledge. However, there can be a limit to what students can do, what Vygotsky (1978) 
cals the ?Zone of Proximal Development?. The zone of proximal development is the area 
bounded by the student?s ability to solve problems independently and the students? ability 
to solve problems with the help of someone more capable (Vygotsky, 1978). The upper 
boundary changes with the student?s competence (Vygotsky, 1978). ?In the context of 
mathematics task involving new content for a learner, the zone can represent the level of 
succes a student can achieve on that task with and without adult guidance or in 
collaboration with more capable peers? (Bay-Wiliams & Herrera, 2007, p. 29). Once the 
student can solve a particular problem independently, the zone of proximal development 
shifts to a new upper boundary (Vygotsky, 1978). 
As stated earlier, ?No one questions the idea that what a teacher knows is one of 
the most important influences on what is done in clasrooms and ultimately on what 
 
 
 
 
16 
students learn? (Fennema & Franke, 1992, p. 147). Acording to NCTM (2000) students 
are capable of learning mathematics with understanding, so the responsibility rests on the 
teacher to teach in a way that provides opportunities for students to learn with 
understanding. This requires that teachers have a deep understanding of the mathematics 
they teach and how to teach it (NCTM, 2000). 
Teaching 
The ?Teaching Principle? in NCTM?s (2000) Principles and Standards for School 
Mathematics states that ?Efective mathematics teaching requires understanding what 
students know and need to learn and then chalenging and supporting them to learn it 
wel? (p. 16). However, NCTM (2000) went further and highlighted three requirements 
of efective teaching. 
? Efective teaching requires knowing and understanding mathematics, 
students as learners, and pedagogical strategies. 
? Efective teaching requires a chalenging and supportive clasroom 
learning environment. 
? Efective teaching requires continualy seking improvement. (NCTM, 
2000, pp. 17-19). 
NCTM (2000) insisted that there is ?no one ?right way? to teach? (p.18), as there are 
many strategies and methods of teaching students. However, in order to teach efectively, 
knowledge of mathematics, students, and teaching is a must (NCTM, 2000). Also, 
teachers need to be aware of students? zone of proximal development and keep them 
chalenged and set a clasroom environment that supports student growth (NCTM, 2000). 
 
 
 
 
17 
Further, teachers need to be constantly improving to met the needs of their students, 
since al students need to learn, and no two students are exactly the same (NCTM, 2000). 
 Knowledge in Teaching. Silverman and Thompson (2008) stated that ?it is 
axiomatic that teachers? knowledge of mathematics alone is insufficient to support 
[teachers?] atempts to teach for understanding? (p. 499). Bal?s (2003) example of 
teaching  showed that not only do teachers need to know the mathematics of 
what they are teaching, but also abilities specific to the profesion of teaching 
mathematics. Conceptions of teacher knowledge for teaching mathematics (e.g. An, 
Klum, & Wu, 2004; Fennema & Franke, 1992; Hil, Bal, & Schiling, 2008) were built, 
in part, upon general conceptions of teacher knowledge like Shulman (1987) and Elbaz 
(1983). Shulman (1987) discussed categories of teachers? knowledge base and sources of 
teachers? knowledge base, and Elbaz (1983) focused on ?practical knowledge?, 
knowledge specific to teachers. 
 Elbaz (1983) categorized practical knowledge (knowledge specific to teachers) of 
the teacher (Sarah) she studied into five parts; knowledge of self, milieu, subject mater, 
curriculum, and instruction. However, the structure of teacher?s knowledge has three 
dimensions, rules of practice, principles of practice, and images?what directs decision 
making (Elbaz, 1983). When looking at Sarah?s knowledge of herself, three aspects were 
apparent; her knowledge of her skils and abilities, relationships with others (colleagues, 
administration, students?etc.), and personality traits and limitations (Elbaz, 1983). 
Knowledge of milieu comprised of beliefs about the milieu and ?by the way she 
structures her social experience in the school? (Elbaz, 1983, p. 50). Since Sarah was an 
English teacher, evidence of this category ?subject mater knowledge? was specific to 
0.30.7?
 
 
 
 
18 
English; but the categories of subject mater that appeared were ?the conceptions which 
underlie the diferent facets of content, the ways in which content from diferent subject 
mater areas is selected and combined, and how this content changed as Sarah used it in 
teaching? (Elbaz, 1983, p. 55). Knowledge of curriculum involved ?both the development 
proces and the underlying approach to curriculum development? (Elbaz, 1983, p. 69). 
Lastly, knowledge of instruction referred to theory of learning, students as learners, 
organizing instruction, interacting with students, and asesments of student learning 
(Elbaz, 1983). 
 Similar to Elbaz (1983), Shulman (1987) used excerpts of an English teacher 
conducting a clas, but Elbaz?s (1983) book focused on what the English teacher showed 
to support each category of teacher knowledge and how the categories came about; 
whereas Shulman?s (1987) article used excerpts from an English teacher?s class as 
examples of the many categories of teacher knowledge. However, Shulman?s (1987) 
teacher knowledge categories were based on teachers from diferent disciplines; English 
literature, science, mathematics and history. Shulman (1987) expanded what was 
presented earlier in Shulman (1986) by providing more categories, content knowledge; 
general pedagogical knowledge; curriculum knowledge; pedagogical content knowledge; 
knowledge of learners and their characteristics; knowledge of educational contexts; and 
?knowledge of educational ends, purposes, and values, and their philosophical and 
historical grounds? (p. 8). Again, the categories of teacher knowledge stated in Shulman 
(1986) were content knowledge, curriculuar knowledge, and pedagogical content 
knowledge. 
 
 
 
 
19 
Both Shulman (1987) and Elbaz?s (1983) contributions to defining teacher 
knowledge impacted conceptions of teacher knowledge for teaching mathematics (An, 
Klum, & Wu, 2004; Fennema & Franke, 1992; Hil, Bal, & Schiling, 2008). An, Klum, 
and Wu?s (2004) conceptual framework for pedagogical content knowledge cited 
Shulman (1985) and Elbaz?s (1983) notion of the complex and extensive nature of 
teachers? knowledge and its importance to student learning. An, Klum, and Wu (2004) 
later cited Shulman?s (1987) article for his definition of pedagogical content knowledge 
as a foundation for their framework. Related to Shulman?s (1987) article in Harvard 
Education Review, Shulman?s (1986) presidential addres focused on three areas of 
teacher knowledge; subject mater knowledge, pedagogical content knowledge, and 
curriculum knowledge. Fennema and Franke (1992) atributed Shulman?s (1986) 
categories of teacher knowledge as the foundation of Peterson?s (1988) concepts of 
mathematics teacher knowledge framing teacher knowledge, while Elbaz (1983) was 
discussed as an aspect of teacher knowledge, practical and personal. Fennema and Franke 
(1992) stated that although Elbaz?s (1983) conceptions were based on an English teacher 
and the distinction betwen knowledge and beliefs were not made clear, Elbaz?s (1983) 
study would ?provide a fertile ground for investigation? (Fennema & Franke, 1992, p. 
159). Table 2-1 summarizes Elbaz (1983) and Shulman (1987). 
Table 2-1 Elbaz (1983) and Shulman?s (1987) conceptions of teacher knowledge 
 Elbaz (1983) Shulman (1987) 
Teacher 
Knowledge 
Practical knowledge: 
? knowledge of self 
? milieu 
? subject mater knowledge 
? curriculum 
? teaching 
Categories of knowledge base: 
? general pedagogical knowledge 
? curriculum knowledge 
? pedagogical content knowledge 
? knowledge of learners and their 
characteristics 
 
 
 
 
20 
? knowledge of educational 
contexts 
? ?knowledge of educational ends, 
purposes, and values, and their 
philosophical and historical 
grounds? (Shulman, 1987, p. 8) 
 
Knowledge in Teaching Mathematics. In discussing frameworks of teacher 
knowledge, Fennema and Franke (1992) cited Shulman?s (1986) framework as the basis 
of Peterson?s (1988) framework for mathematics teacher knowledge; knowledge of ?how 
students think in specific content areas, how to facilitate growth in students? thinking, and 
self-awarenes of their own cognitive proceses? (Fennema & Franke, 1992, p.157). 
Peterson (1988) proposed that content knowledge is held within each of the three 
categories mentioned previously. Bal?s (2003) sentiment about more coursework in 
mathematics was similar to Peterson?s (1988) stance on the impact of content knowledge 
to teaching mathematics, that is, content knowledge is meaningful when applied to 
decoding how students think in specific content areas, facilitating growth in students? 
thinking, and understanding their own thinking of mathematics. However, teachers have 
content knowledge that may not directly impact teaching the content (Bal, Thames, & 
Phelps, 2008; Fennema & Franke, 1992). 
Since 2000, Deborah Bal and colleagues of the Learning Mathematics for 
Teaching (LMT) Project designed and piloted surveys that measure teachers? knowledge 
for teaching mathematics. Components of mathematical knowledge for teaching were 
based on the fundamental question of ?What mathematical knowledge is needed to help 
students learn mathematics?? (Hil, Bal, & Schiling, 2004, p. 10). The domain map, in 
Figure 2-1 is a model of mathematical knowledge for teaching with relationships to 
 
 
 
 
21 
pedagogical content knowledge and subject mater knowledge from Shulam?s conception 
(Hil, Bal, Schiling, 2008). There are many conceptions of what knowledge teachers 
need to teach mathematics (e.g. Fennema & Franke, 1992; Peterson, 1988); however, 
what constitutes as mathematical knowledge for teaching ?remains underconceptualized 
and understudied? (Hil, Bal, & Schiling, 2008, p. 395). There are many studies that 
focus on teacher pedagogical content knowledge, but ?[m]ost definitons are perfunctory 
and often broadly conceived? (Bal, Thames, & Phelps, 2008, p. 394). 
 
Figure 2-1 Domain map for mathematical knowledge for teaching as it relates to 
Shulman?s (1987) pedagogical content knowledge.  From ?Unpacking pedagogical 
content knowledge: Conceptualizing and measuring teachers? topic-specific knowledge 
of students.? by H.C. Hil, D.L. Bal, and S.G. Schiling, 2008, Journal for Research in 
Mathematics Education, 39(4), p. 377. Copyright 2008 by Journal for Research in 
Mathematics Education. Reprinted with permision. 
 
The Learning Mathematics for Teaching (LMT) Project aimed to be more specific about 
aspects of pedagogical content knowledge in defining mathematical knowledge for 
teaching (Bal, Thames, Phelps, 2008). Common content knowledge is the subject 
knowledge in teaching mathematics that is the same as mathematics in other jobs or 
disciplines in which mathematics is also used. Specialized content knowledge is teachers' 
 
 
 
 
22 
knowledge of mathematics used in teaching (Bal, Hil, & Bas, 2005) ?but not directly 
taught to students? (Hil, Slep, Lewis, & Bal, 2007). Knowledge of content and 
teaching is the knowledge of teaching the content with emphasis on teaching; e.g. how to 
correct student mistakes so that they wil understand and how to build on what the 
students already know (Hil, Bal, & Schiling, 2008). Separate from both knowledge of 
content and teaching and specialized content knowledge; knowledge of content and 
students is the subset of pedagogical content knowledge that enables teachers to identify 
common student errors, interpret students' understanding of content, identify students' 
developmental sequences, and identify common student computational strategies (Hil, 
Bal, & Schiling, 2008). Knowledge of content and students is also separate from 
knowledge of curriculum and knowledge of content and teaching because elements of 
knowledge of content and students appear in the clasroom independent from teaching 
methods and curriculum choice (Hil, Bal, & Schiling, 2008).  
Table 2-2 lists the definitions of the types of mathematical knowledge that Hil, Bal, and 
Schiling (2008) defined. 
Table 2-2 Definitions of select types of mathematical knowledge for teaching 
Type of Knowledge Definition 
Common Content Knowledge subject knowledge in teaching mathematics (Hil, 
Bal, & Schiling, 2008) 
 
Specialized Content Knowledge ?mathematical knowledge that is used in teaching, 
but not directly taught to students? (Hil, Slep, 
Lewis, & Bal, 2007, p. 132) 
 
Knowledge of Content and Students knowledge that enables teachers to identify common 
student errors, interpret students' understanding of 
content, identify students' developmental sequences, 
and identify common student computational 
strategies (Hil, Bal, & Schiling, 2008) 
 
 
 
 
 
23 
Knowledge of Content and Teaching knowledge of teaching the content with emphasis on 
teaching (Hil, Bal, & Schiling, 2008) 
 
Hil, Bal, and Schiling (2008) described their eforts to conceptualize and 
develop teachers? knowledge of content and teachers? knowledge of content and students. 
They wrote, piloted, and analyzed results from multiple-choice items. Since, Hil, Bal, 
and Schiling (2008) based their multiple choice items on empirical data from literature, 
and the first step was to test the multiple-choice items validity as a measurement of 
knowledge of students and teaching; they did not measure how knowledge of content and 
students related to improving student learning in mathematics. They confirmed that 
?teachers have skils, insights, and wisdom beyond that of other mathematicaly wel-
educated adults? (Hil, Bal, & Schiling, 2008, p. 395). Although most scholars, teachers, 
and teacher educators would agree that teachers? knowledge of students? thinking in 
particular domains is likely to mater, what constitutes such ?knowledge? has yet to be 
understood? (Hil, Bal, & Schiling, 2008, p. 395). 
Profesional Standards for Teaching Mathematics. ?Understanding learning 
as a proces of individual and social construction provides teachers with a conceptual 
framework with which to understand the learning of their students? (Simon, 1993, p. 7). 
Based on how students learn and what type of knowledge they need to learn, the National 
Council of Teachers of Mathematics (NCTM) (1991) produced Profesional Standards 
for Teaching Mathematics. NCTM (1991) listed five major shifts that need to occur in 
practice at the time to improve mathematics teaching: 
? toward clasrooms as mathematical communities?away from clasrooms 
as simply a collection of individuals; 
 
 
 
 
24 
? toward logic and mathematical evidence as verification?away from the 
teacher as the sole authority for right answers; 
? toward mathematical reasoning?away from merely memorizing 
procedures; 
? toward conjecturing, inventing, and problem solving?away from an 
emphasis on mechanistic answer-finding; 
? toward connecting mathematics, its ideas, and its applications?away from 
teaching mathematics as a body of isolated concepts and procedures (p. 3). 
From these suggested shifts, four esentials of teaching were emphasized: 
? Seting goals and selecting or creating mathematical tasks to help students 
achieve these goals; 
? Stimulating and managing clasroom discourse so that both the students 
and the teacher are clearer about what is being learned; 
? Creating a clasroom environment to support teaching and learning 
mathematics; 
? Analyzing student learning, the mathematical tasks, and the environment 
in order to make ongoing instructional decisions (NCTM, 1991, p. 5). 
 From what is suggested of teachers to do and based on what is known about learning, the 
teacher neds to act as the facilitator of the exchange of knowledge in the clasroom and 
not as the sole dispenser of knowledge (NCTM, 1991). Teachers create the environment 
in which students learn. An element of creating a productive learning environment is to 
provide students with the opportunity to learn multiple strategies, so that they may have 
an arsenal with which to approach a problem (Bransford, Brown, & Cocking, 2000). 
Once students have enough knowledge of multiple strategies, students can depend on 
themselves to solve problems or enlist the help of peers (Bransford, Brown, & Cocking, 
2000). This must be acomplished with minimal help from the teacher so that they may 
 
 
 
 
25 
then take charge of their own learning (Bransford, Brown, & Cocking, 2000); self-
reliance. 
 Summary. The section ?Knowledge and Teaching? compared two widely cited 
early conceptions of teacher knowledge, Elbaz (1983) and Shulman (1987). ?Profesional 
Standards for Teaching Mathematics? funneled the diferent types of teacher knowledge 
into shifts teachers needed to make to improve mathematics teaching. Although not 
explicitly stated, in order to comply with the five suggested shifts and four esentials of 
teaching, teachers need to have a firm knowledge base; including the categories 
mentioned by Elbaz (1983), Shulman (1986), Shulman (1987), Peterson (1988), and 
mathematical knowledge for teaching by the Deborah Bal and colleagues in the Learning 
Mathematics for Teaching (LMT) Project. The next section wil delve deeper into what it 
means to learn and teach geometry. 
Learning and Teaching Geometry 
 ?Years ago an informal definition [of geometry] might have been that it was a 
branch of mathematics devoted to the study of shapes and space. Now, however, a more 
apt definition might be ?the branch of mathematics that studies visual phenomena?? 
(Malkevitch, 2009, p. 14). Al four of NCTM?s (2000) standards for geometry, from 
kindergarten to 12
th
 grade, show atention to studying visual phenomena. They are: 
? Analyze characteristics and properties of two- and three- dimensional 
geometric shapes and develop mathematical arguments about geometric 
relationships 
? Specify locations and describe spatial relationships using coordinate geometry 
and other representational systems 
? Apply transformations and use symmetry to analyze mathematical situations 
 
 
 
 
26 
? Use visualization, spatial reasoning, and geometric modeling to solve 
problems (p. 308). 
Apparent from the standards, ?Geometry is more than definitions; it is about describing 
relationships and reasoning? (NCTM, 2000, p.41). This section looks at theories, 
standards, proposed practices, and studies on teaching and learning geometry. 
 Learning. This section explores theories on the learning of geometry, expounding 
on the van Hiele levels and Driscoll?s (2007) geometric habits of mind. The van Hiele 
level theory was developed by wife and husband, Dina van Hiele-Geldof and Pierre 
Marie van Hiele, in separate doctoral disertations in 1957 (Usiskin, 1982). Soon after the 
completion of their disertations, Dina pased away and further work with the van Hiele 
levels was done by Pierre Marie van Hiele (Usiskin, 1982). 
Van Hiele Levels. Level 0 is the recognition or visualization level, where students 
learn the names of figures (van Hiele, 1986). Level 1 is where students identify properties 
of figures, caled the analysis level (van Hiele, 1986). In the informal deduction level, or 
Level 2, students order figures and relationships; this is not where students formaly 
prove yet, but begin to use simple deduction (van Hiele, 1986). In Level 3, the formal 
deduction level, students formaly use deduction as a means of understanding the 
connections betwen postulates and theorems and using them to prove other theorems 
and postulates (van Hiele, 1986). Finaly, Level 4 is rigor, where students understand ?the 
necesity for rigor and are able to make abstract deductions (van Hiele, 1986). Students 
cannot skip levels, as acording to the theory, they atain levels sequentialy (Batista, 
2007a; van Hiele, 1986; Usiskin, 1982). Table 2-3 shows the van Hiele levels by name 
and description. 
 
 
 
 
27 
Table 2-3 The van Hiele theory. (Breyfogle & Lynch, 2010, p. 234) 
Level Name Description 
0 Visualization Se geometric shapes as a whole; do not focus on 
their particular atributes. 
 
1 Analysis Recognize that each shape has diferent properties; 
identify the shape by that property. 
 
2 Informal Deduction Se the interrelationships betwen figures. 
3 Formal Deduction Construct proofs rather than just memorize them; 
se the possibility of developing a proof in more 
than one way. 
 
4 Rigor Learn that geometry needs to be understood in the 
abstract; se the ?construction? of geometric 
systems. 
 
The van Hiele level theory ?has three aspects; the existence of levels, properties 
of the levels, and movement from one level to the next? (Usiskin, 1982, p. 4). Burger and 
Shaughnesy?s (1986) study contributed to confirming the existence of levels by adding 
examples of student thought at the various van Hiele levels. Burger and Shaughnesy 
(1986) cited Fuys et al. (1985) and Mayberry?s (1983) results which supported the 
hierarchal nature of the van Hiele Levels as the foundation of the ability to asign levels. 
Burger and Shaughnesy?s (1986) interview study involved 45 students from grades 1 to 
12 and university mathematics majors. ?The tasks included drawing shapes, identifying 
and defining shapes, sorting shapes, determining a mystery shape, establishing properties 
of paralelograms, and comparing components of a mathematical system? (Burger & 
Shaughnesy, 1986, p. 31). To show how Burger and Shaughnesy?s (1986) data 
supported the van Hiele levels, a summary of data supporting Level 1 reasoning is shown 
below. 
 
 
 
 
28 
 Level 1 
1. Comparing shapes explicitly by means of properties of their components. 
2. Prohibiting clas inclusions among general types of shapes, such as 
quadrilaterals. 
3. Sorting by single atributes, such as properties of sides, while neglecting 
angles, symmetry, and so forth. 
4. Application of a litany of necesary properties instead of determining 
sufficient properties when identifying shapes, explaining identifications, and 
deciding on a mystery shape. 
5. Descriptions of types of shapes by explicit use of their properties, rather than 
by type names, even if known. For example, instead of rectangle, the shape 
would be referred to as a four-sided figure with al right angles. 
6. Explicit rejection of textbook definitions of shapes in favor of personal 
characterizations. 
7. Treating geometry as physics when testing the validity of a proposition; for 
example, relying on a variety of drawings and making observations about 
them, 
8. Explicit lack of understanding of mathematical proof (p. 44). 
The list above was the level indicator for van Hiele Level 1. Burger and Shaughnesy 
(1986) created level indicators for levels 0-3. A complete listing of levels 0-3 is in Burger 
and Shaughnesy (1986) pages 43-45. ?No atempt was made to investigate van Hiele 
Level 4 with these subjects, a level that requires the ability to compare diferent 
geometries? (Burger & Shaughnesy, 1986, p. 34). Students? answers to the clinical 
interview tasks in the study reinforced the discretenes of the van Hiele Levels 0-3 
(Burger & Shaughnesy, 1986). 
Geometric Habits of Mind. Driscoll (2007) suggested that teachers of students in 
grades 5-10 foster geometric habits of mind in preparation for students to take high 
 
 
 
 
29 
school geometry because what students should be learning in high school geometry is 
built upon geometric ideas learned in the middle school. The four geometric habits of 
mind are: reasoning with relationships, generalizing geometric ideas, investigating 
invariants, and balancing exploration and reflection (Driscoll, 2007). 
Reasoning with relationships is ?actively looking for relationships (e.g., 
congruence, similarity, and paralelism) within and betwen geometric objects in one, 
two, and three dimensions? (Driscoll, 2007, p. 12). K?chemann and Hoyles? (2006) 3-
year longitudinal study highlighted the need for students to be actively looking for 
relationships, emphasis on the word ?actively?. Results from K?chemann and Hoyles? 
(2006) study showed that students that did not make progres were also students that did 
not actively explore other connections or relationships. Lawson and Chinnappan?s (2000) 
qualitative study of 36 male students taking geometry showed that 10
th
 grade male 
students were more succesful in problem solving when they had more content 
connections than other 10
th
 grade male students that did not. Without the connectednes 
of content, studies showed that it was dificult for students to use something that is not 
there (Herbst, 2006; Lawson & Chinnappan, 2000). Generalizing geometric ideas is 
?wanting to understand and describe the ?always? and the ?every? related to geometric 
phenomena? (Driscoll, 2007, p. 12). Investigating invariants is students paying atention 
to what is not changing, or staying the same (Driscoll, 2007). Balancing exploration and 
reflection ?is trying various ways to approach a problem and regularly stepping back to 
take stock? (Driscoll, 2007, p. 14). The four habits of mind are not separate, as ?a 
problem solver is likely to draw on several conceptual tools while approaching a 
 
 
 
 
30 
problem? (Driscoll, 2007, p. 15). Table 2-4 below gives indicators of students? geometric 
habits of mind. 
Table 2-4 Geometric Habits of Mind and Their Indicators (Driscoll, 2007, pp.12-15) 
Geometric Habit of Mind Indicators 
Reasoning with 
Relationships 
Basic: Identification of figures presented in a problem and 
correct enumeration of their properties 
Advanced: Relating multiple figures in a problem through 
proportional reasoning and reasoning through symmetry 
 
Generalizing Geometric 
Ideas 
Basic: Uses one problem situation to generate another, or 
when the solver intuits that he or she hasn?t found al the 
solutions 
Advanced: Generate al solutions and make a convincing 
argument as to why there are no more; or wondering what 
happens if a problem?s context is changed 
 
Investigating Invariants Basic: Decides to try a transformation of figures in a 
problem without being prompted, and considers what has 
changed and what has not changed 
Advanced: Consider extreme cases for what is being asked 
by a problem 
Balancing Exploration and 
Reflection 
Basic: Drawing, playing, and/or exploring with occasional 
(though maybe not be consistent) stock-taking 
Advanced: approaching a problem by imagining what a 
final solution would look like, then reasoning backward; or 
making what Herbst (2006) cals ?reasoned conjectures? 
about solutions with strategies for testing the conjectures 
 
 
 
 
 
31 
 Teaching. This section nearly mirrors the previous section in that this section 
focuses on teaching to move students through the van Hiele levels and principles for 
fostering geometric thinking. Teaching requires more than just the knowledge of the 
subject (Bal, 2003). An, Klum, and Wu (2004) highlighted a Chinese saying, if the 
teacher is to give students a cup of water, the teacher must have a pail of water of their 
own. 
Van Hiele Levels. The phases of learning from the van Hiele model as described 
in van Hiele (1984) synthesized by Mistreta (2000) is in Table 2-5 below. 
 
Table 2-5 Phases of Learning from the van Hiele Model (Mistreta, 2000, p. 367) 
Phase Description 
Information Discussions are held where the teacher learns of 
the students' prior knowledge and experience with 
the subject mater at hand. 
 
Directed Orientation The teacher provides activities that alow students 
to become more acquainted with the material being 
taught. 
 
Explication A transition betwen reliance on the teacher and 
students' self-reliance is made. 
 
Free Orientation The teacher is atentive to the inventive ability of 
the students. Tasks that can be approached in 
numerous ways are presented to the students. 
 
Integration The students summarize what was learned during 
the leson. 
 
Breyfogle and Lynch (2010) suggested that since ?movement through [the van Hiele] 
model depends on experiences of the learner[,] [w]el-devised tasks, [such as 
asesments], help move students through the levels? (p. 238). The asesments that 
 
 
 
 
32 
teachers should use must ?alow students to demonstrate their acquired knowledge in 
meaningful way[s] and helps them continue to learn as they go through the proces? 
(Breyfogle & Lynch, 2010, p. 234). Table 2-6 below provides examples of what teachers 
should do for students to move from one level to the next. 
Table 2-6 The van Hiele model of geometric understanding. (Breyfogle & Lynch, 2010, 
p. 234) 
Level Name Description Example Teacher Activity 
 Visualization Se geometric 
shapes as a 
whole; do not 
focus on their 
particular 
atributes. 
A student would 
identify a square but 
would be unable to 
articulate that it has 
four congruent sides 
with right angles. 
Reinforce this level by 
encouraging students 
to group shapes 
acording to their 
similarities. 
1 Analysis Recognize that 
each shape has 
diferent 
properties; 
identify the 
shape by that 
property. 
A student is able to 
identify that a 
paralelogram has 
two pairs of paralel 
sides, and that if a 
quadrilateral has two 
pairs of paralel sides 
it is identified as a 
paralelogram. 
Play the game ?gues 
my rule,? in which 
shapes that ?fit? the 
rule are 
placed inside the 
circle and those that 
do not are outside the 
circle (se 
Russel and 
Economopoulos 
2008). 
2 Informal 
Deduction 
Se the 
interrelationships 
betwen figures. 
Given the definition 
of a rectangle as a 
quadrilateral with 
right angles, a 
student could identify 
a square as a 
rectangle. 
Create hierarchies 
(i.e., organizational 
charts of the 
relationships) or Venn 
diagrams of 
quadrilaterals to show 
how the atributes of 
one shape imply or are 
related to the 
atributes of others.  
3 Formal 
Deduction 
Construct proofs 
rather than just 
memorize them; 
se the 
possibility of 
Given three 
properties about a 
quadrilateral, a 
student could 
logicaly deduce 
Provide situations in 
which students could 
use a variety of 
diferent angles 
depending on what 
 
 
 
 
33 
developing a 
proof in more 
than one way. 
which statement 
implies which about 
the quadrilateral (se 
2.7). 
was given (e.g., 
alternate interior or 
corresponding angles 
being congruent, or 
same-side interior 
angles being 
supplementary). 
4 Rigor Learn that 
geometry needs 
to be understood 
in the abstract; 
se the 
?construction? of 
geometric 
systems. 
Students should 
understand that other 
geometries exist and 
that what is 
important is the 
structure of axioms, 
postulates, and 
theorems. 
Study non-Euclidean 
geometries such as 
Taxi Cab geometry 
(Krause 1987). 
 
The following is an example where students should be able to answer this question using 
formal deduction. (Breyfogle & Lynch, 2010, p. 235) 
Three Properties of a Quadrilateral 
Property D: It has diagonals of equal length. 
Property S: It is a square. 
Property R: It is a rectangle. 
 Which is true? 
a. D implies S, which implies R. 
b. D implies R, which implies S. 
c. S implies R, which implies D. 
d. R implies D, which implies S. 
e. R implies S, which implies D. 
Source: Usiskin (1992) 
Principles for Fostering Geometric Habits of Mind. Driscoll (2007) proposed 
three principles for fostering geometric thinking, the first of which is ?Geometric 
thinking develops with the help of regular problem-solving opportunities? (p. 95). In 
order for students to learn with understanding; new knowledge needs to form connections 
with prior knowledge (Bransford, Brown, & Cocking, 2000), and Driscoll (2007) cited 
Bransford, Brown, and Cocking (2000) as theoretical basis for the first principle. Driscoll 
 
 
 
 
34 
(2007) proposed that from his experienced, ?problem solving provides a rich context for 
fostering understanding, active organization, and connections to prior knowledge? (p. 
96). One of the examples Driscoll (2007) gave was: ?Two vertices of a triangle are 
located at (0, 6) and (0, 12). The triangle has area 12. What are al possible positions for 
the third vertex? And how do you know you have them al?? (p. 98). This example shows 
the connections betwen geometry ideas and the coordinate system from algebra 
(Driscoll, 2007). 
 The second principle is ?Geometry in the middle grades demands special atention 
to teacher-student communication? (Driscoll, p. 95). Two main components of this 
principle are language and geometry and the power of teacher questioning (Driscoll, 
2007). ?Geometric problem solving invites drawn, spoken, and given gestured 
representation of understanding, along with writen verbal and writen symbolic 
representations, and so it invites multimodal mathematical communication? (Driscoll, 
2007, p. 100). Thus, ?teacher questioning can be a powerful tool? (Driscoll, 2007, p. 101) 
for students to learn geometry with understanding. Teachers need to be purposeful in the 
type of questions they are asking (Driscoll, 2007). 
Table 2-7 Driscoll?s (2007) framework for questioning (p. 102) 
Question 
Type 
Purpose Examples 
Orienting To focus students? atention 
on the problem and/or on a 
particular way to approach 
the problem. 
What is the problem asking? 
Would tangrams be useful here? 
Do you think comparing these two sides 
might help? 
Asesing  To gauge students? 
understanding of their 
statements and actions while 
What do you think congruent means in the 
problem statement? 
Why did you fold the paty paper like that? 
 
 
 
 
35 
problem solving. How did you arrive at this answer? 
How did you know this is a rectangle? 
Advancing To help students extend their 
thinking toward a deeper 
understanding of the 
problem 
How could you convince a skeptic that the 
figure you?ve made is a paralelogram? 
What if you didn?t know the measure of this 
angle? 
How would you solve the problem without 
graphing paper? 
What types of triangles wil this work for 
and why? 
 
 The third and final principle is ?Middle-grades geometry is groundwork for high 
school geometry? (p. 95). Since the focus of high school geometry is proving (Batista, 
2009; NCTM, 2000), middle school educators need to place emphasis on convincing 
explanations, ?cognitive demand of tasks, development of a ?geometric eye?, and 
emphasiz[e] metacognitive development? (Driscoll, 2007, p. 109). In collecting student 
work samples for the book, Driscoll (2007) found three main problems: 
? Middle graders appear to have litle experience in putting together 
convincing mathematical explanations?for geometry problems in 
particular. 
? Sometimes, language appears to be a barrier to writing an explanation? 
? In geometry problems, middle graders often use perception as a warrant 
for their claims (e.g., the line ?looks straight,? or the angle is a right angle 
because it ?has an L-shape?)? (p. 110). 
Therefore, to mitigate these problems, middle grades teachers need to: 
provide more opportunities for students to construct convincing 
explanations?[provide opportunities] acesible to students for whom language 
may be a barrier?[and] ask questions that both orient students to places in their 
 
 
 
 
36 
claims where they may be relying too much on perception, and advance their 
thinking to go beyond perception (Driscoll, p. 110). 
Middle grades teachers also need to keep in mind the cognitive demand of tasks given to 
students (Driscoll, 2007). Teachers need to keep in mind that certain questions may lower 
the cognitive demand of the task and hamper the development of geometric habits of 
mind. Students need to develop the ?geometric eye?, that is, the visual thought that ?helps 
in geometric problem solving? (Driscoll, 2007, p. 113). This ability can be developed 
through practice with carefully selected tasks (Driscoll, 2007). 
 Even though the research question of this study focused on high school geometry 
teachers, literature on teaching and learning geometry at the middle school is important as 
wel. From Vygotsky?s Zone of Proximal Development (Bransford, Brown, & Cocking, 
2000; & Vygotsky, 1978) and van Hiele Levels (Usiskin, 1982; & van Hiele, 1986), 
teachers need to met students at students? levels of understanding in order to foster 
learning; and middle school is prior to high school.   
 Studies on Teaching and Learning Geometry. As mentioned in chapter one, a 
search using databases such as Academic Search Premier, Education Research Complete, 
ERIC, Profesional Development Collection, PsycARTICLES, and PsycINFO; 
profesional orgainzations? websites for journal articles such as nctm.org and amte.net; 
and general search engines such as Google Scholar?yielded many articles on 
mathematics teacher knowledge, but few that described what teachers did in the 
clasroom in relation to student achievement. Of those articles that focused on geometry, 
the majority of them were of the elementary grades, some at the middle school level, and 
few at the high school level. Also, the geometry articles could logicaly be organized into 
 
 
 
 
37 
two categories regarding technology; those that focused on the use of technology?
dynamic geometry software, and those that did not. In the next sections, studies related to 
teaching geometry are presented. 
 Dynamic Geometry Software. There are many instances in mathematics 
education literature where students benefited from using dynamic geometry software like 
Geometer?s Sketchpad (Batista, 2007a). Some benefits of dynamic geometry software 
over paper and pencil methods are its eficiency (faster than paper and pencil) (Oner, 
2008) and freedom to experiment (not limited by paper and pencil) (Marrades & 
Guti?rrez, 2000). Some studies such as Ubuz, ?st?n, and Erbas, (2009) also touted the 
wonders of dynamic geometry software as it improved retention of seventh grade 
students. However, they compared the results of a clas that used dynamic geometry 
software to a clas where the teacher taught instrumentaly, thus complicating results for 
the usefulnes of dynamic geometry software with teaching philosophy. Referring back to 
the beginning of this chapter, the use of dynamic geometry software in this case was for 
relational understanding and positive results cannot be fully atributed to dynamic 
geometry software. In contrast, Han (2007) compared two clases as wel, and also one 
with dynamic geometry software and the other without; but both were taught relationaly. 
Han (2007) found that students that had the opportunity to use Geometer?s Sketchpad 
showed higher van Hiele levels for knowledge of quadrilaterals than students that used 
paper and pencil methods 
From clasroom observations, Marrades and Guti?rrez (2000) found dynamic 
geometry software (Cabri version 1.7) to be more useful in teaching students to prove 
than paper and pencil methods. A feature of Cabri was ?dragging?, and this ?lets students 
 
 
 
 
38 
se as many examples as necesary in a few seconds, and provides them with imediate 
fedback that cannot be obtained from paper-pencil teaching? (Marrades & Guti?rrez, 
2000, pp.119-120). The study involved two pairs of fourth grade students over 30 weks 
with two 55 minute clases per wek (Marrades & Guti?rrez, 2000). Data from three 
lesons (beginning, middle, and end) were chosen to iluminate how the students were 
progresing in their reasoning ability (Marrades & Guti?rrez, 2000). 
The main objectives of the teaching unit observed for this study were: 
? To facilitate the teaching of concepts, properties and methods usualy 
found in the school plane geometry curriculum ? 
? To facilitate a beter understanding by students of the need for and 
function of justification in mathematics. 
? To facilitate and induce the progres of students toward types of 
justification closer to formal mathematical proofs ? (Marrades & 
Guti?rrez, 2000, p. 97). 
Activities were generaly structured in three phases; create and/or explore, generate 
and/or verify conjectures, and justify conjectures (some activities did not have this stage). 
In the first stage, sometimes the teacher provided the figure and students had aces to it 
to use it to explore properties instead of creating their own (Marrades & Guti?rrez, 2000). 
Teachers prompted students with questions such as ?why is the construction valid?? and 
?why is the conjecture true?? (Marrades & Guti?rrez, 2000, p. 98). ?As part of the 
didactical contract defined in the clas, pupils knew that requirements like ?justify your 
conjecture? carried the implicit meaning of ?justify why your conjecture or construction is 
true?? (Marrades & Guti?rrez, 2000, p. 98). At the beginning of clas, students were 
 
 
 
 
39 
presented with worksheets by the teacher where they had to ?write their observations, 
comments, conjectures, and justifications? (Marrades & Guti?rrez, 2000, p. 98). 
Answers from the worksheets, history from Cabri software, and semi-structured 
interviews were the basis of conclusions of the study. Both pairs of students improved in 
their justification abilities (Marrades & Guti?rrez, 2000). Marrades and Guti?rrez (2000) 
concluded that students were able to ?progres toward more elaborated types of 
justifications? (p. 120) and showed confidence in their ability of forming deductive 
justifications and formal proofs. Marrades and Guti?rrez (2000) agreed with other studies 
on the benefits of dynamic geometry software (e.g., Batista, 2007a; Han, 2007), and in 
addition, showed improvement in student proving abilities when using Cabri. 
Batista (2007b) documented ?progres and dificulties in constructing meaning 
for a spatial property? (p. 69) of a pair of 5
th
 grade students, Mat and Tom. Mat and 
Tom?s clas had been working in pairs on computers using a feature of Geometer?s 
Sketchpad caled ?Shape Makers? (Batista, 2007b). Shape Makers alows students to 
make certain shapes with diferent sizes and orientations (Batista, 2007b). For example, 
the Paralelogram Maker alows students to manipulate sides and angles of a 
paralelogram to any paralelogram so long as it fits on the screen (Batista, 2007b). One 
of the goals of Shape Makers is to help students move from van Hiele levels 0 and 1 to 2 
(Batista, 2007b). In the task ?What shapes can a rectangle maker make?? (Batista, 
2007b, p. 69) the clas was given six diferent shapes and were asked to decide which 
were possible (Figure 2-2). Mat and Tom correctly predicted which shapes could be 
created (Batista, 2007b). 
 
 
 
 
40 
 
Figure 2-2 Rectangle Maker Task.  From ?Learning with understanding: Principles and 
proceses in the construction of meaning for geometric ideas.? by M. Batista, 2007, The 
Learning of Mathematics, 69th Yearbook of the National Council of Teachers of 
Mathematics, p. 69. Copyright 2007 by The Learning of Mathematics, 69th Yearbook of 
the National Council of Teachers of Mathematics. Reprinted with permision. 
 
?After using the Rectangle Maker to check Shapes 1 and 2, Mat and Tom had some 
initial dificulty manipulating the Rectangle Maker to make Shape 3? (Batista, 2007b, 
p.70) because of what Tom caled a ?slant?. The teacher chalenged Mat and Tom by 
asking them to define what Tom caled a ?slant? (Batista, 2007b). It was evident that 
Tom and Mat tried again and made Shape 3 (Batista, 2007b). Tom and Mat had 
?sufficiently acurate mental models for rectangles and how the Rectangle Maker moves? 
due to their correct predictions and sufficient manipulation to show subsequent shapes 
were not able to be made by the Rectangle Maker (Batista, 2007b). 
 The following is an excerpt betwen Tom, Mat, and the teacher as to why Shape 
4 cannot be made by the Rectangle Maker (Batista, 2007b). 
 
 
 
 
41 
 Tom: I?m positive it can?t do this one. 
Mat: It [the Rectangle Maker] has no slants? It can?t make a slant? The 
Rectangle Maker can?t make something like that slanted here [motioning 
along the bottom then left side of Shape 4]. 
Teacher: You mean the angle there? 
Mat: Yeah, it has an angle. [Shape] 4 is no because the Rectangle Maker can?t 
make a slant. 
Tom: Cause the Rectangle Maker can only make a square and a rectangle and 
that?s not a square or a rectangle? (Batista, 2007b, p. 70). 
Even though Mat?s ?reasoning was stil extremely imprecise? (Batista, 2007b, p.70), 
being able to describe spatial relationships betwen parts of shapes shows that he was 
moving from van Hiele level 1 towards level 2. Batista (2007b) also made the 
connection betwen what Mat was doing to Bransford, Brown, and Cocking, 2000; 
where new knowledge is constructed based on existing knowledge. Mat created new 
concepts based on the common words ?slant? and ?angle? (Batista, 2007b). In contrast, 
?Tom?s last comment indicat[ed] Level 1 reasoning?he was thinking about shapes 
strictly as wholes? (Batista, 2007b, p. 70). Explaining why the Rectangle Maker could 
not make Shape 4 was dificult for Mat and Tom (Batista, 2007b). 
Thus, to facilitate shifting their focus from parts of a shape to the whole shape, the 
teacher asked Mat and Tom ?What is diferent about the angle here [motioning to the 
upper left angle on the Rectangle Maker] and the angle here [motioning to the upper left 
angle on Shape 4]?? (Batista, 2007b, p. 71). This got Mat and Tom to start noticing 
parts more, but not to where they needed to be to understand why the Rectangle Maker 
could not make Shape 4 (Batista, 2007b). So the teacher drew on their screen Shape 3 
and Shape 4 next to each other and asked Mat and Tom what the diferences were 
betwen Shape 3 and Shape 4 (Batista, 2007b). Batista (2007b) saw that the teacher 
 
 
 
 
42 
knew exactly what the diferences were to situate the question in that manner, but Mat 
and Tom stil did not se what the teacher saw (Batista, 2007b). The teacher recognized 
that Mat and Tom needed some more time with Shape Makers and pointed them to 
looking at the measures of the sides and angles that were on the screen of the shapes 3 
and 4; and when they looked at that, the ?aha? moment happened (Batista, 2007b). The 
teacher?s ability to facilitate learning was an important aspect of moving students through 
the van Hiele Levels (Batista, 2007b). In order for learning to occur, acording to 
Bransford, Brown, and Cocking, 2000; Tom and Mat of Batista?s (2007b) study ?had to 
mentaly construct a new way of thinking about shapes? (p. 77). Within the interaction 
betwen the teacher and Mat and Tom, the teacher had to determine the students? 
dificulties and help them through the dificulty within a reasonable amount of time in 
order for them to learn.  
 Han?s (2007) disertation investigated the use of Geometer?s Sketchpad with 97 
8
th
 grade students in Minneapolis, of which 57 students used Geometer?s Sketchpad 
(GSP) and the other 40 by paper and pencil methods. The researcher set up both activities 
where students learned about properties of quadrilaterals by observing characteristics of 
diferent quadrilaterals, and the only diference was that one group had paper and pencil, 
and the other group used GSP (Han, 2007). Both activities required students to ?explore 
geometric concepts with hands-on experiences, to develop conceptual understanding, and 
to stimulate higher level of thinking and mathematical reasoning ability? (Han, 2007, p. 
53). Teachers facilitated learning through questioning to get students to come to their 
own correct conjectures of quadrilateral properties (Han, 2007). Both groups of students 
were taught relationaly, and students that had the opportunity to use Geometer?s 
 
 
 
 
43 
Sketchpad showed higher van Hiele levels for knowledge of quadrilaterals than students 
that used paper and pencil methods (Han, 2007). 
A qualitative segment from a bigger study by Jones (2000) showed that 12 year 
old students in middle school using Cabri I (a dynamic geometry software similar to 
Geometer?s Sketchpad), through an instruction sequence on clasifying quadrilaterals, 
indicated upward movement in van Hiele levels. A sample of initial dialogue betwen the 
teacher and the students is as follows: 
Teacher: What sort of shape is that? 
Karol: It?s a rhombus. 
Teacher: How do you know it?s a rhombus? 
Karol: Our old maths teacher used to cal a rhombus a drunken square, because 
it?s like a square, only sick. 
Teacher: What do you know about a rhombus, from what you have done? 
Heather: It?s got a centre. 
Karol: It?s like a diamond . . .. But it?s not a square. 
Teacher: What can you say about the sides or the angles . . . or the diagonals? 
Karol: Those two angles [indicating the angles at one pair of opposite vertices] 
are the same, and those two are the same . . . [indicating the other pair of 
opposite angles] But they are not al the same [indicating adjacent angles] 
And . . . the sides are al the same length . . . I think. 
Heather: It?s the same distance across each side. 
Teacher: What can you say about how the diagonals cross? 
Karol: A right angle. 
Teacher: How do you know? 
Karol: It looks straight (Jones, 2000, pp. 74-75). 
A sample of the dialogue betwen student and teacher at the end of the leson is as 
follows: 
 
 
 
 
44 
Teacher: Why that arrow? [indicating that a rhombus is a special form of 
paralelogram]. 
Karol: It?s just like the rhombus and the square because . . . because al the sides . 
. . the sides are . . . the opposite sides are of equal length, but there [in the 
rhombus] they [the diagonals] cross at 90 degrees and there [indicating a 
paralelogram that is not a rhombus] they don?t. 
Initialy, students used everyday language for description and relied on perception, but by 
the end of the unit, were able to explain what they were doing mathematicaly and used 
mathematical reasoning to justify claims (Jones, 2000). Jones (2000) stated that this shift 
was ?mediated by the software environment? (p. 80). However, the teacher set up the 
environment with computer software to facilitate learning (Jones, 2000). 
The teacher set up ?carefully designed? (Jones, 2000, p. 81) problem based tasks 
for students to work in pairs or smal groups of 3-6. The teacher questioned students and 
redirected them to pertinent features of the geometry software, encouraged conjecturing, 
and focused students towards forming mathematical explanations (Jones, 2000). 
?Without such factors, the meditational impact of the software could be such that it may 
distract students from the geometry of the problem situation or possibly reduce the 
perceived need for deductive proof? (Jones, 2000, p. 81). 
Using dynamic geometry software in conjunction with teachers teaching for 
understanding has shown to benefit students? van Hiele levels on a topic (Han, 2007), as 
wel as improved abilities to verbalize and reason mathematicaly (Jones, 2000). 
Technology is certainly useful for students to learn geometry; however, not al students 
have aces to technology (Bishop & Forgasz, 2007). 
Non Dynamic Geometry Software studies related to the van Hiele Levels. Senk 
(1989) asesed van Hiele Levels of reasoning and proof writing abilities of 241 high 
 
 
 
 
45 
school students at the beginning and at the end of their geometry course. The asesment 
for proofs had two short answer items and four formal proofs (Senk, 1989). The 
asesment for van Hiele Levels of reasoning was with Usiskin?s (1982) Van Hiele 
Geometry Test. The Van Hiele Geometry Test had 25 multiple-choice questions, with 5 
questions for each level. ?High school students? achievement in writing geometry proofs 
is positively related to van Hiele levels of geometric thought and to achievement on 
standard nonproof geometry content? (Senk, 1989, p. 318). Further, Senk?s (1989) 
findings aligned to the van Hiele theory that students who were not at levels 3 or 4 could 
not produce proofs on the proofs asesment. 
 A closer look at the relationship betwen teaching and learning proofs showed 
that certain teacher moves corresponded to student actions that eventualy resulted in 
students being able to prove on their own (Martin, McCrone, Walace, Bower, & 
Dindyal, 2005). The diagram in Figure 2-3 shows teacher and student verbal interactions 
for one of the cases studied provided by Martin et al. (2005, p. 116). 
 
