THE RELATIONSHIP OF TEACHERS? CONTENT KNOWLEDGE AND
PEDAGOGICAL CONTENT KNOWLEDGE IN ALGEBRA, AND
CHANGES IN BOTH TYPES OF KNOWLEDGE AS A
RESULT OF PROFESSIONAL DEVELOPMENT
Except where reference is made to the work of others, the work described in this
dissertation is my own or was done in collaboration with my advisory committee.
This dissertation does not include proprietary or classified information.
Delilah Joy Willoughby Black
Certificate of Approval:
Marilyn Strutchens W. Gary Martin, Chair
Associate Professor Professor
Curriculum and Teaching Curriculum and Teaching
Kimberly L. King-Jupiter Wlodzimierz Kuperberg
Associate Professor Professor
Educational Foundations, Leadership Mathematics and Statistics
and Technology
Joe F. Pittman
Interim Dean
Graduate School
THE RELATIONSHIP OF TEACHERS? CONTENT KNOWLEDGE AND
PEDAGOGICAL CONTENT KNOWLEDGE IN ALGEBRA, AND
CHANGES IN BOTH TYPES OF KNOWLEDGE AS A
RESULT OF PROFESSIONAL DEVELOPMENT
Delilah Joy Willoughby Black
A Dissertation
Submitted to
the Graduate Faculty of
Auburn University
in Partial Fulfillment of the
Requirements for the
Degree of
Doctor of Philosophy
Auburn, Alabama
August 4, 2007
iii
THE RELATIONSHIP OF TEACHERS? CONTENT KNOWLEDGE AND
PEDAGOGICAL CONTENT KNOWLEDGE IN ALGEBRA, AND
CHANGES IN BOTH TYPES OF KNOWLEDGE AS A
RESULT OF PROFESSIONAL DEVELOPMENT
Delilah Joy Willoughby Black
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon request of individuals or institutions and at their expense.
The author reserves all publication rights.
Signature of Author
Date of Graduation
iv
VITA
Delilah Joy (Willoughby) Black, daughter of William Eldridge and Pauline
(Storey) Willoughby, was born August 11, 1952, in Lafayette, Alabama. She graduated
from Daviston High School as salutatorian in 1970. She attended Southern Union State
Junior College for two years, then entered the University of Alabama in June, 1973 and
graduated with a Bachelor of Science degree in Secondary Mathematics Education in
May, 1975. While working as a mathematics teacher at Berry High School, she entered
Graduate School, University of Alabama, in January 1976, and graduated with a Master
of Arts degree in Secondary Mathematics Education in August, 1978. After teaching
mathematics at Berry High School, Daviston High School, Edward Bell High School and
Horseshoe Bend High School for twenty-seven years, she entered Graduate School,
Auburn University, in June, 2001. She married Walter William (Bill) Black on December
19, 1970 and they have two children Shannon Ashley (Black) Allen and Damion Heath
Black.
v
THE RELATIONSHIP OF TEACHERS? CONTENT KNOWLEDGE AND
PEDAGOGICAL CONTENT KNOWLEDGE IN ALGEBRA, AND
CHANGES IN BOTH TYPES OF KNOWLEDGE AS A
RESULT OF PROFESSIONAL DEVELOPMENT
Delilah Joy Willoughby Black
Doctor of Philosophy, August 4, 2007
(M.A., University of Alabama, 1978)
(B.S., University of Alabama, 1975)
355 Typed Pages
Directed by W. Gary Martin
In seeking to improve the mathematics education of all students, it is important to
understand the connection between the content knowledge and pedagogical content
knowledge of mathematics teachers, and how professional can provide growth in both of
these types of knowledge. We do not have an answer about the interplay of content
knowledge and pedagogical content knowledge in successful instructional practices in the
mathematics classroom. For a period of a year and a half surveys were administered to a
number of secondary mathematics teachers and surveys, observations, and interviews
focused on four secondary mathematics teachers. All of the teachers taught mathematics
and particularly the four teachers all taught Algebra I at their respective schools.
vi
Assessment of the teachers? content knowledge, pedagogical content knowledge, and
changes of instructional practices also included interviews with professional development
presenters.
vii
ACKNOWLEDGEMENTS
The author would like to thank her Lord and Savior Jesus Christ who provided
healing from sickness and the fortitude to complete this study. The author is reminded
that ?With men this is impossible; but with God all things are possible.? (Matthew
19:26).
The author would also like to thank Dr. Martin for his many hours of advisement,
comments to ?expand,? and also for abandoning his red ink for a more soothing color. I
would also like to thank my other committee members, Dr. Strutchens, Dr. King-Jupiter,
and Dr. Kuperberg for the hours they spent in reading, editing, and commenting to
improve this paper.
The author would also like to thank my mother and remember my father for the
faith they had in me and the encouragement they gave me. Even though my father passed
away before the completion of this project, I still feel the presence of his spirit urging me
forward. Mother, now we have the answer to ?Are you finished yet?? Thank you to my
daughter Shannon, son Damion, and sisters Gaye and Merry for continually giving me
encouragement. Last I would like to thank my special friend Kathy for all of the
encouragement and being there when I needed to vent my frustrations. Without all of
these people in my life, I would not have attempted this project!
viii
Style manual or journal used Publication Manual of the American Psychological
Association (5
th
editon)
Computer software used Microsoft Word, Microsoft Excel, Statistical Package of
Social Sciences (SPSS), Version
ix
TABLE OF CONTENTS
LIST OF TABLES ..................................................................................................... xvii
LIST OF FIGURES ................................................................................................... xix
I. STATEMENT OF THE PROBLEM ............................................................. 1
II. REVIEW OF RELATED LITERATURE ..................................................... 7
Student Knowledge ........................................................................................ 7
The Algebra Strand ............................................................................ 9
Algebra Content in High School .............................................. 10
Conceptual and Procedural Knowledge ................................... 12
Mathematical Processes ........................................................... 13
Algebra I Course????????????????????. 15
Conclusion??????????????????????.. 16
Teacher Knowledge ....................................................................................... 17
Content Knowledge ........................................................................... 17
Definition of Content Knowledge ............................................ 18
Algebra Content Knowledge.................................................... 19
Synthesis .................................................................................. 20
Pedagogical Content Knowledge ................................................................... 21
Definitions of Pedagogical Content Knowledge ................................ 21
Synthesis ............................................................................................ 26
Instruction to Promote Student Success in Algebra ....................................... 27
Instrumental Understanding and Relational Understanding .............. 27
Conclusion ............................................................................. 30
Projects to Improve Algebra Instruction????????. ........... 30
x
The Algebra Project???????????????? 30
The QUASAR Project??????????????... . 32
Conclusion ........................................................................ 33
Research Related to Content Knowledge
and Pedagogical Content Knowledge ................................................ 33
Content Knowledge Studies with Elementary Teachers????? 34
Content Knowledge Studies with Secondary Teachers ..................... 37
Measuring Mathematical Knowledge for Teaching .......................... 41
Changes in Teacher Knowledge ........................................................ 43
Professional Development ............................................................................. 45
Conclusion ......................................................................................... 49
Research Questions ........................................................................................ 49
III. THEORETICAL UNDERPINNINGS AND METHODOLOGIES .............. 51
Theoretical Underpinnings............................................................................. 51
Study Design .................................................................................................. 55
Context for the Study ..................................................................................... 56
Multi-District Mathematics Systemic
Improvement Program (MDMSIP) .................................................... 56
Survey Research ............................................................................................. 60
Subject Selection ................................................................................ 61
Instrumentation .................................................................................. 61
Algebra Content Knowledge Instrument (ALCKIN) ............ 62
Content Knowledge and Pedagogical
Content Knowledge .......................................................... 62
Test Format ....................................................................... 63
Item Development ............................................................. 64
Field Testing ..................................................................... 64
Teacher Attitude Survey ........................................................ 66
Procedure for Survey Study ............................................................... 67
Administration of Surveys ..................................................... 67
Data Analysis ......................................................................... 69
xi
Multi-Individual Case Studies ....................................................................... 70
Subject Selection ................................................................................ 71
Data Sources and Procedure .............................................................. 74
Pre-Professional Development ...................................................... 77
Algebra Content Knowledge Instrument and Teacher Attitude
Survey ...................................................................................... 77
Classroom Observations .......................................................... 77
Teacher Interviews ................................................................... 79
Professional Development ............................................................ 82
Document Analysis .................................................................. 82
Professional Development Presenter Interviews ..................... 83
Quarterly Meetings .................................................................. 83
Post-Professional Development .................................................... 84
Classroom Observations .......................................................... 84
Interviews ................................................................................ 84
Journaling ...................................................................................... 85
Data Analysis ..................................................................................... 86
Pre-Professional Development ...................................................... 86
Professional Development ............................................................ 86
Post-Professional Development .................................................... 86
Conclusion ..................................................................................................... 87
IV. RESULTS OF THE SURVEY RESEARCH ................................................ 89
Coding ............................................................................................................ 89
Question One-Functions ................................................................................ 92
Analysis of the Results from Question One ....................................... 92
Conclusion ......................................................................................... 100
Question Two-Simplifying Algebraic Expressions ....................................... 101
Analysis of Results of Question Two ................................................ 102
Conclusion ......................................................................................... 106
xii
Question Three-General Algebraic Knowledge ............................................. 106
Conclusion ......................................................................................... 108
Question Four-Understanding of Number Systems ....................................... 109
Conclusion ......................................................................................... 110
Question Five-Solving an Equation ............................................................... 111
Analysis of the Results from Question Five ...................................... 112
Conclusion ......................................................................................... 115
Question Six-Simplifying Algebraic Expressions ......................................... 115
Analysis of the Results from Question Six ........................................ 116
Conclusion ......................................................................................... 118
Question Seven-Using Algebraic Reasoning in Relationship to Geometry .. 119
Analysis of the Results from Question Seven.................................... 121
Conclusion ......................................................................................... 124
Question Eight-Rate of Change ..................................................................... 125
Analysis of the Result from Question Eight ...................................... 127
Conclusion ......................................................................................... 131
Question Nine-Family of Functions............................................................... 131
Analysis of Results of Question Nine ................................................ 133
Conclusion ......................................................................................... 136
Question Ten-Algebraic Representations within an Equation ....................... 136
Analysis of Results of Question Ten ................................................. 137
Conclusion ......................................................................................... 139
Teacher Attitude Survey ................................................................................ 140
Conclusion ..................................................................................................... 142
Content Knowledge ........................................................................... 142
Pedagogical Content Knowledge ....................................................... 143
xiii
V. CASES ........................................................................................................... 145
Description of Professional Development ..................................................... 146
Coding ............................................................................................................ 147
Mrs. Cotney.................................................................................................... 152