Figure 2-3 Teacher and student verbal interactions for one of the cases.  From ?The 
interplay of teacher and student actions in the teaching and learning of geometric proof.? 
by T. Martin, S. McCrone, M. Bower, and J. Dindyal, 2005, Educational Studies in 
Mathematics, 60(1), p. 116. Copyright 2005 by Springer. Reprinted with permision. 
 
 
 
 
46 
The ?choices and expectations [of the teacher] often led to discussion of student 
reasoning, proof modeling, and refinement of an argument? (Martin, et al. 2005, p. 106) 
for this particular case. For each of the four cases, an interaction flow chart was made 
(Martin, et al. 2005). From those four cases, Martin et al. (2005) concluded that when the 
teacher posed an open-ended task, it led to ?engaging students in verbal reasoning, this 
provides a landscape for proving? (Martin, et al. 2005). When students were engaged in 
verbal reasoning the teacher ?analyz[ed], coach[ed], and revoic[ed] questions back to 
students, [therefore] the teacher [could] monitor and influence students? developing 
ability to construct chains of reasoning? (Martin, et al. 2005, p.121). 
Herbst (2006) used a geometry problem to se what a teacher did to engage 
students, and how what the teacher did to engage students impacted the activity. The 
question posed to students was ?Say you have a triangle ABC, how can you find a point 
O inside ABC so that the three new triangles AOB, BOC, and AOC have the same area?? 
(Herbst, 2006, p. 322). That geometry problem is an area problem that also caled for 
knowledge of the relationship betwen the medians and area of a triangle (Herbst, 2006). 
Only one teacher was observed to se what the teacher did to engage students and what 
the teacher did to engage students impacted the activity. 
This teacher, Megan, liked teaching geometry, and ?had an extensive background 
in mathematics and education? (Herbst, 2006, p. 321). ?She was a strong presence in the 
room and led a rich and logical development of the subject, constantly seking student 
input in the form of focused questions, oriented toward making connections and 
anticipating problems? (Herbst, 2006, p. 321). Megan engaged students in the activity, 
and students worked on aspects of the activity, but never on making a conjecture about 
 
 
 
 
47 
the relationship betwen medians and area of a triangle (Herbst, 2006). In introducing the 
problem; Megan was about to draw a triangle, but stopped and asked students to draw a 
triangle, and reminded them that they knew how to divide a triangle into two equal areas 
from the previous day; and then asked students to divide a triangle into three equal areas 
(Herbst, 2006). Even though there were slight diferences betwen diferent clases in 
how Megan stated the task for the day, students understood what was expected of them 
and worked on the problem (Herbst, 2006). During the leson, Megan was cognizant of 
the leson?s objectives and moved students towards that goal through questioning 
(Herbst, 2006). Some students answered her initial question with ?in the middle?, and 
Megan questioned students to be more specific with what they meant by ?in the middle? 
(Herbst, 2006). From what Herbst (2006) observed, what the students were wiling to do 
while stil engaged in the task made it very dificult for the teacher to push students 
towards making a conjecture on their own without prompting. 
Swaford, Jones, and Thornton (1997) studied the impact that a four wek 
intervention program had on 4
th
 ? 8
th
 grade teachers? content knowledge and instruction 
of geometry. The focus of the intervention was on content knowledge of geometry and 
student cognition in geometry. Al 49 teachers that participated were given a pretest and 
postest, to determine pre-intervention and post-intervention van Hiele levels. Three tasks 
to determine van Hiele levels at the pretest and postest were hour long structured 
interview, 25 question multiple choice test designed for students adapted from the 
Cognitive Development and Achievement in Secondary School Geometry (CDASG) 
project, and a leson plan task where teachers had 20 minutes to write a leson plan on a 
given topic for their grade level. Al teachers increased van Hiele levels by the postest 
 
 
 
 
48 
from the pretest, and 50% of the teachers increased by two levels. Determined from 
achievement through intervention, four of the least improved and four of the most 
improved teachers, eight total, were chosen for follow-up observations. Observations 
were to confirm the van Hiele level shown in their postest leson plan. There was 
improvement in what the teachers taught and how it was taught. Within the group of eight 
that participated in the follow-up, some key changes in what was taught were: increase 
time in clasroom teaching geometry; increase in quality of geometric tasks students were 
asked to engage in; and a shift from role of teacher is to give answers to teacher as a 
facilitator, ?generating questions that probed students? thinking and engaging them to 
discuss? (Swaford, Jones, & Thornton, 1997, p. 480). ?Change in the way geometry was 
taught was further reflected in the higher expectations teachers had for their students? 
thinking in geometry? (Swaford, Jones, & Thornton, 1997, p. 480). 
Herbst and Kosko (2012) documented eforts in developing an instrument to 
measure mathematical knowledge for teaching high school geometry (MKT-g). MKT-g 
focused on four domains from Bal et al. (2008), common content knowledge (CCK), 
specialized content knowledge (SCK), knowledge of content and teaching (KCT), and 
knowledge of content and students (KCS) (Herbst & Kosko, 2012). Scores from items 
piloted showed no statistical relationship with total years of experience teaching in 
general (Herbst & Kosko, 2012). However, scores from collections of items in domains 
showed statisticaly significant correlations with years of experience teaching high school 
geometry (Herbst & Kosko, 2012). Table 2-8 summarizes articles involving geometry. 
Table 2-8 Articles involving geometry 
Author(s) Grade Teacher Atributes Student Achievement 
Marrades 4
th
 - Activities were generaly structured in students were able to 
 
 
 
 
49 
& 
Guti?rrez 
(2000) 
 
grade three phases; create and/or explore, 
generate and/or verify conjectures, and 
justify conjectures 
- provided the figure and students had 
aces to it to use it to explore properties 
instead of creating their own 
- prompted students with questions such 
as ?why is the construction valid?? and 
?why is the conjecture true?? (Marrades 
& Guti?rrez, 2000, p. 98).  
?progres toward 
more elaborated types 
of justifications? (p. 
120) and showed 
confidence in their 
ability of forming 
deductive 
justifications and 
formal proofs 
 
Martin, 
et al. 
(2005) 
High 
School 
- Posed an open-ended tasks 
- Engaged students in verbal reasoning 
- Analyzed, coached, and re-voiced 
questions back to students 
Developed ability to 
construct chains of 
reasoning 
 
Herbst, 
(2006) 
High 
School 
cognizant of the leson?s objectives and 
moved students towards that goal 
through questioning 
students were wiling 
to reason when 
engaged in the task 
Jones 
(2000) 
12 year 
old 
students 
- Used Cabri I 
- teacher questioned students and 
redirected them to pertinent features of 
the geometry software, encouraged 
conjecturing, and focused students 
towards forming mathematical 
explanations 
upward movement in 
van Hiele levels 
 
Han 
(2007) 
Middle 
School 
- Questioning to get students to come to 
their own correct conjectures of 
quadrilateral properties 
- Both groups of students were taught 
relationally 
 
Students that learned 
using dynamic 
geometry software 
scored higher on 
author created 
asesment 
Herbst & 
Kosko 
(2012) 
In-
service 
teachers 
- Looked at CCK, SCK, KCT, KCS 
- General scores were not statisticaly 
significant to years taught 
- Scores on domains were significant to 
years taught high school geometry 
 
 
 
 
 
 
50 
Teacher Knowledge, Classroom Instruction, and Student Achievement 
To organize existing teacher knowledge literature for the purpose of framing the 
problem for this disertation, studies are clasified into four categories: first, teachers? 
knowledge and its impact on student achievement; second, teachers? knowledge and how 
it relates to instruction; clasroom instruction and how that impacts student achievement; 
and finaly, studies of teacher knowledge without clasroom instruction or student 
achievement. 
Teacher Knowledge and Student Achievement. Many studies linking teacher 
knowledge and student achievement are of elementary grades (e.g. Carpenter, Fennema, 
Peterson, Chi-Pang, & Loef, 1989, first grade teachers teaching problem solving skils in 
addition and subtraction; Hil, Rowan, & Bal, 2005, first and third grade teachers; Saxe, 
Gearhart, & Nasir, 2001, upper elementary teachers teaching fractions). Some studies 
made the connection betwen teacher knowledge to student achievement to show positive 
efects of profesional development (e.g. Jacobson & Lehrer, 2000; Saxe, Gearhart, & 
Nasir, 2001). Others (e.g. Darling Hamond, 2000), like Monk (1994) looked at diferent 
sources of teacher knowledge and/or other possible factors in student achievement to se 
what predicted student achievement beter. However, Monk?s (1994) study wil be 
expounded upon in this disertation over Darling-Hamond?s (2000) study due to its 
relevancy to mathematics. The following studies expounded upon either related to this 
disertation more than others in its group mentioned and/or were major studies in 
mathematics education. 
Data on 483 secondary science teachers and 608 mathematics teachers? subject 
knowledge and knowledge of pedagogy were collected to se the efect on student 
 
 
 
 
51 
achievement in Monk?s (1994) study. The instruments used for collecting data were 
researcher generated surveys and achievement tests (Monk, 1994). As a part of a larger 
study, surveys were filed out by teachers, administrators, parents, and students; while the 
achievement tests were administered to the students (Monk, 1994). Coursework 
completed in the subject area represented the teachers? subject knowledge while courses 
in pedagogy represented teacher knowledge of pedagogy. Monk (1994) sampled 2,829 
tenth grade public school students during the base year from schools randomly selected 
al over the nation over the course of three years; 1987, 1988, and 1989. In regards to 
results from student achievement in comparison to characteristics found on teacher 
surveys, Monk (1994) found that student achievement positively related to the number of 
courses the teacher took in the subject. However, teacher college coursework in pedagogy 
also had positive efects on student achievement, and at times, had a greater impact on 
student achievement than subject mater courses (Monk, 1994). 
Hil, Rowan, and Bal?s (2005) study of 1,190 first graders and 1,773 third 
graders; and 334 first and 365 third grade teachers in 115 elementary schools during the 
2000?2001 through 2003?2004 school years. Hil, Rowan, and Bal?s (2005) aim was to 
se if teachers? mathematical knowledge for teaching could positively predict student 
achievement. Teacher data was gathered from teacher self-report logs, self-report 
questionnaires, and a 25-question multiple choice test (Hil, Rowan, & Bal, 2005). The 
questionnaires gave Hil, Rowan, and Bal (2005) teacher background as wel as teacher 
knowledge for teaching. Their measure for teacher knowledge of teaching represents the 
?knowledge that teachers use in clasrooms, rather than general mathematical 
knowledge? (Hil, Rowan, & Bal, 2005, p. 387). Two examples of items on the multiple 
 
 
 
 
52 
choice test measuring teacher content knowledge of teaching mathematics is shown in 
Figure 2-4 (Hil, Rowan, & Bal, 2005). 
 
Figure 2-4 Two Sample Questions on the Teacher Questionnaire. From ?Efects of 
teachers? mathematical knowledge for teaching on student achievement.? by H.C. Hil, B. 
Rowan, and D.L. Bal, 2005, American Educational Research Journal, 42(2), p. 401-402. 
Copyright 2005 by SAGE Publications. Reprinted with permision. 
 
 
 
 
 
53 
These two items exemplified ?two key elements of content knowledge for teaching 
mathematics: ?common? knowledge of content?. and ?specialized? knowledge used in 
teaching students mathematics? (Hil, Rowan, & Bal, 2005, p. 387). Due to the two key 
elements of the test, items where the correct answer indicated a teaching orientation were 
eliminated.  
Hil, Bal, and Rowan (2005) did not create the asesments for the students like 
they did for the teachers. Instead, 
The measures of student achievement used here were drawn from CTB/ McGraw-
Hil?s Terra Nova Complete Batery (for spring of kindergarten), the Basic 
Batery (in spring of first grade), and the Survey (in third and fourth grades). 
Students were asesed in the fal and spring of each grade by project staf, and 
scores were computed by CTB via item response theory (IRT) scaling procedures. 
(Hil, Rowan, & Bal, 2005, p. 382). 
Hil, Rowan and Bal (2005) found that teachers? mathematical knowledge for teaching 
positively predicted student gains in mathematics achievement for first and third grade 
students. An improvement that Hil, Rowan, and Bal (2005) made on studies involving 
teacher knowledge is that this one involved not just computational fluency, but actual 
knowledge specific to teaching. 
Sumary. Studies in the past quantified teachers? subject area knowledge, like 
Monk (1994), by looking at the coursework of teachers in college, college degree, and to 
some extent, teaching years; and quantifying teacher knowledge through asesments of 
current knowledge has only become mainstream in research starting in the past decade or 
so (Hil, Rowan, Bal, 2005). Coursework, degree(s), and teaching years were found to be 
 
 
 
 
54 
insufficient to determine a teacher?s content knowledge (Hil, Rowan, Bal, 2005). Thus, 
Hil, Rowan, and Bal (2005) used a questionnaire aimed at asesing teachers for their 
knowledge of content and knowledge for teaching students. However, Hil, Rowan, and 
Bal?s (2005) participants took part in a larger study, thus information like courses taken 
and years teaching has been gathered, but was not a part of this particular study using the 
questionnaire for teacher knowledge. With a large sample size and collection of both 
qualitative and quantitative data, both qualitative and quantitative questions can be 
answered. Table 2-9 summarizes the two studies expounded upon in this section linking 
teacher knowledge and student achievement. 
Table 2-9 Teacher Knowledge and Student Achievement 
Author(s) Grade Teacher Knowledge Student 
Achievement 
Monk 
(1994) 
Secondary Content and pedagogy: 
Coursework completed in subject 
area and pedagogy 
Positively related 
to the number of 
subject area 
courses, but 
courses in 
pedagogy had 
similar or greater 
positive efects 
Hill, 
Rowan, & 
Bal (2005) 
1
st
 & 3
rd
 
grade 
Mathematical knowledge for 
teaching: Logs, questionnaires, 
and multiple choice test 
Positively 
predicted student 
achievement 
 
Classroom Instruction and Student Achievement. Hil, Rowan, and Bal (2005) 
atributed origins of research in predicting student achievement from teacher 
characteristics to proces-product literature on teaching; citing Brophy and Good (1986), 
Gage (1978), and Doyle (1977). Brophy?s (1986) review of research ?summarize[d] the 
 
 
 
 
55 
findings of research linking teacher behavior to student achievement? (p. 1069). Brophy 
(1986) addresed both quantity and quality measures of teaching behaviors that positively 
influenced student achievement. Aspects of teaching behaviors that were documented 
quantitatively that positively impacted student achievement were opportunity to learn and 
content covered; role definition, expectations, and time alocation; clasroom 
management and student-engagement time; consistent succes and academic learning 
time; and active teaching (Brophy, 1986). In addition to how much teachers performed 
the above tasks, Brophy (1986) aserted that how wel teachers performed instructional 
tasks were also addresed; the qualitative measures. Those are, giving information, 
questioning the students, and reacting to student responses (Brophy, 1986). One critique 
by Shulman (1986) of proces-product research naming Brophy and Good (1986) and 
Gage (1978) among others was the lack of reference to subject specific knowledge and 
clasroom and student contexts. Shulman (1986) atributed this void to ?the quest for 
general principles of efective teaching? (p. 6). 
Mindful of Shulman?s (1986) criticisms of proces-product research, I chose to 
expound upon studies like Jacobson and Lehrer?s (2000) study as the scope of what the 
teacher does in the clasroom is not limited to general teaching behaviors. Their approach 
to studying teacher knowledge and student achievement was not solely based on teacher 
behaviors, but on specific teachers? mathematical knowledge for teaching geometry 
(Jacobson & Lehrer, 2000). Unlike both Monk (1994) and Hil, Rowan, and Bal (2005) 
which had large numbers of participants; Jacobson and Lehrer (2000) studied only four 
teachers. 
 
 
 
 
56 
This study, on a smaler scale alowed the researchers to examine on a deeper 
level what teachers did in the clasroom to promote discourse for student learning in 
conjunction to student achievement as a result of the discourse. Jacobson and Lehrer?s 
(2000) study involved second grade teachers and their ?knowledge about students? 
thinking? (p. 90). This case study was of four teachers and their clasrooms, two of which 
are in the third year as participants in a research program on teaching and learning of 
geometry, and the other two were not (Jacobson & Lehrer, 2000). Al four teachers had 
gone through Cognitively Guided Instruction (CGI) training, where instructional 
decisions are based on research of student thinking. The two teachers that participated in 
the research program were more knowledgeable in the area of ?students? thinking about 
space and geometry? (Jacobson & Lehrer, 2000, p. 90) than the other two. 
At the end of the 4 wek unit, al students ?comprehended and produced rotations, 
reflections, and compositions of motions,? but students of the two more student-
knowledgeable teachers ?learned more than did their counterparts, and this diference in 
learning was maintained over time? (Jacobson & Lehrer, 2000, p.90). ?Although al 
teachers elicited students? thinking, the two teachers who were more knowledgeable 
about students? thinking about space orchestrated clasroom talk in ways that refined, 
elaborated, and extended students? thinking, albeit in diferent ways? (Jacobson & 
Lehrer, 2000, p.86). One of the two teachers that were in the research program on 
teaching and learning of geometry facilitated discourse ?By posing questions and re-
voicing students? comments that focused, refined, or ?lifted out? important ideas? 
(Jacobson & Lehrer, 2000, p. 86). Having participated in profesional development, the 
teachers that were more knowledgeable about student thinking of geometry tailored their 
 
 
 
 
57 
clasroom discourse acordingly (Jacobson & Lehrer, 2000). Evidence in the case study 
showed that efective discourse was happening often in the two experimental clases and 
not in the other two control clases (Jacobson & Lehrer, 2000). 
Han (2007); Marrades and Guti?rrez (2000); and Pesek and Kirsher (2000) (described 
earlier in this chapter) were also studies where instruction in the clasroom impacted 
student achievement. Han (2007) was a disertation that focused on one unit where both 
clases were taught relationaly, but the one clas that used dynamic geometry software 
scored higher on author created asesment than the one without using dynamic geometry 
software. Marrades and Guti?rrez (2000) students were able to ?progres toward more 
elaborated types of justifications? (p. 120) and showed confidence in their ability of 
forming deductive justifications and formal proofs when the teacher thoughtfully created 
activities, scafolded, and asked questions that promoted student learning. Pesek and 
Kirsher (2000) compared students who were taught relationaly as opposed to 
instrumentaly, and found that students that were taught relationaly were beter able to 
apply what was learned. Table 2-10 summarizes the studies that discuss clasroom 
instruction and student achievement. 
Table 2-10 Clasroom Instruction and Student Achievement 
Author(s) Grade Teacher Knowledge Student Achievement 
Han 
(2007) 
Middle 
School 
- Questioning to get students to 
come to their own correct 
conjectures of quadrilateral 
properties 
- Both groups of students were 
taught relationaly 
 
Students that learned 
using dynamic 
geometry software 
scored higher on 
author created 
asesment 
 
Jacobson 
& Lehrer 
(2000) 
2
nd
 
grade 
Knowledge about students? thinking: 
Observation ? more knowledgeable 
teachers orchestrated clasroom talk in 
Learned more with 
permanence 
 
 
 
 
58 
ways that refined, elaborated, and 
extended students? thinking 
 
Marrades 
& 
Guti?rrez 
(2000) 
 
4
th
 
grade 
- Activities were generaly structured in 
three phases; create and/or explore, 
generate and/or verify conjectures, and 
justify conjectures 
- provided the figure and students had 
aces to it to use it to explore properties 
instead of creating their own 
- prompted students with questions such 
as ?why is the construction valid?? and 
?why is the conjecture true?? (Marrades 
& Guti?rrez, 2000, p. 98). 
 
students were able to 
?progres toward 
more elaborated types 
of justifications? (p. 
120) and showed 
confidence in their 
ability of forming 
deductive 
justifications and 
formal proofs 
 
Pesek & 
Kirshner 
(2000) 
5
th
 
grade 
Taught relationaly  Able to beter apply 
what was learned 
 
 Teacher Knowledge and Classroom Instruction. Some studies like Cohen 
(1990) and Heaton (1992) observed teachers during instruction in the clasroom and 
based teacher knowledge off of what was observed; while other studies like Borko et al. 
(1992), Thompson and Thompson (1994), and Hil et al. (2008) both asesed teacher 
knowledge separately from instruction, and then asesed level of instruction. Cohen 
(1990) and Heaton (1992) were two studies among many in the literature that helped 
shape Hil et al.?s (2008) Mathematical Quality of Instruction observation protocol about 
significant mathematical errors in instruction. The following paragraphs of this section 
wil expound on Borko et al.?s (1992), Cohen?s (1990), Heaton?s (1992), Hil et al.?s 
(2008), and Thompson and Thompson?s (1994) articles. 
Cohen?s (1990) article served the purpose of looking at the relationship betwen 
educational policy and teaching practice. Mrs. Oublier, the elementary teacher in the 
study, believed that she was teaching what educational policy had suggested, teaching for 
 
 
 
 
59 
understanding, but Cohen (1990) found that Mrs. Oublier taught very traditionaly. 
Further, she did not promote discourse, acepted inacurate student answers to her 
questions, and did not take opportunities that Cohen (1990) deemed necesary to develop 
student understanding. One example of this occurring was on a leson about estimation, 
and she acepted inacurate answers and failed to follow up on why those answers were 
wrong. In addition, she did not lead students to the correct answer based off of what the 
students talked about; she provided the rule for why estimation was the way it was. 
Another study that helped shape Hil et al.?s (2008) Mathematical Quality of 
Instruction observation protocol about significant mathematical errors in instruction was 
Heaton?s (1992) case study of a fifth grade teacher. Heaton?s (1992) study found that 
there were significant problems in instruction in the areas of mathematics content, 
facilitation of student learning, leson choice and focus, and learning outcomes. 
However, students were engaged in learning and were working on real-world problems 
(Heaton, 1992). On one observation, the leson was a part of a 5-day unit that had 
students designing a park on a $5000 budget. It was not explicitly stated in the article 
what the teacher thought the goal of the leson was, but Heaton (1992) stated that the 
goal should have been reasonablenes of answers. From observation of the leson, it was 
clear to Heaton (1992), that this was not the case. The teacher had students measure a 
sandbox at the school to get a sense of scaling, but two students measured in yards and 
the other student, that measured the height, measured it in fet. The sandbox was 46 yards 
x 10 yards x 1 foot. The teacher instructed students to multiply 46x10x1 and did not catch 
this error throughout the leson. In this case, teacher self-asesment of teaching ability 
did not match reality (Heaton, 1992). Hil et al. (2008) atributed ?policymakers? 
 
 
 
 
60 
concerns about the quality of clasroom work? (p. 433) to studies like Cohen (1990) and 
Heaton (1992). 
Borko et al. (1992) studied a student teacher?s failed atempt to teach division of 
fractions conceptualy from many diferent vantage points. Borko et al. (1992) looked at 
coursework that Ms. Daniels, the sixth grade student teacher in the study, as basis for 
teacher knowledge. Ms. Daniels was a math major, with her math credits in courses 
modern algebra and computer science courses, but did not take many elementary 
mathematics courses; especialy one in elementary number concepts and operations. 
When diferent aspects like coursework, beliefs about teaching, and beliefs about 
learning were considered; Borko et al. (1992) suggested that the reason for Ms. Daniels? 
failed atempt to teach division of fractions conceptualy was because of the mised 
opportunity to explore number concepts and operations at the elementary level. 
Bil, a middle school teacher, in an article by Thompson and Thompson (1994) 
?was very adept at reasoning proportionaly, whether relationships were direct or inverse? 
(p. 282). In a paper and pencil test developed by the Rational Number Project (Post, 
Harel, Behr, & Lesh, 1991), Bil ?meaningfully and creatively solved each proportional 
reasoning item? (Thompson & Thompson, 1994, p. 282). An example given by 
Thompson and Thompson (1992) was: 
?A pan balance is off center. An object put on one side weighs 10 lb. The same 
object put on the other side weighs 40 lb. How much does the object weigh?? Bil 
saw easily that the two pans? distances from the fulcrum had to be inversely 
proportional to the ratio of the weights. So, if d1/d2 is the ratio of the distances 
from the fulcrum, where d1 is the shorter distance, and if we let x stand for the 
 
 
 
 
61 
object?s weight, then the ratios x/40 and 10/x must be equal to d1/d2, and 
therefore must be equal to each other (p. 282). 
However, in a tutoring sesion where the student was confused about making sense of 
what the numbers mean when rates were represented in fraction form and what to do with 
them, Bil was not able to facilitate conceptualy, what the student needed to know. Bil 
had conceptual knowledge of the topic, but was not able to facilitate the student to learn 
conceptualy.  
An exploratory study conducted by Hil et al. (2008) of mathematical knowledge 
for teaching and the mathematical quality of instruction showed that ?there is a 
significant, strong, and positive asociation betwen levels of MKT [mathematical 
knowledge for teaching] and the mathematical quality of instruction? (Hil, et al., 2008, p. 
430). This sample was of ten teachers, betwen 2
nd
 and 6
th
 grade, of which 5 of the 
strongest cases were chosen to be ilustrated in the article (Hil, et al., 2008). They 
completed pencil and paper asesment of mathematical knowledge for teaching, were 
videotaped for nine lesons, and participated in post observation debriefings and 
interviews. The mathematical knowledge for teaching instrument asesed teachers? 
common and specialized content knowledge (Hil, et al., 2008). The observation protocol 
used was the Mathematical Quality of Instruction (MQI) developed by the Learning 
Mathematics for Teaching Project. Areas of focus for Mathematical Quality of 
Instruction when observing the teachers were: 
? Mathematics errors?the presence of computational, linguistic, representational, 
or other mathematical errors in instruction; 
? Contains subcategory specificaly for errors with mathematical language 
 
 
 
 
62 
? Responding to students inappropriately?the degree to which teacher either 
misinterprets or, in the case of student misunderstanding, fails to respond to 
student utterance; 
? Connecting clasroom practice to mathematics?the degree to which clasroom 
practice is connected to important and worthwhile mathematical ideas and 
procedures as opposed to either non-mathematical focus, such as clasroom 
management, or activities that do not require mathematical thinking, such as 
students following directions to cut, color, and paste, but with no obvious 
connections betwen these activities and mathematical meaning(s); 
? Richnes of the mathematics?the use of multiple representations, linking 
among representations, mathematical explanation and justification, and 
explicitnes around mathematical practices such as proof and reasoning; 
? Responding to students appropriately?the degree to which the teacher can 
correctly interpret students? mathematical utterances and addres student 
misunderstandings; 
? Mathematical language?the density of acurate mathematical language in 
instruction, the use of language to clearly convey mathematical ideas, as wel as 
any explicit discussion of the use of mathematical language (Hil, et al., 2008, p. 
437). 
Further, Hil, et al. (2008) looked at other possible factors that influence teacher 
knowledge and quality of instruction as wel: ?teacher beliefs about how mathematics 
should be learned and how to make it enjoyable by students; teacher beliefs about 
curriculum materials and how they should be used; and the availability of curriculum 
materials to teachers? (Hil, et al., 2008, p. 497); and found teacher knowledge was itself, 
a factor of the factors that influence teacher knowledge and quality of instruction. Thus 
?the inescapable conclusion of this study is that there is a powerful relationship betwen 
what the teacher knows, how [the teacher] knows it, and what [the teacher] can do in the 
context of instruction? (Hil, et al., 2008, p. 496). Table 2-11 below summarizes articles 
on teacher knowledge and clasroom instruction. 
 
 
 
 
63 
Table 2-11 Teacher Knowledge and Clasroom Instruction 
Author(s) Grade 
Level 
Purpose 
Borko et al. 
(1992) 
6
th
 grade Diferent vantage points of why a student teacher failed to 
teach fraction division conceptualy 
 
Cohen (1990) elementary Educational policy and clasroom instruction 
Heaton 
(1992) 
5
th
 grade Importance of subject mater knowledge in teaching 
mathematics and what happens when there is an apparent 
lack of knowledge from observations 
 
Hil, et al. 
(2008) 
2
nd
 -6
th
 
grade 
teachers 
 
Exploratory study of mathematical knowledge for teaching 
and the mathematical quality of instruction 
Thompson & 
Thompson  
Middle 
School 
Documented a teacher?s dificulty conceptualy explaining 
rates when the teacher understands rates conceptualy 
 
Teacher Knowledge. Articles like Garet, Porter, Desimone, Birman, and Yoon, 
(2002); Hil et al. (2007); and Hil, Schiling, and Bal (2004); looked at profesional 
development and its efect on teacher knowledge. Hil, Bal, and Schiling?s (2008) study 
took data from teachers that were going through profesional development, but the data 
taken was to help define an aspect of mathematical knowledge for teaching. Studies like 
An, Klum, and Wu (2004); Ma (1996); Schmidt, Tato, Bankov, Bl?meke, Cedilo, 
Cogan, Han, Houang, Hsieh, Paine, Santilan, and Schwile, (2007); compared diferent 
teachers? knowledge in diferent countries. Most of the work that has been done to ases 
teacher content knowledge in geometry using the Van Hiele Geometry Test by Usiskin 
(1982) was by Erdo?an Halat in Turkey. Herbst and Kosko (2012) published a paper that 
documented eforts in developing an instrument that measured mathematical knowledge 
for teaching high school geometry (MKT-G). Details of that study were expounded upon 
earlier in this chapter on page 48.  
 
 
 
 
64 
Hil, Bal and Schiling (2008) used asesment documents to ases teachers? 
mathematical knowledge for teaching. Part of the data was from California?s 
Mathematics Profesional Development Institutes for their efectivenes. There were 
more than 370 teachers who responded for each of their three forms. The three forms 
were ?number and operations common and specialized content knowledge; number and 
operation KCS [knowledge of content and students]; and paterns functions, and algebra 
common content knowledge? (Hil, Bal, & Schiling, 2008, p. 382). They found that 
knowledge of content and students is distinct form pure pedagogical or content 
knowledge (Hil, Bal, & Schiling, 2008). 
In a comparison study, An, Klum, and Wu (2004) used an author-constructed 
teaching questionnaire, beliefs questionnaire, interviews, and observations to compare 
pedagogical content knowledge of middle school teachers in China and the United States. 
The sample size for this study at the questionnaire phase was 28 mathematics teachers in 
Texas and 33 mathematics teachers in China; five teachers from each country were 
observed to represent a diverse range of educational background and teaching experience. 
An, Klum, and Wu (2004) reported that they followed the Instructional Criteria 
Observation Checklist adapted from a framework set forth by the American Asociation 
for the Advancement of Science for analyzing instructional quality of mathematics 
textbooks. Acording to An, Klum, and Wu (2004). 
The observation criteria included specific activities in the categories: building on 
student ideas in mathematics, being alert to students? ideas, identifying student 
ideas, addresing misconceptions, engaging students in mathematics, providing 
first-hand experiences, promoting student thinking about mathematics, guiding 
 
 
 
 
65 
interpretation and reasoning, and encouraging students to think about what they 
had learned (pp. 151-153). 
As An, Klum, and Wu (2004) stated in their methodology section, the use of interviews 
and observations was to provide validity and reliability for the data collected from 
questionnaires. A sample pedagogical content problem from the questionnaire is shown 
in Figure 2-5. 
 
Figure 2-5 Sample problem from the Mathematics Teaching Questionnaire (An, Klum, & 
Wu, 2004, p. 152). 
 
For the problem in Figure 2-5, 46% of the responses of the U.S. teachers thought that 
Adam forgot the prerequisite knowledge of finding common denominators, while 55% of 
the responses of the Chinese teachers thought that Adam did not understand the 
prerequisite knowledge of finding common denominators (An, Klum, & Wu, 2004). 
?Forgeting? and ?not understanding? do not have the same meaning (An, Klum, & Wu, 
2004). ?Forgeting? implies that the teacher did not know of the chalenges students face 
in learning addition of fractions, whereas ?not understanding? implied that the teacher is 
aware of the student misconception (An, Klum, & Wu, 2004). 
Results from the teaching questionnaire and beliefs questionnaire showed that 
teachers sampled from the two countries varied greatly, and the results from the 
observations and interviews showed consistency in the results from the questionnaire ? 
pedagogical content knowledge has an impact on teaching practice (An, Klum, & Wu, 
 
 
 
 
66 
2004). Teachers sampled in the United States gave students a lot of diferent activities 
using manipulatives to promote creative thinking to develop concept mastery but lacked 
making the connection betwen what the students were doing and the abstract concept 
(An, Klum, & Wu, 2004). ?The Chinese system emphasizes gaining correct conceptual 
knowledge by reliance on traditional, more rigid development of procedures, which has 
been the practice of teaching and learning mathematics content for many years? (An, 
Klum, & Wu, 2004, p. 169). 
Cankoy (2010) is another example of teacher in teacher knowledge literature. 
Unlike An, Klum, and Wu (2004); Cankoy?s (2010) study focused on a specific topic. 
Cankoy (2010) aimed to explore teacher topic-specific pedagogical content knowledge in 
asking for what the teacher thought was the best teaching strategy for teaching a
0
 = 1, 0! 
= 1 and a?0 where a?0 to high school student. Of the 58 teachers in this study, the range 
of years of experience was from 1 ? 17 years with the mean of 5.9 and standard deviation 
of 4.65. Cankoy (2010) categorized the teachers into two groups of teaching experience, 
?novice? having taught 1-4 years (n=33) and ?experienced? having taught 5-17 (n=25). 
Qualitative data analysis through coding was used to sort the diferent teacher responses, 
and ?
2
 test was used to determine if experience was related to strategy. For each a
0
 = 1, 0! 
= 1 and a?0 where a?0, there were three main approaches; rule, patern, and algebraic or 
limit for a?0 where a?0. Surprisingly, for 0! = 1 and a?0 where a?0, some teachers had 
no answer, and for a?0 where a?0, there were some undeterminable approaches. An 
average of 27% of experienced teachers suggested patern approach for al three 
statements, and only an average of 1% of novice teachers preferred that approach. 
However, most teachers in both groups relied heavily on the procedure and memorization 
 
 
 
 
67 
to teach students. The separation betwen novice and experienced teachers was not as 
clear cut as expected. Cankoy?s (2010) teaching strategies for the topic were literature 
based. 
In several studies, Erodgan Halat used the Van Hiele Geometry Test (VGHT) to 
quantatively analyze diferences in geometric understanding of teachers of diferent 
gender, school level (Halat, 2006; Halat, 2008, May), and pre-service compared to in-
service (Halat, 2008; Halat & Sahin, 2008). When the reasoning stages of in-service 
middle and high school mathematics teachers in geometry were investigated, Halat 
(2008) found that the in-service middle and high school mathematics teachers represented 
al the van Hiele levels, visualization, analysis, ordering, deduction, and rigor, and that 
there was no diference in terms of mean reasoning stage betwen in-service middle and 
high school mathematics teachers. Results of which were contrary to preconceived 
notions of the diferences betwen middle school and high school teachers. Moreover, 
from Halat?s 2008 results, there was no significant gender diference found regarding the 
geometric thinking levels. In this study, there was a total of 148 in-service middle and 
high school mathematics teachers. Since the focus of this disertation is on the 
relationships betwen knowledge exhibited before and during clasroom instruction, and 
not geometry teachers? overal knowledge of geometry; the VGHT wil not be used to 
ases teacher knowledge. 
An interesting study outside mathematics education comes from science 
education, where teacher pedagogical content knowledge from the perspective of 
experienced science teachers was studied (Le & Luft, 2008). The teachers in the study 
were chosen from a program where experienced teachers were paired up with beginning 
 
 
 
 
68 
teachers in a mentorship program ? four teachers voluntered to participate in this study, 
and al four had at least 10 years of teaching experience and at least three years of 
mentoring experience in this program. Over a period of two years, the four teachers 
participated in semi-structured interviews, clasroom observations, a collection of leson 
plans, and monthly reflective summaries. Coding teachers? responses of what science 
teachers need to know for teaching yielded seven components under pedagogical content 
knowledge, knowledge of science, goals, students, curriculum and organization, teaching, 
asesment, and resources. Teachers were then asked to organize the seven components 
in a concept map showing how each is related to the other, and al four had diferent 
conceptualizations, each placing importance on diferent components. The results from 
this study (Le & Luft, 2008) showed that experienced teachers? conceptions of 
pedagogical content knowledge was not that diferent from those proposed by Shulman 
(1987) content knowledge; general pedagogical knowledge; curriculum knowledge; 
pedagogical content knowledge; knowledge of learners and their characteristics; 
knowledge of educational contexts; and ?knowledge of educational ends, purposes, and 
values, and their philosophical and historical grounds? (p. 8). 
Since Shulman (1987), researchers, when conducting research, (e.g., Cankoy, 
2010; Hil, Rowan, & Bal, 2005; Le & Luft, 2008; Monk, 1994) focused on diferent 
aspects of pedagogical content knowledge. For example, Cankoy (2010) studied what 
high school mathematics teachers? believed are best instructional strategies for teaching 
a
0
 = 1, 0! = 1 and a?0 where a?0 and named that ?topic specific pedagogical content 
knowledge?. On a broader note, Li Ping Ma?s (1996) disertation looked at teachers in 
China and teachers in the United States to compare their profound understanding of 
 
 
 
 
69 
fundamental mathematics (PUFM) and how PUFM is atained. Similar to An, Klum, and 
Wu?s (2004) study, findings showed that teacher knowledge is very important, and the 
way teachers conduct clas largely depends on teacher pedagogical content knowledge. 
Table 2-12 summarizes the studies that elucidate teacher in teacher knowledge literature. 
 
 
Table 2-12 Teacher in the Teacher Knowledge Literature 
Author(s) Grade 
Level 
Purpose 
Hil, Bal, & 
Schiling 
(2008) 
 
Elementary Measuring the domain of mathematical knowledge for 
teaching, specificaly knowledge of content and students 
Hil, et al. 
(2008) 
2
nd
 -6
th
 
grade 
teachers 
 
Exploratory study of mathematical knowledge for 
teaching and the mathematical quality of instruction 
An, Wu, & 
Klum (2004) 
Middle 
School 
Compare pedagogical content knowledge of middle 
school teachers in China and the United States 
 
Cankoy 
(2010) 
High 
School 
What the teacher thought was the best teaching strategy 
for teaching a
0
 = 1, 0! = 1 and a?0 where a?0 to high 
school student 
 
Halat (2008) Secondary Compare van Hiele levels of in-service middle school 
and high school teachers 
 
Le & Luft 
(2008) 
Secondary ?look at how mentor science teachers conceptualized 
their own PCK that impacted their teaching practice? (p. 
1348). 
 
Ma (1996) Elementary Compared teachers in China and the United States on 
their profound understanding of fundamental 
mathematics 
 
Herbst & 
Kosko 
(2012) 
High 
School 
Measured mathematical knowledge for teaching high 
school geometry, specificaly common content 
knowledge, specialized content knowledge, knowledge 
of content and teaching, and knowledge of content and 
students. 
 
 
 
 
 
70 
Asesing Teacher Knowledge 
Part of the goal of this disertation is to explore teachers? mathematical 
knowledge for teaching. Diferent asesments of diferent teacher knowledge types will 
be necesary (Hil, Bal, & Schiling, 2008). As mentioned in the introduction, acording 
to Hil, Slep, Lewis, and Bal (2007); there are three contemporary presures on 
researchers to develop more coherent approaches to ases teachers? knowledge: political 
environment, evidence of teacher competence, and to help define what makes teaching a 
profesion. Studies in the past quantified teacher?s subject area knowledge by looking at 
the coursework of teachers in college, college degree, and to some extent, teaching years; 
and quantifying teacher knowledge through asesments of current knowledge has only 
become mainstream in research starting in the past decade or so (Bal, & Rowan, 2004; 
Hil, Rowan, Bal, 2005). Usiskin?s (1982) Van Hiele Geometry Test tested students? 
content knowledge in geometry, and several researchers have used it to ases current 
content knowledge of pre-service and in-service teachers. So far, in terms of high school 
geometry and asesing teacher knowledge, Usiskin?s (1982) van Hiele Geometry Test is 
the only one. In this section, various types of teacher asesment wil be discussed. 
 Observation Protocols. In order to ases teacher knowledge in selecting and 
using representations and actual emphasis of key concepts; interviews were proven to be 
insufficient, as what teachers believe that they are doing might not equate to what they 
actualy do in the clasroom (An, Klum, & Wu, 2004). An, Klum, and Wu (2004) used an 
author-constructed teaching questionnaire, beliefs questionnaire, interviews, and 
observations to compare pedagogical content knowledge of middle school teachers in 
China and the United States. Acording to An, Klum, and Wu (2004) 
 
 
 
 
71 
The observation criteria included specific activities in the categories: building on 
student ideas in mathematics, being alert to students? ideas, identifying student 
ideas, addresing misconceptions, engaging students in mathematics, providing 
first-hand experiences, promoting student thinking about mathematics, guiding 
interpretation and reasoning, and encouraging students to think about what they 
had learned (pp. 151-153). 
The sample size for this study at the questionnaire phase was 28 mathematics teachers in 
Texas, and 33 mathematics teachers in China; five teachers from each country were 
observed to represent a diverse range of educational background and teaching experience. 
An, Klum, and Wu (2004) reported that they followed the Instructional Criteria 
Observation Checklist adapted from a framework set forth by the American Asociation 
for the Advancement of Science for analyzing instructional quality of mathematics 
textbooks. 
For a large-scale study focusing on measuring teachers? quality of instruction 
Matsumura, Garnier, Slater, and Boston (2008) based their rubrics in the mathematics 
framework on ?the Mathematical Task Framework developed by Mary Kay Stein, 
Margaret Smith and their colleagues? (p. 276). Matsumura et al. (2008) chose this 
framework because it considered "the potential of a task to support higher level, 
conceptual thinking including students' opportunity to engage with a range of 
representations in their responses, and the implementation of the task in practice (as 
enacted)." (p. 277). In the Mathematical Tasks Framework were three major areas; 
analyzing mathematics instructional tasks, using cognitively complex tasks in the 
clasroom, and learning from cases (Smith et al. 2005). Under "analyzing mathematics 
 
 
 
 
72 
instructional tasks" were "defining levels or cognitive demand of mathematical tasks", 
"matching tasks with goals for student learning", "diferentiating levels of cognitive 
demand", "gaining experience in analyzing cognitive demands", and "moving beyond 
task selection and creation" (Smith et al. 2005). Development of "tasks during a leson" 
and "paterns of task setup and implementation" fel under the umbrela of "Using 
cognitively complex tasks in the clasroom" (Smith et al. 2005). 
Matsumura et al. (2008) inferred from their research that the minimum number of 
asignments from a teacher to ases their beliefs of level of work expected of students is 
four, and minimum number of observations to get an acurate picture of how the teacher 
facilitates discourse is also four. Teachers filed out a form for each asignment 
explaining their reasoning behind the asignment, where the leson fit in the content of 
the course, and the belief of level of dificulty of questions; as wel as provided four 
student samples of graded work for each asignment at the teachers? perception of two 
levels, high and medium. Matsumura?s et al. (2008) analysis showed strong correlation 
betwen items overlapping asignments and observable traits in clas, yielding a 
dependability coeficient of .80 for four asignments; thus some observable traits in clas 
can also be asesed by collecting asignments. 
 Learning Mathematics for Teaching (LMT) created the mathematical quality of 
instruction (MQI Instrument) to look at the predictive validity of their test for 
mathematical knowledge for teaching (Hil, et al. 2008). The five elements of 
mathematical quality of instruction observed in MQI were; mathematics errors, 
responding to students inappropriately, connecting clasroom practice to mathematics, 
richnes of the mathematics, responding to students appropriately, and mathematical 
 
 
 
 
73 
language (Hil, et al., 2008). Hil et al., (2008) used MQI to measure mathematical quality 
of instruction to compare to teacher scores of mathematical knowledge for teaching. A 
more detailed explanation of the study is on page 62.  
An et al. (2004) and Matsumura et al. (2008) used established observation 
protocols to gather the data necesary for their respective studies. Another observation 
protocol widely used in reform mathematics and science clasrooms is caled Reformed 
Teaching Observation Protocol (RTOP) (Sawada & Pilburn, 2000). RTOP should only be 
used by trained observers, and a method in which to be trained is to read through the 
training guide provided by the creators of RTOP. The theoretical and philosophical 
rationale behind the reform mathematics movement was that of constructivism ? where 
atainment of knowledge is through active participation of the learner; and ?RTOP was 
designed to capture the current reform movement, and especialy those characteristics 
that define ?reform teaching?? (Sawada & Pilburn, 2000, p. 2). There are many 
observation protocols available, but for the purpose of this disertation, I created an 
observation protocol specific to capturing data for the types of teacher knowledge. 
Observation alone is not sufficient to gather data on teacher knowledge (Hil, Slep, 
Lewis, & Bal, 2007). 
 Concept Maps. Concept maps are tools that researchers may use to ases 
conceptual understanding of a concept (Novak & Ca?as, 2008; Wiliams, 1998). 
Concepts are typicaly enclosed in planar geometric shapes with few words that signify 
the concept, sometimes caled semantic units or units of meaning (Novak & Ca?as, 
2008). Then betwen concepts are arrows that signify two concepts are related, and 
relation may be writen on that arrow; with the option of cross-links, or arrows crossing 
 
 
 
 
74 
in a concept map (Novak & Ca?as, 2008). Concept maps are also hierarchical in nature, 
depending on the directions in which the arrows point, some concepts may be a few 
concepts away from another concept; thus creating an order in which concepts are related 
(Novak & Ca?as, 2008). Below, Figure 2-6, is a concept map that shows key features of 
concept maps. ?Concept maps tend to be read progresing from the top downward" 
(Novak & Ca?as, 2008, p. 2). 
 
Figure 2-6 Sample concept map (Novak & Ca?as, 2008, p. 2). 
 Carol G. Wiliams used concept maps to ases students' conceptual knowledge of 
function (Wiliams, 1998). Her subjects were 28 students in third quarter calculus; 14 
took al three semesters of reformed calculus, and 14 took al three semesters traditional 
calculus; and eight profesors of mathematics. There was a special sesion on how to 
construct a concept map. The main goal of the study was to use concept maps as a 
research tool to reflect conceptual understanding. " The general homogeneity of the 
experts' concept maps and their distinct variance from students' maps lend credibility to 
 
 
 
 
75 
the conclusion that concept maps do capture a representative sample of conceptual 
knowledge and can diferentiate among fairly disparate levels of understanding" 
(Wiliams, 1998, p. 420). 
The purpose of Jin and Wong?s (2010) study was to explore several diferent 
training durations to help secondary level students acquire sufficient concept mapping 
skils to use when learning mathematics. The time increments were 10 minutes, 15 
minutes, and 50 minutes. They found that teaching how to concept map meaningfully 
took 50 minutes and is best done alongside current curriculum (Jin & Wong, 2010). Jin 
and Wong (2010) defined ?meaningfully? as meaningful placement of given concepts, 
labeled connections, and acuracy of descriptions. However, a document like a teachers? 
concept map of the topic without the teacher explaining gives limited insight into what 
teachers know about teaching; even with clasroom observations of the teachers teaching 
the topic. 
Chinnappan and Lawson (2005) interviewed two experienced teachers who were 
involved in a larger study of five experienced teachers and five novice teachers in 
Australia. The purpose of this study was to test the framework in with Chinnappan and 
Lawson (2005) proposed to study teacher knowledge for teaching. These two teachers 
each had taught for at least 15 years at the high school level and were recommended by 
peers or other sources as exemplary teachers (Chinnappan & Lawson, 2005). These two 
teachers did not teach at the same school (Chinnappan & Lawson, 2005). Subjects were 
asesed through structured interviews where making concept maps were the interview 
tasks to gauge geometric content knowledge and geometric knowledge for teaching 
(Chinnappan & Lawson, 2005). Results showed that the mapping procedure was useful 
 
 
 
 
76 
for purposes where teacher content knowledge and content knowledge for teaching are 
both required for comparison (Chinnappan & Lawson, 2005). 
Interviews. Chinnappan and Lawson (2005) conducted three (one hour long) 
interviews with each teacher. The first interview focused on what the teacher knew about, 
and how to teach the content (Chinnappan & Lawson, 2005). Typical question during the 
first interview were ?Tel me about what you know about squares? and ?Tel me how you 
would teach square to your students? (Chinnappan & Lawson, 2005, p. 205). Teachers 
responded to four problems for the second interview (Chinnappan & Lawson, 2005). 
These problems were within the content area of what they taught; and in this case, about 
squares (Chinnappan & Lawson, 2005). Some questions asked during the second 
interview were ?How would you expect your students to tackle this problem? and ?What 
type of dificulties would you expect your students to experience if they are given these 
problem, Why?? (Chinnappan & Lawson, 2005, p. 205). The purpose of the final 
interview was to gather any more pertinent information not gathered in previous 
interviews (Chinnappan & Lawson, 2005). A sample statement to get the teacher to talk 
was ?You haven?t said anything about the symmetry of a square yet? (Chinnappan & 
Lawson, 2005, p. 205). Chinnappan and Lawson used interviews with and without 
concept maps as an interview task to gather evidence of teacher knowledge (Chinnappan 
& Lawson, 2005). 
An Klum and Wu (2004) used interviews to triangulate data from other data 
collection sources. As mentioned in the previous section, An Klum and Wu (2004) used 
interviews in conjunction with surveys and observations to compare pedagogical content 
knowledge of middle school teachers in China and the U.S. The set of interview 
 
 
 
 
77 
questions examined teachers? beliefs of how teacher beliefs impact teaching practices, 
teaching approaches, preparation for instruction, and interpretation of student thinking. 
These interviews followed clasroom observations ?explored further the teachers? 
pedagogical content knowledge and its importance in their teaching? (An, Klum, & Wu, 
2004, p. 153). 
Some tasks in interviews for teachers are the same tasks given to students (Hil, 
Slep, Lewis, & Bal, 2007). For example, Graeber, Tirosh, and Glover (1989) asesed 
prospective elementary teachers? misconceptions about multiplication and division with 
the same asesment for students. Burger and Shaughnesy (1986) devised tasks to 
interview students and were able to ases their van Hiele Levels, but did not ases in-
service teachers. However, two college mathematics majors were included in the Burger 
and Shaughnesy (1986) study. 
Burger and Shaughnesy (1986) determined van Hiele Levels of reasoning of 45 
students from grades 1 to 12 and university mathematics majors through experimental 
interviews. ?Experimental tasks were administered to each student by one of the four 
researchers in an audiotaped clinical interview? (Burger & Shaughnesy, 1986, p. 33). 
There were eight problems about geometric shapes for each individual (Burger & 
Shaughnesy, 1986). One of the tasks was for students to identify and define diferent 
quadrilaterals (Burger & Shaughnesy, 1986). Students were given a sheet of paper with 
many diferent types of quadrilaterals on it, and they were asked to write an ?S? by the 
square, ?P? by the paralelogram, and so forth (Burger & Shaughnesy, 1986). 
Researchers recorded and analyzed tapes to gather evidences of levels of thought (Burger 
& Shaughnesy, 1986). From those, van Hiele level of reasoning was determined based 
 
 
 
 
78 
on the individual?s predominate level of reasoning (Burger & Shaughnesy, 1986). Table 
2-13 summarizes literature on asesing teacher knowledge by type of asesment; 
observation protocols, concept maps, and interviews. 
Table 2-13 Summary of Literature Asesing Teacher Knowledge 
Type Researcher Purpose 
Observation 
protocols 
An, Klum, & 
Wu (2004) 
Compare teachers in China to teachers in United 
States 
 Matsumura, 
Garnier, Slater, 
& Boston (2008) 
 
Efficient number of observations and asignments 
to collect from teachers for a large scale study 
 Hil, et al. (2008) Exploratory study of mathematical knowledge for 
teaching and the mathematical quality of instruction 
 
Concept Maps Wiliams (1998) 
 
Assess students' conceptual knowledge of function 
 Chinnappan & 
Lawson (2005) 
 
Determine efectivenes of concept maps to ases 
teacher knowledge 
 Jin Wong Determine appropriate amount of time to teach 
efective creation of concept maps 
 
Interviews Burger & 
Shaughnesy 
(1986) 
 
Determine student van Hiele Level of reasoning 
 Chinnappan & 
Lawson (2005) 
 
Determine efectivenes of concept maps to ases 
teacher knowledge 
 An, Klum, & 
Wu (2004) 
Compare teachers in China to teachers in United 
States 
 
Summary of Literature Review 
 Chapter one gives reasons for why it would be pertinent to conduct this research 
study, and chapter two gives the literature background. Specificaly, this chapter started 
with defining what it means for students to learn and teachers to teach. In discussing 
 
 
 
 
79 
knowledge in teaching mathematics, I discussed Peterson?s (1988) framework and Hil, 
Bal, and Schiling?s (2004) domain map for mathematical knowledge for teaching. 
Figure 2-7 is a domain map for mathematical knowledge for teaching as it relates to 
Shulman?s (1987) pedagogical content knowledge (Hil, Bal, Schiling, 2008, p. 377). 
 
 
 
Figure 2-7 Domain map for mathematical knowledge for teaching as it relates to 
Shulman?s (1987) pedagogical content knowledge.  From ?Unpacking pedagogical 
content knowledge: Conceptualizing and measuring teachers? topic-specific knowledge 
of students.? by H.C. Hil, D.L. Bal, and S.G. Schiling, 2008, Journal for Research in 
Mathematics Education, 39(4), p. 377. Copyright 2008 by Journal for Research in 
Mathematics Education. Reprinted with permision. 
 
Specialized content knowledge is ?mathematical knowledge that is used in 
teaching, but not directly taught to students? (Hil, Slep, Lewis, & Bal, 2007, p. 132). 
Hence, specialized content knowledge is not categorized under pedagogical content 
knowledge. Hil, Slep, Lewis, and Bal (2007) gave Figure 2-8, on the next page, as an 
example to ilustrate specialized content knowledge. 
 
 
 
 
80 
 
Figure 2-8 Sample SCK question for Multiplication. From ?Efects of teachers? 
mathematical knowledge for teaching on student achievement.? by H.C. Hil, B. Rowan, 
and D.L. Bal, 2005, American Educational Research Journal, 42(2), p. 401-402. 
Copyright 2005 by SAGE Publications. Reprinted with permision. 
 
In answering the question in Figure 2-8, ?If the teacher understands these explanations 
for the students? methods, she then must make a determination about whether each 
method generalizes to the multiplication of other whole numbers, perhaps by referencing 
the commutative or distributive properties of multiplication? (Hil, Slep, Lewis, & Bal, 
2007, pp. 132-133). Knowledge of content and students is the subset of pedagogical 
content knowledge that enables teachers to identify common student errors, interpret 
students' understanding of content, identify students' developmental sequences, and 
identify common student computational strategies (Hil, Bal, & Schiling, 2008). In 
general, knowledge of content and students is ?the amalgamated knowledge that teachers 
 
 
 
 
81 
posses about how students learn content? (Hil, et al., 2007, p. 133). Knowledge of 
content and teaching is the knowledge of teaching the content with emphasis on teaching; 
e.g. how to correct student mistakes so that they wil understand and how to build on 
what the students already know (Hil, Bal, & Schiling, 2008). Hil et al. (2007) 
summarizes knowledge of content and teaching as ?mathematical knowledge of the 
design of instruction? (p. 133). Specificaly, knowledge of content and teaching ?includes 
how to choose examples and representations, and how to guide student discussions 
toward acurate mathematical ideas? (Hil, et al., 2007, p. 133). 
Specialized content knowledge, knowledge of content and students, and 
knowledge of content and teaching; the middle slice of Bal and colleagues? domain map 
(Hil, Bal, & Schiling, 2008) of mathematical knowledge for teaching, are the three 
areas of mathematical knowledge for teaching that are the focus of this disertation. I 
renamed these three areas collectively as Teacher Applied Content Knowledge (TACK) 
because I se them as applications of content knowledge. 
 TACK. Teachers? Applied Content Knowledge (TACK) is based from a subset of 
Bal and colleagues? categories of mathematical knowledge for teaching; knowledge of 
content and teaching, knowledge of content and students, and specialized content 
knowledge. The following subsections of this section describe these three categories of 
mathematical knowledge for teaching in reference to the data that wil be collected for 
each. 
Knowledge of Content and Teaching. Knowledge of content and teaching (KCT) 
is the knowledge of teaching content with emphasis on teaching (Hil, Bal, & Schiling, 
2008). For example, an aspect of KCT is how to correct student mistakes so that teachers 
 
 
 
 
82 
understand how to build on what students already know (Hil, Bal, & Schiling, 2008). 
Hil et al. (2007) summarized knowledge of content and teaching as ?mathematical 
knowledge of the design of instruction? (p. 133). Specificaly, knowledge of content and 
teaching ?includes how to choose examples and representations, and how to guide 
student discussions toward acurate mathematical ideas? (Hil, et al., 2007, p. 133). 
Based off of literature from Bal and colleagues on knowledge of content and teaching, I 
simplified it to four categories to look at for this disertation: 1) sequence of information 
or concepts taught, 2) time it takes to cover each item of the sequence of information, 3) 
explanations to teach, and 4) questions to teach. 
Knowledge of Content and Students. Knowledge of content and students is 
knowledge of content specificaly involving students (Bal, Thames, & Philips, 2008). 
Hil, Bal, and Schiling (2008) defined knowledge of content and students as knowledge 
that enables teachers to identify common student errors, interpret students' understanding 
of content, identify students' developmental sequences, and identify common student 
computational strategies. Specificaly for this study, knowledge of content and students is 
demonstrated by the teacher in three ways: modifications to what is being taught based on 
student dificulties; student misconceptions and how to help students overcome them; and 
knowledge of student prerequisite skils. 
Specialized Content Knowledge. Specialized content knowledge, ?mathematical 
knowledge that is used in teaching, but not directly taught to students? (Hil, Slep, 
Lewis, & Bal, 2007, p. 132) and was asesed by Hil, et al. (2008) in a multiple choice 
paper and pencil test. For the purpose of specificity in coding, sections of transcript data 
were coded as specialized content knowledge if it exemplified any of the mathematical 
 
 
 
 
83 
tasks of teaching from Bal, Thames, and Phelps (2008), with the exclusion of 
?Explaining mathematical goals and purposes to parents? as shown in Table 2-14 below. 
Table 2-14 Mathematical Tasks of Teaching 
Presenting mathematical ideas 
Responding to students? ?why? questions 
Finding an example to make a specific mathematical point 
Recognizing what is involved in using a particular representation 
Linking representations to underlying ideas and to other representations 
Connecting a topic being taught to topics from prior or future years 
Explaining mathematical goals and purposes to parents 
Appraising and adapting the mathematical content of textbooks 
Modifying tasks to be either easier or harder 
Evaluating the plausibility of students? claims (often quickly) 
Giving or evaluating mathematical explanations 
Choosing and developing useable definitions 
Using mathematical notation and language and critiquing its use 
Asking productive mathematical questions 
Selecting representations for particular purposes 
Inspecting equivalencies (p. 10) 
 
In short, I condensed specialized content knowledge as the knowledge of concepts, 
definitions, and procedures for teaching. However, specialized content knowledge sems 
to overlap with knowledge of content and students and knowledge of content and 
teaching. Thus the next section aims to clarify the relationships. 
SCK, KCT, and KCS Relationships. Bal, Thames, and Phelps (2008) noted that 
even though distinguishing betwen common content knowledge, specialized content 
 
 
 
 
84 
knowledge, knowledge of content and students, and knowledge of content and teachers; 
may be subtle, it is possible given this example: 
recognizing a wrong answer is common content knowledge (CCK), while sizing 
up the nature of the error may be either specialized content knowledge (SCK) or 
knowledge of content and students (KCS) depending on whether a teacher 
draws predominantly from her knowledge of mathematics and her ability to 
carry out a kind of mathematical analysis or instead draws from experience with 
students and familiarity with common student errors. Deciding how best to 
remediate the error may require knowledge of content and teaching (KCT) 
(p.11). 
However, I contend, for the purpose of this study that the kind of diference, 
highlighted above, betwen SCK and KCS cannot be determined within this study 
and is not necesary to determine the diference. In the way that TACK is defined, 
KCS, SCK, and KCT are not necesarily mutualy distinct in their manifestations in 
the data for data analysis, but were defined separately as a way to explicate the 
origins of TACK. 
Teaching and Learning Geometry. After discussion of literature on learning and 
teaching, literature regarding teaching and learning of geometry were expounded upon. 
Specific to teaching and learning geometry, separate from other topics in mathematics, 
are the van Hiele Levels (van Hiele, 1986), phases of learning based on the van Hiele 
Model (Mistreta, 2000), and geometric habits of mind (Driscoll, 2007).  
Through discussing literature with evidence of student achievement, certain 
atributes of teachers surfaced. Many teachers used technology in the clasroom, in the 
form of dynamic geometry software (Han, 2007; Jones, 2000; Marrades & Guti?rrez, 
2000). In Marrades and Guti?rrez (2000) activities were generaly structured in three 
 
 
 
 
85 
phases; create and/or explore, generate and/or verify conjectures, and justify conjectures. 
The teacher provided the figure and students had aces to it to use it to explore 
properties instead of creating their own (Marrades & Guti?rrez, 2000). Teachers 
questioned students with questions such as ?why is the construction valid?? and ?why is 
the conjecture true?? (Han, 2007; Jones, 2000; Marrades & Guti?rrez, 2000, p. 98). Also, 
teachers analyzed, coached, and re-voiced questions back to students (Martin, et al., 
2006). 
Teacher Knowledge, Classroom Instruction, and Student Achievement. 
Studies discussed in reference to general teaching and learning, teaching and learning of 
geometry, and other studies involving teacher knowledge, clasroom instruction, and 
student achievement was further detailed and synthesized into four categories. These four 
categories were: teacher knowledge and student achievement, clasroom instruction and 
student achievement, teacher knowledge and clasroom instruction, and teacher 
knowledge. From research on literature of studies in the four categories, atributes of 
teachers of students that showed achievement were similar to those described in teaching 
and learning geometry. However, in additional teacher atributes that surfaced were group 
work for students, persistence in student asking students questions, and knowledge 
related to teaching mathematics. 
 
 
 
 
 
86 
 
Asesing Teacher Knowledge. Discussing studies involving teacher knowledge 
and student achievement, clasroom instruction and student achievement, teacher 
knowledge and clasroom instruction, and teacher knowledge exposed methods in 
which those researchers asesed teacher knowledge. The ?Asesing Teacher 
Knowledge? section emphasized three diferent asesments of teacher knowledge 
to expound upon; observation protocols, concept maps, and interviews. Some 
studies from the previous chapter were further discussed in relation to the methods 
in which they asesed teacher knowledge for each asesment method, while other 
studies that supported the use of those asesment methods were discussed as wel.  
The focus of this disertation is on teacher knowledge during planning phase 
and execution of the leson. Simon and Tzur (1999) stated that researchers need 
rich data to ases the complex relationships of teacher?s beliefs, knowledge, 
values, intuitions, felings, and practices. This chapter showed the literature behind 
choice of the aspects of teacher knowledge, teaching and learning geometry, area of 
need in literature on geometry teacher knowledge and clasroom instruction, and 
literature on asesing teacher knowledge. The next chapter describes the theoretical 
perspective and methodologies used for this disertation.  
 
 
 
 
 
 
87 
Chapter 3: Theoretical Perspectives and Methodologies 
In the first chapter of this disertation, I presented evidence to support that student 
achievement in geometry is an area of concern (Lappan, 1999; Mullis, Martin, & Foy, 
2008) and why improving student achievement in geometry is necesary (National 
Research Panel, 1989; Shaughnesy, 2011). Thus, in order to improve student 
achievement in geometry, I chose to look at teacher knowledge of teaching geometry. In 
the second chapter I expounded on theories of learning and teaching; in general and 
specific to geometry; before delving into literature on teacher knowledge. In transition to 
this chapter, how to conduct this study of teacher knowledge of teaching geometry, the 
second chapter ended with a summary of diferent ways to ases teacher knowledge. 
Since this study aims to add to the body of research pertaining to teacher 
knowledge, this paragraph describes the alignment that I se betwen this study and 
Silver and Herbst?s (2007) idea of scholarship as a theory-making endeavor. From the 
standpoint of scholarship as a theory-making endeavor, there exist ongoing relationships 
betwen research, problems, theory making, and practices (Silver & Herbst, 2007). 
Although many relationships betwen research, problems, theory making, and practices 
have been addresed in chapters 1 and 2, the focus of this disertation?s place in the 
theory-making endeavor is the bidirectional relationship betwen research and practices. 
Data collection of teachers? knowledge while in practice is based on research. Findings 
from data collection of teachers? knowledge while in practice wil hopefully inform 
 
 
 
 
88 
research; which in turn, improves practice. In order to unite the various aspects of the 
statement of the problem of this disertation, multiple frameworks wil be used. 
The first section of this chapter sets the background for the frameworks and 
methodologies of this study by addresing the theoretical perspective and philosophical 
asumptions. Then research design of this study is detailed; specificaly, participants, 
sources of data, and procedures for conducting this research study. Once the background 
for the research design is explained, analysis of TACK and analysis with the geometry 
filters are discussed.  
Theoretical Perspective 
This study looked at Teachers? Applied Content Knowledge, a subset of Hil, 
Bal, and Schiling?s (2008) Mathematical Knowledge for Teaching. Hil, Bal, and 
Schiling?s (2008) measures of a subset of Mathematical Knowledge for Teaching, 
knowledge of content and students, were based ?on empirical evidence regarding how 
students learn? (p. 375). They stated that they did not base it on theory because ?several 
theories of student learning legitimately compete? (p. 375). Since this study looked at 
more of Hill, Bal, and Schiling?s (2008) Mathematical Knowledge for Teaching than 
knowledge of content and students, giving this disertation a theoretical perspective based 
on a theory of learning is pertinent (Lester, 2010). 
 The theoretical perspective of this disertation is grounded in constructivism as a 
learning theory. 
Constructivism derives from a philosophical position that we as human beings 
have no aces to an objective reality, i.e. a reality independent of our way of 
knowing it. Rather, we construct our knowledge of our world from our 
 
 
 
 
89 
perceptions and experiences which are themselves mediated through our previous 
knowledge (Simon, 1993, p. 5). 
A tenet of constructivism is that knowledge is constructed by an active participant, when 
existing knowledge interacts with new experiences (Simon, 1993; Stefe & D?Ambrosio, 
1995). Constructivism is a theory of learning (Simon, 1993), and thus, as stated above, 
does not prescribe a particular model of teaching (Stefe & D?Ambrosio, 1995). Models 
of teaching based on constructivism are constructed from tenets of constructivism (Stefe 
& D?Ambrosio, 1995). Teachers that view student learning from the constructivist 
perspective ?study the mathematical constructions of students and interact with students 
in a learning space whose design is based, at least in part, on a working knowledge of 
students? mathematics? (Stefe & D?Ambrosio, 1995, p. 148). D?Ambrosio and Stefe 
(1995) caled these teachers ?constructivist teachers?. 
 In Herbst (2006), Megan, a high school teacher who taught geometry, was 
?powered by an interest in connecting students? mathematical experiences to other 
aspects of life? (p. 321). She often started clas with open-ended problems, and then a 
whole clas discussion before students worked in groups to solve the problem. In the 
whole clas discussion, Herbst (2006) observed Megan move students towards the 
objective of the leson through questioning. For example, some students answered her 
initial question with ?in the middle?, and Megan questioned students to be more specific 
with what they meant by ?in the middle? (Herbst, 2006). Describing Megan?s general 
design for each day of her clas showed that she was a constructivist teacher since she 
designed her leson in order for students to come to their own conclusions through 
interactions with the teacher and other students. 
 
 
 
 
90 
Philosophical Asumptions 
 ?The research design proces in qualitative research begins with philosophical 
asumptions that the inquirers make in deciding to undertake a qualitative study? 
(Creswel, 2007, p. 15). I referred to Creswel?s (2007) philosophical asumptions with 
implications for practice for my stance on nature of reality (ontological), relationships 
with this study?s participants (epistemological), the language I use to discuss this research 
(rhetorical), the proces of research (methodological), and the role of my values in this 
study (axiological). 
 The nature of reality (Creswel, 2007) for this study was subjective. The 
implication for practice from this ontological asumption (Creswel, 2007) was the use of 
quotes and themes in the words of the teachers, and evidence of teacher knowledge from 
multiple perspectives. I collaborated with teachers through interviews and observations to 
study their geometry TACK. From the epistemological asumption (Creswel, 2007), 
time in the schools with the teachers before, during, and after observed lesons was my 
atempt to lesen the distance betwen me and the teachers. The language I used the 
language of qualitative research to describe each aspect of this study. My methodological 
approach (Creswel, 2007) looked at details before making generalizations, described 
details in the context of the study, and continualy revised interview questions from 
experiences observing.  The axiological asumption (Creswel, 2007) was that I 
acknowledge that this study is value-laden and that biases are present. Thus, this chapter 
served to discuss the values that shape this study and includes disclosure of my biases.  
 
 
 
 
91 
Research Design 
At the beginning of this chapter, I described how this study fits within research as 
a theory-making endeavor (Silver & Herbst, 2007), and the theoretical perspective and 
philosophical asumptions that undergirds this study. In this section, I wil describe the 
research design. Since this study aimed to look at connections of geometry Teachers? 
Applied Content Knowledge, planned and executed; methodology is based on 
mathematics education literature for studying teachers knowledge and the nature of the 
research question. Simon and Tzur (1999) stated that researchers need rich data to ases 
the complex relationships of teacher?s beliefs, knowledge, values, intuitions, felings, and 
practices. To study the connections betwen teacher knowledge shown during the 
planning and execution of a leson required data that is ?wel grounded, rich descriptions 
and explanations of proceses in identifiable local contexts? (Miles & Huberman, 1994, 
p. 1). Thus, the appropriate research design is qualitative. Since this study is on teacher 
knowledge, I decided that the best course of action would be to study each teacher as a 
case in a multiple case study. The next paragraph details further the choice of case study 
as an approach to collecting and analyzing data. 
Creswel (2007) suggested the following procedures for conducting a case study: 
determine if the approach is appropriate, identify the case or cases, determine data 
collection, determine type of analysis, and interpreting the findings. Since this study is 
about high school geometry Teachers? Applied Content Knowledge (TACK) shown 
during the planning of the leson and the teachers? actual executions of the leson, there 
are two components to the question: first, TACK planning and the other TACK 
execution. It makes sense to explore the two components to this question within a teacher 
 
 
 
 
92 
as a case. The research question ?seks to provide an in-depth understanding of the 
cases? (Creswel, 2007, p. 74). Since there were two teachers; this case study has two 
cases, making this a multi case study. The next section describes each teacher in further 
detail. 
Participants. Case studies ?can look at many individuals connected with the case 
or at just a few or even one individual? (Lichtman, 2011, p. 109). For this study, each 
participant, or teacher, is a case.  A key element of case studies is the ?focus on an 
extensive examination of a particular group, program, or project? (Lichtman, 2011, p. 
109). This study focused on geometry teacher knowledge of two teachers. In choosing 
teachers for this study, I needed teachers who were currently teaching high school 
geometry, who have taught geometry multiple times recently, and who I have observed 
teach in a manner for students to learn with relational understanding. I decided that these 
exemplary teachers would give me a greater base of teacher knowledge to look at the 
connections betwen teacher knowledge shared during planning and observed during 
execution of the leson. Initialy there were three teachers chosen for the study, with the 
hope that I would be able to study al three, but with the understanding that two teachers 
for the study would be sufficient. One of the teachers did not respond with permision 
despite prompting. Two teachers participated in this study.  
My involvement with profesional development in a southeastern state, 
supervising student teachers, and supervising practicum students alowed for interaction 
with a vast network of teachers in a southeastern state. From this network of teachers, 
two with whom I have closely worked satisfied these criteria and were able to participate 
in the study. Acording to Creswel (2007), this is caled purposeful sampling. Types of 
 
 
 
 
93 
sampling include theory based, criterion, and convenience (Miles & Huberman, 1994). 
Both of these teachers are examples of constructivist teachers, which also means they 
were selected based on theory ? theory based (Miles & Huberman, 1994). Each teacher 
met the set of criterion listed above ? criterion, which is ?useful for quality asurance? 
(Miles & Huberman, 1994, p. 28). Also, as previously stated, these teachers were selected 
from a group of teachers I already knew, so this is also a convenience sample with the 
above conditions. 
Mrs. Lotus* and Mrs. Orchid* both teach in the same district, but Mrs. Lotus 
teaches at the junior high school, and Mrs. Orchid at the high school. Given the timing in 
relation to the implementation of the Common Core State Standards and textbook 
revisions within the district, both teachers stated that they had to change from what they 
taught last year. Table 3-1 displays information about their college major and years 
taught. 
Table 3-1 Teacher information 
Teacher Major/Degrees Total Years Taught, 
of those how many 
geometry 
How many years taught 
geometry in the past 10 
years? 5 Years? 
Mrs. 
Orchid 
BS, M.Ed., and Ed.S 
Mathematics Education 
10, 10 10, 5 
Mrs. 
Lotus 
BS Education Major in Math, 
minor in physical science 
33, 15 10, 5 
 
Both Mrs. Lotus and Mrs. Orchid completed bachelor?s degrees that required coursework 
in education and mathematics. At the time of data collection, Mrs. Orchid recently 
completed an educational specialist degree in mathematics education, and was currently 
working on a doctorate in mathematics education. Both teachers taught at least one high 
 
 
 
 
94 
school geometry clas of students for the most recent 10 years of their teaching. Mrs. 
Orchid taught the high school geometry course for 10 years, which was her entire 
teaching career. Mrs. Lotus taught mathematics for 33 years, and of those 33 years, she 
taught 15 years of geometry. Thus, both teachers had recent experience teaching high 
school geometry.   
Mrs. Orchid and Mrs. Lotus was each considered a case. One clas was observed 
throughout the study for each teacher. The content observed was high school geometry. 
The clasroom observation length of time was determined by the preset clas schedule. 
The data collection time period was set within a semester, and each teacher was at a 
diferent school, so there were two diferent locations. Creswel (2007) suggested that 
cases in case studies need to be within a bounded system, clearly defined boundaries for a 
case that form a whole. ?Boundaries [are] often bounded by time and place? (Creswel, 
2007, p. 244). For this study, clas of students, content, length of time for data collection, 
and location set the boundaries for each case (Creswel, 2007). During data collection, 
there were a variety of sources of data. The next section describes the sources of data. 
Sources of Data. At the end of chapter two, I summarized literature on diferent 
ways to ases teacher knowledge; observation protocols, concept maps, and interviews. 
In collecting data for the two components of the question for this study, TACK planned 
and the TACK executed; I needed to know what the teacher planned to conduct clas and 
what actualy happened in clas. Thus, sources of data were collected in the form of 
interviews and observations. Concept maps were drawn during interviews. 
Interviews occurred before and after observations as wel as before and after 
units. ?Interviewing is an active proces where interviewer and interviewe through their 
 
 
 
 
95 
relationship produce knowledge? (Kvale & Brinkmann, 2009, p. 17). For this study, 
aspects of knowledge of Teachers? Applied Content Knowledge (TACK), was created 
from the interaction betwen interviewer and interviewe (Kvale & Brinkmann, 2009). 
So much of what a teacher plans to teach and what happens when a teacher teaches a 
leson is not sen; and thus, without the interviewer, this knowledge would not be 
revealed (Kvale & Brinkmann, 2009). Interviews prior to lesons and units were used to 
gather data on TACK planned. Other interviews and observations where the teacher 
referred to what they had planned also were added to the body of data for TACK planned, 
but were categorized as: data collected about planning but not collected prior to 
execution. 
Prior to data collection, I created an observation protocol (Appendix C) to collect 
data during observations and videotaped the teachers as they taught their lesons. The 
observation protocol consisted of six questions: 
1. How did the teacher explain concepts, definitions, and procedures? 
 
2. What are areas of teacher dificulties in explanations? 
 
3. What was the sequence of information presented? 
 
4. What was the time spent on phases of the leson? 
 
5. What student dificulties arose? If so, how was it shown? What did the teacher do 
in response? 
 
6. What were actual student levels of prerequisite skils and how did the teacher 
teach to those levels? 
 
These questions were generated from definitions of each aspect of TACK; specialized 
content knowledge, knowledge of content and students, and knowledge of content and 
 
 
 
 
96 
teaching. The choice for number of observations as the minimum of three days is similar 
to Pesek and Kirshner?s (2000) choice of number of observations. 
 Decisions for the design of this research, qualitative, multi-case study, 
participants, and sources of data were based on literature and discussions with my 
commite members. The basis of these decisions contributes to the quality of this study 
(Corbin & Strauss, 2008). The next section further discusses the ?quality? of this study. 
Quality of Research 
 This is a qualitative study, and as the researcher, I am the key instrument of data 
collection. This section details how the findings are reliable and valid. ?Reliability 
pertains to the consistency and trustworthines of research findings? [and] Validity 
refers in ordinary language to the truth, the correctnes, and the strength of the statement? 
(Kvale & Brinkmann, 2009, p. 245-246). There are many perspectives to evaluating 
qualitative research. Diferent perspectives come with diferent key terms to describe 
them. Some, like Kvale and Brinkmann (2009) use reliability and validity, while others, 
like Corbin and Strauss (2008) prefer the phrase ?quality of research?. Lincoln and Guba 
(1985) used the terms: credibility, transferability, dependability, and conformability. 
Creswel (2007) summarized many diferent perspectives, and I found that the term 
?quality? best fits this study. In the following paragraphs, quality of research of this study 
was discussed in general as wel as it pertains to key stages of the study; reviewing the 
literature, gaining permisions for the study, collecting data, data analysis, and reporting 
the study.  
Creswel (2007) provided several suggestions for establishing ?good? research; 
utilize peer review, provide rich thick description, make researcher bias known, ensure 
 
 
 
 
97 
member checking, and include triangulation. I asked two individuals to code portions of 
the data for inter-rater reliability. These two individuals are graduate students within my 
program of secondary mathematics education. In writing this disertation for both data 
collected and analysis, I used rich thick description to provide enough context to show 
how I arrived at my conclusions. Following this section, I have a section labeled 
?Researcher Bias?, in which I described the experiences that I had which informed my 
theoretical perspective. Since this is a disertation, my commite provided peer review 
and expert opinion. Video and audio recordings served to make sure that transcriptions 
were acurate and documented things in the clasroom pertaining to research that I may 
have mised when observing. 
For inter-rater reliability, Miles and Huberman (1994) suggested that the equation 
to calculate reliability is the number of agreements divided by the total number of 
agreements and disagreements. Using this formula, Miles and Huberman (1994) stated 
that the first time reliability is not beter than 70%, second time a few days later with 
beter definitions should be around 80%; and eventualy, final check-coding about two-
thirds of the way through the study should be about 90% reliability. Portions of 
interviews (generaly 5 minutes) and observations (generaly chosen for the multiple 
codes that showed up within a short amount of time. During the first round, reliability 
was about 60% for both raters. For the second round, reliability was up to 16/21, roughly 
76% for one and 80% for the other. For the final round, reliability was 17/18, or roughly 
94% for one, and 100% for the other. 
The literature review, impetus, methods for data collection, and methods for data 
analysis were approved by a commite of four profesors; two mathematics educators, 
 
 
 
 
98 
one educational researcher, and one mathematician. Following approval from the 
commite, approval from the school district was obtained, followed by the Institutional 
Review Board. As stated above, during data collection, I checked often with individual 
members of my commite. Al identifying aspects on physical information like writen 
leson outlines were removed prior to my possesion of them. During transcription, 
student names were dealt with in one of two ways: numbered for events like S, S1, S2, 
and so on; or pseudonyms were given to students whose names were repeated a lot and 
out of context of events. No actual student names were typed into transcriptions. During 
data analysis al formal levels of analysis were revisited multiple times to ensure nothing 
was left behind. Also, when coding, as mentioned above, I had two individuals to 
establish inter-rater reliability.  
Data on what teachers planned and executed were collected from many diferent 
sources. Unit interviews and pre-observation interviews collected data on what the 
teachers planned to teach. Audio from the teachers? perspective, and video and audio 
recording from an observer?s perspective of teacher knowledge during a clasroom 
observation. During observations, when possible, I also wrote notes of teacher knowledge 
I observed and questions to ask teachers about their knowledge shown during the 
observation. After observations, teachers reported rationale for some of the clasroom 
decisions, giving further details on teacher knowledge during clasroom instruction. 
These multiple sources at multiple times provided triangulation to get a more reliable 
acount of teacher knowledge. In the next section, I present my researcher bias. 
Researcher Bias. Besides the need I saw in the literature for further investigation, 
my profesional and personal background was the impetus of my need to know about the 
 
 
 
 
99 
diferences in pedagogical content knowledge (PCK) in general. I was a high school 
mathematics teacher for 4.5 years in California and taught high school geometry among 
other mathematics courses. As semesters went on, I picked up more geometry clases to 
teach because those were the clases that the other teachers did not want to teach. I come 
from the perspective that one is never done learning. Teachers should acknowledge the 
responsibility that their job entails, and improve their practice day-to-day, year-to-year. 
My parents were educated in Taiwan and my maternal grandmother, whom I 
caled by her Chinese title is Popo (maternal grandmother) was educated in China ? one 
of the first women in her vilage to be educated, was a high school mathematics 
curriculum specialist in Taiwan. Growing up, I was greatly influenced by her teaching 
methodology since I did not speak English fluently until the third grade, and the English 
as a Second Language (ESL) program did not provide the services for the language I 
spoke; thus learning was not done in the clasroom, but rather at home. My memory of 
mathematics learning in the early grades was highly conceptual as Popo driled math 
facts and taught me how to memorize procedures, but also taught me why those 
procedures are the way they are. I was not exposed to the discovery approach, a type of 
conceptual teaching, until later on in school, but did not find the two methods 
contradictory in terms of how much I learned. 
Through teaching, I discovered that my students learned more and retained more 
knowledge when I taught conceptualy versus proceduraly. The time it took to cover the 
same material also decreased when I taught conceptualy. Due to the lack of studies that 
are similar to this study, I relied on the conceptions from research related to this study to 
guide me as I conducted this study. 
 
 
 
 
100 
So far, theoretical perspectives, philosophical asumptions, and participants and 
sources of data in the research design were based on literature and discussions with my 
commite members. Since I was the instrument for data collection, it was necesary for 
me to make my biases transparent. In order to make this study replicable, the data 
collection procedures are also made transparent. The procedures for data collection were 
also based on literature and discussions with my commite members.  
 Procedures for Collection of Data. Once al necesary approvals were met, I e-
mailed the teachers with a description of the study and possible dates to set up initial 
metings. Below I described procedures in relation to each stage of metings. 
Initial Meting. This meting was for the purpose of introducing the teacher to 
the study, seting up a timeline for completing data collection and geting the necesary 
forms determined for students. Teachers were informed that this study looks at teacher 
knowledge. To prepare the teachers for the interview task of making a concept map, a 
sample concept map (Appendix E) was given and discussed at this meting. 
Interview for Unit Plan and Concept Map. In some cases, this interview was 
combined with the initial meting when the teacher was ready. The purpose of this 
interview was to get the teachers? overal plan for the main concept(s) covered and 
provide a conceptual background for the consecutive lesons to be observed. Two 
interview tasks were asked of the teachers; concept map (Appendix D) and unit plan 
interview (Appendix B). Teachers were reasured that the concept maps could be 
modified each time they were interviewed for the duration of the study. I provided a copy 
of Appendix D, the concept map; and paper for the teachers to write on for the unit plan 
interview. Al of these interviews were video and audiotaped. 
 
 
 
 
101 
Interviews for Leson Plans. The purpose of these interviews was to get a closer 
look at exactly what the teacher planned to teach in the preselected consecutive days 
within the unit. These interviews were conducted after the interview for the unit plan and 
concept map, but before each observation. Teachers were able to modify their concept 
maps at any time during these interviews. These interviews were also audio and video 
taped. Interviews for leson plans also followed the same format as Appendix B. 
Observations of Lesons. Teachers chose the dates of a minimum of three 
observations on the basis that the dates chosen were to best show the development of the 
concept or skil. Al consecutive days were within a five-day wek, which alowed for 
opportunity to observe changes made in planning and execution of lesons. For each 
teacher, observations of both units were with the same group of students. I used the 
observation protocol (Appendix C) to collect data during observations and videotaped the 
teachers as they taught their lesons. Due to the necesity of following the teacher around 
to video record interactions, minimal note were taken via the Observation Protocol during 
the observation. For privacy purposes, care was taken to record only student work during 
teacher-student interactions and not their faces; so when I walked around the room, the 
video camera was pointed at the floor. 
Interview for Reflections of Lesons. These interviews were conducted to 
provide details to what was observed when the teacher was teaching. Teachers were 
asked questions about their thoughts while watching video and audio taped evidence to 
provide context. The purpose of that was to provide the ?why? and ?how? behind the 
decisions and judgments made in the clasroom. Questions asked in post-execution 
interviews stemed from similarities and diferences in the planning and executing of the 
 
 
 
 
102 
leson. Teachers were asked to clarify relationships betwen items planned and executed 
and why they stayed the same or changed. 
Teacher Self-Reflections of Lesons. For the lesons within the unit for which I 
was not in the clasroom to observe, teachers e-mailed me their reflections of what they 
planned on teaching and how it unfolded in the clasroom. These were for the purpose of 
giving beter context for the consecutive lesons. Thus, teachers answered two main 
questions for self-reflection. First, what and how do you plan on teaching the leson? 
Second, what happened as you expected and what changed, and why? Over the course of 
the unit, answers to these questions gave additional context to what I observe for the 
preset consecutive leson. 
Interview for Reflections of Units. The purpose of these interviews was to 
provide a context for the consecutive lesons further than daily self-reflections for days 
on the unit I did not observed. These interviews atained the teacher?s perspective on how 
the unit was executed in comparison to what was planned. This was also an opportunity 
for me to ask additional questions from the unit; and was an opportunity for teachers to 
share any remaining details about the unit that was not shared in previous interviews, 
observations, or teacher self-reflections. Table 3-2 is a table of the sequence of data 
collection. 
Table 3-2 Sequence of Data Collection 
- Initial Meting 
- Interview for Unit Plan and Concept Map 
- Reflections for the days not observed in the unit prior to consecutive 
observations 
 
 
 
 
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- Minimum of three consecutive observations: 
Interview of leson 
Observation of leson 
Interview after leson 
- Reflections for the days not observed in the unit after consecutive 
observations but before the end of the unit 
- Unit reflection interview 
 
 In this section I explicated the quality of this research study. The introduction to 
this section detailed how quality of this study was upheld in each aspect of this study. 
The next section discusses analysis of TACK and analysis through the geometry filter.  
Analysis of TACK 
Teachers? Applied Content Knowledge was based on a subset of Mathematical 
Knowledge for Teaching from Bal and colleagues described in Chapter 2 section titled 
?TACK? on page 81.  The three domains of Mathematical Knowledge for Teaching I 
rename TACK are Knowledge of Content and Teaching (KCT), Knowledge of Content 
and Students (KCS), and Specialized Content Knowledge (SCK). Specialized Content 
Knowledge (SCK) is the knowledge of concepts, definitions, and procedures for teaching 
(Bal, Thames, & Phelps, 2008). Knowledge of content and teaching (KCT) is the 
sequence of information or concepts taught, time it takes to cover each item of the 
sequence of information, explanations to teach, and questions to teach (Bal, Thames, & 
Phelps, 2008). Knowledge of content and students (KCS) is leson modifications, student 
dificulties, and student prerequisite skils (Bal, Thames, & Phelps, 2008). From this, the 
eight areas of teacher knowledge that make up TACK are: 
 
 
 
 
 
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? Knowledge of concepts, definitions, and procedures for teaching 
? Sequence of information to be presented 
? Timing for phases of the leson 
? Ways to explain concepts, definitions, and procedures 
? Ways to ask questions to guide student thinking 
? Modifications to lesons based on students? needs. 
? Student misconceptions and dificulties and how to addres them 
? Knowledge of level of prerequisite skils of students 
When asking about SCK planning during interviews, the main question asked is: 
What knowledge of concepts, definitions, or procedures for teaching did the teacher 
exhibit? The question in connection to the corresponding observation is: What 
knowledge of concepts, definitions, or procedures for teaching did the teacher 
exhibit? Similar questions were asked of the other seven areas. Table 3-3 lists the 
areas of focus for teacher knowledge that are components of TACK separated by 
the two aspects of the research question; planning and execution. 
Table 3-3 Planning and Execution areas of focus of TACK 
 
Planning Execution 
Specialized 
Content 
Knowledge 
Knowledge of concepts, 
definitions, and procedures for 
teaching 
Knowledge of concepts, definitions, 
and procedures used in teaching 
 
Knowledge 
of Content 
and 
Teaching 
Plan for sequence of information 
to be presented. 
 
Sequence of information presented. 
Planned time for phases of the 
leson. 
 
Time spent on phases of the leson. 
How to explain the concepts, 
definitions, and procedures. 
 
Explained concepts, definitions, and 
procedures. 
Planned key questions to guide 
student thinking. 
 
Questions asked of students to guide 
their thinking. 
 