Pre-Professional Development........................................................... 152
Classroom Environment ........................................................ 152
Student Ability ....................................................................... 153
Teacher Attitude Survey ........................................................ 155
Content Knowledge ............................................................... 156
Procedural knowledge ....................................................... 156
Conceptual knowledge ...................................................... 157
Mathematical processes .................................................... 157
Pedagogical Content Knowledge ........................................... 158
Classroom Instruction ............................................................ 161
Student questioning ........................................................... 164
Analyzing student errors ................................................... 165
Multiple methods of solving problems ............................. 166
Conclusion ............................................................................. 168
Professional Development ................................................................. 169
Post-Professional Development ......................................................... 172
Classroom Environment ........................................................ 172
Student Ability ....................................................................... 172
Changes in Content Knowledge............................................. 173
Changes in Pedagogical Content Knowledge ........................ 174
Reflection of Changes in Knowledge in Instructional Practices 177
Conclusion ............................................................................. 179
Conclusion of Case ............................................................................ 179
Mrs. Willoughby ............................................................................................ 180
Pre-Professional Development........................................................... 180
Classroom Environment ........................................................ 181
Student Ability ....................................................................... 181
Teacher Attitude Survey ........................................................ 182
Content Knowledge ............................................................... 182
Procedural knowledge ....................................................... 182
Conceptual knowledge ...................................................... 184
xiv
Mathematical processes .................................................... 184
Pedagogical Content Knowledge ........................................... 185
Classroom Instruction ............................................................ 186
Student questioning ........................................................... 188
Analyzing student errors ................................................... 190
Multiple methods of solving problems ............................. 191
Conclusion ............................................................................. 197
Professional Development ................................................................. 198
Post-Professional Development ......................................................... 202
Classroom Environment ........................................................ 203
Student Ability ....................................................................... 203
Changes in Content Knowledge............................................. 204
Changes in Pedagogical Content Knowledge ........................ 205
Reflection of Changes in Knowledge in Instructional Practices 206
Conclusion ............................................................................. 207
Conclusion of Case ............................................................................ 208
Mrs. Pitchford ................................................................................................ 208
Pre-Professional Development........................................................... 209
Classroom Environment ........................................................ 209
Student Ability ....................................................................... 210
Teacher Attitude Survey ........................................................ 211
Content Knowledge ............................................................... 211
Procedural knowledge ....................................................... 211
Conceptual knowledge ...................................................... 212
Mathematical processes .................................................... 213
Pedagogical Content Knowledge ........................................... 213
Classroom Instruction ............................................................ 214
Student questioning ........................................................... 217
Analyzing student errors ................................................... 217
Multiple methods of solving problems ............................. 218
Conclusion ............................................................................. 218
Professional Development ................................................................. 219
Post-Professional Development ......................................................... 219
Classroom Environment ........................................................ 220
Student Ability ....................................................................... 220
xv
Changes in Content Knowledge............................................. 221
Changes in Pedagogical Content Knowledge ........................ 222
Reflection of Changes in Knowledge in Instructional Practices 223
Conclusion ............................................................................. 224
Conclusion of Case ............................................................................ 225
Mrs. Colley ................................................................................................ 225
Pre-Professional Development........................................................... 226
Classroom Environment ........................................................ 226
Student Ability ....................................................................... 227
Teacher Attitude Survey ........................................................ 227
Content Knowledge ............................................................... 229
Procedural knowledge ....................................................... 229
Conceptual knowledge ...................................................... 230
Pedagogical Content Knowledge ........................................... 231
Classroom Instruction ............................................................ 233
Student questioning ........................................................... 235
Analyzing student errors ................................................... 237
Multiple methods of solving problems ............................. 238
Conclusion ............................................................................. 238
Professional Development ................................................................. 239
Post-Professional Development ......................................................... 243
Classroom Environment ........................................................ 243
Student Ability ....................................................................... 243
Changes in Content Knowledge............................................. 244
Changes in Pedagogical Content Knowledge ........................ 247
Reflection of Changes in Knowledge in Instructional Practices 247
Conclusion ............................................................................. 250
Conclusion of Case ............................................................................ 251
Conclusion ................................................................................................ 251
VI. SUMMARY AND RECOMMENDATIONS................................................ 256
Summary ........................................................................................................ 258
Limitations ..................................................................................................... 267
xvi
Implications of the Study ............................................................................... 269
Implications for Teachers .................................................................. 269
Implications for Teacher Education ................................................... 271
Implications for Professional Development....................................... 272
Implications for Policy Makers .......................................................... 272
Implications for Future Research ................................................................... 273
Conclusion ..................................................................................................... 274
REFERENCES .......................................................................................................... 276
APPENDICES ........................................................................................................... 288
Appendix A: Algebra Content Knowledge Instrument (ALCKIN) ............ 289
Appendix B: Teacher Attitude Survey ........................................................ 298
Appendix C: RTOP Classroom Observation Survey .................................. 308
Appendix D: First Teacher Interview Protocol ........................................... 314
Appendix E: Protocol for Second Teacher Interview ................................. 322
Appendix F: Professional Development Presenters Interview Protocol ..... 325
Appendix G: Code List for the Algebra Content Knowledge Instrument ... 327
Appendix H: Code List for the Case Studies ............................................... 331
xvii
LIST OF TABLES
Table 1 Item Format of the Algebra Content Knowledge
Instrument (ALCKIN) ............................................................................. 65
Table 2 Years of Teaching Experience ................................................................. 69
Table 3 School Information Regarding Demographics and Results of Alabama
High School Graduation Exam ................................................................ 72
Table 4 Demographics on Case Study Participants .............................................. 74
Table 5 Case Study Time Frames and Data Collection Points ............................. 76
Table 6 Codes for the Algebra Content Knowledge Instrument ........................... 90
Table 7 Responses to Question One on Functions ................................................ 94
Table 8 Explanations for Questions 1A and 1B .................................................... 95
Table 9 Question Two Responses to Simplifying Algebraic
Expressions .............................................................................................. 103
Table 10 Question Three Responses to General Algebraic
Knowledge ............................................................................................... 108
Table 11 Question Four Responses to Understanding of Number
Systems .................................................................................................... 110
Table 12 Question Six Responses ........................................................................... 116
Table 13 Question Seven Responses on Using Algebraic
Reasoning in Relationship to Geometry .................................................. 121
Table 14 Question Seven, Part C Responses .......................................................... 122
Table 15 Question Eight Responses on Rate of Change ......................................... 127
xviii
Table 16 Group with the Best Understanding of Slope .......................................... 129
Table 17 Question Nine Responses to Families of Functions ................................ 134
Table 18 Question Ten Responses to Algebraic Representations
within an equation .................................................................................... 138
Table 19 Teacher Attitude Survey Summary .......................................................... 141
Table 20 Codes for Case Studies ............................................................................ 149
LIST OF FIGURES
Figure 1. Item measuring specialized content knowledge (Ball et al., 2005, p. 43) 25
Figure 2. Incorrect mathematical procedure for multiplying two whole
Numbers. (Ball et al., 2005, p. 17) ........................................................... 26
Figure 3. First Mathematics Task in the Teacher Interview .................................... 80
Figure 4. Question One from the Algebra Content Knowledge Instrument ........... 93
Figure 5. Teacher? Incorrect Sketch for the Graph y =
x
1
...................................... 98
Figure 6. Example of a graph not passing the vertical line test ............................... 100
Figure 7. Question Two from the Algebra Content Knowledge Instrument ........... 102
Figure 8. Participant?s simplification of the algebraic expression by factorization and
division ..................................................................................................... 104
Figure 9. Question three from the Algebra Content Knowledge Instrument .......... 107
Figure 10. Question four from the Algebra Content Knowledge Instrument ........... 109
Figure 11. Question five from the Algebra Content Knowledge Instrument ............ 112
Figure 12. Question six from the Algebra Content Knowledge Instrument ............. 116
Figure 13. Participant?s pictorial representation to show two algebraic expressions are
equivalent ................................................................................................. 118
Figure 14. Question seven from the Algebra Content Knowledge Instrument ......... 120
Figure 15. Question eight from the Algebra Content Knowledge Instrument .......... 126
Figure 16. Question nine from the Algebra Content Knowledge Instrument ........... 132
xix
xx
Figure 17. Question ten from the Algebra Content Knowledge Instrument ............. 137
Figure 18. Mrs.Cotney?s work for the wheels and vehicles problem. ....................... 157
Figure 19. Student?s non-procedural method for solving x
2
= 2x + 8 ....................... 160
Figure 20. Mrs. Willoughby?s setup to solve the wheels and vehicles problem ....... 183
Figure 21. Mrs. Pitchford?s work on the quadratic problem. .................................... 221
Figure 22. Mrs. Colley?s work on the wheels and vehicles problem ........................ 229
Figure 23. Mrs. Colley?s work for the quadratic problem ......................................... 245
1
I. STATEMENT OF THE PROBLEM
Successful competition by the United States in the global economy depends on
having adults who are well prepared in mathematics and science (National Center for
Education Statistics, 2001). Adults of the twenty-first century need to be mathematically
proficient in order to be productive members of our society (Ball, 2003a), and the need
for mathematics in everyday life has never been greater and will indeed continue to
increase (National Council of Teachers of Mathematics [NCTM], 2000). Mathematical
competence can open doors to a productive future, while the doors to a productive future
can remain closed for those students lacking in mathematical competence (NCTM, 2000).