 
 
 
105 
Knowledge 
of Content 
and 
Students 
Leson modifications based on 
perceived student dificulties. 
 
Did those student dificulties stil 
arise? If so, how was it shown? What 
did the teacher do in response? 
Other anticipated areas of 
student dificulties and how to 
help students overcome. 
 
Actual presented areas of student 
dificulties and how the teacher 
helped students overcome. 
Anticipated student levels of 
prerequisite skils and how the 
teacher plans acordingly. 
Actual student levels of prerequisite 
skils and how the teacher taught 
acordingly. 
 
This leads to the separation of the research question ?What connections are there betwen 
the high school geometry TACK shown during the planning of the leson and the 
teachers? actual executions of the leson?? into eight more explicit questions. 
? What are the connections betwen planned and executed specialized 
content knowledge? 
? What are the connections betwen planned and executed sequence of 
information to be presented? 
? What are the connections betwen planned and executed time spent on 
phases of the leson? 
? What are the connections betwen planned and executed explanations of 
concepts, definitions and procedures? 
? What are the connections betwen planned and executed key questions 
to guide student thinking? 
? What are the connections betwen planned and executed leson 
modifications? 
? What are the connections betwen planned and observed teachers? 
acknowledgement of student dificulties and how to overcome them? 
? What are the connections betwen planned and observed teachers? 
knowledge of student prerequisite skils? 
 
 
 
 
106 
I answered these questions following Creswel?s (2007) approach to data analysis and 
representation for a case study. Answers to these questions within the cases were 
described in context. I used categories to establish themes or paterns. When interpreting 
the data, I used direct interpretation and developed naturalistic generalizations. When 
describing the findings from the cases, I used narratives, tables, and figures.  
As stated in the Research Design section, the research design follows that of a 
case study. However, to get a look at the relationships betwen the aspects of TACK, I 
used approaches from grounded theory. Acording to Creswel (2007), describing open 
coding categories is the first step to describing data analysis.  
For coding using the grounded theory approach, I determined eight a priori codes 
from literature of KCT, KCS, and SCK. These a priori codes follow what Kele (2007) 
cals ?common sense categories?. These codes followed directly from literature (Bal, 
Thames, & Phelps, 2008) and what I described as aspects of TACK at the beginning of 
this section. Table 3-4 shows a priori codes in tabular form. A copy of the a priori codes 
in tabular form can also be found in the Coding Guide in Appendix H on page 242. 
Table 3-4 a priori codes 
 
Planning Execution 
Specialized 
Content 
Knowledge 
SCK-P 
Knowledge of concepts, 
definitions, and procedures 
for teaching 
SCK-E 
Observable knowledge of concepts, 
definitions, and procedures used in 
teaching 
 
Knowledge 
of Content 
and 
Teaching 
KCTseq-P 
Plan for sequence of 
information to be presented. 
 
KCTseq-E 
Sequence of information presented. 
KCTtime-P 
Planned time for phases of 
the leson. 
KCTtime-E 
Time spent on phases of the leson. 
 
 
 
 
107 
 
KCTexplain-P 
How to explain the 
concepts, definitions, and 
procedures. 
 
KCTexplain-E 
Explained concepts, definitions, and 
procedures. 
KCTquestions-P 
Planned key questions to 
guide student thinking. 
 
KCTquestions-E 
Questions asked of students to guide their 
thinking. 
Knowledge 
of Content 
and 
Students 
KCSmod-P 
Leson modifications based 
on perceived student 
dificulties. 
 
KCSmod-E 
Did those student dificulties stil arise? If 
so, how was it shown? What did the 
teacher do in response? 
KCSdificulties-P 
Other anticipated areas of 
student dificulties and how 
to help students overcome. 
 
KCSdificulties-E 
Actual presented areas of student 
dificulties and how the teacher helped 
students overcome. 
KCSprerequisite-P 
Anticipated student levels 
of prerequisite skils and 
how the teacher plans 
acordingly. 
KCSprerequisite-E 
Actual student level s of prerequisite 
skils and how the teacher taught 
acordingly. 
 
When approaching data analysis through grounded theory, open codes; in the case 
of this study, a priori codes and emergent codes; were described before engaging in axial 
coding.  During axial coding, one code at a time was chosen to focus on the causal 
condition, context, intervening conditions, strategies, and consequences. Categories were 
then interrelated to provide a story of how the diferent aspects of TACK interacted. 
Conclusions needed the combination of both approaches to data analysis from case study 
and grounded theory. Themes that came out through data analysis through case study 
approach were compared to codes and findings from axial coding from grounded theory. 
 
 
 
 
108 
This method gave the most cohesive report for the answer to the research question: What 
connections are there betwen the high school geometry TACK shown during the 
planning of the leson and the teachers? actual executions of the leson? 
In this section I described the data analysis of TACK through both the case study 
approach and grounded theory approach. Findings from both approaches brought together 
in the Conclusions chapter were the best method to answer the research question. 
However, since this study looked at geometry Teachers? Applied Content Knowledge, 
knowledge of teaching and learning geometry was also an important aspect to consider. 
The next section discusses how I applied geometry filters as a lens to look through what 
the teacher planned and what was executed.   
Analysis through Geometry Filter 
Since this study focused on teachers? knowledge related to high school geometry, 
there were three pieces of the geometry filter for which to sort the data for geometry 
content; Breyfogle and Lynch?s (2010) geometric understanding with examples of 
teacher activities, Mistreta?s (2000) phases of learning from the van Hiele Model, and 
Driscoll?s (2007) geometric habits of mind. For the geometry filter, three questions arise. 
First, what van Hiele level did the teacher?s activity support? Table 3-5 shows only the 
van Hiele levels and description from Table 2-6 on page 32. Van Hiele levels were 
identified based on what the teacher planned to teach and what actualy happened in the 
clasroom. Van Hiele levels identified for what a teacher planned was based on what 
teachers described they were going to conduct in the clasroom; and levels for activities 
that actualy happened in the clasroom were based on what the teachers actualy did with 
 
 
 
 
109 
the students during clas. Table 3-5 is an abridged version of Breyfogle and Lynch?s 
(2000) van Hiele Levels. 
Table 3-5 Abridged version of Breyfogle and Lynch?s (2000) van Hiele Levels 
Level Name Description 
 Visualization Se geometric shapes as a whole; do not focus on 
their particular atributes. 
1 Analysis Recognize that each shape has diferent properties; 
identify the shape by that property. 
2 Informal Deduction Se the interrelationships betwen figures. 
3 Formal Deduction Construct proofs rather than just memorize them; 
se the possibility of developing a proof in more 
than one way. 
4 Rigor Learn that geometry needs to be understood in the 
abstract; se the ?construction? of geometric 
systems. 
 
From the van Hiele levels, Mistreta (2000) proposed phases of learning that 
supports movement from one van Hiele level to the next. Thus, the second question, what 
phases of learning did the teacher support? Table 3-6 on the next page displays phases of 
learning from the van Hiele model proposed by Mistreta (2000). Similar to analysis of 
the van Hiele levels, phases were determined for what the teacher planned to teach and 
what actualy happened in the clasroom. Phases determined for what a teacher planned 
was based on what teachers described they were going to conduct in the clasroom; and 
phases for what actualy happened in the clasroom were based on what the teachers 
actualy did with the students during clas.  
 
 
 
 
 
110 
Table 3-6 Phases of Learning from the van Hiele Model (Mistreta, 2000, p. 367) 
Phase Description 
Information Discussions are held where the teacher learns of 
the students' prior knowledge and experience 
with the subject mater at hand. 
 
Directed Orientation The teacher provides activities that alow 
students to become more acquainted with the 
material being taught. 
 
Explication A transition betwen reliance on the teacher and 
students' self-reliance is made. 
 
Free Orientation The teacher is atentive to the inventive ability of 
the students. Tasks that can be approached in 
numerous ways are presented to the students. 
 
Integration The students summarize what was learned 
during the leson. 
 
The third piece of the geometry filter was through Driscoll?s (2007) geometric 
habits of mind. Thus, this filter provided insights to the geometric habits of mind that 
were supported by the teacher; and, of those, what the indicator(s) showed; basic or 
advanced. Table 3-7 on the next page shows the geometric habits of mind and their 
indicators. 
Table 3-7 Geometric Habits of Mind and Their Indicators (Driscoll, 2007, pp.12-15) 
Geometric Habit of Mind Indicators 
Reasoning with 
Relationships 
Basic: Identification of figures presented in a problem and 
correct enumeration of their properties 
Advanced: Relating multiple figures in a problem through 
proportional reasoning and reasoning through symmetry 
 
 
 
 
111 
Generalizing Geometric 
Ideas 
Basic: Uses one problem situation to generate another, or 
when the solver intuits that he or she hasn?t found al the 
solutions 
Advanced: Generate al solutions and make a convincing 
argument as to why there are no more; or wondering what 
happens if a problem?s context is changed 
Investigating Invariants Basic: Decides to try a transformation of figures in a 
problem without being prompted, and considers what has 
changed and what has not changed 
Advanced: Consider extreme cases for what is being asked 
by a problem 
Balancing Exploration and 
Reflection 
Basic: Drawing, playing, and/or exploring with occasional 
(though maybe not be consistent) stock-taking 
Advanced: approaching a problem by imagining what a 
final solution would look like, then reasoning backward; or 
making what Herbst (2006) cals ?reasoned conjectures? 
about solutions with strategies for testing the conjectures 
 
Below, Table 3-8 is a summary of the components of the three pieces of the geometry 
filter. I used this as a quick reference of al three pieces of the geometry filter on one 
page. 
Table 3-8 Geometry Filter 
Van Hiele Level Phase Geometric Habits of Mind 
Visualization Information Reasoning with Relationships (B/A) 
Analysis Directed Orientation Generalizing Geometric Ideas (B/A) 
 
 
 
 
112 
Informal 
Deduction 
Explication Investigating Invariants (B/A) 
Formal Deduction Free Orientation Balancing Exploration and Reflection (B/A) 
Rigor Integration  
 
This section described the three pieces of the geometry filter; van Hiele Levels, 
phases of learning from the van Hiele Model, and geometric habits of mind. The 
geometry filter adds to findings from each TACK analysis, as teaching and learning 
specific to geometry is highlighted in this lens for analysis. Analysis was an iterative 
proces, but there were two main categories of time when data analysis occurred; during 
data collection and after the observation or unit. The following section describes the 
methods to this proces in more detail.  
Analysis During Data Collection. Betwen pre-observation interviews and the 
actual observation of a leson, characteristics pertaining to what the teacher planned and 
what the teacher executed were noted in order to ask the teacher about it in the post-
observation interview. These characteristics are specific to the subcategories within 
TACK. Answers to pre-observation questions (Appendix B) were jotted down on either 
paper or noted in my electronic notepad. During the leson, when possible, notes were 
filed out on my electronic notepad as set up in the observation protocol (Appendix C). 
This section describes the proces in which pre-observation interviews and observations 
were analyzed in reference to its respective TACK aspect.  
For each day of observation, two lists of sequence of information were generated; 
one of what the teacher planned, and the other of what happened in clas. During data 
 
 
 
 
113 
collection, these lists were created informaly, sometimes without first transcribing, due 
to limited time betwen pre-observation interview, observation, and post-observation 
interview. An ?event? within sequence of information was when the teacher moved on to 
another example, problem, or activity. These statements were then grouped into ?events?. 
Often times during pre-observation interviews, the teachers did not mention going over 
homework or checking homework, but it was something that they did during the leson. 
Post-observation interviews revealed that the teachers planned for those activities, but did 
not fel like it was necesary to tel me, they were more interested in teling me what the 
leson was going to be for the day. Thus, when comparing what the teacher planned and 
what was executed, the sequence of events for executed was compiled first, and what 
teachers planned to do were noted next to what happened. Anything extra that the teacher 
planned to do was also noted next to where it should have been in the leson, or below if 
it did not fit; and this comparison document guided the post-observation questions. 
Teachers were asked how much time they planned to spend on events and phases. 
In general, phases for both teachers were belringer/work, checking homework, going 
over belringer/work, going over homework, leson, and summary. Using recordings after 
the observation before the end of the unit, I complied two lists; planned and executed, 
demarked by phases. Within each phase were the appropriate events, timing, 
explanations, questions, misconceptions, modifications, student prerequisite skils, and 
evidence of specialized content knowledge. This comparison alowed for a deeper look at 
the data so that I could ask teachers for clarification before the end of data collection. Al 
follow up questions from this proces have timestamps of where in the recording the 
 
 
 
 
114 
question arose for the teacher to watch to refresh their memory of what happened 
surrounding the moment in question.  
 Analysis After Data Collection for Each Observation/Unit. In general, analysis 
occurred in an organic iterative manner. Of Carney?s (1990) thre levels of analytical 
abstraction, at the level of summarizing and packaging data, I transcribed interviews and 
clasroom observations so I could just work with text. Also at this level, initial coding 
took place, where I highlighted segments of transcriptions and coded for the component 
of TACK. The second level of Carney?s (1990) three levels of analytical abstraction, 
repackaging and aggregating the data, was when I looked over al the highlighted 
segments and determined ways to sort within each aspect of TACK; the KCT, KCS, and 
SCK; that made the most sense for that aspect. Finaly, the third level, synthesis of data 
(Carney, 1990), was when I took what was determined in level two, and created the 
comparison of the two cases within data collected for each teacher; what was planned and 
what was executed during the leson. At each level were diferent ways in which I 
displayed data for the next level of analysis, and finaly drew conclusions. Before 
describing what was done to analyze data at each step, I briefly expounded on how I 
displayed the data in the following paragraphs. 
 Miles and Huberman (1994) described two main formats for displaying 
qualitative data; matrices and networks. Depending on what was needed to se 
connections betwen planning and execution of a leson; for some elements of TACK, 
matrices were used, and others, networks. Matrices have defined rows and columns, 
while networks have a series of nodes that link betwen them (Miles & Huberman, 1994). 
Matrices are helpful in understanding the ?flow, location, and connection of events? 
 
 
 
 
115 
(Miles & Huberman, 1994, p. 93). Within a network, the nodes are points and the links 
are the lines that connect them (Miles & Huberman, 1994). Networks are useful when the 
focus is ?on more than a few variables at a time? (Miles & Huberman, 1994, p. 94).  
Level 1. At this first level of formaly analyzing data, I used a two-column matrix 
for each data source; one column for the transcription and the other for notes on the 
transcription. What was in the notes column depended on the source of data. The sources 
of data being transcripts of concept map interviews, unit interviews, pre/post observation 
interviews, and observations. Each row of the matrix for interviews contained either a 
question and answer; or if the answer involved multiple events, like when a teacher goes 
through what they were going to teach that day describing multiple problems and 
activities, then a row for each event. For observations, rows separated phases, or when 
multiple events occurred in a phase, I separated those events into separate rows. These 
rows were an organizational tactic for data reduction (Miles & Huberman, 1994). For 
each row of notes, the first sentence gives a summary of what was in the transcription of 
that row, then explanations of highlighted portions of the transcription. Segments of 
transcription data that pertained to any of the eight a priori codes were highlighted, and 
the corresponding code was marked in the ?notes? column with a short description of the 
highlighted portion as it pertained to the code. Some highlighted portions have multiple 
codes, so descriptors distinguished them. 
 At this level of analysis, anything that could be taken as related to one of the eight 
a priori codes (Appendix A) were highlighted and coded. When the teacher mentioned 
order of examples, problems, or activities; those statements were coded as sequence of 
information executed. So during clasroom observations, when the teacher stated, ? next 
 
 
 
 
116 
we wil go on to?? that was coded as ?KCTseq-E?, for knowledge of content and 
teaching sequence executed. Whenever the teacher mentioned amount of time for a phase 
in the pre-leson interview; that was coded as ?KCTtime-P?, for time for phases planned. 
Time for phases of the leson for observations were coded from video and audio 
recording timestamps. The remaining codes follow the same proces. For full codebook 
including directions on unitizing and coding, refer to Appendix H. Table 3-9 shows the 
codes. 
Table 3-9 TACK with subdomains and listed codes 
 
Planning Execution 
Specialized 
Content 
Knowledge 
SCK-P 
Knowledge of concepts, 
definitions, and procedures 
for teaching 
 
SCK-E 
Knowledge of concepts, definitions, and 
procedures used in teaching 
Knowledge 
of Content 
and 
Teaching 
KCTseq-P 
Plan for sequence of 
information to be presented. 
 
KCTseq-E 
Sequence of information presented. 
KCTtime-P 
Planned time for phases of 
the leson. 
 
KCTtime-E 
Time spent on phases of the leson. 
KCTexplain-P 
How to explain the 
concepts, definitions, and 
procedures. 
 
KCTexplain-E 
Explained concepts, definitions, and 
procedures. 
KCTquestions-P 
Planned key questions to 
guide student thinking. 
KCTquestions-E 
Questions asked of students to guide their 
thinking. 
Knowledge 
of Content 
and 
Students 
KCSmod-P 
Leson modifications based 
on perceived student 
dificulties. 
 
KCSmod-E 
Did those student dificulties stil arise? If 
so, how was it shown? What did the 
teacher do in response? 
 
 
 
 
117 
KCSdificulties-P 
Other anticipated areas of 
student dificulties and how 
to help students overcome. 
 
KCSdificulties-E 
Actual presented areas of student 
dificulties and how the teacher helped 
students overcome. 
KCSprerequisite-P 
Anticipated student levels of 
prerequisite skils and how 
the teacher plans 
acordingly. 
KCSprerequisite-E 
Actual student level s of prerequisite skils 
and how the teacher taught acordingly. 
 
 Level 2. This level of analysis enabled me to take what was found in level one, 
and decide how to display and organize the data to beter se connections (Miles & 
Huberman, 1994). Since I already had matrices from level one, I decided to extend 
observation matrices by two more columns to the left to compare what the teacher 
planned with what actualy happened in the clas; one column for corresponding notes, 
and the other for the parts of the original transcription. So for each clasroom 
observation, I took the matrix from level one analysis and added al relevant parts of the 
corresponding pre-observation interview to the two newly created columns. The setup of 
columns is shown below: 
Planned Executed 
Parts of interview Notes Notes Clasroom Transcription  
    
 
I also looked through transcriptions of interviews and observations where the teacher 
stated anything relevant to the planning of the clasroom observation. For example, when 
I started with the level one analysis matrix of unit 1, leson 2, I looked back at al 
interviews and observations prior to unit 1, leson 2 for specific references to the 
 
 
 
 
118 
planning or execution of unit 1, leson 2. Those were also added to the planning columns, 
but they were noted for which source they came from as wel. Once this was complete, I 
added another two columns to the right; notes and transcriptions of corresponding 
reflection comments pertaining to what happened in clas. A similar proces was used in 
looking at other references to the observation after the post-observation interview. So the 
document containing most of the data for the leson consisted of three main columns; 
planned, executed, and reflection. Each of the columns were then divided into two 
columns, notes and transcription. The setup of the columns is shown below: 
Planned Executed Reflected 
Transcription Notes Notes Transcription Transcription Notes 
      
 
In some cases, multiple highlighted segments, when combined together, form a 
complete idea, like an explanation of a concept. These occurred within the multiple 
codes, creating multiple variables to display. With multiple variables, I knew from Miles 
and Huberman (1994) that I needed to generate a network containing nodes and links to 
se relationships. Nodes for diferent subdomains of TACK have diferent definitions, 
because of their diferent focuses. Below is a list of definitions for nodes specific to the 
subdomain of TACK. An additional copy of these definitions is available in Appendix G. 
Definitions 
? Nodes for sequences ? examples, problems, definitions, and theorems; i.e. each 
separate belwork problem as a node 
 
 
 
 
119 
? Nodes for explanations ? when a student asks a question, the teacher explains 
rather than caling on another student, or asks a question in return, then that is a 
node for explain. If the same explanation is given for the same question, whether 
or not asked by the same student; both count within one node. If the same 
explanation is given to another question, it is a diferent node. Explanations can 
also be unsolicited, each topic of explanation counting as a node. 
? Nodes for questions ? There are two situations where Mrs. Lotus poses questions 
related to content, to engage in student interaction or in response to student 
interaction. Each question eliciting a specific response from students is considered 
a node. Multiple questions eliciting the same specific response is also considered 
one node. Questions in response to student questions follow the same counting 
algorithm. 
? Nodes for modifications ? planned leson modifications are modifications like the 
choice of one example over another or a purposeful choice of a problem in order 
to bring out or avoid certain perceived student dificulties. Each modification 
counts as one node. 
? Nodes for dificulties ? each anticipated dificulty that the teacher does not plan 
on specificaly addresing, but acknowledges existence, is considered a node. 
Further, al dificulties that arise during the execution of the leson that was not 
previously verbalized to me is also considered a node. Same dificulty on multiple 
students at multiple times stil counts as one node. 
? Nodes for prerequisite skils ? each skil a teacher mentioned that the student 
should know either in interviews or during clas time, counts as a node. Same 
 
 
 
 
120 
prerequisite skils mentioned for multiple students at multiple times stil counts as 
one node. 
Level 3. Once nodes were established, level three was actualy putting the nodes 
into a network. For this level, notes from level two and one were constantly revisited. 
One method in which I used to analyze sequence data was with nodes on a two-column 
relationship graphic like what is shown below in Figure 3-1. 
 
Figure 3-1 Sample node diagram 
 
Nodes like this were similarly used to analyze teacher knowledge of explanations, 
questions, modifications, anticipated areas of student dificulties, and anticipated student 
prerequisite skils. This excludes teachers? knowledge of timing. Idealy, this sems like 
it should work for timing, however, in interviews the teachers repeatedly stated that they 
do not have specific times for problems, phases, or sometimes even days. 
Planned!
Bellringer!problem!1!
Bellringer!problem!2!
Bellringer!problem!3!
Homework!review!problem!4!
Homework!review!problem!13!
Homework!review:!review!of!
included!angle!and!included!side!
Executed!
Bellringer!problem!1!
Bellringer!problem!2!
Address!what!it!means!to!be!
corresponding!
Bellringer!problem!3!
Homework!review!problem!7!
Homework!review!problem!13!
Homework!review:!review!of!
included!angle!and!included!side!
 
 
 
 
121 
 Also using node terminology, matrices of the days of observations were 
summarized with transcriptions for easier comparison. For analysis levels 1 and 2, I kept 
transcriptions in the matrices as not to mis any important data. For ease of viewing and 
conclusion making as not to overload with text (Miles & Huberman, 1994), I left out 
original transcriptions and kept summary statements only. 
 
Sumary of Order of Analysis. First level of analysis involved a lot of coding. It 
was the initial examination of data to focus in on TACK. With the elements of TACK 
found in level one, level two was defining and determining how to find connections 
betwen planned and executed subdomain of TACK. Level three was putting the matrices 
and networks together in a way that is useful to draw conclusions. Each level was 
revisited multiple times so that data would not be mised. From matrices and networks 
created in levels two and three, conclusions about geometry content could be made as 
wel.  
Summary of Theoretical Perspectives and Methodologies 
This chapter detailed the theoretical perspectives and methodologies of this study. 
To set the background for the frameworks and methodologies for this study, I described 
how I saw this study contribute to scholarship as a theory-making endeavor (Silver & 
Herbst, 2007).  The theoretical perspective of this study is constructivism, where an 
active participant constructs knowledge when existing knowledge interacts with new 
experiences (Simon, 1993; Stefe & D?Ambrosio, 1995). Constructivism, as a theory of 
learning, aligns with why teachers need to teach in a way that supports students to learn 
 
 
 
 
122 
relationaly. Further, constructivism aligned with many of the teacher atributes of 
teachers of students that showed achievement. Thus, an important criterion for teachers 
for this study was that they were constructivist teachers.  
To study these teachers in a meaningful way, philosophical asumptions; 
ontological, epistemological, axiological, rhetorical, and methodological; were outlined. I 
used direct quotes and paraphrased in the words of the teachers to provide evidence of 
geometry Teachers? Applied Content Knowledge (TACK). I collaborated with teachers, 
spending time interviewing and observing, to create an acurate picture of geometry 
TACK at the time of data collection. In this chapter of the study, I openly discussed the 
methods used for data collection and analysis, complete with making my biases known. 
The language used in the following chapters follows that of qualitative research; 
narratives with occasional use of first-person pronouns. As detailed in the data collection 
and analysis, I worked with data collected from observations and interviews before 
drawing conclusions; and this was an iterative proces, continuously going back to the 
data.  
There were two main lenses for data analysis; TACK and geometry filter. TACK 
looked at specialized content knowledge, knowledge of content and students, and 
knowledge of content and teaching. Although high school geometry was the context for 
discussing TACK in this study, data analysis through TACK would not provide a 
comprehensive depiction of geometry TACK. Thus, data analysis through the geometry 
filter focused on the teacher?s knowledge of teaching and learning geometry, and the 
connections betwen planned and executed lesons. The next chapter discusses findings 
 
 
 
 
123 
from TACK and geometry filter first for the case of each teacher, and then I summarize 
findings from both teachers.  
 
 
 
 
 
 
124 
Chapter 4: Research Findings 
Over the course of two months, with a total of twenty school visits, I collected 
about 40 hours of data in the form of video and audio recordings. I observed Mrs. Orchid, 
a teacher at the high school, in one clas for four consecutive days of one unit and then 
for three consecutive days of a second unit. I observed Mrs. Lotus, a teacher at the middle 
school, in one clas for three consecutive days of two units. Each observation lasted about 
ninety minutes. Below is a tabular representation of data collected for each teacher. 
Table 4-1 Data Collected 
Teacher  Unit Observation 
Mrs. Orchid 
1 Pre, Post 
1 Pre, Post 
2 Pre, Post 
3 Pre, Post 
4 Pre, Post 
 
2 Pre 
1 Pre, Post 
2 Pre, Post 
3 Pre, Post 
Mrs. Lotus 
1 Pre, Post 
1 Pre, Post 
2 Pre, Post 
3 Pre, Post 
 
2 Pre, Post 
1 Pre, Post 
2 Pre, Post 
3 Pre, Post 
 
Preliminary data analysis occurred while data was stil being collected. Data from 
the pre-observation interview and the corresponding observation were analyzed in an 
informal manner before the post-observation interview with Mrs. Orchid. Circumstances 
 
 
 
 
125 
in scheduling did not alow for Mrs. Orchid to debrief with me within a month after the 
second unit. I decided that data collected after each day of observation and teacher self-
reflection for each day of a unit after my observations sufficed to give an acurate picture 
of what happened during the unit as a whole.  
Since each consecutive leson in a unit built upon the previous leson, I analyzed 
data as I was collecting it. Teachers? Applied Content Knowledge (TACK) and geometry 
filter, described in chapter 3, were used for analysis during data collection. As mentioned 
in the previous chapter, there were eight explicit questions regarding Teacher Applied 
Content Knowledge (TACK) shown during planning and execution of a leson. 
? What are the connections betwen planned and executed specialized 
content knowledge? 
? What are the connections betwen planned and executed sequence of 
information to be presented? 
? What are the connections betwen planned and executed time spent on 
phases of the leson? 
? What are the connections betwen planned and executed explanations of 
concepts, definitions, and procedures? 
? What are the connections betwen planned and executed key questions 
to guide student thinking? 
? What are the connections betwen planned and executed leson 
modifications? 
? What are the connections betwen planned and observed teachers? 
acknowledgement of student dificulties and how to overcome them? 
? What are the connections betwen planned and observed teachers? 
knowledge of student prerequisite skils? 
In this chapter, these questions and the geometry filter is discussed in the context of each 
of the units observed. 
 
 
 
 
126 
This chapter is organized in two main sections? Mrs. Orchid and Mrs. Lotus. For 
each teacher, I describe the context in which they taught before going into detail about 
the units. Discussions of each unit start with general information about the unit, and then 
the analysis of the lesons observed within the unit through aspects of TACK and the 
geometry filter, and findings that surfaced from the analysis.  
Mrs. Orchid 
Mrs. Orchid taught geometry twice a day, and the clas that I observed was the 
earlier of the two, imediately following the lunch period. Clas duration was about 90 
minutes. Due to compatibility of time, I generaly had an hour to conduct interviews 
before an observation. At times, when scheduling alowed, we scheduled post-
observation interviews on the same day as the observation. When this was not possible, 
we combined post-observation interviews with pre-observation interviews of the next 
observation. 
Her geometry clas was comprised of students from ninth through twelfth grade in 
high school. There were twenty-four students observed for each of the lesons of both 
units. Student desks were arranged in groups of four al front facing, but when it was time 
for group work, students turned desks to face each other. Her general routine for clas 
was belringer, check homework, go over bel ringer, go over homework, leson, and 
summary. Belringers were problems for students to work on independently at the 
beginning of clas with limited teacher instruction. Their purpose was to review 
prerequisite skils needed for the leson or to clarify something done the previous day.  
Mrs. Orchid graduated from a mathematics education program where one of the 
emphases of the program was effective use of technology, and she took it upon herself to 
 
 
 
 
127 
incorporate as much useful technology as possible. The technology that I saw in her 
clasroom included her teacher?s computer, laptop, document camera, SmartBoard, and 
graphing calculators. She also had some remaining whiteboard space that was not 
covered by the SmartBoard, for which she used to write some notes and the date.  
This introductory section described Mrs. Orchid?s clas I observed, time of day, 
the students, clasroom procedures, and technology available in the clasroom. The rest 
of this section is dedicated to Mrs. Orchid reporting findings from Mrs. Orchid?s units. 
First, the context for each unit wil be given, then concept maps, what Mrs. Orchid 
planned for the unit, and then what was observed and planned for each observed leson is 
discussed. Teacher?s Applied Content Knowledge (TACK) and findings through the 
geometry filter are discussed within the context of each unit. At the end of each unit is a 
summary of findings from the unit. At the end of both units is a summary of findings 
from both of Mrs. Orchid?s units and themes from her units.    
Unit 1. Mrs. Orchid was eager for me to observe the first unit. The two main 
reasons for this were that it included an activity for congruent triangles that she had used 
every year since she started student teaching, and the activity in which students 
discovered paralelogram properties showcased efective use of technology, which Mrs. 
Orchid considered an important part of the mathematics clasroom. However, this was 
the first time she had combined congruent triangles and quadrilateral properties in the 
same unit, using congruent triangles to prove paralelogram properties. So Mrs. Orchid 
was apprehensive about how the unit would work as a whole. The following sections 
describe the concept maps and the general plan for the unit before delving into the 
breakdown of what was planned and executed for lesons observed. 
 
 
 
 
128 
Concept Maps. Since the unit was about using congruent triangles to prove 
properties of quadrilaterals, she ended up making two separate concept maps, one for 
congruent triangles and the other for quadrilaterals. She started with describing to me 
what each category title meant to her and then added nodes in that category. Early on in 
the interview for this concept map, Mrs. Orchid asked how much she should share about 
her knowledge, and she stated that there would not be enough time for her to tel me 
everything. I asured her that she could add to it during any interview time asociated 
with the unit. So Mrs. Orchid explained concepts about congruent triangles that were 
directly related to the unit. She ended up not adding anything to this concept map 
throughout the interview, but referred to concepts when discussing parts of the unit. 
Figure 4-1 is the first of the two concept maps, congruent triangles. 
 
Figure 4-1 Mrs. Orchid?s concept map for congruent triangles 
 
 
 
 
129 
Mrs. Orchid started with the defining features of congruent triangles. She stated 
that congruent triangles have the same perimeter and have the same area and that 
corresponding sides and angles are congruent. Then she moved on to related features. She 
started with listing the triangle congruence shortcut theorems, side-side-side (SS), 
angle-angle-side (AAS), angle-side-angle (ASA), and side-angle-side (SAS). Then she 
wrote that the triangle congruence shortcuts were the ?minimum information to 
determine congruency? and connected the nodes. Also a part of related features was the 
side-side-angle (SSA) ambiguous case which she connected to the hypotenuse-leg 
theorem. Mrs. Orchid stated that the side-side-angle is the ambiguous case because it 
only works, statisticaly, half the time. Related to that is hypotenuse-leg, specific to right 
triangles. During discussion of related features, Mrs. Orchid constantly referred back to 
an application from architecture, which involved trusses. A truss is a structure with 
triangles sharing a side (Ching, 2011).  An example of a truss is shown in Figure 4-2. 
Figure 4-2 An example of a truss for the roof of a house 
 
She stated that this was the only application at the time of the first observation, because 
she asigned bonus points for students to share other real-world examples of congruent 
triangles, and the students had not turned in any yet. She planned on adding students? 
examples to discussions as students brought them. By the end of the unit, no student had 
brought extra examples. 
 
 
 
 
130 
Under related concepts, Mrs. Orchid started talking about quadrilateral properties 
and stated that she needed to make another concept map on quadrilaterals on a separate 
page. On the concept map for triangles, she listed the diagonal in a rectangle and special 
paralelograms. She stated that the diagonal in a rectangle creates two congruent 
triangles. Before moving on to creating the concept map for quadrilaterals, she added that 
the altitude from the vertex of an isosceles triangle creates two congruent triangles. 
For the concept map on quadrilaterals, Mrs. Orchid started talking about defining 
features and then the quadrilateral clasifications. She stated that there are a lot of 
defining features. Mrs. Orchid started defining features by stating that quadrilaterals have 
four sides. Then she stated that there are alternate definitions: rectangles, rhombi, and 
squares. Then she stated that the trapezoid and its special case, the isosceles trapezoid, 
are not like the rectangle, rhombus, or square, in that the later shapes required two pairs 
of paralel sides, whereas trapezoids only required one pair of paralel sides. The two 
related features were that there could be constructions of quadrilaterals based on one or 
more properties and that there were other alternate definitions of quadrilaterals. A related 
concept that Mrs. Orchid shared was that isosceles triangle properties were similar to 
isosceles trapezoid properties. She stated that she used the idea of cutting an isosceles 
triangle paralel to the base to show students that for an isosceles trapezoid, the base 
angles and nonparalel sides are congruent. Figure 4-3 is Mrs. Orchid?s concept map for 
quadrilaterals.  
 
 
 
 
131 
 
Figure 4-3 Mrs. Orchid?s concept map for quadrilaterals 
As Mrs. Orchid expanded the concept map for quadrilaterals, she began talking 
about the connections betwen using congruent triangles and quadrilaterals. She also 
discussed the new textbook being used and new content standards, which then led to her 
talking about how she taught and planned on teaching some of these concepts and how 
books and past content standards addresed those concepts.  
Unit 1 Planned. This unit was titled ?Using congruent triangles to prove 
properties of quadrilaterals.? Mrs. Orchid planned to start off the unit on the first day 
with an introduction to congruent triangles, continue on days two and three with an 
activity that built triangle congruence theorems, introduce quadrilaterals on day four, and 
continue to paralelogram properties on day five. She planned to discuss minimum 
 
 
 
 
132 
properties or characteristics of paralelograms on day six, special paralelograms with a 
focus on proofs on day seven, and trapezoids on day eight. She planned to discuss 
minimum properties or characteristics of paralelograms on day six, special 
paralelograms with a focus on proofs on day seven, and trapezoids on day eight. The unit 
would conclude with review and a test on days nine and ten, respectively. Mrs. Orchid 
stated that, in general, she plans with the knowledge that the number of days for a unit 
may be extended by a day or two, depending on the needs of her students. 
When asked why she sequenced the unit the way she did, she stated that since the 
point of the unit was to use congruent triangles to prove the properties of quadrilaterals, it 
would first be necesary to spend time establishing what it means for two triangles to be 
congruent. Mrs. Orchid stated that, from past experience, she knew it took several days 
for students to learn this wel. She expanded further by stating that the focus was not just 
on figuring out whether two triangles were congruent, but on proving that they were 
congruent by using triangle congruence theorems. Mrs. Orchid felt that triangle word 
problems in the book would solidify the idea that corresponding parts of congruent 
triangles were congruent and that it was esential to understand this idea before using it to 
explore and prove properties of quadrilaterals. 
Mrs. Orchid noted that even though the concept of congruent triangles was 
something students should have learned in eighth grade, she expanded their knowledge 
by focusing on reasoning and proving the congruence of triangles. Since the rest of the 
unit depended on student knowledge of congruent triangles, corresponding parts, and 
how to mark triangles as congruent, Mrs. Orchid decided that it would be beneficial to 
spend a full day on congruent triangles. The prerequisites for this unit that Mrs. Orchid 
 
 
 
 
133 
already covered earlier in the semester were basic familiarity with Geometer?s Sketchpad, 
triangle properties, and some basic algebra skils like being able to solve an equation for a 
variable (e.g., 3x + 7 = 10). The two examples of triangle properties that Mrs. Orchid 
gave were (1) the ability to reason that equilateral triangles are also equiangular and (2) 
that the sum of the interior angles of a triangle is 180?. As a general rule, Mrs. Orchid 
stated that she covered al prerequisite skils needed for a leson prior to the leson. This 
either occurred right before the leson, in the form of a belringer, or was something that 
was covered earlier in the semester. 
Acording to Mrs. Orchid, the general area of anticipated dificulty for this unit 
was the idea of sufficiency. From past experience with students, Mrs. Orchid stated that 
students generaly had a dificult time determining what was a sufficient amount of 
information for a definition. Mrs. Orchid pointed out that students usualy added 
redundant information. In terms of triangle congruency, Mrs. Orchid stated that it was 
easy for students to determine whether two triangles were congruent. However, in 
determining congruency of two triangles, understanding the minimum amount of 
information needed, and why it was the minimum amount was dificult for students. In 
order to develop the idea of sufficiency, Mrs. Orchid?s plan was to coach correct phrasing 
and ask whether information was useful. The other area of anticipated dificulty within 
this unit was proofs. In preparation for student areas of dificulty, Mrs. Orchid stated that 
working in groups and sharing results as a whole clas generaly helped struggling 
students. Other areas of anticipated dificulty were activity-specific and wil be discussed 
in the context of the observation days on which the dificulties became apparent. 
 
 
 
 
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Observed Lesons: Planned and Executed. In discussions with Mrs. Orchid, we 
decided that it would be best for me to observe the two-day activity (beginning with day 
two of the planned unit) on building triangle congruence theorems, as wel as the 
subsequent two days, during which the connections from triangle congruence to 
quadrilaterals were to be established. The activities of the first two days of observation 
were designed to build on the idea of identifying congruent parts of a triangle and the 
triangle congruence shortcut theorems. At the time of the unit interview, Mrs. Orchid had 
the two-day activity and paralelogram properties (days one, two, and four of the 
observation) planned. The third day of observation was slated to be an introduction to 
quadrilaterals, but she expected to plan that leson closer to when it was going to be 
taught. This unit is the only unit that I observed for four consecutive days. Al other units 
were only three days each. 
Mrs. Orchid caled the two-day activity developing triangle congruence shortcuts 
?Triangle in a Bag.? The Triangle in a Bag activity started with the premise that Mrs. 
Orchid had a triangle hidden in a bag. Students could not se the triangle. They were 
supposed to recreate that triangle by asking Mrs. Orchid questions. The answers to those 
questions were supposed to help students recreate the hidden triangle. Mrs. Orchid 
planned to repeat this proces five times, each time with a diferent triangle. Mrs. Orchid 
referred to the proceses with each triangle as ?rounds.? 
For each round, Mrs. Orchid had prepared a piece of paper that listed the 
atributes of the triangle: angle measures and side lengths. Also in the bag, were 
cardstock cutouts of the triangles to the specified dimensions stated on the cardstock. 
Vertices were labeled on the cutout. She used the cardstock to compare to student 
 
 
 
 
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drawings. Se Figure 4-4 for the first two triangles in the order prepared for presentation 
on Observation Day 1. 
 
Figure 4-4 First two triangles of Triangle in a Bag activity 
The goals for the first day of this activity was to get students to start asking 
questions and get a fel for what were productive questions and what were not. For 
example, one student asked if the sum of the interior angles added to one hundred eighty 
degrees. Anticipated dificulties also were observed, like asking for more information 
than what was necesary. For example, one group correctly drew triangle CAB, and their 
questions are listed next to Figure 4-5.  
 
 
  
Figure 4-5 Triangle and asociated student questions. 
 
C B 
7.3 cm 
8.8 cm 
7.7 cm 
72? 
52? 56? 
A 
What degres are each angle of the triangle? 
hat type of triangle is it? 
What is each side length? 
hat side is the longest side? 
 
 
 
 
 
136 
Mrs. Orchid stated that the question ?What type of triangle is it?? was unnecesary. The 
question ?What side is the longest side??  helped students label the vertices.   
Additionaly for the first day, Mrs. Orchid was hoping that students would be able 
to start reducing the number of questions to the minimum of three. She planned for 
students to go through about two rounds. Observations for Triangle in a Bag for this day 
of clas occurred as planned.  
For the second day of the Triangle in a Bag Activity, Mrs. Orchid planned to start 
the day by summarizing what students learned from the two rounds the day before and 
then continuing with the rounds. After the three remaining rounds, Mrs. Orchid planned 
on having students work on claswork from the Triangle in a Bag activity. The two 
questions on the claswork printout were: 
1. What must be true about two triangles in order for them to be congruent? 
2. What did the activity ?Triangle in a bag? have to do with congruent triangles? 
For the first question, Mrs. Orchid hoped that students would be able to state in their own 
words that in order for two triangles to be congruent, al parts have to be equal. For the 
second question, Mrs. Orchid wanted students to draw the connection that the triangles 
they drew and the triangle hidden in the bag were supposed to be congruent.  
After the first two questions were answered, Mrs. Orchid planned for groups to 
present diferent triangle shortcuts and fil out a chart summarizing the shortcuts. Mrs. 
Orchid postulated that there would not be enough time for students to present al five of 
the triangle congruence shortcuts (side-side-side, side-angle-side, angle-side-angle, angle-
angle-side, and hypotenuse leg for right triangles) if al five shortcuts were present. Due 
 
 
 
 
137 
to lack of time, Mrs. Orchid only had three groups present: side-angle-side, side-side-
angle, and angle-side-angle. 
When groups led the clas in filing out the summary boxes, Mrs. Orchid 
instructed students to highlight the parts of the triangle they were talking about to help 
other students understand the diferences betwen the shortcuts. Mrs. Orchid knew which 
groups to cal up because as students were geting the correct triangle with the correct 
three questions, she marked their papers with the triangle congruence shortcut they used 
without teling them why she did so. For each presentation, Mrs. Orchid reiterated, 
expanded, and questioned student statements to prompt them for proper vocabulary and 
development of concepts. Mrs. Orchid asked and explained al that she planned, in 
addition to eliciting student specific statements. 
 The third day of observation started off with the belringer addresing the side-
side-angle ambiguous case. Mrs. Orchid planned to start this day with a belringer for 
students to work on while Mrs. Orchid checked homework. The planned belringer posed 
two questions: 
1. How many non-congruent triangles can you create with the following 
measures? 
Angle A = 37? 
AB = 7cm 
BC = 4.5 cm 
2. Can this combination of measures be used to decide if two triangles are 
congruent? Why or why not? 
 
Mrs. Orchid also planned on providing students with the following Figure 4-6 to use. 
 
 
 
 
138 
 
Figure 4-6 Belringer Observation Day 3 
After discussing the belringer, Mrs. Orchid planned on discussing the reflexive 
property, that something is congruent to itself. The four problems Mrs. Orchid selected 
from the book for claswork were selected for content and context variations. Students 
solved these problems mainly by using the ?corresponding parts of congruent triangles 
are congruent? (CPCTC) relationship. The context for these problems ranged from 
courier service routes to airplane runway lengths. Mrs. Orchid wanted students to be 
exposed to something other than construction examples. During the pre-observation 
interview, Mrs. Orchid talked about how to solve each problem. The last problem asked 
for students to write a flow chart proof, and Mrs. Orchid stated that she had not covered 
that with her students and that she wanted her students to write a paragraph proof instead. 
There was only enough time left in clas for students to work on the first problem, the 
one in the context of couriers. Planned questions and explanations were observed in 
addition to others Mrs. Orchid stated while interacting with what students said and did. 
 The fourth day of observation also started with a belringer and review of the 
belringer. The belringer reviewed relationships of angle measures when two parallel 
lines are cut by another set of paralel lines not paralel to the first set. This belringer 
reminded students of the necesary prerequisite skils for the activity of the day. The 
activity was for students to discover properties of paralelograms via Geometer?s 
 
 
 
 
139 
Sketchpad. Mrs. Orchid introduced the activity with having students atempt to construct 
a paralelogram with Mrs. Orchid leading them to the correct steps before they went to 
the computer lab and discovered properties in pairs. After most students were done in the 
lab, Mrs. Orchid led discussion back in the clasroom on summarizing properties of 
paralelograms. Timing occurred as anticipated, Mrs. Orchid executed al that she 
planned. 
Unit 1 TACK. This section addreses three main ideas: specialized content 
knowledge, knowledge of content and teaching, and knowledge of content and students. 
These are the three aspects of Teachers? Applied Content Knowledge (TACK). For each 
aspect of TACK, I discuss connections betwen what was planned and what was 
executed for the unit as a whole. Connections betwen planned and executed aspects of 
TACK for each day of Mrs. Orchid?s Unit 1 are in Appendix J.  
Criteria for determining specialized content knowledge (SCK) are in the list of 
mathematical tasks of teaching in Appendix H under the ?Codes? section of the Coding 
Guide. However, for convenience, I list it below as wel. 
? presenting mathematical ideas, 
? responding to students? ?why? questions, 
? finding an example to make a specific mathematical point, 
? recognizing what is involved in using a particular representation, 
? linking representations to underlying ideas and to other representations, 
? connecting a topic being taught to topics from prior or future years, 
? explaining mathematical goals and purposes to parents, 
? appraising and adapting the mathematical content of textbooks, 
? modifying tasks to be either easier or harder, 
? evaluating the plausibility of students? claims (often quickly), 
 
 
 
 
140 
? giving or evaluating mathematical explanations, 
? choosing and developing useable definitions, 
? using mathematical notation and language and critiquing its use, 
? asking productive mathematical questions, 
? selecting representations for particular purposes, 
? inspecting equivalencies (Bal, Thames, & Phelps, 2008, p. 10). 
Modifying tasks to be easier or harder, and explaining mathematical goals and purposes 
to parents were not planned or observed throughout the days that I observed Mrs. Orchid. 
Interviews were always scheduled, so during interview time Mrs. Orchid did not 
communicate to parents in my presence. Thus, this occurred outside the realm of data 
collection for this study.   
During observations, Mrs. Orchid stated additional responses to students? ?why? 
questions. For example, during rounds with the Triangle in a Bag activity, Mrs. Orchid 
generaly responded to students? ?why? questions with explanations and questions. When 
Mrs. Orchid responded with questions, she was guiding students to answer their own 
questions. A group of students had what they thought was enough information to draw a 
triangle, but when they started drawing, they noticed it did not look like a triangle. They 
had the measures of two sides and an angle, but when drawn, it only looked like a very 
slanted ?v?. One student asked why it did not sem like a triangle. Mrs. Orchid?s 
response was ?How could you make that a triangle? What would you have to do?? A 
response like this was not stated during the interview prior to observation of the leson. 
For al other mathematical tasks of teaching, what Mrs. Orchid planned was observed.  
For knowledge of content and teaching (KCT) I looked at the sequence of 
information to be presented within a leson, timing of phases in the leson, explanations, 
 
 
 
 
141 
and questions. In looking at what was planned and observed, sequences of phases of 
lesons and timing of phases of the leson occurred as planned. For each observed day of 
the unit, additional explanation and questions were observed. For example, during the 
Triangle in a Bag activity, Mrs. Orchid gave additional explanations for geting students 
to determine the usefulnes of questions to minimize the amount of information needed. 
Some of these were specific to student questions, like the question of whether the interior 
angles of a triangle added up to 180?. Below is the conversation that followed after Mrs. 
Orchid saw the writen question ?Do the angles add up to 180??? 
Mrs. Orchid: no, you're not going to make me answer those questions 
{Students in the group laugh} 
Mrs. Orchid: I'm not going to let you ask me that question 
{Students in the group laugh} 
Student 1: wel, why not? 
Mrs. Orchid: Why am I not going to let you ask that question? 
Student 1: because its always going to be 
Mrs. Orchid: Yes, no, it?s always 
Student 1: 180 
Mrs. Orchid: Cause it?s a 
Student 1: Triangle 
Mrs. Orchid: yeah 
{Students in the group laugh} 
Student 1: oh 
Student 2: oh oh  
Mrs. Orchid: oh yeah  
rs. Orchid: okay, so what you two are doing look totaly diferent, so you might 
want to start talking and se, yeah? yeah? 
 
Mrs. Orchid responded to this situation later in clas by asking the students if knowing 
the interior angles of a triangle added up to 180? helped recreate the triangle in the bag.  
For knowledge of content and students (KCS), I looked at modifications to the 
leson, student misconceptions and or dificulties, and prerequisite knowledge students 
needed for the leson. For the first three days of observation, there were no planned or 
 
 
 
 
142 
observed modifications to the leson. On the fourth day of observation, the only 
modification to the leson was not alowing this clas of students to use trial and error on 
their own in the computer lab to figure out on Geometer?s Sketchpad how to construct a 
paralelogram. In the pre-observation interview, Mrs. Orchid stated: 
There have been some clases in the past where um I can send them to the 
computer lab and tel them to figure out how to construct a paralelogram [Me: 
Um hm] and give them hints walking around and they?ll work on it. I'm not too 
confident with their ability to do that, and the reason is I don?t have the time, so 
we?re going to talk about it as a clas. [Me: Um hm] And they?ll jot down notes 
on how to do it, [Me: Um hm] and then they?ll go to the computer lab and do it. 
For al observed days, additional student misconceptions and dificulties were observed, 
but Mrs. Orchid stated that she had sen those dificulties before. Al prerequisite 
knowledge that Mrs. Orchid stated that was needed for the lesons were observed.  
In Mrs. Orchid?s first observed unit, al TACK aspects that were shared during 
interviews before the unit and observations were observed. However, additional 
explanations, questions, student misconceptions, and student dificulties were observed. 
Mindful that this section explicated planned and executed TACK, and the focus of this 
study is on geometry teachers? applied content knowledge, the next section uses the 
geometry filter as a magnifying glas to examine Mrs. Orchid?s knowledge of teaching 
geometry from planned to executed.   
 Unit 1 Geometry Filter. This section expands on Mrs. Orchid?s teacher 
knowledge shown in Unit 1 through the geometry filter. The three aspects of the filter are 
van Hiele levels (Breyfogle & Lynch, 2010), phases of learning based on the van Hiele 
levels (Mistreta 2000), and geometric habits of mind (Driscoll, 2007). Teachers did not 
label van Hiele levels (Breyfogle & Lynch, 2010), phases of learning based on the van 
 
 
 
 
143 
Hiele levels (Mistreta 2000), and geometric habits of mind (Driscoll, 2007) in their 
interviews with me nor did they explicitly state it during observations. I identified each 
through descriptions in literature compared to clasroom observation and interview 
statements. Tables of each of the geometry filters are provided below in their respective 
sections. A complete list of the geometry filters is in Appendix I. Expanded explanations 
for the geometry filters are in Chapter 3. Appendix K provides daily acounts of van 
Hiele levels (Breyfogle & Lynch, 2010), phases of learning based on the van Hiele levels 
(Mistreta 2000), and geometric habits of mind (Driscoll, 2007) of observed lesons for 
this unit. The geometry filter provides the geometry context to TACK as the research 
question focused on geometry teachers? applied content knowledge of teaching geometry.  
Van Hiele Levels. For convenience, Table 4-2 shows the van Hiele Levels 
modified from Breyfogle and Lynch (2010, p. 234). Expanded explanations of the van 
Hiele Levels are in Chapter 3.  
Table 4-2 Van Hiele Levels abridged from Breyfogle and Lynch (2010, p. 234) 
Level Name Description 
 Visualization Se geometric shapes as a whole; do not focus 
on their particular atributes. 
1 Analysis Recognize that each shape has diferent 
properties; identify the shape by that property. 
2 Informal Deduction Se the interrelationships betwen figures. 
3 Formal Deduction Construct proofs rather than just memorize 
them; se the possibility of developing a proof 
in more than one way. 
4 Rigor Learn that geometry needs to be understood in 
the abstract; se the ?construction? of geometric 
systems. 
 
 
 
 
 
144 
From interviews, Mrs. Orchid stated that students should practice informal proofs 
by justifying their answers throughout the geometry course. This belief was based on 
research and profesional experience in mathematics education. In this unit, Mrs. Orchid 
planned for several opportunities for students to write formal proofs, one of them a 
question on claswork for the third observed day of the unit. However, due to time, that 
problem was asigned for homework. For the remaining days, I determined that planned 
and observed activities were at van Hiele Level 2, informal deduction. For example, in 
the Triangle in a Bag activity, students saw relationships betwen the hidden triangle and 
the triangle being created. Mrs. Orchid led students in discussions where they stated their 
reasoning and justified why the triangle they constructed was congruent to the triangle in 
the bag. Throughout the observations in the unit, Mrs. Orchid asked students questions to 
get them to justify their answers.  
Planned and executed informal deduction connected to various aspects of TACK 
from interviews and observations. During interviews and observations, Mrs. Orchid?s 
planned and observed key questions to guide student thinking, for the most part, were 
questions chalenging students to justify their answers. Some students that were having 
dificulty with the Triangle in a Bag activity initialy had problems recognizing properties 
of triangles, this being a problem I identified as van Hiele Level 1. Mrs. Orchid facilitated 
discussion about properties of triangles, before asking students to justify why asking 
about these properties of triangles were useful to create a congruent triangle; which I 
categorized as first discussing at van Hiele Level 1, and then Level 2. This showed Mrs. 
Orchid?s knowledge of van Hiele Levels and how to move students through the distinct 
levels for topics in geometry. Conversations with Mrs. Orchid after clasroom 
 
 
 
 
145 
observations verified my interpretation of her understanding of van Hiele Levels and 
moving students through the Levels.  
Phases of Learning Based on the van Hiele Model. For convenience, Table 4-3 
shows the phases of learning from the van Hiele model. Expanded explanations of the 
phases of learning based on the van Hiele model are in Chapter 3. 
Table 4-3 Phases of Learning from the van Hiele Model (Mistreta, 2000, p. 367) 
Phase Description 
Information Discussions are held where the teacher learns of 
the students' prior knowledge and experience with 
the subject mater at hand. 
Directed Orientation The teacher provides activities that alow students 
to become more acquainted with the material being 
taught 
Explication A transition betwen reliance on the teacher and 
students' self-reliance is made. 
Free Orientation The teacher is atentive to the inventive ability of 
the students. Tasks that can be approached in 
numerous ways are presented to the students. 
Integration The students summarize what was learned during 
the leson. 
 
From interviews and observations, I identified planned and executed phases of 
learning based on the van Hiele Model. All identified phases of learning based on the van 
Hiele Model planned were observed within the four days of observation. For the first 
three days of observation, I identified four phases planned and observed: directed 
orientation, explication, free orientation, and integration. Within interview and 
observation data on the four days of observation, I did not find evidence of the 
information phase. In discussions with Mrs. Orchid, I learned that an activity typical of 
 
 
 
 
146 
the information phase was planned and executed on the first day of the unit, the day 
before I started observing. On the first day of the unit, Mrs. Orchid asesed student prior 
knowledge of triangle congruence through discussions and activities where knowledge of 
triangle congruence, or lack of, would be apparent. Students worked on activities 
involving triangle congruence on the first three days of observation for this unit, thus the 
purpose of the activities representative of the information phase for this topic already 
occurred before I started observing. On the fourth day of observation, Mrs. Orchid asked 
students what they knew about defining and constructing paralelograms before starting 
the activity in the computer lab. I viewed this activity as showing the information phase 
for and how to construct quadrilaterals using Geometer?s Sketchpad.  
Looking at what Mrs. Orchid planned and executed through the phases of learning 
iluminated Mrs. Orchid?s knowledge of presenting mathematical ideas from the 
perspective of teaching geometry. Although Mrs. Orchid did not explicitly refer to the 
phases of learning when describing planned and executed, evidence of the phases 
emerged when Mrs. Orchid discussed sequence of information to be presented within a 
leson, presenting mathematical ideas, and connecting a topic being taught to topics from 
prior or future years. When Mrs. Orchid discussed the overal plans for the unit, I learned 
that the information phase would only be present in two days of the unit; triangle 
congruence and quadrilaterals. Al other phases were present in each observed day of the 
unit.  
The Triangle in a Bag activity, application problems, and paralelogram activities 
alowed students to become more acquainted with the material being taught. Each day, 
through various times of the day, Mrs. Orchid transitioned betwen reliance on her and 
 
 
 
 
147 
students? self-reliance. Evidence of this was revealed in Mrs. Orchid?s directions for what 
students should do throughout the activities and responses to students? questions. Each 
activity contained tasks that students could approach in a number ways. At the end of 
each day, Mrs. Orchid took at least five minutes to have students summarize what was 
learned during the leson.  
 Geometric Habits of Mind. For convenience, Table 4-4 shows Driscoll?s (2007) 
geometric habits of mind. Expanded explanations for the geometric habits of mind are in 
chapter 3.  
Table 4-4 Geometric Habits of Mind and Their Indicators (Driscoll, 2007, pp.12-15) 
Geometric Habit of Mind Indicators 
Reasoning with 
Relationships 
Basic: Identification of figures presented in a problem and 
correct enumeration of their properties 
 
Advanced: Relating multiple figures in a problem through 
proportional reasoning and reasoning through symmetry 
 
Generalizing Geometric 
Ideas 
Basic: Uses one problem situation to generate another, or 
when the solver intuits that he or she hasn?t found al the 
solutions 
 
Advanced: Generate al solutions and make a convincing 
argument as to why there are no more; or wondering what 
happens if a problem?s context is changed 
 
Investigating Invariants Basic: Decides to try a transformation of figures in a 
problem without being prompted, and considers what has 
changed and what has not changed 
 
Advanced: Consider extreme cases for what is being asked 
by a problem 
 
 
 
 
148 
Balancing Exploration and 
Reflection 
Basic: Drawing, playing, and/or exploring with occasional 
(though maybe not be consistent) stock-taking 
 
Advanced: approaching a problem by imagining what a 
final solution would look like, then reasoning backward; or 
making what Herbst (2006) cals ?reasoned conjectures? 
about solutions with strategies for testing the conjectures 
 
Table 4-4 included basic and advanced indicator for each of the geometric habits of mind. 
For this unit, at diferent portions of the leson, both basic and advanced were observed. 
In the daily observation acount of geometric habits of mind, unles otherwise stated, 
both basic and advanced were planned and executed.  
For al four days observed during this unit, I observed evidence of Reasoning with 
Relationships and Balancing Exploration and Reflection during planned and observed. 
Mrs. Orchid?s plan for the first day of observation only showed basic, but when Mrs. 
Orchid asked students to justify answers, some students used reasoning through 
symmetry, which I identified to be advanced. Mrs. Orchid stated that this discussion was 
not something she had planned for because students had not been exposed to 
transformations in her clas yet. However, she was pleasantly surprised. On the second 
day of observation, evidence from interviews and observations showed generalizing 
geometric ideas as both planned and observed. An unplanned discussion about triangle 
congruence that showed ?generalizing geometric ideas? occurred at the end of group 
work in the computer lab.  
Student: Mis Orchid? Are the triangles inside are they congruent? Because they 
have the same measurements from the diagonals, this line in this line 
Mrs. Orchid: So what shortcut trick are you using, what theorem are you using 
Student: side side side? 
Mrs. Orchid: Hm, very interesting, you're two days ahead of us 
Student: that?s shocking 
Mrs. Orchid: very good 
 
 
 
 
149 
 
Even though this was not planned, Mrs. Orchid showed the knowledge to answer 
students? questions. In addition, on the second day of observation, questions that I 
identified that fostered investigating invariants were not stated during planning but were 
observed. On the fourth day of observation, activities that I considered that fostered 
investigating invariants was both planned and observed. However, on the fourth day of 
observation, I did not se evidence to determine generalizing geometric ideas during 
planning but I saw evidence when the leson was observed. Observations of geometric 
habits of mind that were observed but not planned occurred in the form of additional 
questions Mrs. Orchid asked of some students who finished ahead of the majority of the 
clas. Once these questions surfaced during group work, Mrs. Orchid brought those 
discussions into the full clas discussions as wel. 
Even though Mrs. Orchid did not state geometric habits of mind that she was 
addresing in her leson, the fact that she was teaching geometry opens the possibility 
that instruction would adhere to some geometric habits of mind. Evidence for Mrs. 
Orchid?s plans for fostering geometric habits of mind emerged when she discussed 
presenting mathematical ideas, asking productive mathematical questions, linking 
representations to underlying ideas and to other representations, connecting topic being 
taught to topics from prior or future years, sequence of information to be presented, and 
key questions to guide student thinking. During observations, evidence of Mrs. Orchid 
fostering geometric habits showed in the TACK aspects in addition to responding to 
students? ?why? questions. 
 
 
 
 
150 
In general, what I identified during planning was observed during the executed 
leson. From planned to executed, additional identified phases of learning based on the 
van Hiele model and geometric habits of mind were observed.  For the third day of 
observation, Mrs. Orchid planned for informal and formal deduction, which I determined 
as van Hiele levels 2 and 3, but during clas, only van Hiele level 2 was observed.  
Findings from Mrs. Orchid Unit 1.  The general finding for Mrs. Orchid?s first 
unit was that geometry TACK was observed as planned. Through both TACK analysis 
and the geometry filter; sequence of activities, timing, questions, explanations, areas of 
student dificulties, modifications, student prerequisite knowledge, van Hiele Levels, 
phases of learning based on the van Hiele Level, and geometric habits of mind were 
observed as planned. Additional questions, explanations, student dificulties, phase of 
learning based on the van Hiele model, and geometric habit of mind beyond what was 
planned were observed. Additional questions and explanations semed to be a fairly 
strong trend for this unit, consistently appearing in each observed leson. Findings from 
the second unit add to the understanding of Mrs. Orchid?s geometry TACK planned and 
executed.  
Unit 2. Mrs. Orchid was excited to share this unit with me because she created a 
trigonometry activity that she had not taught before. She had not taught Law of Sines and 
Law of Cosines in high school geometry until Common Core State Standards, which Mrs. 
Orchid incorporated the year I observed her. Mrs. Orchid stated that she was eager for me 
to observe the activity she created but was apprehensive about the fact that she had not 
taught this unit before. She created this activity through verbal collaboration with other 
mathematics educators, research, standards, and textbooks. 
 
 
 
 
151 
Concept Map. For the concept map for the second observed unit, Mrs. Orchid 
recorded her voice in tandem with the construction of the concept map on her 
SmartBoard, but it was lost due to a computer virus. She waited a couple of days to try to 
send it once the problem was fixed, but there were stil technical isues. Therefore I only 
received the map without audio. Figure 4-7 is a re-creation of the concept map that Mrs. 
Orchid sent me. She did not use the categories of defining features, related features, 
related concepts, and applications like the concept maps she created for the first unit. I 
stil included this concept map to elucidate the concepts pertinent to this unit.  
 
Figure 4-7 Mrs. Orchid Concept Map Unit 2 
The focus of this unit was trigonometry. Coming from the main idea of 
trigonometry as shown on the map are right triangle ratios, right triangle relationships, 
Law of Sines, and Law of Cosines. These are concepts that Mrs. Orchid planned to teach 
in the trigonometry unit.  Concepts related to right triangle ratios are indirect measure, 
 
 
 
 
152 
similar triangles, sine, cosine, and tangent. Topics coming from right triangle 
relationships are geometric mean, 30? 60? 90? triangles, and 45? 45? 90? triangles. Since 
this diagram was not in front of Mrs. Orchid and me at the time of this interview, 
descriptions of the groups of nodes came from Mrs. Orchid?s explanation of concepts to 
be covered in the unit. 
Unit 2 Planned. The main focus of this unit was trigonometry. Due to a change in 
the school holiday schedule, Mrs. Orchid planned for more students to be absent and mis 
the day of the test for the previous unit. Thus, for the first day of this unit, Mrs. Orchid 
planned for some students to be taking a test from the previous unit and other students to 
be working on a worksheet reviewing simplifying radicals with a focus on square roots. 
For the second day of the unit, Mrs. Orchid planned for 45?-45?-90? triangles and for the 
third day of the unit, 30? -60? -90? triangles. On the fourth day, identifying opposite and 
adjacent legs were planned to be the belringer in introduction to tangent ratio. The fifth 
day of the unit was sine and cosine ratio. The sixth day was an activity caled ?A Pigpen 
for Monica,? calculating area of a triangle using the sine ratio. The seventh and eight day 
of the unit was Law of Sines and Law of Cosines. The ninth day was review, and the 
tenth day was the unit test. The three days I observed for this unit was the sixth, seventh, 
and eighth days of the unit. 
 Observed Lesons: Planned and Executed. When discussing this unit with Mrs. 
Orchid, she was very excited about teaching and sharing with me the Pigpen for Monica 
activity. Thus, she wanted me to observe the sixth, seventh, and eighth days of the unit, 
starting with Pigpen for Monica on the sixth day. Activities on the seventh and eighth 
days of the unit, Law of Sines and Law of Cosines, related to knowledge from Pigpen for 
 
 
 
 
153 
Monica activity. From here on out, I refer to days of observation, one through three, 
rather than the days of the unit.   
The first observation of this unit started with a belringer that consisted of four 
problems where variations on base and height were given to calculate the area of a 
triangle, which is half the base times the height. She decided to use four diferent 
triangles, as each one of them offered diferent points of discussion. The first point Mrs. 
Orchid wanted to deal with is the misconception students have that the base is on the 
bottom, so some of the triangles had the base on top. For the first and second problem, 
the base was not at the bottom, and the base and height measurements were both labeled 
and given. Figure 4-8 shows the orientations of the triangles in the belringer problems 
Mrs. Orchid planned. 
1. 2. 3.  4.  
Figure 4-8 Orientations of the triangles in the belringer problems 
 
For the third problem, the length of the base and the side measurement of the left side 
were given. Also, the angle in betwen the two was given. For this problem, the 
anticipated misconception was that students might confuse what looked congruent versus 
what is marked congruent. The height for triangle number three looked like it bisected the 
base, but it realy did not. Students were expected to use the tangent ratio in order to get 
the other side of the base, but Mrs. Orchid expected that some students would use sine 
ratio, cosine ratio, or the Pythagorean Theorem. The fourth problem had the height 
 
 
 
 
154 
outside the triangle, and Mrs. Orchid hoped that students would be able to notice that the 
triangle was obtuse. Each of these triangles had diferent talking points in order for Mrs. 
Orchid to addres possible misconceptions that students might have had that were crucial 
to the understanding of the Pigpen for Monica activity.  
Mrs. Orchid?s ? A Pigpen for Monica? activity posed that Sofia, a girl who is 
building a pigpen for her pig Monica, has two pieces of fencing of eight fet and six fet 
long, and the side of a barn of 24 fet long. The question is posed, ?What angle betwen 
the fences would provide Monica with the most area in her pen? How can Sofia find the 
area? How can Sofia be sure that she has determined the greatest possible area for 
Monica?? 
The first part of the activity planned for students to draw the shape of the pen and 
calculate the area given certain angles of atachment of the fences to each other. Mrs. 
Orchid wanted students to draw the pen before discussing the questions posed by the 
activity so that they could think about the application of the problem. She wanted 
students to discuss if they needed al 24 fet of the barn, leading to a discussion about 
triangle inequality. Mrs. Orchid wanted students to keep their belringer out so that they 
could make the connection betwen number three on the belringer and how to find the 
height to calculate the area for the problems in Part 1. She planned on discussing with the 
students that both eight and six were possible bases and that the area, regardles of the 
base chosen to calculate, would be the same. Having students calculate the area three 
times for diferent angles in Part 1 helped them to se the patern and gave them a reason 
to arrive at a formula. Three guiding questions at the end of Part 1 aided in the 
development of the concept of rewriting the formula for the area of a triangle, ? base x 
 
 
 
 
155 
height, to ? base x aSin C. Figure 4-9 show what the leters ?a? and ?C? represent on a 
triangle, in the rewriten formula for area of a triangle ?base x aSin C?. 
 
 
Figure 4-9 Triangle with angles and sides labeled for reference to the formula for area 
Part 2 of the activity explored whether or not the formula always works. The first 
question asks students to sketch the pigpen using 120? and draw the altitude. The second 
question asks students to determine the height. This is where the students could not draw 
the altitude inside the triangle, and where Mrs. Orchid hoped that students would draw 
the connection to number four on their belringer. In order to calculate the height, since 
students were using a trigonometry chart that only goes up to 89?, Mrs. Orchid had 
students use the supplement of 120?, 60? to find the height. Figure 4-10 shows the 
relationship betwen the supplements and the height. 
 
Figure 4-10 Measure of angle A is 60?, the supplement of 120?. Height is side a. 
 
a 
b
c 
A 
a 
 
 
 
 
156 
Mrs. Orchid planned for students to make the connection that the sine ratio of 
supplements are the same, but she was not prepared to delve deeper into understanding 
why they are the same due to time. Part 3 has four questions: 
1. What angle measurement would yield the greatest area for the pigpen? 
Explain. 
2. How can Sofia be sure that it is the greatest area possible? Explain. 
3. Sofia discovered that she has only enough hay to cover 22 ft
2
 and Monica 
must have hay covering the entire area of her pigpen. What angle should Sofia 
use to atach the fences so that she can cover the entire area of Monica?s pen 
with hay? 
4. Sofia?s dad said that she can use only 1/3 of the barn wal for the pigpen. Does 
this change your solution? If yes, explain and find the new solution. If no, 
explain why it would be the same solution. 
The question posed at the beginning of the activity finaly is posed in question 1 of part 3: 
?What angle measurement would yield the greatest area for the pigpen? Explain.? 
Question two: ?How can Sofia be sure that it is the greatest area possible? Explain.? This 
one is an informal proof for the first question. The third question was created, because 
after this whole activity, Mrs. Orchid stated that she realized that the answer did not 
require students to use the sine relationship. She changed the third question so that 90? 
could not be the answer. Mrs. Orchid stated that due to time, she thinks it is aceptable 
not to cover number four when doing this activity. She had not taught this leson before 
so she was unsure of the pacing. 
 
 
 
 
157 
 After the first day of observation, Mrs. Orchid realized that the Pigpen for Monica 
activity was going to take longer than expected. Thus, she decided to spend an extra day 
on Pigpen for Monica, the second day of observation, and move on to Law of Sines for 
the third day. The Law of Sines activity started with Mrs. Orchid discussing with the 
clas the requirements of a triangle in order to use the sine ratio and what they knew they 
could do if an altitude was drawn. Students were given triangle ABC, and then they were 
instructed to draw height k from vertex B, shown in Figure 4-11. 
 
Figure 4-11 Figure of Triangle Labeled for students to set up Law of Sines 
 
Then Mrs. Orchid led students through seting up the sine ratio for sin A and sin C, which 
is sin A= 
!
!
 and sin C= 
!
!
 . Solving for k for each of them yielded k = c(sin A) and k = 
a(sin C), which then meant that c(sin A) = a(sin C), or 
!"#!!
!
 = 
!"#!
!
 . Similar steps are 
taken with B. Deriving the Law of Sines this way connected to the Pigpen for Monica 
activity and helps develop a conceptual understanding of the Law of Sines. For the first 
and third days of observation, what was planned was observed. Discussions ran longer 
than expected on the second day, so students finished the Pigpen for Monica activity on 
the third day of observation.  
k 
a 
c 
b 
 
 
 
 
158 
Students experienced al the activities that were planned for this unit. The major 
changes to plans were that A Pigpen for Monica took three days instead of one, Law of 
Sines only took half a day, and review of the unit shared a day with Law of Cosines, so 
the unit test was given on the day that was originaly planned. Throughout each day, 
homework took longer than anticipated, and student participation was much lower than 
Mrs. Orchid expected. Mrs. Orchid viewed this clas as the wel-behaved clas, in that 
they usualy did their work and turned in homework. However, for this unit, they were 
les engaged, and Mrs. Orchid believed it was because the semester was about to end.  
Unit 2 TACK. Of the three days of this unit, only the first day was executed as 
planned during the unit interview. The two other observed days of the unit each took 
more time to do the activities anticipated. Interviews before the observation each day 
gave Mrs. Orchid the opportunity to share changes made to the observation day from the 
initial plan during the unit interview. Compared to what Mrs. Orchid planned for each 
day of observation right before the observation, the first and third days of observation 
ocurred as planned. This section addreses three main ideas: specialized content 
knowledge, knowledge of content and teaching, and knowledge of content and students. 
These are the three aspects of Teachers? Applied Content Knowledge (TACK). For each 
aspect of TACK, I discuss connections betwen what was planned and what was 
executed for the unit as a whole. Connections betwen planned and executed aspects of 
TACK for each day of Mrs. Orchid?s Unit 2 are in Appendix J.  
From each day?s pre-leson interview, observation, and reflection interview, al 
planned sequences, explanations, and questions were observed. Timing for the first and 
third observations was observed as planned, but homework review went too long for the 
 
 
 
 
159 
second day. Additional explanations, questions were observed. These additional 
explanation and questions were in response to students? questions and additional areas of 
dificulty. 
Al planned student dificulties and prerequisite knowledge were observed 
throughout the observed days of the unit. Additional student dificulties were also 
observed. For example, on the second observation day, Mrs. Orchid did not anticipate to 
go over how to use the calculator for trigonometry until students asked about it. 
Modifications to the leson for Observation Day 3 were made based on student behavior 
and limited time. The Pigpen for Monica activity took longer than expected, so she 
created a way to help students review what was learned the previous days without taking 
more time than necesary.  
Similar to Mrs. Orchid?s first unit, explaining mathematical goals and purposes to 
parents and modifying tasks to be easier or harder were not planned or observed. Even 
though Mrs. Orchid modified the activity on the third day, the modification did not 
change the dificulty of the activity. Since Mrs. Orchid developed her own activities for 
the days of observation for this unit, she did not discuss appraising and adapting the 
mathematical content of textbooks. For the remaining mathematical tasks of teaching, 
what was planned was observed.  
For this unit, al of what was planned was observed. Additional student 
dificulties, explanations, and key questions were observed. This section looked at 
planned and observed TACK. The next section expounds upon the geometry aspects of 
planned and observed TACK using the geometry filter.   
 
 
 
 
160 
 Unit 2 Geometry Filter. Since this unit was on trigonometry, only relevant 
aspects of the geometry filter wil be discussed.  Similar to the discussion for Mrs. 
Orchid?s Unit 1 geometry filter, I determined van Hiele levels (Breyfogle & Lynch, 
2010), phases of learning based on the van Hiele levels (Mistreta, 2000), and geometric 
habits of mind (Driscoll, 2007) through descriptions in literature compared to clasroom 
observation and interview statements.  
Throughout the observed lesons, from interviews and observations, Mrs. Orchid 
asked students to justify their answers in discussions, which I identified as van Hiele 
Level 2, informal deduction (Breyfogle & Lynch, 2000). Evidence that led me to 
determine van Hiele Level 2 in relation to evidence for TACK appeared in planned and 
executed presenting mathematical ideas, responding to students? ?why? questions, and 
questions to guide student thinking. On the first observed day, during the Pigpen for 
Monica activity, Mrs. Orchid asked students to identify the shape of the pigpen. I 
determined that identifying the shape of the pigpen showed van Hiele Level 1, analysis. 
TACK aspects that provided evidence for this van Hiele Level was in planned and 
executed presenting mathematical ideas and questions to guide student thinking.  
As for phases of learning based on the van Hiele Model (Mistreta, 2000), 
evidence from interviews and observations led me to conclude that al three observed 
days showed directed orientation, explication, and free orientation. The Pigpen for 
Monica activity and Law of Sines activity provided students with the opportunity to 
become more acquainted with the material being taught. Evidence of orientation in 
relation to TACK aspects was presenting mathematical ideas and sequence of 
information to be presented. I determined that explication was planned and executed in 
 
 
 
 
161 
al three days of observation, and evidence of this from TACK was presenting 
mathematical ideas, directions for the activities, and key questions to guide student 
thinking. Most of the problems in A Pigpen for Monica activity were atentive to the 
inventive ability of students. Evidence of this in relation to TACK aspects appeared when 
Mrs. Orchid discussed presenting mathematical ideas and sequence of information to be 
presented. I determined that integration was planned for al three days of observation, but 
it was only observed on the first and third days of observation. On the second day of 
observation, Mrs. Orchid ran out of time, and students were not prompted to summarize 
what they had learned that day. In relation to TACK evidence, Mrs. Orchid did not 
question for students to summarize, and the planned sequence of summarizing what was 
learned that day was not observed.  
Evidence from interviews and observations led me to determine that each day of 
observation emphasized a diferent geometric habit of mind (Driscoll, 2007). On the first 
day, Mrs. Orchid prompted students to reason with relationships, primarily in geting 
students to draw the shape of the pigpen. Also during the Pigpen for Monica activity, 
Mrs. Orchid asked students if they could think of other angle measures where the sines of 
those angles are equivalent. This showed the geometric habit of mind caled generalizing 
geometric ideas (Driscoll, 2007). The plan for the second observation day during the Law 
of Sines activity was for students to make reasoned conjectures about what the 
relationship would be, but this plan did not occur. This exemplified balancing 
explorations and reflections. However, on the third observation day of this unit, during 
the Law of Sines activity, the activity was planned and observed. Al geometric habits of 
mind for this unit were connected to TACK in presentation of mathematical ideas, 
 
 
 
 
162 
responding to students? ?why? questions, sequence of information to be presented, and 
key questions to guide student thinking.  
Findings from Mrs. Orchid Unit 2.  The general finding for Mrs. Orchid?s 
second unit was that geometry TACK was observed as planned; with the exception of 
timing and sequence not occurring as planned for the second day of observation. Through 
both TACK analysis and the geometry filter for al lesons observed; questions, 
explanations, areas of student dificulties, modifications, student prerequisite knowledge, 
van Hiele Levels, phases of learning based on the van Hiele Level, and geometric habits 
of mind were observed as planned. Additional questions, explanations, and student 
dificulties, were observed. Additional questions and explanations semed to be a prety 
strong trend for this unit as wel, consistently appearing in each observed leson.  
 Summary of Unit 1 and Unit 2. For both units, in general, what Mrs. Orchid 
planned was observed. Additional explanations, questions, student dificulties, and 
prerequisite knowledge were observed. In reflecting with Mrs. Orchid, these were due to 
student input. Additional student dificulties were observed with additional explanations 
and questions from Mrs. Orchid. Otherwise, in general, what students said, did, or asked 
caused Mrs. Orchid to add explanations and questions. Most lesons did not have 
modifications. The modifications to the lesons that were made were due to time and 
student involvement. The modification to a leson in the first unit was due to student 
behavior. Part of the decision for modification in the second unit was due to the fact that 
students did not have the prerequisite knowledge that Mrs. Orchid had asumed in her 
plans, and thus she was wiling to stray from the plans in order to make sure students 
understood before moving on to the next topic. 
 
 
 
 
163 
 Modifications to lesons in both units did not change planned or executed aspects 
of the geometry filter. Throughout both units, Mrs. Orchid showed evidence that she 
fostered al geometric habits of mind and used al phases of learning based on the van 
Hiele Model. Al van Hiele Levels planned were observed. Through interactions with 
students, Mrs. Orchid was observed facilitating additional van Hiele Levels. Consistency 
in planned and executed, without sacrificing cognitive demand, for pieces of the 
geometry filter showed Mrs. Orchid?s deep knowledge of teaching and learning 
geometry. Throughout data analysis of both of Mrs. Orchid?s units, some themes 
emerged.  
 Themes from Unit 1 and Unit 2. Certain paterns specific to Mrs. Orchid 
appeared through the multiple iterations of combing through data. These paterns were 
related to themes consistent with both teachers: timing, questioning, familiarity with 
activities used, areas of apprehension, and sources of knowledge. In six of the seven 
observed lesons, at least one phase within the leson exceded the time Mrs. Orchid?s 
anticipated. In every observed leson, Mrs. Orchid restated planned key questions at least 
three times, with many scafolding questions, to get students to give answers that 
?showed understanding?. From interviews with Mrs. Orchid about timing and teacher-
student dialogues in clas, she stated that her students needed more wait time in order to 
verbalize their understanding. From my observations, this showed Mrs. Orchid?s 
persistence to get students to make those connections of new material to what they know, 
showing relational understanding.  
Mrs. Orchid was very familiar with many of the activities used in the first unit. 
However even in those lesons, there were stil areas of apprehension. Even though she 
 
 
 
 
164 
used the activity Triangle in a Bag many times before, she had not used a document 
camera with students presenting before. Further, in the first unit, she had not taught the 
activities within the first unit together before, and was apprehensive about how the unit 
would fit together as a whole.  Mrs. Orchid was apprehensive about sharing the second 
unit with me, because she had not taught the content or the activity before; but wanted to 
share the unit because it contained a leson she created, A Pigpen for Monica. 
Throughout al areas of apprehension, her vast sources of knowledge provided her with 
multiple resources to use in planning and execution of the activities, lesons, and units.  
Mrs. Lotus 
 Similar to Mrs. Orchid, Mrs. Lotus taught the high school geometry course, but 
she taught the course to eighth graders in a junior high school. She taught geometry twice 
a day, once at the beginning of the day and the other after lunch. The clas that I observed 
was the one after lunch. The geometry clas I observed was also about ninety minutes 
long. Her available time to met me was within two hours after the time for the clas I 
observed. Thus, pre-observation interviews were conducted about 20 hours before each 
observation. I was able to code pre-observation interviews prior to the observation, but I 
had to take notes during observation and imediately compare pre-observation data to 
what just happened in the clasroom in order to conduct the post-observation interview. 
Fortunately, I was able to ask in subsequent reflections what I was not able to ask right 
away. 
Throughout al my observations, there were 21 students. Students sat in groups of 
four at the long sides of two long tables placed together. Her general routine for clas was 
also belwork, check homework, go over belwork, go over homework, leson, and 
 
 
 
 
165 
summary. However, belwork problems were review of skils for students to work on 
independently at the beginning of clas with limited teacher instruction. Usualy Mrs. 
Lotus used belwork to refresh skils from two lesons prior, but sometimes it was to 
review a concept that semed important, but not necesarily tied to the leson.  
Technology that was available in her clasroom was an interactive whiteboard, 
document camera, and teacher computer. Also, through a program at the school, each 
student had an asigned laptop from the beginning of the semester, so they used it on a 
daily basis in her clasroom. Mrs. Lotus used al available technology on a daily basis, 
from students adding to notes provided by Mrs. Lotus on their laptops to Mrs. Lotus 
showing students geometric constructions on a dynamic geometry program on the 
interactive whiteboard while they construct using the same program on their laptops. Mrs. 
Lotus stated that technology was a big emphasis at her school.  
This introductory section gave the context for Mrs. Lotus?s clas I observed, time 
of day, students, clasroom procedures, and technology available in the clasroom. The 
remainder of the section focuses on findings from Mrs. Lotus? two observed units. First, 
the context for each unit wil be given, then concept maps, what Mrs. Lotus planned for 
the unit, and then what was observed and planned for each observed leson is discussed. 
Teacher?s Applied Content Knowledge (TACK) and findings through the geometry filter 
are discussed within the context of each unit.  
 
Unit 1. Mrs. Lotus wanted me to observe this unit because of her familiarity with 
the content and the interesting activities that she typicaly does with the students for this 
unit. In the unit interview, Mrs. Lotus stated that she looked for new projects for some of 
 
 
 
 
166 
the days within this unit, but upon further reflection, she decided that the original project 
that she used in years past beter addresed what students needed to learn.  
 Concept Map. During the initial meting with Mrs. Lotus, she understood the 
example of the sample concept map I gave her, and she was able to produce a sample one 
for the subject squares. For the actual study, she atempted a concept map for the first 
unit. She started with an oval with the word ?triangles? in the middle, and from that oval, 
links were drawn to other ovals, each connecting to the oval ?triangles? only. These other 
ovals contained the words inequalities, special segments, congruence tests, and 
clasifications. She stated that it was too mesy and did not want to continue on that sheet 
of paper. With encouragement to continue, she politely stated that she would think about 
it and redraw it later. She took the paper and asked for the concept map not to be 
recorded. During subsequent interviews, when I asked her to add to the concept map, she 
stated that she had nothing to add. 
Unit 1 Planned. The main idea of this unit was to introduce students to properties 
of triangles and triangle congruence. For the first day of the unit, Mrs. Lotus planned to 
learn students? prior knowledge, establish vocabulary terms for triangles by sides and 
angles, and work on proving conjectures related to triangle sum conjecture and interior 
and exterior angles. On the second and third day of the unit, Mrs. Lotus planned for 
students to work on an activity that developed conceptual understanding of diferent 
points of concurrency on a triangle. Mrs. Lotus stated that this activity might be a two-
day leson. The next concepts before the unit test were triangle inequality and triangle 
congruence. Triangle inequality was supposed to take a day, and triangle congruence was 
supposed to take two days. Mrs. Lotus decided that it would be best for me to observe the 
 
 
 
 
167 
activity that develops conceptual understanding of diferent points of concurrency of a 
triangle and triangle inequality. By the start of the first day of observation, Mrs. Lotus 
noticed that there were some things from the first day of the unit that needed to be 
addresed on the second day of the unit. Thus, the first day of observation was going to 
be dedicated to notes and proofs, and the second and third days of observation were going 
to be spent doing the activity that developed conceptual understanding of diferent points 
of concurrency on a triangle. 
Observed Lesons: Planned and Executed. On the first observed day of the unit, 
Mrs. Lotus reviewed the notes from the day before on definitions and naming parts of 
triangles, proved some conjectures related to the triangle sum theorem, and started on the 
Equaly Wet problem. The triangle sum theorem states that the sum of the interior angles 
of a triangle is 180?. The corollaries that Mrs. Lotus asked students to prove were: 
1. The acute angles of a right triangle are 
complementary  
 Given: ?ABC with m?C = 90? 
  
Prove: m?A + m?B = 90? 
 
2. What is the sum of the measures of the exterior angles of a triangle? Prove 
your answer.  
 
 
 
 
168 
 
The Equaly Wet problem posed that a girl named Leslie planted three flowers, and 
needed to place a sprinkler so that al three flowers could get equaly watered by being 
equaly spaced from the sprinkler. The three questions that the problem posed were: 
1. Determine which arrangement of flowers wil make this possible and which are 
impossible. 
2. For the arrangements found for which it wil be possible, describe how Leslie can 
find the correct locations for the sprinkler. 
3. Can you generalize this problem? 
Mrs. Lotus gave students a few minutes at the end of clas to discuss the first two 
questions from this problem and asked students to spend at most 15 minutes working on 
the first two problems for homework.  
 For the next day, Mrs. Lotus started the clas by discussing their ideas about 
problem numbers 1 and 2 from Equaly Wet. During student discussion, diferent 
atempts at constructing the middle from three points alowed Mrs. Lotus to transition to 
an activity on Geometer?s Sketchpad in which students constructed orthocenters, 
incenters, and circumcenters. On the third day of observations, students explored the 
other points of concurrency in a triangle, summarized the diferent points of concurrency, 
and practiced with some real world and textbook problems. Each day of the observation 
was executed as planned.  
A 
B 
C 1 2 
3 
4 
6 
5 
 
 
 
 
169 
 Unit 1 TACK. Similar to both of Mrs. Orchid?s unit Teachers? Applied Content 
Knowledge (TACK) analysis, codes for specialized content knowledge, knowledge of 
content and teaching, and knowledge of content and students were determined from 
literature prior to collecting data for this study. Analysis was conducted in the procedure 
discussed in chapter 3. For the observed lesons in the unit, timing and sequencing went 
exactly as planned by Mrs. Lotus. Al planned explanations, questions, and student 
dificulties were observed. Additional student dificulties were observed, and Mrs. Lotus 
provided corresponding additional explanations and questions that had not been planned. 
Also, due to students? specific questions related to how they were justifying an answer or 
writing a proof, Mrs. Lotus showed additional explanations and questions. Al specialized 
content knowledge shown during planning was observed during the execution of the 
leson.   
For this unit, al of what was planned was observed. Additional student 
dificulties, explanations, and key questions were observed. The additional explanations 
and questions were in response to additional student dificulties. This section focused on 
planned and observed TACK. The next section expands on the geometry aspects of 
planned and observed TACK using the geometry filter.   
 Unit 1 Geometry Filter. From what I gathered on interviews and observations and 
put through the geometry filter; evidence for levels, phases, and habits of mind that I 
identified in what Mrs. Lotus planned was observed for each observation day. 
Throughout diferent phases of instruction in each observed day, Mrs. Lotus asked 
students to justify their answers, which showed van Hiele Level 2, informal deduction. 
For example, on the second day of observation, during the discussion of the findings for 
 
 
 
 
170 
the Equaly Wet problem, when students presented their solutions on the board, Mrs. 
Lotus asked students why they thought their solution worked in al cases. Evidence from 
TACK that showed van Hiele Level 2 during observations explanations and key 
questions to guide student thinking. Mrs. Lotus stated that she planned on having students 
prove conjectures formaly when talking about presenting mathematical ideas and 
sequence of information to be covered during the day. During the observation, Mrs. Lotus 
asked students to prove two conjectures formaly. Thus, this showed van Hiele Level 3, 
formal deduction.  
 Throughout the three observed lesons, the phases of learning based on the van 
Hiele Levels that I determined were both planned and observed through interviews and 
observations were directed orientation, explication, free orientation, and integration. 
From initial interviews with Mrs. Lotus, her philosophy of what phases of learning 
should be present each day of instruction corresponded with the phases of learning based 
on the van Hiele Levels. For example, on the first day of observation, problems during 
notes and discussion for the homework activity provided students with the opportunity to 
become more familiar with the material. Then on the second and third days of 
observation for directed orientation, Mrs. Lotus?s diferent construction activities 
provided students with opportunities to become more acquainted with the material. The 
phases of learning were stil present in each day, even though the activities were 
diferent. This showed Mrs. Lotus?s specialized content, knowledge in presenting 
mathematical ideas, and knowledge of content and teaching, sequence of information to 
be presented.  
 
 
 
 
171 
For each observed day of this unit, I determined that Mrs. Lotus consistently 
showed fostering three geometric habits of mind: reasoning with relationships, 
generalizing geometric ideas, and balancing exploration and reflection. Like the activities 
for the phases of learning, activities that exemplified the geometric habits of mind were 
diferent, but those habits of mind were stil consistently present in al three days of 
observation. For example, on the first day of observation, students reasoned with 
relationships of the sum of interior angles of a triangle, interior and exterior angles of a 
triangle, exterior angles of a triangle, and possible points equidistant from two and three 
points. On the second day of observation, students reasoned with relationships of the 
intersection of the three perpendicular bisectors of a triangle and the intersection of three 
angle bisectors of a triangle. Similar to phases of learning, this showed knowledge in 
presenting mathematical ideas and sequence of information to be presented; specialized 
content knowledge and knowledge of content and teaching respectively. However, in 
addition, basic and advanced levels of the geometric habits of mind were planned and 
observed, and evidence betwen basic and advanced were in Mrs. Lotus?s activity 
directions and key questions to guide student thinking. The phases of learning based on 
the van Hiele Model and geometric habits of mind stayed consistent for al three days of 
observation. 
Findings from Mrs. Lotus Unit 1.  The general finding for Mrs. Lotus?s first unit 
was that geometry TACK was observed as planned. Through both TACK analysis and 
the geometry filter for al lesons observed; sequence of activities, timing, questions, 
explanations, areas of student dificulties, modifications, student prerequisite knowledge, 
van Hiele Levels, phases of learning based on the van Hiele Level, and geometric habits 
 
 
 
 
172 
of mind were observed as planned. Additional questions, explanations, and student 
dificulties, were observed. For every leson of every observed day of this unit, Mrs. 
Lotus planned and executed every phase of learning based on the van Hiele Model.  
 