As a nation, however, the U. S. is not providing its students with the mathematical
preparation needed to be successful. According to the results of the Third International
Mathematics and Science Study (TIMSS), our students in the United States only achieve
at average levels when compared to students in other countries (NCES, 2003a). In
addition, according the National Assessment of Educational Progress (NAEP) results,
less than 20% of twelfth grade students and about one-third of eighth grade students had
achieved mathematical proficiency (Pehle et al., 2004).
A critical juncture in the mathematical preparation of students is the high school
algebra course. Algebra serves as a gatekeeper, offering differential opportunities for
entry into advanced mathematics courses (Ball, 2003a), for preparation for college
2
(Pascopella, 2000, Lawton, 1997, Chevigny, 1996, Silver, 1997, Olson, 1994), and for
preparation for the world of work (Silver, 1997). Olson (1994) pointed to a study by the
College Board, which suggested that students who take Algebra in high school are two
and one-half times more likely to attend college than those students who do not. When
minority students complete Algebra in high school, the gap between the percentages of
minority and non-minority students who attend college virtually disappears. Students
who are not proficient in Algebra do not have access to a full range of educational and
career opportunities (Ball, 2003a).
Algebra should be embedded throughout the K-12 curriculum in order to provide
the opportunities for students to develop a solid foundation for the content (NCTM,
2000). Students need to develop a deep understanding of the algebra content, and
mathematics in general. Students need to develop fluency with procedures as well as
conceptual understanding for why those procedures work (RAND, 2003). In addition a
students? content knowledge must include the processes of problem solving, reasoning
and proof, communication, making connections, and using representations (NCTM,
2000). Content knowledge provides the ability to know, understand, and have the ability
to use mathematics (NCTM, 2000).
Teachers play a key role in ensuring that all students have the experiences needed
to learn the mathematics necessary for success in future educational opportunities and
careers (Mewborn, 2003). In particular, we must consider the types of knowledge
mathematics teachers need to provide all students with equitable opportunities to learn
algebra. While all might agree teachers need content knowledge of the subject they will
be teaching, there is not a common definition of content knowledge on which everyone
3
agrees. Content knowledge has often been defined by the number of university level
courses taken (Even, 1993, No Child Left Behind Act [NCLB], 2001), the grade point
average at the collegiate level (Even, 1993), or scores on a state mandated test (NCLB,
2001). This, however, is limited.
Teachers need a deep understanding of the mathematics they will teach (CBMS,
2001). Like students, teacher?s content knowledge should include both procedural
knowledge and conceptual knowledge, an understanding how this knowledge is
structured and generated throughout the domain of mathematics (Shulman, 1986). If
students are expected to develop mathematical proficiency and to apply mathematics in
real world situations, no less can be expected of their teachers (CBMS, 2001). However,
studies have found teachers? content knowledge is often thin and inadequate to provide
instruction for students in today?s classrooms (Ball, 1988a, 2003b; Ball & Bass, 2000;
Fuller, 1996; Ma, 1999; Mewborn, 2001; Stacey, et al, 2001).
However, good content knowledge alone is not sufficient to successfully teach
students mathematics (Even, 1993). Teachers also need knowledge of mathematics used
specifically to facilitate the learning of mathematics by students (Sherin, 2002). At the
turn of the twentieth century Dewey (1974) proposed the importance of a proper
relationship between knowledge of content and pedagogy, and this tension still remains at
the turn of the twenty-first century. Shulman (1987) first referred to this mixture of
content knowledge and knowledge of pedagogy that belongs exclusively to teachers as
?pedagogical content knowledge.? Pedagogical content knowledge, also referred to as the
?mathematical knowledge for teaching? (Ball et al. 2003), bundles mathematics
knowledge with the knowledge teachers have about learners, learning, and pedagogy
4
(Ball et al. 2001). Pedagogical content knowledge can help teachers anticipate where
students will have difficulties and be ready with alternative methods, explanations and
representations related to a mathematical topic (Ball et al. 2001). In addition, pedagogical
content knowledge includes representations that are most useful for teaching mathematics
content (Ball et al. 2003).
We need to understand both teacher?s content knowledge as well as their
pedagogical content knowledge. Some research has been conducted in the areas of
teacher content knowledge and pedagogical content knowledge of elementary teachers.
The Learning Mathematics for Teaching project found that elementary teachers who
scored higher on instruments measuring content knowledge and pedagogical content
knowledge produced larger gains in student achievement in mathematics (Ball et al.
2005). However, far fewer studies have focused on these same types of knowledge for
secondary teachers including teachers of algebra. Stump (1997), Even (1993), and
Llinares (2000) conducted studies related to measuring teachers? content knowledge of
specific algebraic topics and its relationship to their pedagogical content knowledge.
These studies suggest that teachers? content knowledge and pedagogical content
knowledge do make a difference in students learning mathematics. However, much work
remains to be done in understanding the content knowledge and pedagogical content
knowledge of teachers and how they effect student learning.
The RAND report (Ball, 2003a) calls for additional research to further explore
content-specific knowledge and pedagogical content knowledge for teaching
mathematics, and how and where such knowledge makes a difference in successful
teaching. Efforts to improve mathematics education for all students will be limited
5
without a better understanding of both content knowledge and pedagogical content
knowledge, as well as the connection between these types of knowledge.
We also need to have a better understanding of how content knowledge and
pedagogical content knowledge are used during mathematics instruction. If content
knowledge and pedagogical content knowledge do make a difference in student
achievement, understanding how these knowledge types are used in instruction remains
an important issue to be solved (Ball, 2003a). What ultimately matters is how teachers are
able to use their knowledge in the course of teaching (Ball et al., 2001). More research is
needed to understand the relationship of content knowledge and pedagogical content
knowledge and how it is used in instructional contexts (Ball, 2003a).
An additional question lies in the extent to which professional development can
change both content knowledge and pedagogical content knowledge of teachers.
Professional development should provide opportunities that support learning both content
knowledge and pedagogical content knowledge as well as how to effectively use them
(Ball, 2003a), and efforts have been made over the past decade to organize professional
development to develop better mathematical knowledge (Ball, 2003a). Methods for
developing content knowledge and pedagogical content knowledge for teaching remains
an unsolved problem for the improvement of mathematics teaching and learning (Ball et
al., 2001).
Furthermore, if professional development can facilitate changes in both content
knowledge and pedagogical, we do not fully understand how these types of knowledge
might influence changes in teachers? instructional practices. Certainly possessing content
6
knowledge and pedagogical content knowledge will not automatically translate into
successful mathematics instruction (Ball et al. 2001).
In order to better understand the impact of teacher knowledge on mathematics
instruction, which in turn affects the future success of students, the following questions
will be addressed in this study: What content knowledge and pedagogical content
knowledge of algebra do high school algebra teachers possess? How are high school
algebra teachers? content knowledge and pedagogical content knowledge reflected in
their instructional practices? What growth in content knowledge and pedagogical content
knowledge can be seen from participating in professional development that includes
attention to increasing both content knowledge and pedagogical content knowledge?
How are changes in content knowledge and pedagogical content knowledge reflected in
instructional practice?
7
II. REVIEW OF RELATED LITERATURE
Mathematics is important in ensuring the future success of all students (RAND,
2003) and algebra serves as a gatekeeper course to differing opportunities for advanced
mathematics courses (RAND, 2003), to college preparation (Chevigny, 1996; Lawton,
1997; Olson, 1994; Pacopella, 2000; Silver, 1997) and for the preparation for the world
of work (Silver, 1997). Without algebra, students have limited chances of success
(RAND, 2003). We will begin this chapter by examining student knowledge, first more
broadly, then more specifically related to secondary students. Given that the success of
students largely depends on the quality of the classroom teacher (Mewborn, 2003) we
will next turn our attention to teacher knowledge necessary for teaching, focusing on
content knowledge, including both procedural and conceptual knowledge and the
processes , and the pedagogical content knowledge needed to teach algebra. Finally, we
will consider professional development and how it can be used to promote growth in both
of these types of knowledge.
Student Knowledge
This section will begin by addressing the importance of algebra for all students,
followed by a broad overview of what is meant by algebra and the specific content that
high school students need to learn. We next address the need for conceptual
8
understanding, along with procedural understanding, mathematical processes, and how
they should learn it. Finally, we focus on what is meant by the Algebra I course.
Principles and Standards for School Mathematics, released by the National
Council of Teachers of Mathematics (NCTM) in 2000, declares, ?Algebra competence is
important in adult life, both on the job and as preparation for postsecondary education?
(p. 37). Without algebraic proficiency, students do not have access to a full range of
educational and career opportunities (RAND, 2003). Many schools in the United States
require students to demonstrate proficiency in algebra (RAND, 2003), in part due to
many states adopting higher standards resulting from public pressure for more
accountability and higher standards (RAND, 2003). The No Child Left Behind Act has
also reinforced these higher standards (RAND, 2003).
The National Assessment of Educational Progress (National Center for
Educational Statistics [NCES], 2000) shows that students who have taken more advanced
classes such as algebra and geometry scored higher than students who had not taken these
courses. Data on the 1996 NAEP assessment also affirmed that achievement was strongly
related to the coursetaking of students (Strutchens et al., 2000). Clearly algebra is an
important part of students? mathematical preparation, and it should be available to all
students. Moses and Cobb (2001) argued that the access of algebra to all students should
be ?the new civil right.? They stated:
?once solely in place as the gatekeeper for higher math and the priesthood who
gained access to it, [algebra] now is the gatekeeper for citizenship, and people
who don?t have it are like the people who couldn?t read and write in the industrial
age?.[Lack of access to algebra] has become not a barrier to college entrance,
but a barrier to citizenship. That?s the importance of algebra that has emerged
with the new higher technology. (Moses & Cobb in Ball et al., 2003, p. 47)
9
High school Algebra serves as a gatekeeper course, offering different
opportunities for entry into advanced mathematics courses (RAND, 2003), for
preparation for college (Pascopella, 2000, Lawton, 1997, Chevigny, 1996, Silver, 1997,
Olson, 1994), and for preparation for the world of work (Silver, 1997). Students who take
Algebra are more likely to attend college than those student who do not (Olson, 1994).
When minority students complete Algebra in high school, the gap between the
percentages of minority and non-minority students who attend college virtually
disappears. Students who are not proficient in Algebra do not have access to a full range
of educational and career opportunities (RAND, 2003).
The Algebra Strand
Principles and Standards for School Mathematics (PSSM), the national standards
for school mathematics released by the National Council of Teachers of Mathematics
(NCTM) in 2000 recommended that all students should learn algebra and it should be
embedded in mathematical topics from kindergarten through twelfth grade. Through the
algebra strand, ?teachers can help students build a solid foundation of understanding and
experience as a preparation for more sophisticated work in algebra in the middle grades
and high school? (NCTM, 2000, p. 37).
Let us first look at the broad overview of the mathematics students should learn,
contained within four major areas of its algebra strand (NCTM, 2000). These areas are:
Understand patterns, relations, and functions; represent and analyze mathematical
situations and structures using algebraic symbols; use mathematical models to
represent and understand quantitative relationships, and analyze change in various
contexts. (p. 296)
10
Experiencing algebra across the grade levels will not necessarily resemble a
formal course in algebra. For example, consider the approach of analyzing changes in
various contexts at different grade levels. ?In prekindergarten through grades 2, students
can, at first, describe qualitative change (?I grew taller over the summer?) and then
quantitative changes (?I grew two inches taller in the last year?)? (NCTM, 2000, p. 40).
Students in grades 3-5 should be able to ?look at sequences and be able to distinguish
between arithmetic growth (2, 5, 8, 11, 14, ?) and geometric growth (2, 4, 8, 16, ?)?
(NCTM, 2000, p. 40). ?With a strong middle-grades focus on linearity, students should
learn about the idea that slope represents the constant rate of change in linear functions?
(NCTM, 2000, p. 40) and be ready in high school to learn ?about classes of functions
with nonconstant rates of change? (NCTM, 2000, p. 40).
Algebra Content in High School
In addition to the general picture of algebra across grades K-12 presented in the
previous section, specific algebra content has been specified for grades 9-12. I will draw
on the recommendations from two sources: the RAND report (2003) and Principles and
Standards for School Mathematics (NCTM, 2000).
The RAND report (2003) is a strategic plan for research and development of
mathematics education in the United States for developing the mathematical proficiency
of all students. Four areas are specified in this report (RAND, 2003): working flexibly
and meaningfully with formulas or algebraic relations, operating with and representations
of numbers, understanding of function, and identifying and using variables in quantitative
contexts as well as patterns. The Principles and Standards for School Mathematics
11
(NCTM, 2000) also contains specific recommendations for grades 9-12 falling within the
four major areas within its algebra standard.
First, students should understand patterns, relations, and functions (NCTM,
2000); the RAND report (2003) refers to this having a ?robust understanding? of
functions. In grades 9-12, students should be able to use various representations for
functions and consider the advantages and disadvantages of each representation relative
to the context of the problem (NCTM, 2000). Students should explore different classes of
functions and how the parameters affect the graphs for each of these functions (NCTM,
2000).
Second, students at the 9-12 grade band should be able to represent and analyze
mathematical situations and structures using algebraic symbols (NCTM, 2000). This
includes having the ability to work flexibly and meaningfully with algebraic symbolism
in representing and solving problems (NCTM, 2000, RAND, 2003). Students should be
able to operate with algebraic symbols as well as be able to understand where to use them
(NCTM, 2000). Students should be able to work flexibly and meaningfully with formulas
or algebraic relations, representing situations, manipulating them, and solving equations
(RAND, 2003). In addition, students should be able to use symbols, formulas, and
functions in representing quantitative context (RAND, 2003).
Third, students at the 9-12 grade band should be able to use mathematical models
to represent and understand quantitative relationships (NCTM, 2000). Students should be
able to identify the class or classes of functions that might model a situation, use
symbolic representations, and draw conclusions about a situation being modeled.
12
Last, students should be able to analyze change in various contexts (NCTM,
2000). Students should be able to approximate change and interpret rates of changes
using graphs and numerical data (NCTM, 2000). Students should be able to use functions
in analyzing the change of one quantity in relation to another (RAND, 2003).
Thus, the algebra that students in grades 9-12 should learn is much broader than
the traditional emphasis on symbolic manipulation. Their knowledge should include: a
robust understanding of functions, should develop fluency in using algebraic structures in
relationship to expressions, equations, and inequalities, using algebraic reasoning in
relationship to other mathematical fields, and analyze change in various contexts.
Conceptual and Procedural Knowledge
In order to achieve these content goals, students need to have a deeper
understanding of mathematics. They must know more than procedural knowledge, which
Hiebert and Lefevre (1986) defined as knowing the rules and algorithms of mathematics.
Procedural knowledge includes a series of actions (Hiebert & Carpenter, 2002) and
within the context of school mathematics often consists of a series of symbolic
manipulations in performing step-by-step procedures. They must also have conceptual
knowledge, which includes the understanding of how and why mathematical algorithms
work (RAND, 2003). The RAND report (2003) explicitly includes attention to both
procedural and conceptual fluency, and this can also be inferred from Principles and
Standards (2000).