 Unit 2 Concept Map and Overview. For the concept map interview of this unit, 
Mrs. Lotus showed me a list of lesons to be covered for this unit. When I inquired about 
how this would fit into a concept map, she stated that the list was the concept map. They 
were in sequence of lesons in the unit. When I asked her to elaborate, she started talking 
about the unit and how each leson builds on the topic in the unit. 
 Unit 2 is titled ?Similar Triangles?. Day one is on ratios and proportions, day two 
similarity, day three and four proportions in similar triangles, day five and six indirect 
measurement, and day seven right triangle proportions. I observed days two, three and 
four; similarity and proportions in similar triangles. Mrs. Lotus wanted me to observe 
those days because of the diferent types of application problems in these sections.  
Observed lesons: Planned and executed. The main activity on the first day of 
observation was the dilations activity. For this activity, Mrs. Lotus gave directions, and 
then had students work in groups on the activity. Within each group, two students started 
with triangles of their own creation, and two students started with trapezoids of their own 
creation. Students were given time to work in groups on constructing the dilations and 
making conjectures about observations. When Mrs. Lotus noticed that most students were 
done, she led the clas in discussing their findings. Then Mrs. Lotus lectured for a bit 
about similar polygons by giving a real world example and noted the specificity of 
language for mathematics and real world. Then Mrs. Lotus led the whole clas through 6 
 
 
 
 
173 
problems, starting with one where they had to se if two triangles were similar and 
progresing in dificulty. For each problem, students were alowed time to work 
individualy and discuss as a group before sharing their answers with the teacher. They 
also progresed in a way to lead to a theorem about perimeters of proportional or similar 
figures. Mrs. Lotus showed students this proof to expose them to more proofs and their 
structure. Then Mrs. Lotus introduced the similarity triangle conjecture by having 
students compare it to what they knew about similar figures. She then introduced the first 
test for similarity, al angles congruent.  
 Leson two was diferent from the other lesons in that Mrs. Lotus?s computer 
caught a virus, and so al of her PowerPoints that she had from before were inacesible. 
She stil knew what she needed to teach and had the activity handout already printed so 
she was going to stil use it. Instead of using saved sketches on GSP, Mrs. Lotus had to 
construct using compas and ruler under the document camera. Also, due to technical 
dificulty, she decided not to have belwork.  
Continuing with the dilation activity, Mrs. Lotus led students by giving them 
instructions for dilating a triangle and then had them measure it for similarity. Each set of 
directions was a similarity test. The first test was sides without angles, and then the next 
test focused on two angles and a side. After the discussion summarizing al the ways to 
determine if two triangles are similar, including the definition, Mrs. Lotus gave a problem 
where students had to determine if two of the three given triangles were similar. Most of 
the students said no, and then one student actualy did the calculation and found that the 
triangles were similar. When Mrs. Lotus asked why the students said no, most of them 
answered that the problem required too much work and that they were too lazy to do it. 
 
 
 
 
174 
After that example, students had to write a proof on how to determine if two triangles 
were similar. After the proof, three examples were given, and then clas ended. What 
Mrs. Lotus planned was executed despite not having technology. The pre leson 
interview was done earlier that morning so Mrs. Lotus was already prepared for the 
unavailability of technology (SmartBoard with al its functions, computer, and aces to 
school-wide hard drive).  
 Since certain parts of technology were stil unavailable, Mrs. Lotus used other 
programs to run clas as she normaly did. So, clas started with belwork and review of 
belwork. Instead of notes being on a hard drive for students to aces, Mrs. Lotus set up 
another digital location to where al students could go to retrieve notes for the day. Thus 
the order of the application problems was presented as follows; pantograph cat, 
pantograph daisy, rock wal, and river. For each progresive application problem, Mrs. 
Lotus gave les information for what students needed to use to solve the problem. While 
students were working on the river problem, Mrs. Lotus walked around and checked 
homework. Then Mrs. Lotus went over homework, talked about the river problem, and 
tied in student responses to the next theorem. Then Mrs. Lotus focused on students 
creating a formal proof of a corollary of the theorem. After going over the proof, Mrs. 
Lotus showed another theorem and had students prove a corollary to that theorem. After 
that, there were two straightforward problems before the last problem, which was an 
application problem, before the end of clas. What Mrs. Lotus planned for, happened in 
clas, and student interactions added to what Mrs. Lotus had to say and explain, but none 
were points of dificulty or misconceptions of content. 
 
 
 
 
175 
Unit 2 TACK. Mrs. Lotus relied heavily on her PowerPoint slides to be used on 
her SmartBoard. Within this unit, technical dificulties within the school made Mrs. 
Lotus unable to use her slides at al. Thus, she had to resort to using the document camera 
like a whiteboard for two lesons. This technical dificulty, however, did not change what 
was taught and what she planned. She had copies of the slides in a file cabinet just in case 
something like this happens. Student dificulties that came up within this unit were 
converting units of measure. This was something that Mrs. Lotus had not planned for, but 
was able to addres. One dificulty that she planned for was students not reading through 
long word problems. Her planned modification was to not use long word problems on a 
test. However, she stated that sometimes, standardized tests ases in a way that long 
word problems are used. She stated that although it is important for students to be able to 
read through long word problems eventualy, due to the fact that her students are eighth 
graders now, she did not think that their grades should be penalized for their imature 
behavior at the moment. 
Unit 2 Geometry Filter. Similar to Unit 1, from what I gathered from interviews 
and observations, through the geometry filter, al that Mrs. Lotus planned were observed. 
The geometry filter highlights certain areas of TACK specific to teaching geometry.  
Throughout the three days of observation, in homework review and discussion of 
problems throughout the leson, Mrs. Lotus chalenged students to informaly prove why 
the answers they came up with were true. This showed van Hiele Level 2, informal 
deduction. On the first and third days of observation, Mrs. Lotus set up opportunities for 
students to write formal proofs, van Hiele Level 3. The decision to set up opportunities 
for students to write formal proofs relates to specialized content knowledge, presentation 
 
 
 
 
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of mathematical ideas and connecting a topic being taught to topics from prior or future 
years.  
Al phases of learning, that I identified, showed evidence in planned and observed 
throughout the three days of observation. For example, on the second day of observation, 
Mrs. Lotus learned of students? prior knowledge and experience with the leson of the 
day during homework review from the previous day. I decided that this showed the 
information phase, where the teacher learns of students? prior knowledge and experience 
with the subject through discussions (Mistreta, 2000). The construction activities showed 
?directed orientation?, where these activities were provided for students to become more 
familiar with the material being taught (Mistreta, 2000). Discussion of the conjectures 
from constructions and problems summarizing what was learned showed explication and 
free orientation phases, because there was a transition of reliance on teacher to student 
self reliance, and the questions themselves were open ended and were atentive to the 
inventive abilities of the students (Mistreta, 2000). Also, the two problems at the end of 
the leson showed integration phase because students were using what they learned 
throughout the day to solve those problems (Mistreta, 2000). 
Throughout the observed days of the unit, three of the geometric habits of mind 
were planned and observed; reasoning with relationships, investigating invariants, and 
balancing exploration and reflection (Driscoll, 2007). From what I observed, during the 
dilation activity, students were reasoning with relationships of the original figure and the 
dilated figure. They made observations of the similarities and diferences betwen the 
original figure and dilated figure. ?Students talking with group members to generalize 
 
 
 
 
177 
observations of their figures?, I identified this to be ?generalizing geometric ideas? 
(Driscoll, 2007).  
Findings from Mrs. Lotus Unit 2. Despite technological dificulties and Mrs. 
Lotus? inability to aces PowerPoint slides that she usualy used, al planned geometry 
TACK was observed. Through both TACK analysis and the geometry filter for al lesons 
observed; sequence of activities, timing, questions, explanations, areas of student 
dificulties, modifications, student prerequisite knowledge, van Hiele Levels, phases of 
learning based on the van Hiele Level, and geometric habits of mind were observed as 
planned. Additional questions and explanations were observed. Based on observations, 
for each observed day of this unit, van Hiele Model phases of learning that I identified 
during planning were also observed.   
 Summary of Unit 1 and Unit 2. For both of Mrs. Lotus? units, geometry TACK 
planned was observed. For the first unit, additional explanations, questions, and student 
dificulties, were observed. For the second unit, only additional explanations and 
questions were observed. Mrs. Lotus credits the lack of student dificulties shown in clas 
to the fact that they are self-motivated. ?If they don?t understand something, they?ll ask 
each other. If it?s something big, they?ll usualy go home and ask a parent or someone 
else.? The only modification was lack of use of the interactive whiteboard with 
PowerPoint slide. The activities were executed as planned.  
Al aspects of the geometry filter were observed as planned. However, in the first 
unit for geometric habits of mind, Mrs. Lotus fostered reasoning with relationships, 
generalizing geometric ideas, and balancing exploration and reflection. In the second 
unit, Mrs. Lotus fostered reasoning with relationships, investigating invariants, and 
 
 
 
 
178 
balancing exploration and reflection. Investigating invariants was not in the first unit, and 
generalizing geometric ideas was not in the second unit. Plans and how Mrs. Lotus 
executed them showed comprehensive knowledge of teaching and learning geometry. For 
Mrs. Lotus, al geometry TACK planned was observed. Additional context for geometry 
TACK for Mrs. Lotus appeared when themes through data analysis emerged.  
 Themes from Unit 1 and Unit 2. Themes surfaced in relation to timing, 
questioning, familiarity with activities used, areas of apprehension, and sources of 
knowledge. For al of Mrs. Lotus? observed lesons, timing occurred as anticipated. There 
were two lesons where activities took slightly les time than anticipated, and students 
had more time to work on homework for the next day. In questioning students, Mrs. 
Lotus usualy gets multiple students to answer one question; and if there was a wrong 
answer, students discussed as a whole clas what the correct answer should be. Thus, for 
big key questions that lead students into discussion, Mrs. Lotus was observed facilitating 
the discussion.  
 Mrs. Lotus was very familiar with al the activities for al six observations in both 
units. Despite her familiarity with these activities, her area of apprehension was when 
technological dificulty arose, and she was not sure of exact timing for phases of the 
leson. Another area of apprehension was having me observe an activity she had not 
taught before. She thought that it would be interesting for me to observe this activity, but 
later changed the activity to one that she used before because it matched the standards 
beter. The new activity came from one of Mrs. Lotus? periodical searches of new 
activities to use with students. To search for new activities, she stated that she used 
Internet searches, profesional development, and collaboration with colleagues. Mrs. 
 
 
 
 
179 
Lotus stated that sometimes, she learns that an old activity can be used in a new way, 
especialy from collaborating with other teachers. Her decision in what to teach comes 
from state standards, textbook, standardized asesments, profesional development, 
college clases, personal education experiences, and profesional experience as a teacher. 
 Connections betwen Planned and Executed TACK 
 For each leson, and consequently for each unit, TACK components were 
compared from planned and executed. In general, al four units showed similar findings. 
What was planned was observed. In the next three sections, I describe findings from both 
teachers in the context of their category of TACK; knowledge of content and teaching, 
knowledge of content and students, and specialized content knowledge.  
 Knowledge of Content and Teaching. Knowledge of content and teaching 
(KCT) involves the knowledge of sequence of phases within a leson or unit, timing for 
phases, explanations for concepts, and key questions to guide student thinking. For both 
teachers, in general, al KCT planned was executed. For both teachers, additional 
explanations and questions were observed. Sequences and timing occurred as planned as 
wel, but this was dependent on each teacher?s definition of time. Mrs. Lotus gave timing 
in minute estimates for each phase of a leson, whereas Mrs. Orchid gave general times 
like ?the rounds wil take most of the clas time?.  
Knowledge of Content and Students. Knowledge of content and students is 
knowledge of content specificaly involving students (Bal, Thames, & Philips, 2008). 
For this study, knowledge of content and students is demonstrated by the teacher in three 
ways: modifications to what is being taught based on student dificulties; student 
 
 
 
 
180 
misconceptions and how to help students overcome them; and knowledge of student 
prerequisite skils. 
 Both Mrs. Orchid and Mrs. Lotus had a leson within their units where there was 
a planned and observed modification. Modifications were due to student dificulties and 
technical dificulty. Standards for student learning was not compromised in their 
modifications. For both teachers, al anticipated student dificulties were observed. 
Additional unanticipated areas of student dificulties were student dificulties that both 
teachers had the knowledge to support student learning through their vast source of 
knowledge. Al anticipated prerequisite skils were observed for students of both 
teachers. Both Mrs. Orchid and Mrs. Lotus were prepared for the mixed ability in 
prerequisite knowledge. Prior to the focus activity of the day, they had other notes and 
activities that addresed the prerequisite skils for those students that lack the skils. 
Further, they both believed that discussions among students who understand and who 
may not understand yet, are more meaningful than the teacher teling the student what to 
do. 
 Specialized Content Knowledge. To discuss aspects of specialized content 
knowledge (SCK) specificaly, in my analysis of SCK, I looked at planned and executed 
SCK through mathematical tasks for teaching (Hil, Bal, Schiling, 2008). In presenting 
mathematical ideas, both teachers presented the way that they planned. Al planned 
responses to students? ?why? questions were observed, but with student interaction, 
additional responses to students? ?why? questions were also observed. For each leson 
observed, both teachers found examples to make specific mathematical points. For each 
leson, each teacher shared her selection of representations for particular purposes. 
 
 
 
 
181 
Generaly, these were based on knowledge of content and students from previous 
experience. In discussing representations, both teachers discussed their links to 
underlying ideas and other representations. Findings were similar for other mathematical 
tasks for teaching. 
For both teachers, al knowledge shown of mathematical tasks for teaching during 
planning were shown during the observation. Throughout the units for both teachers, I 
did not observe ?explaining mathematical goals and purposes to parents? and ?modifying 
tasks to be easier or harder?. Some mathematical tasks for teaching like ?evaluating the 
plausibility of students? claims? had more evidence in the clasroom than during 
interviews. However, al SCK planned were also observed. Additional evidence for SCK 
that were not stated as planned was also observed.  
Geometry Filter for Mrs. Orchid and Mrs. Lotus 
 For both teachers, though the geometry filter; levels, phases, and habits of mind I 
saw during planning was observed during the execution of the leson. For Mrs. Lotus, the 
van Hiele Levels, phases of learning, and geometric habits of mind that I identified were 
al executed exactly as planned. Since Mrs. Orchid?s second unit was on trigonometry, 
applicable aspects of the geometry filter were addresed. Both teachers? first unit focused 
on informal deduction, van Hiele Level 2, and developing formal deduction, van Hiele 
Level 3 within some activities. Mrs. Lotus?s second unit had formal justification each day 
observed.  
When only looking at observed days of the unit for Mrs. Orchid, I found evidence 
to support the later four phases of instruction based on the van Hiele Levels, excluding 
?information? phase until the fourth observation of the first unit. When looking at both of 
 
 
 
 
182 
Mrs. Orchid?s units as a whole, evidence for al phases of learning based on the van Hiele 
Model were present. Mrs. Lotus?s lesons showed al five phases for each observation. Of 
the four geometric habits of mind, ?investigating invariants? was the geometric habit of 
mind for which I found the least occurrence. Through the geometry filter, very litle 
diferences appeared betwen planned and observed. 
Connections of Geometry Filter to TACK 
I used the geometry filter to iluminate connections betwen planned and executed 
Teachers? Applied Content Knowledge. Similar in format to how I discussed the 
geometry filter for each unit, the connections betwen geometry filter and TACK wil be 
discussed by van Hiele Levels, phases of learning based on the van Hiele Model, and 
geometric habits of mind. 
When looking at the van Hiele Level in planning of a leson, aspects of TACK 
that were highlighted were: specialized content knowledge, specificaly, presenting 
mathematical ideas and connecting a topic being taught to topics from prior or future 
years; and knowledge of content and teaching, specificaly, sequence of information to be 
presented, explanations, and key questions to guide student thinking. In determining the 
van Hiele Level during the observation of a leson, aspects of TACK that were 
highlighted in addition to those when looking during the planning of a leson was 
specialized content knowledge, specificaly, responding to students? ?why? questions, 
finding an example to make a specific mathematical point, and asking productive 
mathematical questions.  
 
 
 
 
183 
Both teachers? knowledge of how students learn determined their structure of 
instruction to be very similar to phases of learning based on the van Hiele Model. Since 
the information phase is where discussions are held where the teacher learns of the 
students? prior knowledge and experience with the subject mater at hand (Mistreta, 
2000, p. 367), connections to TACK were: knowledge of content and teaching, 
specificaly, sequence of information to be presented, timing (how long to alow for and 
how much time was used in clas), and key questions to guide student thinking; and 
knowledge of content and students, specificaly, student prerequisite knowledge.  
Directed orientation is where the teacher provides activities that alow students to become 
more acquainted with the material being taught (Mistreta, 2000, p. 367), connections to 
TACK were: presenting mathematical ideas, an aspect of specialized content knowledge 
and sequence of topics to present, an aspect of knowledge of content and teaching. 
During explication, a transition betwen reliance on the teacher and students? self-
reliance is made (Mistreta, 2000, p. 367).  Aspects of TACK iluminated when looking at 
explication were: presenting mathematical ideas, sequence of topics to be presented, 
explanations, and questions to guide student thinking. During observations, explication 
phase was apparent when there was a step drop in number of questions and explanations 
teachers gave, and instead, both Mrs. Orchid and Mrs. Lotus prompted students to ask 
each other questions or gave some points for the students to consider and then 
imediately walk away. Free orientation is where teachers gave students tasks that can 
be approached in numerous ways (Mistreta, 2000, p. 367). The connection to aspects of 
TACK were: specialized content knowledge, specificaly, presenting mathematical ideas, 
connecting a topic being taught to topics from prior or future years, and asking 
 
 
 
 
184 
productive mathematical questions; knowledge of content and teaching, specificaly, 
sequence of information to be presented and key questions to guide student thinking; and 
knowledge of content and students, students? prerequisite knowledge. For integration, 
where students summarize what was learned during the leson (Mistreta, 2000, p. 367), 
connections to TACK were: specialized content knowledge, specificaly, presenting 
mathematical ideas and asking productive mathematical questions; and knowledge of 
content and teaching, specificaly, sequence of information to be presented, explanations, 
and key questions to guide student thinking. Interesting note, however, for Mrs. Orchid, 
when there was limited time; integration phase was left out of one observation day.   
Al four geometric habits of mind (Driscoll, 2007) were fostered betwen the two 
units for each teacher. Evidence that helped me determined geometric habits of mind 
during planning connected to aspects of TACK were: presentation of mathematical ideas, 
an aspect of specialized content knowledge; sequence of ideas to be presented, 
explanations, and key questions to guide student thinking, aspects of knowledge of 
content and teaching; and student prerequisite knowledge, an aspect of knowledge of 
content and students. 
This section was not an exhaustive list of connections, but rather, only the 
strongest paterns of connections for aspects of the geometry filter to aspects of TACK 
were discussed. Some other connections may exist, but through my analysis, evidence 
from the aspects of TACK described above helped determine the aspect of geometry 
filter each teacher addresed. Some connections were specific to one leson, like for 
homework review one leson, Mrs. Orchid fostered the geometric habit of mind 
 
 
 
 
185 
?investigating invariants? to help students clear up a misconception about corresponding 
parts of congruent triangles.  
Comparison of the Cases: Mrs. Lotus and Mrs. Orchid 
The context of the two cases, Mrs. Lotus and Mrs. Orchid were diferent. The 
clas observed of Mrs. Orchid was high school 9
th
 through 11
th
 graders, while Mrs. 
Lotus?s students were 8
th
 graders in an advanced high school geometry course. The 
number of students was similar, Mrs. Orchid had 24 students, and Mrs. Lotus had 25. 
Both teachers were defined as exemplary teachers, and looking at similarities can 
iluminate what both share as exemplary teachers. Looking at diferences sheds light to 
the diferent contexts in which each of these exemplary teachers work. 
 Similarities. Both teachers had a variety of knowledge, specificaly specialized 
content knowledge when discussing the overal plans for the unit. They shared 
knowledge of curriculum, general student misconceptions, philosophy of questioning and 
explaining things to students, and overal where the concepts they were about to teach fit 
into the course and future math clases. For both teachers, generaly what was planned 
was observed. 
In emergent codes, both teachers gave activity directions before starting an 
activity. Within both units for each teacher, there was birdwalking, conversation not 
related to clasroom procedures or content, (Hunter, 2004) about the temperature of the 
room and what students were wearing. Clasroom administration were observed for both 
teachers as wel, and within that, both teachers had intercom interruptions at some point 
during at least one of the units. Once students were working in groups, both teachers 
 
 
 
 
186 
walked around, and at times and showed verification - statements made to students from 
the teacher leting them know that what they are doing is either correct or incorrect 
without questioning or explaining. 
 Differences. The main diference betwen Mrs. Orchid and Mrs. Lotus was the 
presence of activity directions and clasroom administration. Mrs. Orchid had at least 
three times the number of units-of-analysis that were coded clasroom administration and 
at least twice the number of units-of-analysis that were coded activity directions. Of the 
units-of-analysis that were coded clasroom administration, most were discipline related; 
asking students to be quiet, put away their phone, work on the asignment, and not 
sleping during clas. Also, the second most number of units-of-analysis within 
clasroom administration for Mrs. Orchid was related to students using the restroom. 
Also, Mrs. Orchid repeated similar activity directions at least four times for each activity, 
while Mrs. Lotus repeated similar activity directions at most, twice. 
 Even thought both teachers stated that they did not have plans for timing, upon 
further questioning, both gave rough estimates of times that they thought would be 
necesary for each phase of the leson. Mrs. Lotus followed her timing almost perfectly, 
even though she did not look at the clock in the room very often. She went on to the next 
phase of the leson when students semed like they were ready to move on to the next 
topic. Mrs. Lotus stated that she based moving on to the next topic on students? facial 
expresions, correct answers, and ability to explain. Thus, her observed timing followed 
planned timing almost exactly. However, due to dificulties in the clasroom, Mrs. 
Orchid?s timing for the second unit did not go as planned. The same planned sequence, 
 
 
 
 
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explanations, and questions were eventualy addresed; but timing was not exactly 
observed as planned. 
 During discussions of students? answers in clas, for Mrs. Lotus?s observations, 
students usualy answered with the correct answer complete with justification. Thus, 
many of Mrs. Lotus?s remaining follow-up questions were ?Did anyone get anything 
diferent??; ?Does everyone agree??; and ?Questions?? Student responses to these 
questions typicaly were yes, and no to the last two questions. For ?Did anyone get 
anything diferent?? students that did get something diferent usualy spoke up and were 
correct. If they were not correct, other students explained what was incorrect, and the 
clas self-corrected. For Mrs. Orchid?s clas, students often answered with the correct 
answer as wel, but Mrs. Orchid asked several questions to help students explain their 
answers and justify correctly. If a student was incorrect, other students waited for Mrs. 
Orchid to tel them whether or not it was correct, and why it was incorrect. Mrs. Orchid  
facilitated students explaining ?why? questions by asking students questions. 
Adresing the Research Question from Analysis 
The research question was ?What connections are there betwen the high school 
geometry TACK shown during the planning of the leson and the teachers? actual 
executions of the leson?? To discuss connections betwen teacher knowledge planned 
and executed specific to the aspects of TACK, I separated the research question into eight 
more explicit questions. 
? What are the connections betwen planned and executed specialized 
content knowledge? 
? What are the connections betwen planned and executed sequence of 
information to be presented? 
 
 
 
 
188 
? What are the connections betwen planned and executed time spent on 
phases of the leson? 
? What are the connections betwen planned and executed explanations of 
concepts, definitions and procedures? 
? What are the connections betwen planned and executed key questions to 
guide student thinking? 
? What are the connections betwen planned and executed leson 
modifications? 
? What are the connections betwen planned and observed teachers? 
acknowledgement of student dificulties and how to overcome them? 
? What are the connections betwen planned and observed teachers? 
knowledge of student prerequisite skils? 
The following are results to each of the eight explicit questions based on the data 
collected and analysis described in the previous chapter.  
? Specialized content knowledge ? Al planned were observed during 
execution of the leson. Additional responses to students? ?why? questions 
and examples to make specific mathematical points were observed.  
? Sequence of information to be presented ? All planned sequences of 
information were observed during execution of the leson. For Mrs. 
Orchid, in one clas, an additional sequence of ?calculator procedures? 
was added to observation that was not planned when more students that 
anticipated had dificulty using their calculators.  
? Time spent on phases of the leson ? Most of what was planned was 
observed during the execution of the leson. In the instances where 
planned timing was not observed, observed phases took longer than 
planned.   
? Explanations of concepts, definitions and procedures ? Al explanations of 
concepts, definitions and procedures were executed as planned. However, 
 
 
 
 
189 
additional explanations of concepts, definitions and procedures were 
observed during the execution of the leson.  
? Key questions to guide student thinking ? Al key questions to guide 
student thinking were executed as planned. However, additional questions 
to guide student thinking were observed during the execution of the 
leson.  
? Leson modifications ? Al planned leson modifications were observed 
during the execution of the leson.  
? Teachers? acknowledgement of student dificulties and how to overcome 
them ? Al student dificulties and strategies to overcome them that were 
stated prior to the observed leson were observed during the execution of 
the leson. Additional student dificulties and strategies to overcome them 
were observed during the execution of the leson.  
? Teachers? knowledge of student prerequisite skils ? Al student 
prerequisite skils stated during leson planning were observed in 
execution of the leson.  
Al of teacher knowledge shown during leson planning was observed during the 
execution of the leson. Through student interactions, additional teacher knowledge was 
observed during the executed leson. These were the results from analysis of the data 
collection before axial coding.  
Axial Coding  
 Axial coding gave a diferent perspective on individual TACK and emergent 
codes. Unitizing for units of analysis and coding followed the coding guide in Appendix 
H. Units of analysis in general adhered to Miles and Huberman?s (1994) definition of a 
unit of analysis; ?a sentence or multisentence chunk? (p. 65) where a single code can be 
 
 
 
 
190 
easily asigned. Garret?s (2010) definition of topic and unit in relation to units of 
analysis was more applicable to this study. Topic, in relevant to this study, based off of 
Garret?s (2010) was defined as a complete idea, aspect of TACK, or topic of 
conversation upon which the atention of the speaker or speakers is focused. Garrett 
(2010) defined a unit as ?A word, sentence, paragraph or several consecutive sentences or 
paragraphs focusing on the same topic? (p. 264). 
Before starting data collection, there were eight a priori codes; specialized content 
knowledge, knowledge of content and teaching ? sequences, knowledge of content and 
teaching ? time, knowledge of content and teaching ? explanations, knowledge of content 
and teaching ? questions, knowledge of content and students ? modifications, knowledge 
of content and students ? dificulties, and knowledge of content and students ? 
prerequisites. Emergent codes, codes that showed up as themes in the units of analysis 
that are not a priori codes were activity directions, birdwalking (Hunter, 2004), clasroom 
administration, curriculum, researcher technical interactions, verification, and other. 
Activity directions are discussions of what students are about to do and how to do 
it. Birdwalking is defined as teachers engaging in discussion with students about 
something not related to the leson or any of the above codes (Hunter, 2004). Clasroom 
administration, for this study, is defined as discussions pertaining to taking atendance, 
intercom interruptions, bels sounding for other clases, students asking to use school 
facilities outside the clasroom, discipline, and other clasroom procedures. Curriculum, 
in the context of this study, is defined as discussions pertaining to opinions and ideas of 
textbook being used or standards for instruction. ?Researcher technical interactions? were 
discussions regarding administrative aspects of the study like camera angle and 
 
 
 
 
191 
scheduling. Verification is defined as statements made to students from the teacher leting 
them know that what they are doing is either correct or incorrect without questioning or 
explaining. Since everything is coded, ?other? is a code that was used to represent a 
segment that did not fit anywhere else. For a detailed list with examples of emergent 
codes, se Appendix H. 
Once unitized, themes began to appear. During pre-observation interviews, 
conversation topics that appeared were regarding administration of the study, activity 
directions, curriculum, specialized content knowledge, sequencing of activities, timing 
for phases of lesons, explanations of content, questions to guide student thinking of 
topic, modifications to lesons, student dificulties, and student prerequisite skils. During 
observations, activity directions, birdwalking (Hunter, 2004), clasroom administration, 
curriculum, verification, specialized content knowledge, sequencing of activities, timing 
for phases of lesons, explanations of content, questions to guide student thinking of 
topic, modifications to lesons, student dificulties, and student prerequisite skils were 
observed. The following sections describe the relationships betwen the codes when each 
a priori code underwent axial coding, starting with specialized content knowledge. 
 Specialized Content Knowledge. From the defining features of specialized 
content knowledge (SCK), many aspects of SCK overlap with knowledge of content and 
teaching, knowledge of content and students, and some emergent codes. For example, an 
aspect of specialized content knowledge is ?responding to students? ?why? questions?. 
When students asked Mrs. Orchid and Mrs. Lotus ?why? type of questions, both teachers 
responded with key questions and explanations. Table 4-5 lists SCK and codes related to 
specific aspects. 
 
 
 
 
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Table 4-5 Aspects of SCK and related codes 
Aspect of SCK Related codes 
presenting mathematical ideas Explanations, questions, modifications, 
dificulties, and prerequisites 
responding to students? ?why? questions Explanations, questions, dificulties, and 
prerequisites 
finding an example to make a specific 
mathematical point 
Explanations, questions, and dificulties 
recognizing what is involved in using a 
particular representation 
Explanations and questions 
connecting a topic being taught to topics 
from prior or future years 
Explanations, questions, modifications, 
dificulties, and prerequisites 
explaining mathematical goals and 
purposes to parents 
not stated in interviews or observed 
appraising and adapting the 
mathematical content of textbooks 
Explanations, questions, modifications, 
dificulties, prerequisites, and curriculum 
modifying tasks to be either easier or 
harder 
Explanations, questions, modifications, 
dificulties, and prerequisites 
evaluating the plausibility of students? 
claims (often quickly) 
explanations, questions, and verification 
giving or evaluating mathematical 
explanations 
explanations, questions, and verification 
choosing and developing useable 
definitions 
Explanations, questions, modifications, 
dificulties, and prerequisites 
using mathematical notation and 
language and critiquing its use 
Explanations, questions, modifications, 
dificulties, and prerequisites 
asking productive mathematical 
questions 
Explanations, questions, dificulties, and 
prerequisites 
selecting representations for particular 
purposes 
Explanations, questions, modifications, 
dificulties, and prerequisites 
inspecting equivalencies explanations, questions, and verification 
 
Thus from the table, it is apparent that SCK very closely related to KCT in aspects like 
responding to students? why questions and asking productive mathematical questions. 
 
 
 
 
193 
SCK is very closely related to KCS in aspects like presenting mathematical ideas and 
selecting representations for particular purposes.  
 Knowledge of Content and Teaching. During pre-observation interviews, both 
teachers used the sequence of information to discuss timing, explanations, questions, 
dificulties, and prerequisite skils. In analyzing data, observed geometry TACK 
depended on sequence. During observations, teacher?s mentioning of sequence was 
usualy acompanied by a comment about time. Although it is unclear if sequence causes 
change from planned to executed time, or if time causes change from planned to executed 
sequence; time and sequence are closely related.  
 During interviews when teachers did not explicitly state time alotment for 
activity within the leson, I asked about specific timing for phases. In response to how 
much time is taken, both teachers talked about timing and then included student 
prerequisite skils and dificulties, and the questions and explanations necesary for 
student dificulties and prerequisite skils. Also, because both teachers were geting 
adjusted to new curriculum in the form of standards and textbooks, both teachers 
discussed what they did in the past and how much time these phases would have taken 
and what had changed since then. As stated for sequences, during observations, teachers 
made note of timing whether it took longer or shorter than expected when the teacher 
moved on to the next phase of the leson. 
 Explanations and questions were nearly inseparable. Both teachers discussed key 
questions with explanations and vice versa, both in interviews and clasroom 
observations. Also, during observations, when teachers asked a question, and student 
response semed confusing, teachers rephrased and explained the student?s explanation. 
 
 
 
 
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Also during observations, when students asked questions, many times, the teachers 
responded with a question or series of questions to explain. 
 Knowledge of Content and Students. For both teachers, most of the lesons 
observed were lesons that they taught many times before. Both teachers stated that 
modifications that they decided were necesary were already changed to fit their students 
the first couple of times they taught the lesons. Thus, modifications were rare. The only 
modifications that occurred during the observations that I conducted were due to 
technical dificulties or student dificulties. These modifications changed timing, 
explanations, and questions. 
 During interviews, student dificulties were discussed in the context of discussing 
sequences, explanations, questions and prerequisites. When student dificulties were not 
explicitly discussed, after prompting from me, teachers discussed student dificulties with 
prerequisite knowledge, explanations, and questions. During observations, unplanned 
student dificulties resulted in additional explanations and questions from the teachers. In 
one case, student dificulties changed timing, and caused the need for leson modification 
the next day. 
 During interviews, prerequisite knowledge needed was discussed in the context of 
discussing sequences, explanations, questions and student dificulties. When teachers did 
not discuss prerequisite skils explicitly, when I asked, they responded with what 
prerequisite skils they thought students needed in addition to student dificulties that 
arose due to the lack of the prerequisite knowledge. 
 
 
 
 
195 
Summary of Themes from Mrs. Orchid and Mrs. Lotus  
 Looking through interviews and observations from Mrs. Orchid and Mrs. Lotus, 
some themes surfaced: reasons for choices of units to share, source of knowledge, use of 
technology, areas of apprehension, and familiarity with activities. Through interviews 
and clasroom observations, I found that both teachers conducted clas with the basis that 
students need to construct their own knowledge in order to learn, students need to learn 
conceptualy, and therefore, they felt that they needed to teach conceptualy. When asked 
why they conducted clas the way they intended and executed, they stated that it was due 
to a mixture of state standards, textbook, standardized asesments, profesional 
development, college clases, personal education experiences, and profesional 
experience as a teacher.  
Both teachers chose lesons where topics and activities would be interesting to 
watch. Mrs. Lotus?s chosen units were topics that she was very familiar with and had 
interesting activities to share. Mrs. Orchid?s first unit had a very interesting activity that 
she wanted to share, and the second unit was an activity that she created that she wanted 
to share. Elements within the lesons that both teachers chose for me to observe were 
engaging activities, open ended tasks, collaboration among students, and expecting 
students to justify their answers. Prerequisite knowledge required of students for 
activities were always addresed right before the activity or had been addresed at some 
point within the semester prior to the activity. Thus, student learning was based on a 
foundation of prerequisite knowledge that each teacher already built earlier in the 
semester.    
 
 
 
 
196 
Additional questions asked of students and explanations were observed for every 
leson. Both teachers stated that it would be impossible for them to list every student 
question, dificulty, or misconception prior to the observation of the leson. However, 
Mrs. Orchid and Mrs. Lotus had experienced al observed additional student questions, 
areas of dificulty, and misconceptions previously in other lesons. They stated that they 
were confident in their knowledge to met any student dificulty. Both of them also 
added that if they did not know the answer, then they told the students to look it up, and 
then they looked it up too, and discuss the problem the next day. This was mainly due to 
their vast source of knowledge: college coursework, textbook, profesional development, 
teaching experience, and collaboration with colleagues.  
Both teachers had interactive whiteboards and they used them to present 
information on a daily basis; and during the course of this study, both experienced 
technical dificulties where saved items could not be acesed, especialy items saved 
specificaly for the interactive whiteboards. Both teachers stil taught the lesons as 
planned, as the only change in the leson was that information was printed for students to 
se instead of having it on the interactive whiteboard where the teachers could write over 
what was printed. For Mrs. Lotus, instead of having the steps already on slides to present 
to students, she presented step-by-step constructions using the document camera. Prior to 
the leson, Mrs. Lotus stated that the leson may take a litle bit longer because she had to 
take time to draw out the steps, but after the leson, she was pleasantly surprised by the 
minimal change in predetermined time.  
Both Mrs. Orchid and Mrs. Lotus discussed how they planned to teach activities 
that they had not previously taught. Mrs. Lotus ended replacing the new activity with an 
 
 
 
 
197 
activity that she taught before because she felt that it fit beter with what the students 
needed to know. However, some common themes arose from how they planned to teach 
the activity and how they expected for student dificulties and misconceptions. Since they 
both had not taught the activities before, there were no modifications based on previous 
experience. Student areas of dificulties and misconceptions were based on previous 
experiences with other students. For both teachers, areas of dificulties and 
misconceptions that showed up during the activity that matched with what they 
anticipated were the dificulties and misconceptions in prerequisite knowledge. However, 
additional dificulties and misconceptions that were observed mostly were related to the 
concepts learned during the activity.  Figure 4-12 ilustrates this point. 
Planned      Observed  
 
  
 
 
Content source: 
 Prerequisite       Additional 
 Current 
Figure 4-12 Content source of student dificulties and misconceptions 
Student dificulties and misconceptions stated during planning were also observed. Of the 
additional student dificulties and misconceptions that were observed, more were related 
to the current activity than prerequisite knowledge.  
 
 
 
 
198 
Summary of Research Findings 
In the first half of this chapter, I organized data collected by teacher, and then by 
unit, starting with Mrs. Orchid. For each teacher, I discussed overal data collection with 
that teacher, overal plans for the unit, concept map(s) for the unit, and planned and 
executed teacher knowledge for the observed lesons of the unit. Next I discussed 
analysis of the unit, expounding on TACK, geometry filter, and themes from analysis 
without axial coding. The second half of this chapter compiled findings in a more 
succinct fashion, by discussing connections betwen planned and executed TACK for 
both teachers, the geometry filter for both teachers, connections of the geometry filter to 
TACK. After pulling together results from analysis prior to axial coding, I addresed the 
expanded research question, eight questions each addresing an aspect of TACK. Results 
from addresing the research question based on analysis without axial coding confirmed 
the need to employ axial coding to further explore relationships among the codes. Results 
from axial coding were organized by aspect of TACK, specialized content knowledge, 
knowledge of content and teaching, and knowledge of content and students. Through 
discussion of teachers, TACK, geometry filter, and analysis, some themes began to 
emerge. The last section of this chapter addresed the themes that emerged from analysis.   
 
 
 
 
199 
Chapter 5: Conclusions, Discusions, and Suggestions for Future Research 
My interest in high school geometry is based on my experience as a former high 
school geometry teacher. Coincidentaly, a review of the literature revealed that 
nationaly, student achievement in geometry is an area of concern (Clements, 2003; 
Lappan, 1999; Mullis, Martin, & Foy, 2008; Shaughnesy, 2011); and geometry is one of 
the areas of mathematics that is necesary for many careers (National Research Panel, 
1989; WeUseMath.org, 2011). Moreover, teacher knowledge of geometry has an impact 
on student achievement (Monk, 1994; Hil, Rowan, & Bal, 2005). Thus, the decision to 
study geometry teachers? knowledge, specificaly geometry Teachers? Applied Content 
Knowledge (TACK) was due to my interest followed by an examination of literature.   
Formed from literature, the research question for this disertation was: What 
connections are there betwen the high school geometry Teachers? Applied Content 
Knowledge shown during the planning of the leson and the teachers? actual executions 
of the leson? In the previous chapter, I discussed answers to the research question from 
findings specific to the diferent aspects of TACK: specialized content knowledge, 
sequence of information to be presented, timing for phases of the leson, explanations, 
key questions to guide student thinking, student areas of dificulty and misconceptions, 
modifications to the leson or activities, and student prerequisite skils. For each of these, 
the exemplary teachers showed, what was planned during observations of lesons. 
Student engagement during observations often brought about additional instances of these 
actions than was planned. From axial coding, some aspects of TACK and other teacher 
 
 
 
 
200 
knowledge and skils semed more connected than others. This chapter brings together 
the findings from Chapter 4 and literature from the first three chapters into: conclusions 
for this study, discussions related to the conclusions, and suggestions for further research.  
Before discussing conclusions, I discuss limitations to this study.  
Limitations 
 There are three main limitations to this study; exclusion of students, time alotted 
to interview teachers, and technological dificulties. Students were not interviewed but 
their actions were an integral part of this study. From observations, students influenced 
teachers to give more questions and explanations than what the teachers planned. Certain 
types of student interactions may have elicited certain types of teacher responses. 
Additional interviews with students and video cameras stationary at each table would 
have given more information from students? perspectives. This would have been nice to 
study, but exceded the scope of the research question for this study. Student interactions 
are a possible area for further research.  
Due to the timeline being set for a minimum of three consecutive lesons, and 
what happened one day may impact the next day; there was very limited time to 
interview the teacher on their reflection of the leson and the interview prior to the next 
leson. Due to time being limited, both teachers explained within the time alotted what 
they felt was important for me to know about the unit. If I had more time, I think that the 
data collected from the concept maps would be more robust. In retrospect, for the scope 
of the study, conversations from interviews yielded the information sought from concept 
map interviews. 
 
 
 
 
201 
For the first couple of observations, when the video camera was switched from 
being plugged in to a batery source, video and audio data recording stopped for about 6 
seconds. When this occurred, the recorder atached to the teacher was stil recording 
audio; so there was no lapse in audio recording. There were only six of these instances, 
al of which occurred during transitions betwen activities when the camera was switched 
from being stationary to mobile to follow the teacher around. For one observation, the 
teacher put the audio recorder in her pocket, and the entire recording consisted of rustling 
sounds. Fortunately for that day of data collection, the video camera recorded clas time 
without breaks. Technological dificulties prevented Mrs. Orchid from e-mailing me the 
second concept map with audio data of her explaining the concept map. Despite al these 
limitations, the quality of this study was not compromised. The following section 
expounds upon conclusions for this this study. 
Conclusions 
The exemplary teachers in this study showed ownership of depth and breadth of 
Teachers? Applied Content Knowledge for teaching high school geometry. The purpose 
of this study was to se the connections betwen teacher knowledge shown while 
planning and executing the leson. The simple response is: teacher knowledge shown 
during planning was observed during execution of the lesons for the two exemplary 
teachers I studied. However, due to student engagement in the leson, additional teacher 
knowledge was shown during the execution of the leson. This study elucidated some 
characteristics of exemplary teachers, namely how they: 
? Explained concepts 
? Modified lesons 
 
 
 
 
202 
? Addresed student dificulties 
? Conducted clasroom procedures 
? Utilized technology 
? Discussed planned and executed lesons 
Discussions of these characteristics were based on findings from analysis of TACK, 
geometry filters, areas of apprehension, sources of teacher knowledge, and changes to 
lesons from planned to executed.  
How did exemplary teachers explain concepts to students? Both Mrs. Orchid 
and Mrs. Lotus were knowledgeable of the concepts they were going to teach their 
students.  In interviews prior to observations, both teachers discussed how to explain 
concepts to students along with student areas of dificulties, prerequisite knowledge, 
curriculum (textbook and standards), and questions to ask students to help explain the 
concept. Since the teachers had taught the courses for consecutive years, they were able 
to anticipate student dificulties and plan acordingly. For most lesons, both teachers had 
planned questions that would facilitate student understanding of concepts. They also had 
explanations of concepts prepared.  
During execution of the leson, when students asked for an explanation or the 
teacher felt that it was appropriate to give an explanation, both teachers used questions to 
elicit student understanding of the concepts or skils rather than responding to students? 
questions by teling them the answers. When the teachers utilized series of questions, the 
questions started with acesing knowledge from prior lesons and progresed to current 
content being taught. This line of questioning showed that teachers were considerate of 
students? zone of proximal development (Bay-Wiliams & Herrera, 2007); the area 
bounded by the student?s ability to solve problems independently and the students? ability 
 
 
 
 
203 
to solve problems with the help of someone more capable (Vygotsky, 1978). This 
evidence of teacher knowledge also supports what both teachers stated about their views 
of how students? learn, connecting new knowledge to existing knowledge; showing that 
they were constructivist teachers (Stefe & D?Ambrosio, 1995) and in alignment with the 
Teaching and Learning Principles of Principles and Standards for School Mathematics 
(National Council of Teachers of Mathematics [NCTM], 2000). Counting statements of 
explanations of concepts and questions to explain concepts, both teachers had at least 
double the number of questions than statements for each leson. In some lesons the 
number of questions quadrupled the number of statements. 
How did exemplary teachers modify lesons for students? Some of the 
activities in the lesons that Mrs. Orchid and Mrs. Lotus taught had been used in previous 
years. For those activities, both teachers had modified their lesons throughout the years. 
Therefore, at the point of this study, the teachers were prepared for student areas of 
dificulties by providing appropriate levels of prerequisite knowledge, and carefully 
planned explanations and questions. Thus, for activities that the teachers taught many 
times before, modifications were limited to students? behavioral dynamics and 
technological dificulties. Modifications due to student behavior only changed the level 
of supervision from the teacher during an activity. For example, instead of coming up 
with the steps to construct a paralelogram in pairs of students, students discussed the 
topic as a whole clas led by the teacher.  Technological dificulties were unforesen, 
however both teachers quickly found ways to conduct activities as planned. They showed 
familiarity with how to teach the content using other types of technology or methods.  
Modifications did not afect content taught or cognitive demand.  
 