Skemp (1976) referred to procedural knowledge as instrumental understanding
and to relational understanding as the union of both procedural and conceptual
knowledge. Skemp (1976) pointed out the relative advantages of both relational and
13
instrumental understanding. With instrumental understanding, he asserted that students
find mathematics easier to understand, the rewards are more immediate and apparent, and
correct answers are more quickly obtained. With relational mathematics, however,
students can more easily adapt their skills to new tasks, their skills are easier to
remember, relational mathematics can be a goal within itself, and growth and
explorations into new areas are more likely to occur. Thus to be truly successful, students
need both conceptual knowledge and procedural knowledge of mathematics. Learning
mathematics with understanding enables students to actively build new knowledge from
their experiences and prior knowledge (NCTM, 2000). Thus, students need to develop
both procedural knowledge as well as the conceptual knowledge for why procedures
work.
Mathematical Processes
In looking at the mathematical knowledge that students need, we must also
consider the means through which students should acquire and be able to use
mathematics. The RAND report (2003) says for students to be competent in mathematics
they must have conceptual understanding and procedural fluency, which have already
been addressed in the previous section. In addition, student competency should include
strategic competence, adaptive reasoning, and productive disposition. Productive
disposition is the inclination for students to see the usefulness of mathematics and to
believe in one?s own efficacy.
Principles and Standards (NCTM, 2000) explicates this kind knowledge in its
?process standards.? The document proposes five process standards: problem solving,
reasoning and proof, communication, connections, and representations. These processes
14
highlight the ways of acquiring and using mathematical content, including algebra
(NCTM, 2000). I briefly describe each of this in the following.
First, problem solving should provide students with the means to apply and use a
variety of methods for solving problems (NCTM, 2000). Problem solving refers to
solving a task for which a solution method is not immediately known (NCTM, 2000).
Problem solving is an integral part of mathematics learning (NCTM, 2000). Through
strategic competence, students should have the ability to formulate and solve
mathematical problems (RAND, 2003).
Second, students should be able to use the algebra content in reasoning and proof
(NCTM, 2000). Students should be able to develop, investigate, and evaluate
mathematical conjectures, which are fundamental aspects of mathematics (NCTM, 2000).
Being able to reason is essential to mathematics (NCTM, 2000) and must be developed
through consistent use (NCTM, 2000). Students should have the ability to use logical
thought, be able to reflect, explain, and justify (RAND, 2003).
Third, students should be able to communicate mathematics clearly so that others
can understand their thinking (NCTM, 2000). In addition, communication should also
allow students to evaluate the mathematical thinking of others (NCTM, 2000).
Communication helps build meaning and through communication students will build
better mathematical understanding (NCTM, 2000).
Fourth, students should be able to recognize connections between mathematical
topics (NCTM, 2000) and to other areas of mathematics. They should also be able to
recognize and apply mathematics in contexts outside mathematics. By making
15
connections between mathematical ideas, students will develop a deeper understanding of
the mathematics they are studying (NCTM, 2000).
Last, students need to be about to create, understand, and use a variety of methods
to represent mathematical situations (NCTM, 2000). Students should be able to use
representations in modeling and interpreting phenomena (NCTM, 2000). How students
represent mathematics is fundamental in how they understand and use these ideas
(NCTM, 2000).
Process skills must be interwoven with the mathematics content. Students cannot
solve problems without understanding and using mathematical content. Likewise,
processes can be learned in the content. For example, ?establishing geometric knowledge
calls for reasoning. The concepts of algebra can be examined and communicated through
representations? (NCTM, 2000, p. 31). Thus, students need to be proficient in algebraic
concepts using all of the mathematical processes. These processes enable students to
understand and use the mathematics content in solving problems in various contexts
(NCTM, 2000).
Algebra I Course
In the previous sections a broad view of Algebra was provided, in which it was
presented as a strand that students should learn in all K-12 grade levels, as well as
specific expectations for the 9-12 grade band. At the high school level algebra topics are
typically taught in a specific course, although some states teach high school-level algebra
as part of an integrated curriculum (NCES, 2007). All students in Alabama, the state in
which this study was conducted, must satisfy educational requirements set by the
Alabama State Department of Education (ASDE), which include completing an algebra
16
one course. Algebra I is required because of its importance in the development of
mathematical empowerment (ASDE, 2003). The course is defined in the Alabama Course
of Study: Mathematics (ASDE, 2003) as follows:
Algebra I is a formal, in-depth study of algebraic concepts and the real number
system. In this course students develop a greater understanding of and
appreciation for algebraic properties and operations. Algebra I reinforces concepts
presented in earlier courses and permits students to explore new, more
challenging content which prepares them for further study in mathematics. The
course focuses on the useful application of course content and on the development
of student understanding of central concepts. Appropriate use of technology
allows students opportunities to work to improve concept development. As a
result, students are empowered to perform mathematically, both with and without
the use of technological tools. (p. 49)
This document also outlines two alternatives for meeting this requirement: ?To better
meet the needs of students of varying abilities, school systems may offer Algebra I (140
hours/one credit) or Algebra IA and IB (280 hours/two credits)? (ASDE, 2003, p. 49).
The two year sequence for Algebra I suggests that it was designed students who need
more time in which to develop algebraic proficiency (Paul, 2005).
Conclusion
Students will be limited in their opportunities for future success both in the world
of academics and workplace without algebraic proficiency (RAND, 2003, Chevigny,
1996, Lawton, 1997, Olson, 1994, Pacopella, 2000, Silver, 1997). Learning algebra is
important to all students, and emphasis should be placed on the development of algebra
17
skills throughout the K-12 curriculum (NCTM, 2000, RAND, 2003) Algebra includes not
only the knowledge of the content, but also the processes by which students acquire and
are able to do mathematics (NCTM, 2000). Finally, for students to be successful in
learning algebra, the content should be addressed within contexts where students can
develop procedural knowledge as well as the conceptual underpinnings for these
procedures.
Teacher Knowledge
The classroom teacher is the key factor in influencing the mathematical success of
students (Mewborn, 2003). Given the complexities of teaching, it is difficult to simply
create a list of attributes and types of knowledge that an effective mathematics teacher
should possess, at least one on which everyone would agree. For example, school age
students were asked to provide their views of the qualities effective teachers should
possess (Ball, 1988a). Students replied they felt teachers need to be enthusiastic, helpful,
and strict, but they also need to be knowledgeable of their subject matter (Ball, 1988a).
Teachers need to know the mathematics content they will be teaching; Lee (et al, 2003)
noted that if a teacher does not know mathematics, he/she will not teach it. We will focus
on the aspects of teacher knowledge in the following sections. We will first look at
content knowledge, how it is defined, and algebra content knowledge in particular. Next
we will focus on pedagogical content knowledge and its definition.
Content Knowledge
Teachers should be knowledgeable in the content areas for which they are
responsible to teach. This must include a deep understanding of the mathematics they are
18
teaching (NCTM, 2000), including both mathematical concepts and procedures
(Conference Board of the Mathematical Sciences [CBMS], 2001). It will difficult for any
teacher to teach others about a subject if the teacher does not know the content
himself/herself. The same is true for mathematics teachers. Mathematics teachers should
know the mathematics they are teaching. But what exactly is it that they should know and
how should they know it? The following section will address the definition of content
knowledge, algebra content knowledge and a synthesis of the two types of knowledge.
Definition of Content Knowledge
There is considerable variation in what people mean when they refer a teacher?s
subject matter knowledge, what some also refer to as content knowledge. Earlier
definitions measured subject matter knowledge of pre-service and in-service teachers in
terms of the number of mathematics courses taken at the university level and/or the
teacher?s grade point average (Even, 1993). Some may argue that the number of
mathematics courses measures the content knowledge of teachers, and this mathematical
knowledge can be increased by taking more college-level mathematics courses
(Noddings, 1998). The belief that a teacher?s content knowledge is determined by the
number of college level mathematics courses taken may guide policy decisions
(Mewborn, 2001). For example, with the passage of the No Child Left Behind legislation
the number of college mathematics courses taken or score on a state mandated test, is
used as a determining factor in defining what a ?highly qualified? teacher is (NCLB,
2001). This, however, can be misleading. For example, Noddings (1998) suggested that
having adequate subject matter knowledge is not always insured by the presence of a
mathematics major.
19
The CBMS report (2001) contains specific recommendations for the preparation
of mathematics teachers at each grade level. Emphasis from these recommendations
recognized the need for teachers to develop a deep understanding of mathematics,
develop a connection of mathematical topics, and develop basic mathematical ideas
(CBMS, 2001).
Teachers should possess at least the depth of mathematical knowledge that
students need to learn (RAND, 2003). Teachers must have a deep understanding of the
mathematical ideas in the 9-12, including both procedural knowledge as well as the
conceptual knowledge underlying why the procedures work (CBMS, 2001).
Teachers? content knowledge should also contain the mathematical proficiency in
using mathematical processes and applying mathematics in real world situations (NCTM,
2000, RAND, 2003). Teachers should be able to recognize and apply mathematics in
contexts outside of mathematics (NCTM, 2000).