 
 
 
204 
Lesons in one unit for one teacher contained activities that the teacher had not 
taught before because the content was new to the high school geometry course as a result 
of new state standards. From how the teacher planned during the unit interview, activities 
were modified due to student dificulties and students? behavioral dynamics. Despite the 
modifications, students were stil held to the same cognitive demand and content 
standards; al within the same amount of overal time, days alotted for the unit.  
How did exemplary teachers addres student difficulties? Similar to the 
discussion on modifications, through the years of teaching, these two teachers have sen 
most students? dificulties to the activities that they used before. Thus, they have made 
the necesary changes to explanations and questions, and made modifications to activities 
when appropriate. When multiple students showed the same dificulty, teachers 
readdresed the clas with either explanations with statements and/or questions, or a 
modification to the leson where the teacher taught procedures for what was needed; in 
the case of this study, how to use a calculator for a specific function. When I observed 
that a student was at a lower van Hiele Level than the activity required, I noticed that the 
teachers questioned at the level the student was on, and gradualy brought the student to 
where he/she needed to be; mindful not to skip levels and work within their zone of 
proximal development (Bay-Wiliams & Herrera, 2007). Both teachers stated multiple 
times during interviews that they have sem al of the dificulties that occurred in clas 
before when they taught the leson.  
Both teachers noted that occasionaly, there have been new student dificulties 
that they had not sen before. With new student dificulties that they have not 
encountered before, if they knew the solution to the dificulty, they guided students to the 
 
 
 
 
205 
correct solution. However, if the solution was not something they understood readily, 
they asked students to take it home and research the solution for extra credit. If the 
teachers did not arrive at the solution within a day through research on their own with 
colleague or literature, they contacted profesors at the local university for help. Although 
none of this was observed during this study, discussions with their colleagues and 
profesors showed this was a reoccurring cycle throughout the years.  
How did exemplary teachers conduct classroom procedures? Both teachers 
knew the sequence of content to cover within an observation day very wel. At times, 
without looking at the clock, they made comments about whether or not the clas was on 
track for timing for the activity. I believe that the teachers? ability to acomplish 
clasroom administrative procedures and conduct student learning was largely due to 
their respect for time and what they, the teacher, and students should be doing at any 
given time. For example, while students were working on warm-up activities, the 
teachers checked homework, took atendance, and redirected off-task behavior; while 
walking around and evaluating the plausibility of student?s claims on the warm-up 
problems. When students exhibited off-task behavior, like talking about something 
unrelated or sleping during an activity; teachers stated activity directions and what 
students were supposed to do within the activity. Teachers diseminated materials that 
were needed for the day while giving activity directions or while students worked on 
something else. From the time students walked into these teachers? clasrooms to the time 
they left the clasroom, students always had something to do related to the content. Al of 
clas time was utilized for activities planned for the day.  Not only were these two 
teachers comfortable with the content and how to teach it, they were also very familiar 
 
 
 
 
206 
with conducting everyday clasroom procedures in a way that was supportive of teaching 
the content.  
How did exemplary teachers utilize technology? Technology; in terms of 
interactive whiteboard, document camera, teacher computer, and student computers; was 
readily available to both teachers. Both teachers exhibited knowledge of how to use al 
available technology, not only for themselves, but was able to facilitate student learning 
using the technology. Teachers provided students opportunities to use technology to aid 
in learning the content, but also helped students learn how to use technology when 
needed. Throughout the entire study, use of technology supported learning, and did not 
detract from learning.  
How did exemplary teachers discus planned and executed lesons? When the 
teacher discussed what was planned al aspects of TACK were discussed together. Their 
discussions were guided by the sequence of phases of the leson, taking time to talk about 
other aspects of TACK as they related to phases of the leson. They showed knowledge 
of how to teach geometry from planning and enacting appropriate Hiele Levels 
(Breyfogle & Lynch, 2010), phases of learning based on the van Hiele Model (Mistreta, 
2000), and geometric habits of mind (Driscoll, 2007) even though they did not mention 
these by their proper names or titles. A theme that emerged from their discussion of 
planned lesons were sources of knowledge for content and how to teach the content. 
Their plans for what to teach and how to teach it were based on many sources of 
knowledge; teaching experience, coursework, textbook, profesional development, and 
collaboration. The source of knowledge most often referred to was previous experience 
teaching the activity or topic.  
 
 
 
 
207 
Summary. Many characteristics in the way the exemplary teachers in this study 
talked about and taught geometry was similar. They often explained concepts with 
questions, modified lesons when necesary without sacrificing the integrity of the 
leson, addresed student dificulties at the appropriate level so students could learn, 
showed efective use of time management for both teaching content for student learning 
and clasroom administrative tasks, and shared multiple sources of knowledge that were 
frequently used to continue to improve their skils as a teacher. These characteristics 
showed depth and breadth of their knowledge of mathematics and how to teach it.  
Comparison of the teachers? characteristics with literature on teaching and learning 
geometry showed that they were very knowledgeable in how to teach geometry in ways 
that helped students to learn with understanding.  
Findings compared to literature on teaching and learning geometry 
From observations of Mrs. Orchid and Mrs. Lotus, many similarities were 
observed betwen their clasroom atributes and those shown by teachers of students 
showing achievement in geometry  (Han, 2007; Jones, 2000; Marrades & Guti?rrez, 
2000). These atributes were use of technology (Han, 2007; Jones, 2000; Marrades & 
Guti?rrez, 2000); questioning to get students to justify their answers (Han, 2007; Jones, 
2000; Marrades & Guti?rrez, 2000); and employed the proces of analyzing, coaching, 
and re-voicing questions back to students (Martin, et al. 2005). 
Similar to Han (2007), Jones (2000), and Marrades and Guti?rrez (2000); Mrs. 
Orchid and Mrs. Lotus used technology in the clasroom, specificaly dynamic geometry 
software (DGS). Both teachers used DGS in at least one of the lesons they shared with 
me throughout the study. For some activities involving DGS, both Mrs. Orchid and Mrs. 
 
 
 
 
208 
Lotus provided students with figures to use to explore properties instead of creating their 
own (Marrades & Guti?rrez, 2000). While using DGS, similar to Marrades and Guti?rrez 
(2000), activities were structured in three phases; create and/or explore, generate and/or 
verify conjectures, and justify conjectures. For Mrs. Orchid, this was the format for the 
paralelogram properties activity; and for Mrs. Lotus, this was the points of concurrency 
in a triangle activity. While using DGS, both teachers encouraged collaboration; Mrs. 
Orchid?s students in pairs, and Mrs. Lotus? students in groups of four. 
Throughout group work during activities and student presentations, Mrs. Lotus 
and Mrs. Orchid questioned students with questions such as ?why do you think this is 
true?? and ?can you prove it?? (Han, 2007; Jones, 2000; Marrades & Guti?rrez, 2000). 
For Mrs. Orchid, throughout the clas time, she used these questions to check for 
understanding. Both teachers chose units with lesons that had engaging activities and 
open-ended tasks. Students were encouraged to collaborate in groups and justify their 
answers.  
From what I determined regarding the geometry filter, both Mrs. Orchid and Mrs. 
Lotus showed knowledge of teaching and learning geometry as they planned and enacted 
appropriate van Hiele Levels (Breyfogle & Lynch, 2010), al phases of learning based on 
the van Hiele Model (Mistreta, 2000), and fostered geometric habits of mind (Driscoll, 
2007). In addition to knowledge of teaching and learning geometry, they both had vast 
knowledge of teaching and learning mathematics. Through interviews and observations, 
it was apparent that they had many sources of knowledge. These sources were: teaching 
experience, coursework, textbook, profesional development, and collaboration. When 
interviewed, both teachers were very secure in their plans except in the area of timing 
 
 
 
 
209 
when it involved new curriculum, new textbook material, and changing availability of 
clasroom technology.  
A recent paper explicating initial results from developing an instrument to 
measure mathematical knowledge for teaching high school geometry (MKT-g) looked at 
MKT-g scores compared to number of years teaching geometry, number of years 
teaching total, and number of college courses taken in geometry (Herbst & Kosko, 2012). 
They found that pilot data showed correlations betwen scores of domains of MKT-g, 
knowledge of content and teaching, knowledge of content and students, common content 
knowledge, and specialized content knowledge; and years of experience teaching 
geometry. Thus, years of teaching experience in geometry semed to be a good indicator 
of teachers? knowledge on MKT-g, encompasing al the areas of focus for this study; 
specialized content knowledge, knowledge of content and teaching, and knowledge of 
content and students. I would predict from this that both of the teachers I chose for this 
study, with at least 10 years of teaching geometry in the past 10 years, would do wel on 
MKT-g. Evidence from interviews and clasroom observations substantiated this 
asumption. 
Some of these characteristics of these exemplary teachers overlap with Stein and 
Smith?s (2011) five practices for facilitating efective inquiry-oriented clasrooms: 
anticipating how students wil approach a task, monitoring students as they work on tasks 
in clas, selecting students? strategies to discuss during clas, sequencing order of 
students? presentations, and connecting ideas in ways so that students can learn. In order 
to facilitate efective inquiry-oriented clasrooms using Stein and Smith?s (2011) 
practices, teachers need the knowledge base to do so. This study showed that the 
 
 
 
 
210 
exemplary teachers in this study had the knowledge to facilitate efective inquiry-oriented 
clasrooms.  
For every leson that the teacher has taught before, when asked about 
misconceptions and planned explanations and questions, teachers referred to reflections 
from previous experiences teaching the leson. For example, Mrs. Orchid?s Triangle in a 
Bag activity has been done at least a few times. She stated that when she taught this 
leson the first few times, she was learning diferent unanticipated student responses. 
Now, she has taught the leson enough times to where an unanticipated student response 
is rare. This is in alignment with Stein and Smith?s (2011) strategy of anticipation 
supported by documenting student responses year to year. Some student misconceptions 
and areas of dificulty were in other areas of math, but both teachers stated how these 
dificulties lie in a deeper misconception that afects a current topic. Both teachers? plans 
for sequencing (Stein & Smith, 2011) were heavily based on anticipation (Stein & Smith, 
2011).  
For both teachers, many responses to students? questions, that were content 
related, were in the form of a question. Through a series of questions eliciting student 
responses, eventualy, the student answers their own question. For example, a student 
asked Mrs. Orchid if the order of stating two congruent triangles could be changed. Mrs. 
Orchid replied ?only if what?? The student replied with ?if they are both matching?, and 
Mrs. Orchid asked the student to name it diferently like she stated. Mrs. Lotus frequently 
responded to her students in the same manner. This relates into Stein and Smith?s (2011) 
practice of monitoring, which is supported by anticipating student responses beforehand. 
 
 
 
 
211 
Both teachers shared extensive knowledge of anticipated student responses before the 
execution of the leson, and they were observed during the execution of the leson.  
Implications for Teaching and Learning 
 Of the various sources of knowledge that the exemplary teachers in this study 
shared, and in alignment with Herbst and Kosko?s (2012) study; experience teaching 
geometry is an important source of knowledge for teaching geometry. For the teachers in 
this study, when there was no prior experience of a new activity, much of geometry 
TACK knowledge came from experience teaching related content and activities, and 
other sources. This produces implications for teaching geometry, preparing preservice 
teachers, and providing profesional development for inservice teachers. Aspects of 
characteristics of exemplary teachers important to implications for teaching and learning 
are: explaining concepts, asking key questions to guide student learning, modifying 
lesons, predicting student dificulties, addresing student dificulties, conducting 
clasroom procedures, and utilizing technology. 
Teaching Geometry. In order to support relational understanding, teachers 
teaching geometry need to know explanations of concepts and the potential pitfals in the 
explanations related to student prerequisite knowledge and student areas of dificulty or 
misconceptions. During clas, when asked by students to give an explanation, teachers 
teaching geometry should have a supply of questions to ask the student asking the 
question or other students. When students give answers or explanations, teachers teaching 
geometry need to ask students to justify their answers; as ability to justify is not only a 
good indication of understanding, but it provides students with experience in proving 
informaly (Breyfogle & Lynch, 2010; Driscoll, 2007; Mistreta, 2000).  
 
 
 
 
212 
It is advisable for teachers teaching geometry to have a variety of sources of 
knowledge including but not limited to teaching experience, coursework, textbook, 
profesional development, and collaboration. Teachers stated that recommendations from 
multiple sources help in identifying areas of student dificulties or misconceptions, and 
appropriate modifications to activities without sacrificing cognitive demand (Stein, 
Engle, Smith, & Hughes, 2008). Once areas of student dificulties or misconceptions are 
identified, ways to guide students to learn through the dificulties or misconceptions need 
to be primed prior to execution of the leson. If the use of technology is necesary in an 
activity, teachers need to know how to use the technology themselves, be able to help 
students use it, and facilitate student learning using the technology without taking away 
from the concepts to be learned (Hughes, 2010). In the event of technological dificulties, 
teachers need to have back-up plans such as paty paper or compas and straight edge for 
constructions.  
Another aspect of being wel prepared involves teachers knowing sequence of 
phases of the leson and timing for them very wel. This helps with time management 
when teaching. Both teachers in this study used times when students were working to 
walk around to students that needed individual help, check students for understanding, or 
carry out clasroom administrative tasks like taking atendance.  
Preparing Preservice Teachers. Preservice teachers have limited teaching 
experience. Preparing them with a good foundation of explanations of concepts, key 
questions to guide student learning, aceptable modifications to lesons, possible student 
dificulties and how to addres them, efective use of time during clas, and use of 
technology is important. Providing them with meaningful experiences to learn these can 
 
 
 
 
213 
include, but is not limited to, exposure to literature in mathematics education that 
expounds on these, collaboration with experienced exemplary teachers, and guided 
teaching experience to start their own mental portfolio of geometry TACK based on 
teaching experience.  
Providing Profesional Development. Since the previous section explained 
implications for preparing preservice teachers, this section focuses on providing 
profesional development for inservice teachers.  Teachers in this study most often 
referred to previous teaching experience as a source of knowledge, teachers with diferent 
previous teaching experience would benefit from experiences these exemplary teachers 
have to offer (van Driel & Berry, 2012). Results from this study showed that knowledge 
of explaining concepts, asking key questions to guide student learning, modifying 
lesons, addresing student dificulties, conducting clasroom procedures, and utilizing 
technology for activities teachers are familiar with can be shared with other teachers that 
are les familiar with the activity. In order to continue to improve as teachers, there is a 
fundamental need to collaborate with other teachers to share ideas and brainstorm 
solutions for areas of dificulty. There are always areas for improvement and ideas to 
share. 
Summary. Since the most important source of knowledge for geometry TACK 
for an activity is experience teaching that activity; for those that have not had experience 
teaching the activity, I think it is important to provide meaningful experience teaching the 
activity. For both preservice and inservice teachers, I think that a meaningful experience 
connects geometry TACK for the activity to whatever their knowledge base may be for 
teaching the activity, and provides support for them to teach the activity to have at least 
 
 
 
 
214 
one experience teaching the activity.  
Implications for Research 
Due to the amount of additional data on Teachers? Applied Content Knowledge 
(TACK) observed in the executed leson when compared to the planned leson, I think it 
is important to addres implications for research on teacher knowledge. Clearly, 
observations are necesary to se what teachers do in context when studying teacher 
knowledge. When given the choice betwen paper and pencil asesment and observing a 
teacher, due to the student component, observation gives more TACK data. Looking at 
pre-observation and post-observation interviews, both were equaly important when 
studying TACK planned and TACK executed. Pre-observation interviews gave a good 
baseline of what the teacher thought of prior to student interactions. Post-interviews 
where teachers watched parts of their leson brought to light what teachers ?forgot? to tell 
me about the leson or did not think was important to share. From interviews with both 
teachers, there were many remarks made to the importance of student interaction in 
showing their knowledge that they otherwise did not think to share. 
Further Areas of Study 
 From what I have sen through analysis of data, I saw four imediate areas of 
study; deeper look at questions and explanations, student component, other teachers at 
diferent stages of experience, and other subjects. As sen from both teachers, the 
students caused increased questions and explanations. Thus this is two pronged: looking 
more in depth to student involvement in teacher planning and execution of lesons, and 
types of questions and explanations that were in addition to those planned. A lot of 
 
 
 
 
215 
questions and explanations observed during the execution of the leson not stated prior to 
the execution of the leson showed the connecting practice (Stein & Smith, 2011).  
Also, it would be interesting to look at teachers at diferent stages of experience, 
not necesary number of years teaching, but experience. In my experience with novice 
teachers, specialized content knowledge based on previous experiences teaching the 
content is much les than experienced teachers like Mrs. Orchid and Mrs. Lotus. It would 
be interesting to se what the observable diferences are. Also, the TACK connections 
were observed in relation to high school geometry. My first inclination would be to se 
other levels of geometry since the geometry filter is already in place. However, other 
areas of mathematics can be explored in a similar fashion as wel, like Algebra and 
Statistics. 
 
 
 
 
 
216 
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235 
Appendix 
Appendix A 
 
Planning Execution 
Specialized 
Content 
Knowledge 
Knowledge of concepts, 
definitions, and procedures 
for teaching 
Observable knowledge of concepts, 
definitions, and procedures used in 
teaching 
 Knowledge 
of Content 
and 
Teaching 
Plan for sequence of 
information to be presented. 
 
Sequence of information presented. 
Planned time for phases of 
the leson. 
 
Time spent on phases of the leson. 
How to explain the 
concepts, definitions, and 
procedures. 
 
Explained concepts, definitions, and 
procedures. 
Planned key questions to 
guide student thinking. 
 
Questions asked of students to guide their 
thinking. 
Knowledge 
of Content 
and 
Students 
Leson modifications based 
on perceived student 
dificulties. 
 
Did those student dificulties stil arise? If 
so, how was it shown? What did the 
teacher do in response? 
Other anticipated areas of 
student dificulties and how 
to help students overcome. 
 
Actual presented areas of student 
dificulties and how the teacher helped 
students overcome. 
Anticipated student levels 
of prerequisite skils and 
how the teacher plans 
acordingly. 
Actual student level s of prerequisite 
skils and how the teacher taught 
acordingly. 
 
 
 
 
 
 
 
236 
Appendix B 
Interview Task (videotaped) 
Prior to the interview, the teachers wil be asked to write a list of concepts to be covered 
during the leson/unit. Teachers wil also bring with them any materials that wil be used 
during class, like handouts, pre-determined questions, and homework. 
Initial Questions Regarding the List 
1. Are these in the order of presentation to the clas? If so, why? If not, what would 
it be, and why did you arrange it that way? 
2. How do you plan on teaching this concept? Why? 
3. What prior knowledge is needed for this concept? How much of the prior 
knowledge do your students know? How do you know what they know? 
4. Which concepts wil be harder than others for students to learn, why? Which 
concepts wil be easier, why? Is the answer the same for al students? Why, why 
not? 
5. How wil you know that students understand this concept? 
6. What wil make you decide to go on to the next concept? 
Supplementary Questions if not answered from Initial Questions 
1. What are some student misconceptions? How are you prepared for those? 
2. What are some analogies or comparisons you have prepared in case students need 
more explanations about the concept? 
3. How do you plan on ending the leson? Where does the leson end? 
4. What are some key questions you plan on using to guide students? thinking? 
5. What are some things that I did not ask about that you would like to share? 
 
 
 
 
237 
 
Appendix C 
Observation protocol 
 
7. How did the teacher explain concepts, definitions, and procedures? 
 
8. What are areas of teacher dificulties in explanations? 
 
9. What was the sequence of information presented? 
 
10. What was the time spent on phases of the leson? 
 
11. What student dificulties arose? If so, how was it shown? What did the teacher do 
in response? 
 
12. What were actual student levels of prerequisite skils and how did the teacher 
teach to those levels? 
 
 
 
Appendix D 
Concept Map 
 
Applications       Related Concepts  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Defining Features       Related Features 
 
 
 
Concept(s) to be taught 
 
 
 
 
238 
Appendix E 
Sample Concept Maps 
 
 
(Wiliams, 1998, p. 417) 
 
 
 
 
239 
 
(Lawson & Chinnappan, 2000, p. 211 
 
Appendix F 
Meting Purpose 
Initial Meting Overview of study, concept map 
introduction, set up timeline, forms for 
conducting study (school based). 
Unit Plan and Concept Map Interview Background information on unit and 
TACK 
Interview for Leson 1 Collect data for planning 
Observation 1 Collect data for enacting 
Interview for Reflection of Leson 1 Triangulate data 
Interview for Leson 2 Collect data for planning 
Observation 2 Collect data for enacting 
Interview for Reflection of Leson 2 Triangulate data 
Interview for Leson 3 Collect data for planning 
Observation 3 Collect data for enacting 
Interview for Reflection of Leson 3 Triangulate data 
 
 
 
 
 
 
 
240 
Appendix G 
Definitions 
? Nodes for sequences ? examples, problems, definitions, and theorems; i.e. each 
separate belwork problem as a node 
? Nodes for explain ? when a student asks a question, the teacher explains rather 
than caling on another student, or asks a question in return, then that is a node for 
explain. If the same explanation is given for the same question, whether or not 
asked by the same student; both count within one node. If the same explanation is 
given to another question, it is a diferent node. Explanations can also be 
unsolicited, each topic of explanation counting as a node. 
? Nodes for questions ? There are two situations where Mrs. Lotus poses questions 
related to content, to engage in student interaction or in response to student 
interaction. Each question eliciting a specific response from students is considered 
a node. Multiple questions eliciting the same specific response is also considered 
one mode. Questions in response to student questions follow the same counting 
algorithm.  
? Nodes for modifications ? planned leson modifications are modifications like the 
choice of one example over another or a purposeful choice of a problem in order 
to bring out or avoid certain perceived student dificulties. Each modification 
counts as one node. 
? Nodes for dificulties ? each anticipated dificulty that the teacher does not plan 
on specificaly addresing, but acknowledges existence, is considered a node. 
Further, al dificulties that arise during the execution of the leson that was not 
 
 
 
 
241 
previously verbalized to me is also considered a node. Same dificulty on multiple 
students at multiple times stil counts as one node.  
? Nodes for prerequisite skils ? each skil a teacher mentioned that the student 
should know either in interviews or during clas time, counts as a node. Same 
prerequisite skils mentioned for multiple students at multiple times stil counts as 
one node.  
? Phases ? overal sections of clas where there is an overarching unified goal; 
namely belwork, homework review, leson of the day including notes and 
activity, and working on homework for the next day.  
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
242 
Appendix H 
Coding Guide 
 This guide sets up the units for analysis for both interview transcripts as wel as 
observation transcripts. Interview transcripts included what the teacher what the teacher 
wrote while talking, included preprinted material, and had reproductions of what the 
teacher pointed at while they were talking. For clasroom observations, timing for phases 
of instruction, teacher movement throughout the room, and whom the teacher was 
addresing was noted.   
Unitizing 
  Definitions used to describe the units of analysis for this study follows those set 
forth by Garret (2010).  
Term Definition 
Topic A complete idea, aspect of TACK, or topic of conversation 
upon which the atention of the speaker or speakers is focused. 
Unit A word, sentence, paragraph or several consecutive sentences 
or paragraphs focusing on the same topic (p. 264) 
 
Examples: 
Separation by question in interview 
INTERVIEWER: so what are they supposed to get at by the end of today? 
MRS. ORCHID: by the end of the today, the goal is that they wil have had 
enough opportunities to try and figure out which questions [Me: um hm] are most 
appropriate are most helpful for duplicating the triangle [Me: um hm] um and at 
least some of the groups wil have narrowed it down to three questions that are 
most helpful. Um so that?s that?s just the goal for today, just to try to get that [Me: 
um hm] and we?re not putting a name on any of the, you know, sets of three 
questions or anything. That wil happen tomorrow 
Separation            
INTERVIEWER: great, so you?ll know then that students understand in that way 
when  
MRS. ORCHID: if their groups are able to come up with the right triangle 
questions, then I?ll that groups understand, I won?t know if individual students 
 
 
 
 
243 
necesarily understand, um but hopefully, I don?t know, as I?m encouraging them 
to collaborate together, everybody wil be participating hopefully. [Me: okay] 
Separation within an answer in interview 
INTERVIEWER: Um, so we?re starting, um can you explain to me your list of 
what you're going to cover? 
MRS. LOTUS: okay, I sat down and I was just going to look at I gues 
background. [me: um hm] I've changed books, I gues we changed books [me: um 
hm] plus we changed from our typical course of study to the common core course 
of study [me: um hm] and I um ah al that entails. The new book is very diferent 
from the more traditional book that I've used forever. [me: okay] so I sat down 
and I know I'm going to talk about triangles and I think that?s what the next 
month, two to three months, two months [me: um hm] I?ll be talking about 
triangles so I was looking at al right what do I want what do I need to tel them 
about triangles, what do they need to know and then looking at what do I do what 
have I done that?s in the book and where is it. So I'm, the list is more ?okay, now I 
need to talk about this this this? this is what's in the book [me: um hm] um, the 
new book so that?s my list.  
Within Mrs. Lotus? answer to my question on explaining what she planned on covering, 
she first talked about changing of textbooks and standards required of students. Below 
shows how I separated this portion of the transcript.  
Section 1           
INTERVIEWER: Um, so we?re starting, um can you explain to me your list of 
what you're going to cover? 
MRS. LOTUS: okay, I sat down and I was just going to look at I gues 
background. [me: um hm]  
Section 2           
I've changed books, I gues we changed books [me: um hm] plus we changed 
from our typical course of study to the common core course of study [me: um hm] 
and I um ah al that entails. The new book is very diferent from the more 
traditional book that I've used forever. [me: okay]  
 
 
 
 
 
 
 
244 
Section 3           
so I sat down and I know I'm going to talk about triangles and I think that?s what 
the next month, two to three months, two months [me: um hm] I?ll be talking 
about triangles so I was looking at al right what do I want what do I need to tel 
them about triangles, what do they need to know and then looking at what do I do 
what have I done that?s in the book and where is it. So I'm, the list is more ?okay, 
now I need to talk about this this this? this is what's in the book [me: um hm] um, 
the new book so that?s my list.  
Separation of activity during executed leson (Separation of Phases) 
Mrs. Lotus: ? Now its not something you know because its something somebody 
learned. Okay, any questions before we go on? Okay, so that?s where we 
left yesterday. This is how far we?ve gotten. So today we?re going to pick 
up with the corollary to a theorem. Now a corollary is another statement 
that is related to a theorem that we already learned that we can use that 
theorem to prove. 
Mrs. Lotus was reviewing what students had discussed the previous day, and then she 
went on to the notes for the leson of the day. Below shows the separation: 
 (cont.) Review from the previous day      
Mrs. Lotus: ? Now its not something you know because its something somebody 
learned. Okay, any questions before we go on? Okay, so that?s where we 
left yesterday. This is how far we?ve gotten.  
 Notes for today          
So today we?re going to pick up with the corollary to a theorem. Now a 
corollary is another statement that is related to a theorem that we already 
learned that we can use that theorem to prove? 
Separation within an activity (Separation within Phases) 
Mrs. Orchid: any other groups need more drawing paper? Before we start? 
Student: we do 
{teacher hands drawing paper} 
Mrs. Orchid: anybody else?  
{no one raises their hand} 
Mrs. Orchid: Okay, round three begins right now. What is our goal for the number 
of questions? 
Several students: three 
Mrs. Orchid: three 
 
 
 
 
245 
Mrs. Orchid had just summarized round two within the Triangle in a Bag Activity. The 
natural separation is betwen the summary of round two and the beginning of round three 
of the activity.  
(cont.) Summarized round two        
Mrs. Orchid: any other groups need more drawing paper? Before we start? 
Student: we do 
{teacher hands drawing paper} 
Mrs. Orchid: anybody else? 
{no one raises their hand} 
Beginning of round three for students to work together     
Mrs. Orchid: Okay, round three begins right now. What is our goal for the number 
of questions? 
Several students: three 
Mrs. Orchid: three ? 
Separation of topics within classroom transcription 
Mrs. Orchid: any other groups need more drawing paper? Before we start? 
Student: we do 
{teacher hands drawing paper} 
Mrs. Orchid: anybody else? 
{no one raises their hand} 
Mrs. Orchid: Okay, round three begins right now. What is our goal for the number 
of questions? 
Several students: three 
Mrs. Orchid: three 
Using the same transcript as above, there is separation within an activity by the nature of 
the activity, but also by what was being said.  
(cont.) Sumarized round two ? students needed materials    
Mrs. Orchid: any other groups need more drawing paper? Before we start? 
Student: we do 
{teacher hands drawing paper} 
 
 
 
 
246 
Mrs. Orchid: anybody else? 
{no one raises their hand} 
Beginning of round three for students to work together ? directions to begin  
Mrs. Orchid: Okay, round three begins right now.  
Question asked to help start students on round three    
Mrs. Orchid: What is our goal for the number of questions? 
Several students: three 
Mrs. Orchid: three ? 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
247 
Codes 
A Priori Codes and their Descriptions 
Specialized content knowledge 
Knowledge of concepts, definitions, and procedures for teaching. Specificaly: 
? presenting mathematical ideas,  
? responding to students? ?why? questions,  
? finding an example to make a specific mathematical point,  
? recognizing what is involved in using a particular representation,  
? linking representations to underlying ideas and to other representations,  
? connecting a topic being taught to topics from prior or future years,  
? explaining mathematical goals and purposes to parents,  
? appraising and adapting the mathematical content of textbooks,  
? modifying tasks to be either easier or harder,  
? evaluating the plausibility of students? claims (often quickly),  
? giving or evaluating mathematical explanations,  
? choosing and developing useable definitions,  
? using mathematical notation and language and critiquing its use,  
? asking productive mathematical questions,  
? selecting representations for particular purposes,   
? inspecting equivalencies (Bal, Thames, & Phelps, 2008, p. 10). 
 
Knowledge of Content and Teaching 
? Sequence of information  
? Time for phases of the leson 
? Explanations of concepts, definitions, and procedures 
? Key questions to guide students? thinking 
Knowledge of Content and Students 
? Modifications based on student dificulties 
? Anticipated areas of student dificulties and how to help students overcome 
? Student levels of prerequisite skils and how the teacher plans acordingly 
 
 
 
 
248 
Below is a table of the a priori codes from literature: 
 
Planning Execution 
Specialized 
Content 
Knowledge 
SCK-P 
Knowledge of concepts, 
definitions, and procedures 
for teaching 
SCK-E 
Observable knowledge of concepts, 
definitions, and procedures used in 
teaching 
 
Knowledge 
of Content 
and 
Teaching 
KCTseq-P 
Plan for sequence of 
information to be presented. 
 
KCTseq-E 
Sequence of information presented. 
KCTtime-P 
Planned time for phases of 
the leson. 
 
KCTtime-E 
Time spent on phases of the leson. 
KCTexplain-P 
How to explain the 
concepts, definitions, and 
procedures. 
 
KCTexplain-E 
Explained concepts, definitions, and 
procedures. 
KCTquestions-P 
Planned key questions to 
guide student thinking. 
 
KCTquestions-E 
Questions asked of students to guide their 
thinking. 
Knowledge 
of Content 
and 
Students 
KCSmod-P 
Leson modifications based 
on perceived student 
dificulties. 
 
KCSmod-E 
Did those student dificulties stil arise? If 
so, how was it shown? What did the 
teacher do in response? 
KCSdificulties-P 
Other anticipated areas of 
student dificulties and how 
to help students overcome. 
 
KCSdificulties-E 
Actual presented areas of student 
dificulties and how the teacher helped 
students overcome. 
KCSprerequisite-P 
Anticipated student levels 
of prerequisite skils and 
how the teacher plans 
acordingly. 
KCSprerequisite-E 
Actual student level s of prerequisite 
skils and how the teacher taught 
acordingly. 
 
 
 
 
 
 
249 
Emergent Codes 
Emergent codes, topics and subjects of discussion that showed up in interviews and 
clasroom observations are listed below.  
(AD) Activity directions ? discussions of what students are about to do and how 
to do it. For example, directions for an activity or directions for students to take 
out homework to be checked.  
(B) Birdwalking ? teachers engaging in discussion with students about something 
not related to the leson or any of the above codes, e.g. asking students how their 
day was while checking homework and the student talking to the teacher about 
what their family did that wekend. This was not an apparent atempt at behavior 
modification (e.g. geting student to sit up if they had their head down).  
(CA) Clasroom administration ? discussions pertaining to taking atendance, 
intercom interruptions, bels sounding for other clases, students asking to use 
school facilities outside the clasroom, discipline, and other clasroom 
procedures. 
(Cu) Curriculum ? discussion pertaining to opinions and ideas of textbook being 
used or standards for instruction. 
(RTI) Researcher Technical Interactions ? discussion regarding administrative 
aspects of the study like camera angle and scheduling.  
(V) Verification ? statements made to students from the teacher leting them know 
that what they are doing is either correct or incorrect without questioning or 
explaining. For example: during groupwork, teacher teling students that they are 
doing a good job and on the right track.  
(O) Other ? since everything is coded, this is where a segment that does not fit 
anywhere else is coded. Since the focus is on teacher knowledge, some statements 
that students make that the teacher did not respond to verbaly or nonverbaly 
became categorized as this code.  
 
 
 
 
 
 
 
 
 
250 
Appendix I 
Van Hiele Levels Abridged from (Breyfogle & Lynch, 2010) 
Level Name Description 
 Visualization Se geometric shapes as a whole; do not focus 
on their particular atributes. 
1 Analysis Recognize that each shape has diferent 
properties; identify the shape by that property. 
2 Informal Deduction Se the interrelationships betwen figures. 
3 Formal Deduction Construct proofs rather than just memorize 
them; se the possibility of developing a proof 
in more than one way. 
4 Rigor Learn that geometry needs to be understood in 
the abstract; se the ?construction? of 
geometric systems. 
 
Phases of Learning from the van Hiele Model (Mistreta, 2000, p. 367) 
Phase Description 
Information Discussions are held where the teacher learns of 
the students' prior knowledge and experience with 
the subject mater at hand. 
Directed Orientation The teacher provides activities that alow students 
to become more acquainted with the material being 
taught 
Explication A transition betwen reliance on the teacher and 
students' self-reliance is made. 
Free Orientation The teacher is atentive to the inventive ability of 
the students. Tasks that can be approached in 
numerous ways are presented to the students. 
Integration The students summarize what was learned during 
the leson. 
 
 
 
 
 
251 
Geometric Habits of Mind and Their Indicators (Driscoll, 2007, pp.12-15) 
Geometric Habit of Mind Indicators 
Reasoning with 
Relationships 
Basic: Identification of figures presented in a problem and 
correct enumeration of their properties 
Advanced: Relating multiple figures in a problem through 
proportional reasoning and reasoning through symmetry 
 
Generalizing Geometric 
Ideas 
Basic: Uses one problem situation to generate another, or 
when the solver intuits that he or she hasn?t found al the 
solutions 
Advanced: Generate al solutions and make a convincing 
argument as to why there are no more; or wondering what 
happens if a problem?s context is changed 
 
Investigating Invariants Basic: Decides to try a transformation of figures in a 
problem without being prompted, and considers what has 
changed and what has not changed 
Advanced: Consider extreme cases for what is being asked 
by a problem 
Balancing Exploration and 
Reflection 
Basic: Drawing, playing, and/or exploring with occasional 
(though maybe not be consistent) stock-taking 
Advanced: approaching a problem by imagining what a 
final solution would look like, then reasoning backward; or 
making what Herbst (2006) cals ?reasoned conjectures? 
about solutions with strategies for testing the conjectures 
 
 
 
 
252 
Appendix J 
Mrs. Orchid Unit 1 Observation 1 KCT 
 Observed as Planed Additional Observed 
Sequence Reviewed homework with 
presentations and answering 
questions, review test, and the 
two rounds for Triangle in a Bag. 
None 
Time 
Everything listed in the sequence 
was covered by the end of clas 
time. 
?Homework and test review may have taken a bit 
longer than expected, but discusion for Triangle in a 
Bag was not rushed.? 
Explanations Triangle in a Bag 
? Students made the conection 
that they are drawing 
congruent triangles 
? Used example of constructing 
truses 
? To help determine number of 
questions, reminded students 
that if a question yielded more 
than one answer, then it 
counted as more than one 
question. 
? Explained some congruency notation practices 
during homework presentations 
? Explained how to do a homework problem and 
several test problems 
? Explained several mistakes students made on the 
test 
? Explanations as to why certain student writen 
questions cannot be answered 
? Explanations for drawing 
? Explanations geting students to determine 
usefulnes of questions and to minimize amount 
of information needed 
? Explanations for the necesity of labeling vertices 
 
Questions Triangle in a Bag - Prompted 
students to ask useful questions 
by asking ?how is this question 
useful?? 
Homework and Test Review: 
? ?Can you elaborate on that?? 
? ?Please explain what he/she said.? 
? ?Why do you think you?re answer is 
right/correct?? 
Triangle in a Bag: 
? Questions to determine usefulnes of information 
asked 
? Questions to help minimize information asked 
 
Mrs. Orchid Unit 1 Observation 1 KCS 
 Observed as Planed Additional Observed 
Modifications None None 
Student 
Dificulties/ 
Misconceptions 
Student areas of dificulties for Triangle in a Bag: asking useful 
questions, geting to the minimum number of questions, 
recognizing the conection to congruent triangles, misplaced 
vertices, and construction 
None 
Prerequisite 
Knowledge 
Student prerequisites knowledge for Triangle in a Bag: how to 
use a ruler and a protractor to measure angles and side lengths, 
and create angle measures and side lengths of specified 
measurements 
Triangle inequality 
 
 
 
 
 
 
 
253 
Mrs. Orchid Unit 1 Observation 1 SCK 
Mathematical Tasks of 
Teaching Planed and Executed 
Presenting mathematical ideas ?Triangle in a Bag? ideas were observed as planed. Ideas from 
questions reviewed for homework and test were observed in clas. 
Responding to students? ?why? 
questions 
All planed responses to students? ?why? questions were observed for 
?Triangle in a Bag?. Aditional responses to students? ?why? questions 
were observed for al phases of the leson. 
Finding an example to make a 
specific mathematical point 
For extra credit, Mrs. Orchid asked students to find examples of 
congruent triangles other than construction, with the goal in mind to 
use these as examples. Trusses example was in the textbook. 
 
Selecting representations for 
particular purposes 
Uniform congruence marks minimizes confusion. 
 
Recognizing what is involved in 
using a particular representation 
Mrs. Orchid chose for students to show triangle congruence marks on 
the diagrams in the way that the bok used it or Geometer?s Sketchpad 
to minimize confusion. 
 
Linking representations to 
underlying ideas and to other 
representations 
Mrs. Orchid emphasized to students that coresponding parts had the 
same congruence marks, and vice versa. 
 
Conecting a topic being taught 
to topics from prior or future 
years 
Students learned how to mark coresponding parts of congruent 
triangles congruent in the leson prior. This leson develops the idea of 
triangle congruence shortcuts. Mrs. Orchid planed on using triangle 
congruence to prove parallelogram properties. 
 
Explaining mathematical goals 
and purposes to parents 
None observed 
 
Appraising and adapting the 
mathematical content of 
textboks 
For this leson, other than the homework review from the previous 
day, the textbook was not a part of the leson or homework for this 
observation day. 
Modifying tasks to be either 
easier or harder 
None observed 
Evaluating the plausibility of 
students? claims (often quickly) 
During the observation, she evaluated plausibility of students? claims 
during Triangle in a Bag rounds and when students answered 
questions. 
Giving or evaluating 
mathematical explanations 
Giving mathematical explanations was shown both in planed and 
executed phases of the lesson. Evaluating mathematical explanations 
was observed in al phases of the executed leson. 
 
Chosing and developing 
useable definitions 
None observed 
 
Using mathematical notation 
and language and critiquing its 
use 
Throughout the observation, Mrs. Orchid corected terminology when 
students were not using the corect term, modeled corect terminology, 
and emphasized congruency notation practices. 
 
Asking productive 
mathematical questions 
Observed questions, including planed, incited discusion and led to 
students providing corect answers. 
Inspecting equivalencies For Triangle in a Bag rounds, Mrs. Orchid determined how diferent 
questions could be asking for the same information. 
 
 
 
 
 
 
 
254 
Mrs. Orchid Unit 1 Observation 2 KCT 
KCT Observed as Planed Additional Observed 
Sequence Finish the rounds for Triangle in a 
Bag, group presentations, and 
classwork questions 
None 
Time 
The planed sequence was covered by 
the end of clas time. 
Group discusion went longer than expected due 
to new technology, and claswork discusion 
went shorter than planed 
Explanations Triangle in a Bag 
? Explanations from rounds for 
yesterday 
Classwork 
? Group presentations on how they 
were able to create congruent 
triangles ? diferent groups present 
diferent shortcuts 
? Corect vocabulary wil be coached 
during group presentations 
? Highlight parts students knew 
when presenting 
? Explanations for congruence 
shortcut theorem specific to 
theorem 
 
? Explanations as to why the question canot 
be answered specific to student questions 
? Explanations for drawing 
? Explanations geting students to determine 
usefulnes of questions and to minimize 
amount of information needed 
? Explanations for the necesity of labeling 
vertices 
? Explained how a set of questions can be 
diferent 
? Re-voicing and elaborating student 
explanations for triangle congruence shortcut 
theorems 
Questions Triangle in a Bag 
? Same questions as the day prior 
? Claswork questions 
? Questions to determine usefulnes of 
information asked 
? Questions to help minimize information 
asked 
 
 
 
Mrs. Orchid Unit 1 Observation 2 KCS 
KCS Observed as Planed Additional 
Observations 
Modifications No modifications No modifications 
Student 
Dificulties/ 
Misconceptions 
Student areas of dificulties for Triangle in a Bag: 
asking useful questions, getting to the minimum 
number of questions, recognizing the connection to 
congruent triangles, misplaced vertices, 
construction, triangle JKL, and recognizing 
included side or angle. 
Dificulty for 
recognizing included 
side or angle were not 
observed. 
Prerequisite 
Knowledge 
Prerequisite knowledge for Triangle in a Bag: how 
to use a ruler and a protractor to measure angles and 
side lengths, and create angle measures and side 
lengths of specified measurements 
Triangle inequality 
 
 
 
 
 
 
 
 
 
255 
Mrs. Orchid Unit 1 Observation 2 SCK 
 
 
 
Mathematical Tasks of Teaching Planed and Executed 
Presenting mathematical ideas ?Triangle in a Bag? ideas were observed as planned. 
Responding to students? ?why? 
questions 
During rounds in Triangle in a Bag, Mrs. Orchid responded with 
explanations and questions. 
Finding an example to make a 
specific mathematical point 
Examples specific to situation. 
 
 
Selecting representations for 
particular purposes 
Mrs. Orchid chose for students organize the triangle congruence 
shortcuts into a graphic organizer. It helps students see necessary 
information on specific triangle congruence shortcut easily. 
 
Recognizing what is involved in 
using a particular representation 
Mrs. Orchid created a blank table for students to fil out to organize 
the triangle congruence shortcuts. The graphic organizer had thre 
columns to organize information for the triangle congruence shortcuts: 
shortcut name, picture, and writen description. In each row as a 
diferent triangle congruence shortcut. 
 
Linking representations to 
underlying ideas and to other 
representations 
For shortcut name, Mrs. Orchid also asked students to write the 
abbreviation. When discusing the picture and writing the description, 
Mrs. Orchid introduced the term ?included? where pertinent. 
 
Conecting a topic being taught 
to topics from prior or future 
years 
Students learned how to mark coresponding parts of congruent 
triangles congruent in the leson prior. This leson develops the idea 
of triangle congruence shortcuts. Mrs. Orchid planned on using 
triangle congruence to prove parallelogram properties. 
 
Explaining mathematical goals 
and purposes to parents 
None observed 
 
Appraising and adapting the 
mathematical content of 
textboks 
None observed 
Modifying tasks to be either 
easier or harder 
None observed 
 
Evaluating the plausibility of 
students? claims (often quickly) 
During the observation, she evaluated plausibility of students? claims 
during Triangle in a Bag rounds, statements during presentations, and 
in general when students answered questions. 
 
Giving or evaluating 
mathematical explanations 
Giving mathematical explanations was shown both in planed and 
executed phases of the lesson. 
Chosing and developing 
useable definitions 
Definitions for included angle, included side, and triangle congruence 
shortcuts: ASA, SAS, AS, SS, and HL. 
Using mathematical notation 
and language and critiquing its 
use 
Throughout the observation, Mrs. Orchid corrected terminology when 
students were not using the corect term, modeled corect 
terminology, and emphasized congruency notation practices. 
 
Asking productive mathematical 
questions 
Observed questions, including planed, incited discusion and led to 
students providing corect answers. 
Inspecting equivalencies For Triangle in a Bag rounds, Mrs. Orchid determined how diferent 
questions could be asking for the same information. 
 
 
 
 
256 
Mrs. Orchid Unit 1 Observation 3 KCT 
KCT Observed as Planned Additional Observations 
Sequence Sequence was for students to work on the 
belringer, homework check, belringer review, 
review homework, and then claswork problem 1. 
 
None 
Time No planed timing other than homework wil take 
too long 
There was not enough time for 
classwork problems 2-4. 
Explanations Belringer 
? Directions for the activity 
? Take two measurements and se how many 
triangles can be created 
? It?s not included in the shortcut theorems 
because it?s caled the ambiguous case 
Homework Review 
? Prepared to answer any questions students may 
have 
? Use diferent student heights as example for 
why side-side does not work 
Claswork 
? Using CPCTC to se the importance of using 
congruent triangles 
? Courier ? angle-side-angle 
? Runway problem ? side-angle-side with 
alternate interior angles forcing the lines to be 
paralel 
? Construction ? truses, side-side-side 
Construction ? conecting first flor to second 
flor, perpendicular 
Belringer 
? How to draw and measure 
? Theorems have to work every 
time 
Homework Review 
? How to mark triangles to se 
which parts are congruent 
? Re-voiced student 
explanations to answers to 
homework problems 
? CPCTC 
Claswork 
? How to mark parts congruent 
to se whether or not a side or 
angle is ?included? 
? Re-voiced student 
explanations 
 
Questions Belringer 
? Are you sure there?s not another triangle you 
can draw? 
? Does angle B have to be that big? Can we make 
it smaller? Can we make it larger? What will 
happen if you do? 
Homework Review 
? What is your prof? What evidence do you 
have? 
Claswork 
? How do you know they are congruent triangles? 
? Do you already know the hypotenuse is 
congruent? 
? What other information can we get now about 
the triangles? 
Belringer 
? What the problem is asking 
? Drawing 
? Whether or not this is a 
theorem 
Homework Review 
? What is your prof? What 
evidence do you have? 
? Questions specific to 
homework question 
Claswork 
? How do you know they are 
congruent triangles? 
? Do you already know the 
hypotenuse is congruent? 
? What other information can we 
get now about the triangles? 
 
 
 
 
 
 
 
 
257 
Mrs. Orchid Unit 1 Observation 3 KCS 
KCS Observed as Planed Additional Observations 
Modifications No modifications No modifications 
Student 
Dificulties/ 
Misconceptions 
Planed homework misconceptions 
and misconception of what an 
included angle and what an included 
side is. 
All planed dificulties/misconceptions were 
observed except the homework problem where 
both triangles have two sides and an angle marked 
congruent, but one has an included angle and the 
other does not. 
 
Prerequisite 
Knowledge 
Planed prerequisites for Triangle in 
a Bag: how to use a ruler and a 
protractor to measure angles and 
side lengths, and create angle 
measures and side lengths of 
specified measurements 
All planed prerequisites were observed. 
 
Mrs. Orchid Unit 1 Observation 3 SCK 
Mathematical Tasks of 
Teaching 
Planed and Executed 
Presenting mathematical ideas Belringer, homework review, and claswork ideas were presented as 
planned. Additional homework problems were presented and 
classwork ideas 2-4 were not presented during the observation. 
Responding to students? ?why? 
questions 
Students asked ?why? questions during homework review. Mrs. 
Orchid responded by having other students explain, and coached 
corect vocabulary or ideas where necessary. 
Finding an example to make a 
specific mathematical point 
Examples specific to situation. 
 
 
Selecting representations for 
particular purposes 
For claswork problems, some problems did not have a pre-drawn 
figure so students neded to create their own, and problems that had 
figures were not drawn to scale. 
 
Recognizing what is involved in 
using a particular representation 
Mrs. Orchid chose problems that did not have figures so that students 
can practice creating their own figures from text. Other figures that 
were not drawn to scale were used because students neded to rely on 
information labeled in the figure. These have the potential for student 
dificulties and misconceptions. 
 
Linking representations to 
underlying ideas and to other 
representations 
Understanding how to draw figures from text and use pre-drawn 
figures for claswork problems involved congruence marking from the 
beginning of the unit. This also links to future problems where students 
need to draw and interpret figures. 
Conecting a topic being taught 
to topics from prior or future 
years 
From Triangle in a Bag, students learned triangle congruence shortcut 
theorems. Students used triangle congruence shortcut theorems in 
classwork problems. Classwork problems helped students see triangle 
congruence in diferent contexts. In this unit, students used triangle 
congruence to prove parallelogram properties. 
 
Explaining mathematical goals 
and purposes to parents 
Not observed 
 
 
 
 
 
 
258 
Appraising and adapting the 
mathematical content of 
textboks 
Mrs. Orchid used four problems out of the textbok for claswork. 
However, she changed one of the problems not to require students to 
create a flow chart prof. 
 
Modifying tasks to be either 
easier or harder 
None observed 
 
Evaluating the plausibility of 
students? claims (often quickly) 
During the observation, she evaluated plausibility of students? claims 
during homework review, claswork, and when students answered 
questions. 
 
Giving or evaluating 
mathematical explanations 
Planed mathematical explanations were given during the observation. 
Additional explanations were also observed. Mrs. Orchid evaluated 
student mathematical explanations throughout the observation. 
 
Chosing and developing 
useable definitions 
None observed 
 
Using mathematical notation 
and language and critiquing its 
use 
Throughout the observation, Mrs. Orchid corected terminology when 
students were not using the corect term, modeled corect terminology, 
and emphasized congruency notation practices. 
 
Asking productive 
mathematical questions 
Observed questions, including planed, incited discusion and led to 
students providing corect answers. 
Inspecting equivalencies Mrs. Orchid determined how diferent congruence statements could be 
equivalent. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
259 
Mrs. Orchid Unit 1 Observation 4 KCT 
 
 
 
KCT Observed as Planned Additional Observations 
Sequence Work on the belringer, go over belringer, 
define paralelogram, construct a 
paralelogram using GSP, properties of 
paralelogram activity, summarize 
paralelogram activity, and start on 
homework. 
 
None 
Time Sequence was completed by the end of 
class 
 
Teacher was acidentaly locked out of 
room, so transition time was 10 minutes 
longer than expected. 
Explanations Belringer 
? Want them to focus on angle properties 
inside two sets of intersecting parallel 
lines 
Defining ?Paralelogram? 
? Talk about what a god definition of a 
paralelogram is 
? Talk about what ?al sides are paralel? 
could mean 
? If students say ?square? or ?rectangle?, 
talk about parallelogram clasifications 
Constructing a Paralelogram 
? List step by step 
Paralelogram Activity 
? Compare and measure 
? Remind students of definitions of parts 
of a paralelogram 
Sumary 
? Have students explain answers to 
questions from activity and fil out 
graphic organizer 
 
Belringer: 
? Directions on the belringer and where 
the missing information is for students 
to solve 
? Definition of transversal 
? Alternate interior and alternate exterior 
angle relationships when parallel lines 
are cut by a transversal 
Defining ?Paralelogram?: 
? Symbolic notation for paralel 
? Defined parts of a paralelogram 
Constructing a Paralelogram: Diference 
betwen a sketch or drawing and a 
construction 
Paralelogram Activity: 
? Directions for constructing 
? Diference between ?distance? and 
?length? 
? Explanations specific to student 
questions and answers 
Sumary: How to mark congruence of 
segments 
Questions Belringer 
? What are you suposed to be loking 
at? 
Defining ?Paralelogram? 
? What is a paralelogram? 
? What is a god definition? 
Constructing a Paralelogram 
? How would you do this? 
Paralelogram Activity 
? Are you sure you are loking at the 
corect items? 
Sumary 
Ask students to elaborate on explanations 
Belringer 
? Questions to get students to use the sum 
of the measures of the interior angles of 
a quadrilateral to solve for an angle 
measure. 
? Questions that asked students to 
recognize the relationship of 
consecutive interior angles. 
Paralelogram Activity: 
? What do you cal that point in the 
midle? 
? What hapens when you cut something 
in the middle? What's the math word 
for that? 
 
 
 
 
260 
 
Mrs. Orchid Unit 1 Observation 4 KCS 
 
 
 
 
 
 
 
 
 
 
 
KCS Observed as Planed Additional Observations 
Modification Have students learn how to construct before going 
to the lab instead of learning to construct by trial 
and error on their own. 
None 
Student 
Dificulties/ 
Misconceptions 
Belringer 
? There should be none 
Defining ?Paralelogram? 
? Thinking that a squares or rectangles are the 
only paralelograms 
? Thinking that four paralel lines, or two sets of 
paralel lines is sufficient information for the 
definition of a paralelogram 
? Suficiency for the definition of a paralelogram 
Constructing a Paralelogram 
? Students may not be able to construct a 
paralelogram in GSP, they may just draw it. 
Paralelogram Activity 
? Selecting not enough or to much items on GSP 
? Not knowing definitions of parts of a 
paralelogram 
? No knowing which angles are ?oposite? or 
?consecutive? 
Sumary 
? Explanations for relationships betwen two 
diagonals 
Belringer 
? Definition of transversal 
? Alternate interior and 
alternate exterior angle 
relationships when 
paralel lines are cut by a 
transversal 
? Sum of the measures of 
the interior angles of a 
quadrilateral 
 
Prerequisite 
Knowledge 
Paralel lines, consecutive interior angles theorem, 
knowledge from belringer (calculating mising 
information on angles from given information in a 
diagram), and a working definition of 
paralelograms. 
Definition of a transversal, 
alternate interior and 
alternate exterior angle 
relationships when paralel 
lines are cut by a transversal  
 
 
 
 
261 
Mrs. Orchid Unit 1 Observation 4 SCK 
Mathematical Tasks of 
Teaching 
Planed and Executed 
Presenting mathematical ideas All planed ideas in belringer and paralelogram activity were 
observed. Additional ideas were also observed. 
 
Responding to students? ?why? 
questions 
All planed responses to students? ?why? questions were observed. 
Additional responses to students? ?why? questions were also observed. 
 
Finding an example to make a 
specific mathematical point 
In asking students to define what a paralelogram is, Mrs. Orchid had 
two counterexamples ready: all sides parallel ? four paralel lines to 
each other, and square or rectangle ? talk about quadrilateral 
classifications. 
 
Selecting representations for 
particular purposes 
Mrs. Orchid chose for students to use GSP constructed paralelograms 
to discover properties. 
 
Recognizing what is involved in 
using a particular representation 
Using GSP created paralelograms required students to have aces to 
a computer and knowledge to construct a parallelogram. 
 
Linking representations to 
underlying ideas and to other 
representations 
Constructing the paralelogram was based on a working definition of 
?paralelogram? Mrs. Orchid established in clas. Students were 
familiar with GSP. Students wil be using GSP in future lesons. 
 
Conecting a topic being taught 
to topics from prior or future 
years 
Some students many know some properties of paralelograms from 
previous clases. This leson was for students to discover properties of 
paralelograms. By the end of the unit, students used triangle 
congruence to prove properties of paralelograms. 
 
Explaining mathematical goals 
and purposes to parents 
None observed 
 
 
Appraising and adapting the 
mathematical content of 
textboks 
None observed 
 
 
Modifying tasks to be either 
easier or harder 
None observed 
 
 
Evaluating the plausibility of 
students? claims (often quickly) 
Mrs. Orchid evaluated plausibility of students? claims during each 
phase of the leson. 
 
Giving or evaluating 
mathematical explanations 
Giving and evaluating mathematical explanations was shown both in 
planned and executed phases of the leson. 
 
Chosing and developing 
useable definitions 
Defined paralelogram in order to construct it on GSP. 
  
Using mathematical notation 
and language and critiquing its 
use 
Throughout the observation, Mrs. Orchid corected terminology when 
students were not using the corect term, modeled corect terminology, 
and emphasized congruency notation practices. 
 
Asking productive 
mathematical questions 
Observed questions, including planed, incited discusion and led to 
students providing corect answers. 
Inspecting equivalencies Throughout the leson, Mrs. Orchid determined how diferent student 
questions or statements could be equivalent. 
 
 
 
 
262 
Mrs. Orchid Unit 2 Observation 1 KCT 
KCT Planed as Observed Additional Observations 
Sequence Planed sequence was for students to work on the 
belringer, homework check, belringer review, review 
homework, and then develop the area formula in the 
Pigpen activity. 
 
How to use the calculator during 
homework review 
Time No planed timing other than belwork and homework 
wil take to long 
Belwork and homework tok 
too long but Mrs. Orchid 
planned for that. 
 
Explanations Belringer 
? Dividing by two is the same as multiplying by 0.5. 
? Base of a triangle does not have to be on the 
bottom. 
? If it is not marked to be the midle in the diagram, 
do not asume it is. 
? How to use the area of a triangle formula 
? How to find the height of a triangle 
Homework Review 
? Prepared to answer any questions students may 
have 
? Explanations for why a trigonometry ratio is the 
appropriate one to use for a set of information. 
Pigpen for Monica 
? Explain the major problem for the activity ? 
finding the angle to place the fences so Monica 
wil have the most area in her pigpen. 
? Notice the patern to develop a formula. 
? Explanations to help students verbalize the formula 
? Prompt students to sumarize what they learned 
Belringer 
? Directions for the activity 
? Gave example of measuring a 
person?s height, it needs to be 
perpendicular to the ground 
? Rounding for an answer that 
is not an integer. 
Homework Review 
? How to use the calculator for 
trigonometry functions 
? Answers specific to student 
questions 
Pigpen for Monica 
? Directions for the activity 
? Triangle inequality 
? What it means for a pen to be 
closed 
Questions Belringer 
? Are you sure you?re using the corect trigonometry 
ratio? How do you know? 
? What?s the reason for that? 
? What does this doted segment (height outside the 
triangle) mean? 
Homework Review 
? Are you sure you?re using the corect trigonometry 
ratio? 
? Why is that the corect answer? 
Pigpen for Monica 
? What wil the pigpen lok like? 
? What do you ned to find the area? 
? Was it a god idea to make the angle smaler or 
bigger? 
? Where would the height be? Can you draw a 
segment that would represent the height? Where 
can we put the height so we can calculate it? 
? How would you solve for h? 
? Do you notice a patern? What repeats? 
Belringer 
? What units are we working 
with? 
? Clarifying questions 
Homework Review 
? Why do I not want to spend 
time going over (specific 
homework problem numbers) 
? Questions regarding set up of 
calculators 
Pigpen for Monica 
? What is the problem asking 
you to do? 
? Where to put the angle and 
what kind of angle it would 
be 
? Is there another place to put 
the height? 
? Identifying triangle parts 
 
 
 
 
 
 
263 
Mrs. Orchid Unit 2 Observation 1 KCS 
KCS Observed as Planed Additional 
Observations 
Modifications No modifications No modifications 
 
Student 
Dificulties/ 
Misconceptions 
 
Belringer 
? Base of a triangle has to be on botom 
? If a side loks like it is bisected in the diagram, it is 
bisected. 
? Height canot be calculated outside an obtuse triangle. 
? Use the wrong trigonometry ratio to find the length of 
a mising side. 
Homework Review 
? Determining the corect trigonometry ratio 
? Determining in relation to the angle, which side is 
opposite and which sides is adjacent. 
Pigpen for Monica 
? Drawing a height that can be determined with the 
information given 
? Verbalizing the formula for area using Sine 
 
Dificulty drawing the 
shape of the pigpen 
Prerequisite 
Knowledge 
Planed prerequisite for belringer was how to plug 
numbers into an equation and solve. For al of the 
activities of the day is the diference between sine, 
cosine, and tangent ratios. 
None 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
264 
Mrs. Orchid Unit 2 Observation 1 SCK 
Mathematical Tasks of 
Teaching 
Planned and Executed 
Presenting mathematical ideas All planed ideas in belringer, homework review, and area formula 
for Pigpen for Monica was observed. Aditional ideas were also 
observed. 
Responding to students? ?why? 
questions 
All planed responses to students? ?why? questions were observed.  
Finding an example to make a 
specific mathematical point 
No examples given in pre-observation interview. Measuring a person?s 
height needing to be perpendicular to the ground was observed during 
executed lesson. 
 
Selecting representations for 
particular purposes 
For the belringer, Mrs. Orchid chose to have the triangles draw out to 
feature different orientations of triangles. 
 
Recognizing what is involved in 
using a particular representation 
Mrs. Orchid used triangles in diferent orientations to incite and 
address misconceptions. 
 
Linking representations to 
underlying ideas and to other 
representations 
In Pigpen activity, Mrs. Orchid asked students to continue to draw 
triangles to give a picture to the problem. She stated that being able to 
draw a useful diagram is necesary to solving word problems. 
 
Conecting a topic being taught 
to topics from prior or future 
years 
The Pigpen activity tok what students knew about area formula for a 
triangle and improved upon it to use the sine ratio. This linked to Law 
of Sines. 
 
Explaining mathematical goals 
and purposes to parents 
None observed 
 
 
 
Appraising and adapting the 
mathematical content of 
textboks 
None observed 
 
 
 
Modifying tasks to be either 
easier or harder 
None observed 
 
 
Evaluating the plausibility of 
students? claims (often quickly) 
Mrs. Orchid evaluated plausibility of students? claims during each 
phase of the leson. 
 
 
Giving or evaluating 
mathematical explanations 
Giving and evaluating mathematical explanations was shown both in 
planned and executed phases of the leson. 
Chosing and developing 
useable definitions 
Defined paralelogram in order to construct it on GSP. 
  
Using mathematical notation 
and language and critiquing its 
use 
Throughout the observation, Mrs. Orchid corected terminology when 
students were not using the corect term, and modeled corect 
terminology. 
 
Asking productive 
mathematical questions 
Observed questions, including planed, incited discusion and led to 
students providing corect answers.  
Inspecting equivalencies Throughout the leson, Mrs. Orchid determined how diferent student 
questions or statements could be equivalent. 
 
 
 
 
 
265 
 
Mrs. Orchid Unit 2 Observation 2 KCT 
KCT Observed as Planed Additional Observations 
Sequence Planed sequence was for students to review 
homework problems using area formula ? base x 
aSinC, left over sine and cosine ratio homework; 
finish the Pigpen activity, and then start on the Law 
of Sines activity. 
 
Only homework review and 
part of the Pigpen activity was 
observed. 
Time At least finish sequence up to Pigpen activity. Homework tok to long. Did 
not finish the Pigpen activity 
and did not start Law of Sines. 
 
Explanations  Pigpen for Monica Part 2 
? Directions of what the problem is asking 
? Determine if the formula developed in part 1 
always works 
? It may apear to students that the formula does 
not work for obtuse angles 
? Use the formula for 120? and se if the answer 
is the same for the suplement. 
? Notice patern of values of angles for sine and 
cosine greater than 90? 
? Not going to say ?Sine of suplementary angles 
are the same? 
? Students should then notice that the formula can 
be used with an obtuse angle, as wel as acute. 
Law of Sines - Every time during the Pigpen 
activity, when students are solving for the height, 
and then if they set them up to each other, it ends 
up being the Law of Sines. 
 
All planed observed in 
addition to: 
Homework Review 
? How to use the calculator 
for trigonometry functions 
? Explanation of when to use 
table and when to use 
calculator 
? Answers specific to student 
questions 
Pigpen for Monica Part 2 
? Directions to draw the 
corect height 
? Height neds to be 
perpendicular to the base 
Questions Pigpen for Monica Part 2 
? Does the new area formula always work? Wil it 
work for obtuse angles? 
? Discus why using the area formula for 120? 
yields the same answer as the supplement. 
? Can you fine other angles that have the same 
sine ratio? 
? What's the relationship betwen 60? and 120?? 
? What about right angles? Can we use this with 
90?? 
Law of Sines - During the Pigpen activity, when 
students are solving for the height, and then if they 
set them up to each other, it ends up being the Law 
of Sines. 
Pigpen for Monica 
? Why is that the height? 
? What is the problem with 
this situation? (obtuse with 
height outside) 
 
 
 
 
 
 
 
266 
 
 
 
Mrs. Orchid Unit 2 Observation 2 KCS 
KCS Observed as Planed Additional Observations 
Modifications No modifications During homework Mrs. 
Orchid led students to 
answer homework 
problems. 
 
Student 
Dificulties/ 
Misconceptions 
 
Bellringer 
? Base of a triangle has to be on botom 
? If a side loks like it is bisected in the diagram, 
it is bisected. 
? Height canot be calculated outside an obtuse 
triangle. 
? Use the wrong trigonometry ratio to find the 
length of a missing side. 
Homework Review 
? Determining the corect trigonometry ratio 
? Determining in relation to the angle, which side 
is oposite and which sides is adjacent. 
Pigpen for Monica 
? Drawing a height that can be determined with 
the information given 
? Verbalizing the formula for area using Sine 
 
None 
Prerequisite 
Knowledge 
Planed prerequisite was for students to be able to 
use the area formula ? base x aSinC from the two 
homework problems. 
None 
 
 
 
 
 
 
 
 
 
 
 
 
 
267 
 
Mrs. Orchid Unit 2 Observation 2 SCK 
Mathematical Tasks of 
Teaching 
Planed and Executed 
Presenting mathematical ideas Not al planed ideas were observed. 
Responding to students? ?why? 
questions 
All planed responses to students? ?why? questions were observed. 
Additional responses to students? ?why? questions were also observed. 
Finding an example to make a 
specific mathematical point 
No examples given in pre-observation interview. Measuring a person?s 
height needing to be perpendicular to the ground was observed during 
executed lesson. 
 
Selecting representations for 
particular purposes 
Mrs. Orchid asked students to draw out the triangle for each problem. 
 
Recognizing what is involved in 
using a particular representation 
Mrs. Orchid understod that drawings are not acurate, and asked 
students to label apropriately. 
 
Linking representations to 
underlying ideas and to other 
representations 
In Pigpen activity, Mrs. Orchid asked students to continue to draw 
triangles to give a picture to the problem. She stated that being able to 
draw a useful diagram is necesary to solving word problems. 
 
Conecting a topic being taught 
to topics from prior or future 
years 
The Pigpen activity tok what students knew about area formula for a 
triangle and improved upon it to use the sine ratio. This linked to Law 
of Sines. 
 
Explaining mathematical goals 
and purposes to parents 
None observed 
 
 
 
Appraising and adapting the 
mathematical content of 
textboks 
None observed 
 
 
 
Modifying tasks to be either 
easier or harder 
None observed 
 
 
Evaluating the plausibility of 
students? claims (often quickly) 
Mrs. Orchid evaluated plausibility of students? claims during each 
phase of the leson. 
 
 
Giving or evaluating 
mathematical explanations 
Giving and evaluating mathematical explanations was shown both in 
planned and executed phases of the leson. 
Chosing and developing 
useable definitions 
Defined paralelogram in order to construct it on GSP. 
  
Using mathematical notation 
and language and critiquing its 
use 
Throughout the observation, Mrs. Orchid corected terminology when 
students were not using the corect term, and modeled corect 
terminology. 
 
Asking productive 
mathematical questions 
Observed questions, including planed, incited discusion and led to 
students providing corect answers. 
Inspecting equivalencies Throughout the leson, Mrs. Orchid determined how different student 
questions or statements could be equivalent. 
 
 
 
 
 
268 
 
Mrs. Orchid Unit 2 Observation 3 KCT 
KCT Observed as Planed Additional Observations 
Sequence Planed sequence was for students to watch and 
folow long instructions of a video for the first two 
parts of Pigpen activity, discuss homework (third part 
of Pigpen activity), and then derive Law of Sines 
folowing the Law of Sines activity. 
 
None 
Time No planed time for the day, except to go through al 
of the planned sequence; which is to finish the Law of 
Sines Activity. 
 
None 
Explanations Pigpen for Monica Part 1 
? Directions for the activity 
? Explain the major problem for the activity ? 
finding the angle to place the fences so Monica 
wil have the most area in her pigpen. 
? Triangle inequality 
? What it means for the pen to be closed 
? Notice the patern to develop a formula. 
? Explanations to help students verbalize the formula 
Pigpen for Monica Part 2 
? Directions of what the part 2 is asking 
? Determine if the formula developed in part 1 
always works 
? Directions to draw the corect height 
? Height neds to be perpendicular to the base 
? It may apear to students that the formula does not 
work for obtuse angles 
? Use the formula for 120? and se if the answer is 
the same for the suplement. 
? Notice patern of values of angles for sine and 
cosine greater than 90? 
? Not going to say ?Sine of suplementary angles are 
the same? 
? Students should then notice that the formula can be 
used with an obtuse angle, as wel as acute. 
Pigpen for Monica Part 3 (Homework review) 
? Examine the trig table and se what is the 
maximum value for sine. 
? The suplements match 
Law of Sines 
? Every time during the Pigpen activity, when 
students are solving for the height, and then if they 
set them up to each other, it ends up being the Law 
of Sines. 
? Directions to sketch the triangle to derive Law of 
Sines 
Pigpen for Monica Part 1 
? Explanations during each problem 
where Mrs. Orchid used slightly 
diferent numbers for calculations 
and how it is similar to prior days. 
? Naming sides and angles of a 
triangle 
? Notice the area has not changed 
using diferent heights of the same 
triangle 
? Answers specific to student 
questions 
Pigpen for Monica Part 2 
? Explanation that although the 
relationship betwen 60 and 120 is 
that double of 60 is 120, there may 
be another relationship. 
? Understanding what the 
relationship betwen sine of 60? 
and 120? is ?key?. 
Pigpen for Monica Part 3 
? Explanations specific to each 
problem in this part 
? ?The trig table I gave you is 
designed to show acute angles of a 
right triangle only? 
? Make sure that the calculator is in 
degres mode and not radians 
? Appropriate use of diferent area 
formulas 
Law of Sines 
? ?Oblique is a fancy word for non-
right triangles or non-right? 
? Directions to set up sine ratios for 
diferent angles in a way so that 
students can see that they're equal 
 
 
 
 
269 
Questions Pigpen for Monica Part 1 
? What is the problem asking you to do? 
? Where to put the angle and what kind of angle it 
would be 
? What wil the pigpen lok like? 
? What do you ned to find the area? 
? Was it a god idea to make the angle smaller or 
bigger? 
? Where would the height be? Can you draw a 
segment that would represent the height? Where 
can we put the height so we can calculate it? 
? Is there another place to put the height? 
? Identifying triangle parts 
? How would you solve for h? 
? Do you notice a patern? What repeats? 
Pigpen for Monica Part 2 
? Does the new area formula always work? Wil it 
work for obtuse angles? 
? Why is that the height? 
? What is the problem with this situation? (obtuse 
with height outside) 
? Discus why using the area formula for 120? yields 
the same answer as the suplement. 
? Can you fine other angles that have the same sine 
ratio? 
? What's the relationship betwen 60? and 120?? 
? What about right angles? Can we use this with 
90?? 
Pigpen for Monica Part 3 (Homework review) 
? Preplaned questions in the activity 
? How can you be sure that this is the maximum 
area? 
Law of Sines ? questions to lead students to se 
equivalency of Law of Sines 
Pigpen for Monica Part 1 
? Is every problem you?re going to 
do involve sides of eight and six? 
? What is the problem with this 
situation? (obtuse with height 
outside) 
Pigpen for Monica Part 2 
? Does this only work for 60 and 
120? What else would it work for? 
Pigpen for Monica Part 3 
? Questions to lead students to set up 
the Sine ratios so that the patern of 
them being equal can be sen 
? Why is this posible? Are you 
sure? 
? What do you notice? 
 
 
Mrs. Orchid Unit 2 Observation 3 KCS (no aditional observations) 
KCS Observed as Planed 
Modifications Have students watch a video for the first two parts of Pigpen activity. Al discusions 
involved the teacher and the whole clas, no group work. Students were seated in rows. 
Student 
Dificulties/ 
Modifications 
Mrs. Orchid planed the leson in anticipation for any dificulties that may occur; 
instruction on how where to draw the height that can be used for the problem, using the 
calculator properly, and questions that highlighted the relationship between sines of 
suplementary angles. 
Prerequisite 
Knowledge 
Prerequisite knowledge for students was how to plug numbers into an equation and 
solve. Other than that, Mrs. Orchid planed to give students enough explanation in order 
for them to answer each question in the activities. 
 
 
 
 
 
270 
Appendix K 
Mrs. Orchid Unit 1 Observation 1 Geometry Filter 
Geometry Filter Planed and 
Executed 
Clasrom Evidence 
Van Hiele 
Levels 
Level 2: Informal 
Deduction 
See relationships betwen hiden triangle and the triangle bring 
created. 
Discusions where students state their reasoning and justify. 
 
Phases of 
Learning 
Directed 
Orientation 
 
Explication 
 
Fre Orientation 
 
 
Integration 
The rounds in the Triangle in a Bag activity alowed students to 
become more acquainted with the material being taught. 
With each round of Triangle in a Bag, there was a transition of 
students being les dependent of Mrs. Orchid. 
For each of the rounds of Triangle in a Bag, Mrs. Orchid is 
attentive to the inventive nature of students and the task of 
creating a congruent triangle can be approached in many diferent 
ways. 
At the end of each round, Mrs. Orchid led students to summarize 
what was learned in each round. 
 
Geometric 
Habits of Mind 
Reasoning with 
Relationships 
Balancing 
Exploration and 
Reflection 
Students were reasoning with what were necesary relationships 
betwen two triangles to create duplicates. 
Throughout the rounds in the Triangle in a Bag activity, students 
were balancing exploration and reflection. 
 
Mrs. Orchid Unit 1 Observation 2 Geometry Filter 
Geometry 
Filter 
Planed and 
Executed 
Clasrom Evidence 
Van Hiele 
Levels 
Level 2: 
Informal 
Deduction 
See relationships betwen hiden triangle and the triangle bring created. 
Discusions where students state their reasoning and justify. 
 
 
Phases of 
Learning 
Directed 
Orientation 
 
Explication 
 
 
Fre 
Orientation 
 
 
Integration 
The rounds in the Triangle in a Bag activity alowed students to become 
more acquainted with the material being taught. 
 
With each round of Triangle in a Bag, there was a transition of students 
being les dependent of Mrs. Orchid. 
 
For each of the rounds of Triangle in a Bag, Mrs. Orchid is atentive to the 
inventive nature of students and the task of creating a congruent triangle 
can be approached in many diferent ways. 
 
At the end of al the rounds, students sumarized their findings through 
group presentations and a graphic organizer.  
 
Geometric 
Habits of 
Mind 
Reasoning 
with 
Relationships 
 
Balancing 
Exploration 
Students were reasoning with what were necesary relationships betwen 
two triangles to create duplicates. 
 
 
Throughout the rounds in the Triangle in a Bag activity, students were 
balancing exploration and reflection. 
 
 
 
 
271 
and Reflection 
 
Executed, not 
planned: 
Investigating 
Invariants 
 
 
Mrs. Orchid placed the triangle template in diferent orientations forcing 
students to tel her how to transform the triangle to fit the one they drew. 
 
Mrs. Orchid Unit 1 Observation 3 Geometry Filter 
Geometry 
Filter 
Planed and 
Executed 
Clasrom Evidence 
Van Hiele 
Levels 
Level 2 
Level 3 
Justifying answers through discussion. 
Planed but not observed, students writing a formal prof. 
 
Phases of 
Learning 
Directed 
Orientation 
Explication 
 
Fre 
Orientation 
Integration 
Belringer and aplication problems alowed students to become more 
acquainted with the material being taught. 
Homework review with student presentations and group work on 
application problems showed a transition of students being less dependent 
of Mrs. Orchid. 
The aplication problems can be aproached in many diferent ways. 
At the end of each activity, belringer review, homework review, and 
classwork summary, students summarized what was learned. 
 
Geometric 
Habits of 
Mind 
Reasoning 
with 
Relationships 
Balancing 
Exploration 
and Reflection 
Executed, not 
planned: 
Investigating 
Invariants 
Students were reasoning with what were necesary relationships betwen 
two triangles to create duplicates. 
 
Throughout the rounds in the Triangle in a Bag activity, students were 
balancing exploration and reflection. 
 
Mrs. Orchid placed the triangle template in diferent orientations forcing 
students to tel her how to transform the triangle to fit the one they drew.  
 
 
 
 
 
 
 
 
 
 
 
 
272 
Mrs. Orchid Unit 1 Observation 4 Geometry Filter 
Geometry 
Filter 
Planed and 
Executed 
Clasrom Evidence 
Van Hiele 
Levels 
Level 2: 
Informal 
Deduction 
For belwork review and activity discusions, Mrs. Orchid prompted 
students to state why they believe the answers they came up with were 
true. 
Phases of 
Learning 
Information 
 
Directed 
Orientation 
 
Explication 
Fre 
Orientation 
 
Integration 
Belringer, defining paralelogram, and construction of paralelogram, 
Mrs. Orchid learned of students? prior knowledge through discusions. 
Exploration of paralelogram properties, students were becoming more 
familiar with the material. 
 
During the exploration of paralelogram properties, there was a transition 
of reliance from teacher to student self-reliance. 
Students could use a variety of methods to test properties of 
paralelograms. 
After the exploration, Mrs. Orchid led students in sumarizing their 
findings. 
Geometric 
Habits of 
Mind 
Reasoning 
with 
Relationships 
Balancing 
Exploration 
and Reflection 
Investigating 
Invariants 
Generalizing 
Geometric 
Ideas 
Students were reasoning with relationships of angle measures and lengths 
of sides and diagonals within a paralelogram. 
 
Throughout the activity, students were exploring and reflecting on what 
they found 
 
Mrs. Orchid prompted students to move the paralelograms around to se 
relationships. 
 This was not planed, but observed. For students that were done soner 
than other students, Mrs. Orchid asked them to se if there were other 
observations to be made. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
273 
Mrs. Orchid Unit 2 Leson 1 Geometry Filter 
Geometry 
Filter 
Planed and 
Executed 
Clasrom Evidence 
Van Hiele 
Levels 
Level 1: Analysis 
Level 2: Informal 
Deduction 
Identify the shape of the pigpen. 
 
Discussions where students state their reasoning and justify. 
Phases of 
Learning 
Directed 
Orientation 
 
Explication 
 
 
Fre Orientation 
Integration 
Belwork and homework review ere activities that alowed 
students to become more acquainted with the material that was 
necesary for the Pigpen activity. 
Mrs. Orchid set up the questions in the Pigpen activity so that 
students can transition from reliance on Mrs. Orchid to themselves. 
Belwork, homework, and activity problems had multiple ways to 
get the answer. 
At the end of the Pigpen activity, Mrs. Orchid prompted students to 
sumarize what they learned. 
 
Geometric 
Habits of 
Mind 
Reasoning with 
Relationships 
Identify shape of pigpen, triangle. 
 
 
Mrs. Orchid Unit 2 Leson 2 Geometry Filter 
Geometry 
Filter 
Planed and 
Executed 
Clasrom Evidence 
Van Hiele 
Levels 
Level 2: Informal 
Deduction 
Discusions where students state their reasoning and justify. 
 
Phases of 
Learning 
Directed 
Orientation 
 
Explication 
 
Fre Orientation 
Homework review alowed students to become more acquainted 
with the material that was necesary for the Pigpen activity. 
Mrs. Orchid set up the questions in the Pigpen activity so that 
students can transition from reliance on Mrs. Orchid to themselves. 
Homework and Pigpen activity problems had multiple ways to get 
the answer. 
 
Geometric 
Habits of 
Mind 
Balancing 
Explorations and 
Reflection 
Planed for students during the Law of Sines activity where they 
make reasoned conjectures about what the relationship would be. 
This was not observed during the leson.  
 
 
 
 
 
 
 
 
274 
Mrs. Orchid Unit 2 Leson 3 Geometry Filter 
Geometry 
Filter 
Planed and 
Executed 
Clasrom Evidence 
Van Hiele 
Levels 
Level 2: Informal 
Deduction 
 
Discusions where students stated their reasoning and justified. 
Phases of 
Learning 
Directed 
Orientation 
 
Explication 
 
 
Fre Orientation 
 
Integration 
Pigpen activity alowed students to be more familiar with using the 
sine function in an area formula. Law of Sines activity helped 
students derive the Law of Sines 
Mrs. Orchid wrote the questions in the Pigpen activity so that 
students can transition from reliance on Mrs. Orchid to themselves. 
Students worked on Part 3 of the Pigpen activity on their own for 
homework the night before. 
Part 3 of Pigpen Activity problems had multiple ways to get the 
answer. 
Review at the end of each part of Pigpen activity as wel as Law of 
Sines activity is where Mrs. Orchid led students to sumarize what 
they learned. 
 
Geometric 
Habits of 
Mind 
Generalizing 
Geometric Ideas 
Mrs. Orchid asked students if they could think of other pairs of 
angle measures where the sine of those angle measures are 
equivalent. 
 
 
Mrs. Lotus Unit 1 Observation 1 Geometry Filter 
Geometry 
Filter 
Planed and 
Executed 
Clasrom Evidence 
Van Hiele 
Levels 
Level 2 
 
Level 3 
For belwork review, notes, and homework activity discusion, Mrs. 
Lotus prompted students to state why they believe the answers they 
came up with were true. 
Students were chalenged to prove a corolary 
Phases of 
Learning 
Information 
 
Directed 
Orientation 
Explication 
 
Fre 
Orientation 
Integration 
Belwork and review from the previous day, Mrs. Lotus learned of 
students? prior knowledge. 
Problems during notes and discusion for the homework activity 
provided students with the opportunity to become more familiar with 
the material. 
Problems during notes and student discusion of homework problem 
showed the transition of reliance from teacher to student self-
reliance. 
Students could use a variety of methods prove the corolary and 
approach the homework problem. 
After each activity, Mrs. Lotus led students in sumarizing their 
findings. 
 
 
 
 
275 
Geometric 
Habits of 
Mind 
Reasoning with 
Relationships 
 
Generalize 
Geometric 
Ideas 
 
Balancing 
Exploration and 
Reflection 
Students were reasoning with relationships of the sum of interior 
angles of a triangle, interior and exterior angles of a triangle, exterior 
angles of a triangle, and posible points equidistant from two and 
thre points. 
Proving about sum of interior angles, relationships betwen interior 
and exterior angles, and exterior angles of a triangle chalenged 
students to generalize geometric ideas. 
 
Throughout the leson, students made reasoned conjectures and 
tested them.  
 
 
Mrs. Lotus Unit 1 Observation 2 Geometry Filter 
Geometry 
Filter 
Planed and 
Executed 
Clasrom Evidence 
Van Hiele 
Levels 
Level 2 
 
For belwork review, homework review, and Equaly Wet discusion, 
Mrs. Lotus prompted students to state why they believe the answers 
they came up with were true. 
 
Phases of 
Learning 
Information 
Directed 
Orientation 
 
 
Explication 
 
Fre 
Orientation 
 
Integration 
Belwork and review of Equaly Wet, Mrs. Lotus learned of students? 
prior knowledge. 
Diferent construction activities provided students oportunities to 
become more acquainted with the material. 
Group work during the GSP activity showed the transition of reliance 
from teacher to student self-reliance. 
Students could use a variety of methods to create the constructions to 
form conjectures in the GSP activity. 
 
After each GSP construction, Mrs. Lotus led students in sumarizing 
their findings. 
Geometric 
Habits of 
Mind 
Reasoning with 
Relationships 
 
Generalize 
Geometric 
Ideas 
 
Balancing 
Exploration and 
Reflection 
 
Students reasoned with relationships of the intersection of the thre 
perpendicular bisectors of a triangle and the intersection of three angle 
bisectors of a triangle. 
 
For each of these intersections, Mrs. Lotus asked students if they found 
all posible solutions. 
 
For each problem, students explored and then reflected on the activity 
when Mrs. Lotus led the whole clas in discussion 
 
 
 
 
276 
 
Mrs. Lotus Unit 1 Geometry Filter 
Day Geometry Filter Planed and Executed 
1 Van Hiele Levels Level 2: Informal Deduction, Level 3: Formal Deduction 
Phases of Learning Directed Orientation, Explication, Fre Orientation, Integration 
Geometric Habits of Mind Reasoning with Relationships, Generalizing Geometric Ideas, 
Balancing Exploration and Reflection 
 
2 Van Hiele Levels Level 2: Informal Deduction 
Phases of Learning Directed Orientation, Explication, Fre Orientation, Integration 
Geometric Habits of Mind Reasoning with Relationships, Generalizing Geometric Ideas, 
Balancing Exploration and Reflection 
3 Van Hiele Levels Level 2: Informal Deduction 
Phases of Learning Directed Orientation, Explication, Fre Orientation, Integration 
Geometric Habits of Mind Reasoning with Relationships, Generalizing Geometric Ideas, 
Balancing Exploration and Reflection 
 
Mrs. Lotus Unit 2 Observation 1 Geometry Filter 
Geometry 
Filter 
Planed and 
Executed 
Clasrom Evidence 
Van Hiele 
Levels 
Level 2 
 
Level 3 
In belwork review and textbok homework review, Mrs. Lotus 
challenged students to informally prove why the answers they came up 
with were true; asking students to justify with the question ?why?? 
Construct a prof for a corolary to a theorem 
 
Phases of 
Learning 
Information 
 
Directed 
Orientation 
 
 
Explication 
 
Fre Orientation 
Integration 
Belwork and review homework from the previous day was where Mrs. 
Lotus learned of students? prior knowledge and experience with the 
leson of the day. 
Claswork problems and the dilations activity provided students 
opportunities to become more acquainted with the material. 
Group work during the dilation activity and claswork problems 
showed the transition of reliance from teacher to student self-reliance. 
Students could use a variety of methods to make conjectures in the 
dilation activity as wel as the claswork problems. 
After each claswork problem and group work for the dilation activity, 
Mrs. Lotus led students in sumarizing their findings. 
 
 
 
 
 
277 
Geometric 
Habits of 
Mind 
Reasoning with 
Relationships 
 
Generalize 
Geometric Ideas 
Balancing 
Exploration and 
Reflection 
 
During the dilation activity, students were reasoning with relationships 
of the original figure and the dilated figure. They made observations of 
the similarities and differences betwen the original figure and dilated 
figure. 
 
Students talking with group members to generalize observations of 
their figures. 
 
For each problem and the dilation activity, students explored on their 
own and as a group before sumarizing as a whole class and reflecting 
on how others approached the problem. 
 
 
Mrs. Lotus Unit 2 Observation 2 Geometry Filter 
Geometry 
Filter 
Planed and 
Executed 
Clasrom Evidence 
Van Hiele 
Levels 
Level 2 
 
In homework review and discusion of problems throughout the leson, 
Mrs. Lotus chalenged students to informaly prove why the answers they 
came up with were true 
 
Phases of 
Learning 
Information 
 
Directed 
Orientation 
 
Explication 
 
 
Fre 
Orientation 
 
Integration 
During homework review was when Mrs. Lotus learned of students? prior 
knowledge and experience with the leson of the day. 
The construction activities showed ?directed orientation?, where these 
activities were provided for students to become more familiar with the 
material being taught. 
During discusion of the conjectures from constructions and problems 
showed a transition of reliance on teacher to student self reliance 
The questions and discusion of the conjectures from constructions were 
open-ended and was attentive to the inventive abilities of the students. 
The two problems at the end of the leson showed integration phase because 
students were using what they learned throughout the day to solve those 
problems. 
 
Geometric 
Habits of 
Mind 
Reasoning with 
Relationships 
 
Generalize 
Geometric 
Ideas 
Balancing 
Exploration and 
Reflection 
 
During the dilation activity, students were reasoning with relationships of 
the original figure and the dilated figure. They made observations of the 
similarities and diferences between the original figure and dilated figure. 
 
Students talking with group members to generalize observations of their 
figures. 
 
For each problem and the dilation activity, students explored on their own 
and as a group before sumarizing as a whole clas and reflecting on how 
others approached the problem. 
 
 
 
 
278 
 
Mrs. Lotus Unit 2 Observation 3 Geometry Filter 
Geometry 
Filter 
Planed and 
Executed 
Clasrom Evidence 
Van Hiele 
Levels 
Level 2 
 
Level 3 
For each answer during belwork review and in clas problems, Mrs. 
Lotus chalenged students to justify their answers. 
Mrs. Lotus chalenged students to prove two corolaries related to 
theorems. 
 
Phases of 
Learning 
Information 
 
Directed 
Orientation 
 
 
Explication 
 
Fre Orientation 
 
Integration 
During belwork Mrs. Lotus learned of students? prior knowledge and 
experience with the lesson of the day. 
The aplication problems were activities that alowed students to 
become more acquainted to the topic of the day. 
Progresion of the aplication problems and prof of the two 
corolaries, showed a transition of reliance on teacher to student self 
reliance 
The aplication problems and profs were open ended, thus being 
attentive to the inventive abilities of the students 
At the end of the day, there were two sumary problems that required 
knowledge of material learned throughout the day. Discussion of these 
two problems helped students sumarize what was learned during the 
leson, thus showing integration phase. 
 
Geometric 
Habits of 
Mind 
Reasoning with 
Relationships 
 
Generalize 
Geometric Ideas 
Balancing 
Exploration and 
Reflection 
For each aplication problem, students were reasoning with 
relationships of the figures. Each aplication problem asked a diferent 
aspect of similarity; determining similarity and determining mising 
measures if similar. 
When chalenging students to create profs, Mrs. Lotus asked students 
if the statements in their profs worked in all cases or just the case they 
were trying to make. 
Through working on the aplication problems and profs, students 
explored when working individually and discusion as a group. 
Students reflected also in discusing in smal groups and as a whole 
group. 
 
 
 
 
 
 
 
 
 
279 
Appendix K 
Copyright Permisions 
E-mailed permision response from permisions@nctm.org on November 18, 2013 time 
stamped at 3:45pm EST for figures in: 
Batista, M. (2007b). Learning with understanding: Principles and proceses in the 
construction of meaning for geometric ideas. The Learning of 
Mathematics, 69th Yearbook of the National Council of Teachers of 
Mathematics, 65-79. 
 
Hil, H. C., Bal, D. L., & Schiling, S. G. (2008). Unpacking pedagogical content 
knowledge: Conceptualizing and measuring teachers? topic-specific 
knowledge of students. Journal for Research in Mathematics Education, 
39(4), 372-400. 
 
E-mailed permision response from permisions@sagepub.com on November 18, 2013 
time stamped at 11:37am EST for figures in: 
Hil, H.C., Rowan, B., & Bal, D.L. (2005). Efects of teachers? mathematical 
knowledge for teaching on student achievement. American Educational 
Research Journal, 42(2), 371- 406. 
 
Received license agreement betwen Anna Wan and Springer provided by Copyright 
Clearance Center on November 18, 2013. License number 3271960058767 for figures in: 
 
Martin, T., McCrone, S., Bower, M., and Dindyal, J. (2005). The interplay of 
teacher and student actions in the teaching and learning of geometric 
proof. Educational Studies in Mathematics, 60(1), 95-124.