Algebra Content Knowledge
The CBMS report (2001) specifically addresses the algebra content knowledge
teachers need to teach mathematics to all students. The focus of the five areas of algebra
and number theory included in the section for recommendations for high school teacher
preparation follow:
To be well-prepared to teach such high school curricula, mathematics teachers
need: 1) Understanding of the properties of the natural, integer, rational, real, and
complex number systems. 2) Understanding of the ways that basic ideas of
number theory and algebraic structures underlie rules for operations on
expressions, equations, and inequalities. 3) Understanding and skill in using
20
algebra to model and reason about real-world situations. 4) Ability to use
algebraic reasoning effectively for problem solving and proof in number theory,
geometry, discrete mathematics, and statistics. 5) Understanding of ways to use
graphing calculators, computer algebra-systems, and spreadsheets to explore
algebraic ideas and algebraic representations of information, and in solving
problems. (p. 40)
Synthesis
For the purpose of this study, content knowledge will be defined as both the
procedural and conceptual knowledge as well as the mathematical processes for using
mathematics. This includes the teachers? ability to solve problems using a variety of
methods, adapting to different contexts (NCTM, 2000). In addition, content knowledge
includes the ability to use reasoning and proof to make and investigate conjectures and
evaluate mathematical arguments (NCTM, 2000) and be able to use algebraic reasoning
in relationship to other mathematical topics (CBMS, 2001). Teachers should have the
ability to communicate mathematics so that others can learn and be able to listen to how
others think about mathematics (NCTM, 2000). Teachers should have the ability to make
connections between mathematical topics, between the areas of mathematics, and to real-
world problems (NCTM, 2000). Teachers should be able to access different
representations in organizing mathematics problems and should be able to translate
between the mathematical representations (NCTM, 2000) and be able to model algebra in
real world context (CBMS, 2001). In addition, algebra teachers need an understanding of
the use of technology in solving problems, and technology for exploring algebraic ideas
and representations (NCTM, 2000).
21
Pedagogical Content Knowledge
Content knowledge, however, is not sufficient and there is a difference between
knowing mathematics and being able to teach it (Mewborn, 2001). Teachers also need
knowledge of mathematics specifically used to facilitate the learning of mathematics by
students (Sherin, 2002). Shulman (1987) first referred to this mixture of content
knowledge and knowledge of pedagogy that belongs exclusively to teachers as
?pedagogical content knowledge.? Pedagogical content knowledge (Ball et al. 2003),
bundles mathematics knowledge with the knowledge teachers have about learners,
learning, and pedagogy (Ball et al. 2001). Pedagogical content knowledge can help
teachers anticipate where students will have difficulties and be ready with alternative
methods, explanations and representations related to a mathematical topic (Ball et al.
2001). In addition, pedagogical content knowledge includes representations that are most
useful to teaching mathematics content (Ball et al. 2003).
Definitions of pedagogical content knowledge
The term ?pedagogical content knowledge? was introduced by Shulman (1987) as
the mixture of content knowledge and pedagogy that belongs exclusively to teachers and
has since been used by many authors (Ball, 1988a; Ball, 2000; Borko, et al., 2000; Even,
1993; Fuller, 1996; McGowen, et al., 2002; McNamara, 1991; Sherin, 2002). Shulman
(1987) referred to pedagogical content knowledge as the professional understanding of
how content and pedagogy fit together. Sherin (2002) described pedagogical content
knowledge as the knowledge used specifically for teaching the content, including not
only knowing how to present the domain to facilitate the learning by students, but also
knowing the typical understandings and misunderstandings of the students being taught.
22
Ball (1988a) defined pedagogical content knowledge as the intertwining of
content knowledge and pedagogy to form a subject-specific body of knowledge.
Pedagogical content knowledge also includes the mathematical knowledge needed to
manage both routine and non-routine problems (Ball, 2000). Llinares (2000) felt the
results of his study on functions reaffirmed earlier definitions of pedagogical content
knowledge which included the integration of content knowledge and the knowledge the
teacher had of the students as learners in the classroom environment. Pedagogical content
knowledge bundles the knowledge of learners, learning, and pedagogy with mathematical
content knowledge.
Ma (1999) termed pedagogical content knowledge as the knowledge teachers use
to ?unwrap? mathematical topics in order to present the content in ways to enable
students to be successful in learning mathematics. Ma (1999) studied the difference in
Chinese and U. S. teachers? knowledge of mathematics for teaching and how the Chinese
Based on her conversations with the Chinese teachers, Ma (1999) referred to ?knowledge
packages? as the organization and development of mathematical ideas within a
mathematics domain and involved ?the longitudinal process of opening up and
cultivating such a field in students? minds? (Ma, 1999, p. 114). For example, a
knowledge package for subtraction with regrouping may contain other links such as ?the
composition of numbers within 100, subtraction without regrouping, composing and
decomposing a higher value unit? (Ma, 1999, p. 19).
In addition, teachers have to be aware of the sequence in which mathematical
ideas are developed. For example, in developing the meaning of division of fractions, an
understanding should first be developed from a ?meaning of addition, to the meaning of
23
multiplication of whole numbers, to the meaning of multiplication with fractions, to a
meaning of division with fractions? (Ma, 1999, p. 114). Finally, the Chinese teachers
within the study referred to key elements within a knowledge package as ?concept knots?
that link related mathematical concepts. For example, within the knowledge package of
division of fractions, Chinese teachers referred to the meaning of multiplication,
modeling division of whole numbers, the concepts of a fraction and a whole, and the
meaning of multiplication of fractions as concept knots (Ma, 1999). Teachers must be
able to understand the ?packing? of specifics within a knowledge package as well as the
?unpacking? of these same specifics and how they relate to each other (Ma, 1999).
Procedural knowledge as well as conceptual knowledge are interwoven within the
knowledge packages (Ma, 1999).
In later writings, Ball (CBMS, 2001) referred to pedagogical content knowledge
as the ?mathematical knowledge for teaching.? Ball and Bass (2003) addressed the major
areas that need to be considered in the mathematical knowledge of teaching. First, the
teaching of mathematics involves substantial mathematical work (Ball & Bass, 2003)
such as the following:
? Design mathematically accurate explanations that are comprehensible and useful
for students,
? Use mathematically appropriate and comprehensible definitions;
? Represent ideas carefully mapping between a physical or graphical model, the
symbolic notation, and the operation or process;
? Interpret and make mathematical and pedagogical judgments about students?
questions, solutions, problems, and insights (both predictable and unusual);
24
? Be able to respond productively to students? mathematical questions and
curiosities;
? Make judgments about the mathematical quality of instructional materials and
modify as necessary;
? Be able to pose good mathematical questions and problems that are productive for
students? learning;
? Assess students? mathematics learning and take next steps. (Ball & Bass, 2003, p.
11)
Next, teachers need to be able to unpack mathematical ideas and understand how
mathematical ideas connect across a mathematical domain. For example, when teaching
the multiplication of 35 and 25, this problem can be represented as 35 groups of 25
objects or as a rectangular array with dimensions of 35 and 25 and an area of 875 square
units (Ball & Bass, 2003). Teachers need to be able to access different representations for
mathematical problems that may arise within the instruction. Teachers have to understand
the different representations of and the meaning behind the algorithm for multiplying
whole numbers (Ball et al. 2005), as seen in the previous example.
Teachers need to understand how mathematical ideas change and grow. For example,
teaching elementary students who are subtracting whole numbers that you cannot subtract
a larger number from a smaller number soon becomes false when students began to
consider the set of integers (Ball & Bass, 2003). Teachers must be able to evaluate
different methods for representing and solving mathematical problems whose
generalizability and validity may not be immediately clear (Ball et al. 2005). See figure 1
for an example of the type of item developed for the Learning Mathematics for Teaching
25
project [LMT] to evaluate the ability of teachers to appraise and validate unconventional
methods used by students (Ball et al., 2005).
Imagine that you are working with your class on multiplying large numbers.
Among your students? papers, you notice that some have displayed their work in
the following ways:
Student A
35
x 25
125
+ 75_
875
Student B
35
x 25
175
+ 700
875
Student C
35
x 25
25
150
100
+ 600
875
Which of these students is using a method that could be used to multiply any two
whole numbers?
Method would work
for all whole
numbers
Method would NOT
work for all whole
numbers
I?m not sure
a) Method A
1
2
3
b) Method B
1
2
3
c) Method C
1
2
3
Figure 1. Item for measuring specialized content knowledge (Ball et al. 2005, p. 43)
In addition, teachers need to be able to size up a typical wrong answer and be able to
analyze the source of the student?s error. An example of such an error follows in figure
2.
26
35
x25
175
_70_
245
Figure 2: Incorrect mathematical procedure for multiplying two whole numbers. (Ball et
al. 2005, p. 17)
All of these considerations fall within the domain of the mathematical knowledge
of teaching (Ball & Bass, 2003). The CBMS report (2001) posits that mathematical
knowledge for teaching will allow teachers to better assess student?s work while being
able to understand where and why students make errors, and also to further the students?
creative thinking. With pedagogical content knowledge, teachers are better able to
anticipate where students will have difficulties and be prepared with alternate
explanations or models. Teachers also need to be able to consider the mathematical ideas
of their students and consider the value of different representations.
Synthesis
For the purpose of this study the following definition of pedagogical content
knowledge will be used. Pedagogical content knowledge is viewed as the knowledge of a
teacher to use his/her knowledge of mathematics to ?unwrap? the mathematical topics
and present the content in ways for students to successfully learn the mathematics (Ma,
1999). This knowledge includes the teacher?s ability to use content knowledge to access
different representations, as well as different methods for solving mathematics problems
27
that may arise within the mathematics instruction. Pedagogical content knowledge also
includes the ability of teachers to direct students to make connections between
mathematical topics as well as helping them to see the connectedness of different
representations for those same topics. Pedagogical content knowledge includes the ability
of teachers to understand where and why students make errors and be prepared with
alternative explanations and models. Pedagogical content knowledge also includes the
ability of teachers to respond productively to students? questions and pose problems and
questions that are productive to student learning.
Instruction to Promote Student Success in Algebra
Teaching mathematics to all students needs to address not only ?what you teach
but how you teach it? (Silva et al, 1990, p. 379). Teachers will need to approach teaching
in different ways. In the following sections we will focus on instrumental understanding
and relational understanding and how classroom instruction should promote them. In
addition, we will look at two projects that were successful at teaching algebra to all
students.
Instrumental Understanding and Relational Understanding
Skemp (1976) discussed how the balance of attention to instrumental
understanding and relational understanding can effect classroom instruction. He describes
instrumental understanding as being synonymous with procedural knowledge, while
relational understanding captures both procedural and conceptual understanding and their
interrelationship. Skemp (1976) provided the following analogy. Suppose students were
taught music in two different situations. The first group used pencil and paper with no
28
musical instruments involved. The students studied the notes, how to write them and
what combinations make up a melody. At the end of the study period the students were
asked to write a short melody. The second group began with instruments and related
sound to the notes on paper. This group was also asked to write a short melody. Skemp
(1976) suggested that very few of the first group would be able to successfully write a
melody and most would hate the music class, whereas the majority of the second group
would be successful in the melody writing and enjoy it at the same time. In the same
sense, students who are passive learners of mathematics instruction will not be as
successful as students who are actively involved in their classroom instruction.
Skemp (1976) points out when where relational and instrumental understanding
can cause problems in the classroom. In the first case, teachers want to develop relational
understanding and the students want only instrumental understanding. For example, a
student may say that he/she does not wish to understand how a particular rule works, just
tell me the rule for how to solve a problem. Even though this may be frustrating for the
teacher, Skemp (1976) felt that less damage is done in this situation than when the order
of learning is reversed. If a student wishes to learn relationally and the teacher is teaching
instrumentally, the student may have difficulties in applying rules and formulas simply
because he/she does not understand why he/she is using the particular rules or formulas.
In contrast, Skemp (1976) also points out reasons why teachers may prefer to
teach mathematics instrumentally. They may feel relational understanding takes too long
and is difficult to achieve, that skills must first be developed in order to later develop
relational understanding can later be developed, and that there is little encouragement for
29
newer teachers to teach relationally when everyone else is teaching instrumental
mathematics.
Pesek and Kirshner (2002) designed a study to address the issue that teaching to
develop relational understanding will take too much time (Skemp, 1976). Pesek and
Kirshner (2002) noted that teachers sometimes adopt a two-track strategy for teaching
mathematics. Part of instructional time is devoted to teaching students mathematics for
meaning (relational) while the remainder of the time is spent in teaching mathematics for
recall and procedures (instrumental). This two-track strategy might be the result of
teachers feeling pressured into covering larger amounts of material, along with having an
attitude that teaching students relationally takes more time than simply presenting
material instrumentally (Pesek & Kirshner, 2002).
Pesek and Kirshner (2002) devised a study to investigate whether students who
only learned mathematics relationally could perform as well as a group of students who
were taught instrumentally first and then relationally. The instrumental instruction
consisted of five days of instruction on finding formulas for perimeters and areas of
various geometric shapes and how to use them to solve for area and perimeter. During
this time frame the second group of students reviewed material that was not related to the
unit on perimeter and area. After the first five days of instrumental instruction only, all
students in both groups received three days of relational instruction on the same topics, in
which ?connections were developed through concrete materials, questioning, student
communication, and problem solving? (Pesek & Kirshner, 2002, p. 103). Teachers
providing the relational instruction never used formulas or gave specific strategies for
30
students to find area and perimeter. Student strategies evolved from their own
understanding of area and perimeter (Pesek & Kirshner, 2002).
Following the instruction, students who received only three days of relational
instruction only and were not exposed to the area and perimeter formulas were more able
to explain why the formulas worked than the group that had direct instruction with the
formulas. These students also scored higher on the posttest and retention test than those
students receiving both types of instruction (Pesek & Kirshner, 2002).
Conclusion
It is important for students to develop both the procedures for doing mathematics
as well as the conceptual understanding for why the algorithms work. Developing
relational understanding, which includes both procedural as well as conceptual
understanding, takes less time than instrumental and students who have only relational
understanding do better in mathematics that those students who had received both
relational understanding as well as instrumental.
Projects to Improve Algebra Instruction
The Algebra Project and the Quantitative Understanding: Amplifying Student
Achievement and Reasoning (QUASAR) Project both promoted teaching mathematics in
ways that aligned with the ideas of NCTM (2000). The mathematics achievement of the
students in both projects were higher than the mathematics achievement of students in
traditional mathematics classrooms.
The Algebra Project
Moses (2000) asserted that algebraic thinking skills should be encouraged at
every grade level. Furthermore, by teaching algebraic skills to all students, he felt
31
students will be provided a way of understanding and interpreting occasions that may
arise in daily living. The Algebra Project was begun out of a desire by Robert Moses to
ensure his children would achieve mathematical literacy (Moses et al., 1989).
Components of the Algebra Project included a curriculum that would address the
conceptual leap from arithmetic to algebra, a curricular process which was experimental
and linked familiar physical experiences to abstract mathematics, the expectation for
success which was reinforced by students, parents, teachers, and administrators (Silva et
al., 1990).
There were three broad goals of the Algebra Project (Silva et al. 1990). First, the
Algebra Project had the goal of developing the mathematical ability of middle school
students so that they would be prepared for the study of algebra at the high school level
and could master the skills of college preparation mathematics courses (Silva et al. 1990).
A second goal of the Algebra Project was to change the way mathematics teachers
constructed their learning environments by producing teachers able to facilitate the
learning of algebra related to real life experiences of the students (Silva et al. 1990). A
third goal of the Algebra Project was to build a larger community of individuals including
parents, community volunteers, and administrators and involve them in enhancing
mathematics instructional programs (Silva et al. 1990). Achieving the goals of the
Algebra Project involved a four-fold approach: development of a curriculum at the
middle school level, development of a method of training teachers in both ?what to teach
and how to teach it? (Silva et al, 1990, p. 379), sharing the project with other
communities, and development of a strategy of nurturing support among parents,
organizations within the community and the broader community.
32
As a result of the Algebra Project, the teacher?s role was now that of a facilitator
and the sixth grade curriculum was developed to teach algebraic skills in real world
contexts. Students who were involved in the first year of the project were all placed in a
higher level mathematics course above algebra one when entering high school. About
thirty-nine percent were placed in Honors Geometry or Honors Algebra (Moses et al.,
1989).
The QUASAR Project
A second project has had success in teaching algebra to all students. The
QUASAR project asserted that an emphasis should be placed on students learning
mathematics by engaging in activities that are embedded in the student?s social and
cultural context (Silver et al. 1993). Mathematical instruction must address the
connection between what is taught in school with the social lives of the children asked to
learn it (Silver et al. 1993). The goals of the QUASAR project aimed at producing
students who:
not only can accurately execute algorithms and recall factual knowledge but also
have the capacity to impose meaning and structure on new situations, to generate
hypotheses and critically examine evidence, and to select the most appropriate
from among a repertoire of strategic alternatives. (Silver et al. 1993, p. 14)
The vision of the QUASAR project was for the mathematics classroom to become
a community of collaborative, reflective practice where students are actively engaged in
the mathematics they are learning, as well as challenged to think deeply about
mathematical ideas (Silver et al. 1993). At one of the middle schools involved in the
QUASAR project, which had a reputation for low-performance in mathematics, change
33
was evident in the first grading period when compared to the other twenty middle schools
within the district (Silver, 1995). The QUASAR students in this comparison had the
second highest scores and at the end of the course passed at much higher rates than
students who were at district schools demographically similar to their own (Silver, 1995).
Credit for the exceptional performance was attributed to mathematical instruction that
emphasized thinking, reasoning, communication, and problem solving (Silver, 1995), the
same processes deemed important by NCTM (2000).
Conclusion. All students can learn algebra when the content is developed across
the K-12 curriculum and learned in real world context. Students should be allowed to
develop procedural knowledge as well as conceptual fluency using a variety of methods,
representations, and processes within the learning process. Furthermore, students can be
successful when they are actively engaged in the learning process and challenged to think
deeply about mathematical ideas (Silver et al. 1993).
Research Related to Content Knowledge and Pedagogical Content Knowledge
A better understanding of teachers? content knowledge and pedagogical content
knowledge is necessary to further our ability to provide all students with experiences that
promote success in algebra. In this section, we will look at some research studies
addressing content knowledge and pedagogical content knowledge of teachers, as well as
studies that have focused on changing content knowledge and pedagogical content
knowledge. The first part of the section focuses on elementary teachers, followed by
studies on secondary teachers. Our attention will then turn attempts at measuring teacher
34
knowledge, and this section will end with studies addressing changes in teacher
knowledge.
Content Knowledge Studies with Elementary Teachers
Several studies have focused on the knowledge of elementary pre-service and in-
service teachers in the content areas of multiplication of multi-digit numbers and place
value (Ball, 1988a, Ma, 1999, Ball & Bass, 2000), subtraction and division of fractions
(Ball, 1988a, Ma, 1999, Ball & Bass, 2000), decimal numeration (Stacy et al., 2001), and
geometry (Ma, 1999, Swafford, Jones, & Thornton, 1997, Borko et al., 2000). The
general consensus of these studies is that while teachers could use algorithms to teach
specific topics, they were unable to provide adequate explanations for why those
algorithms work. They were unable to provide concrete examples or other conceptual
approaches that might promote student learning. In addition, teachers had difficulty
recognizing and understanding errors made by students.
Several studies explored teachers? knowledge of multi-digit multiplication, and
how place value related to the multiplication process (Ball, 1988a, Ma, 1999, and Ball &
Bass 2000). Teachers in each of the studies had procedural knowledge of the area, but the
majority of both pre-service and in-service teachers were unable to provide an adequate
explanation for why the multi-digit multiplication algorithm worked. Most explanations
revolved around the placement of zero place holders. Some teachers actually suggested
using pictures of apples or oranges as place holders so the students would not get
confused and think the zero place holder was actually a part of the answer (Ball & Bass,
2000). The majority of the teachers did not provide explanations that included knowledge
about the numbers as multiples of ten and one hundred (Ball & Bass, 2000; Ma, 1999).
35
Ball (1988a) further pointed out when teachers strictly adhere to teaching the shortcut
algorithm, the conceptual understanding of multi-digit multiplication is hidden.
These studies also probed the teachers? knowledge of subtraction and division of
fractions (Ball, 1988a; Ma, 1999; Ball & Bass, 2000). The results were typically the same
as the studies conducted with multi-digit multiplication. Teachers had procedural
knowledge but were generally unable to provide conceptual examples. Furthermore,
teachers were unable to provide an adequate explanation for why the algorithms for
subtraction and division of fractions work. In every area, elementary teachers from the
United States were less likely to give conceptual explanations when explaining the
mathematics in question and were more likely to give further clarification on the rules
and procedures. On the other hand, a majority of the Chinese teachers provided
conceptual explanations of the mathematical topics covered. When interviewed, the
Chinese teachers stressed the importance of knowing not only how to work problems but
also why the procedures that were used worked (Ma, 1999).
Stacy et al. (2001) studied preservice teachers? knowledge of student difficulties
with decimal numeration. Participants were asked to compare decimals and determine
which items would be difficult for students (Stacy et al., 2001). A significant number of
these preservice teachers had an inadequate knowledge of decimals, particularly in
comparing decimal numbers with different numbers of digits and their relationship to
zero. For example, 0.123 is greater than 0.6 because 0.123 has more digits than 0.6. In
addition, children may consider the number with the most digits to be larger, so 0.116
would be larger than 0.2. Stacy et al. (2001) found the explanations of the participants fell
into three distinct categories. First, the participants thought students would have problems
with particular problems but gave no explanations for why they felt this way (Stacy et al.,
2001). Second, participants identified a surface reason for a difficulty, such as one
number has more decimal places but gave no further explanations. Third, participants
explained why the features of the problem would cause difficulty (Stacy et al., 2001).
Thus, many participants could identify problems that would cause students difficulties
but could not adequately explain why.
Ma (1999) and Borko et al. (2000) conducted studies involving teacher
knowledge of geometry. Ma (1999) presented the teachers in her study with the scenario
of a student claiming that as the perimeter of a figure got larger so did the area of the
figure. Only one of the 23 U. S. teachers investigated the claim and provided a
counterexample for the student?s claim. On the other hand, the majority of the Chinese
teachers investigated the student?s claim.
Borko et al. (2000) conducted a case study of a preservice teacher in order to see
how she used her mathematical content knowledge in selecting worthwhile tasks for her
students and in conducting student discourse. The tasks she selected involved circles,
their areas, circumference, and? and involved both hands-on investigations by the
students as well as investigations using Geometer Sketchpad. The teacher in this situation
was confident of her mathematical abilities. Borko et al. (2000) found that teachers who
possess greater content knowledge are more likely to emphasize problem solving, student
inquiry, and conceptual knowledge. Teachers with greater content knowledge promoted
student discourse and asked questions that required higher order thinking skills of their
students (Borko et al., 2000).
36
37
These studies found that elementary teachers had a procedural knowledge of the
mathematics topics but lacked conceptual understanding. Teachers with greater content
knowledge are more likely to emphasize problem solving, student inquiry, conceptual
knowledge, and higher order thinking skills.
Content Knowledge Studies with Secondary Teachers
Relatively few studies have focused on the content knowledge of secondary
mathematics teachers, perhaps because of the belief that content knowledge may not be a
problem at the secondary level because of secondary teacher?s specialized knowledge of
mathematics (Ball et al., 2001). However, the available research, while limited, has
served to reveal the fallacy of this assumption (Ball et al., 2001). At the secondary level,
studies have considered teachers? content knowledge and pedagogical content knowledge
in the areas of slope (Stump, 1997; Sherin, 2002) and functions (Even, 1993; Llinares,
2000; Sherin, 2002). Nathan & Koedinger (2000) studied teachers? perceptions about
algebraic reasoning. These studies included both pre-service and in-service teachers.
Stump (1997) used slope as the basis for studying the teacher?s knowledge of the
subject since slope is a critical topic in algebra. It appears throughout the secondary
mathematics curriculum, and mathematics teachers need to have a deep understanding of
the topic and be able to communicate about slope correctly to the students. The study
explored whether teachers understood the different representations of slope and the
different approaches taken in teaching this topic. Stump (1997) also asked teachers to
include any illustrations, examples, analogies and explanations they would employ while
teaching this concept. The study focused on how the teachers were able to relate different
types of representations of slope, such as algebraic and geometric representations or
38
functional and trigonometric representations. While in-service teachers could reflect on
the difficulties they had observed across the years they had actually taught slope in the
algebra classroom, pre-service teachers could only speculate on what difficulties students
might have with conceptually understanding slope.
Stump (1997) found the majority of teachers experienced difficulties when slope
was related to problems involving functional and trigonometric representations. She
found no difference in the mathematical understanding of slope in pre-service and in-
service teachers. Physically representing slope held more importance than a geometric
representation for most teachers, but none of the teachers mentioned that a physical
representation was a prerequisite of slope. While teaching experience may provide
teachers with some growth in content knowledge, this growth is not insured by
experience alone (Stump, 1997). Stump (1997) found teachers to have a range of levels of
content knowledge, with most weaknesses occurring in the connection between the
different types of representations of slope, such as algebraic and geometric
representations or geometric and trigonometric representations.
Even (1993) studied pre-service secondary teachers? content knowledge and
pedagogical content knowledge of functions. Questionnaires were collected from 152
participants. These questionnaires contained nine nonstandard problems addressing
different aspects of content knowledge about functions, as well as six items related to
students? misunderstandings or misconceptions about functions (Even, 1993). For
example, participants were asked to provide a definition of function and to supply an
alternative definition if students did not understand the first one (Even, 1993). Follow-up
interviews were designed to develop a more accurate and detailed picture of the
39
participants? content knowledge and pedagogical content knowledge (Even, 1993).
Reactions to students? misconceptions were also addressed. For example, ?Is it important
to teach the vertical line test for graphs of functions to students? Why? (What is the
(vertical) line test? What would you teach to your students? How would you teach it?
Can you give me an example?)? (Even, 1993, p. 100). Even (1993) found that most
participants viewed functions as having a ?nice? graph or one that can be described by an
expression. In addition, participants could select between functions and non-functions,
but had difficulty in explaining why it was important to know the difference between
functions and non-functions (Even, 1993). This limited understanding of functions can
put teachers in unfamiliar situations and affect questions they ask, mathematical tasks
they select, and the direction of a student?s mathematical thinking (Even, 1993).
Llinares (2000) explored teachers? knowledge of functions video recordings of
classes, and interviews that called for the analysis of hypothetical situations and analysis
of textbook problems with his case study subjects. The findings of the study were
summed up in three themes: teacher knowledge and the pupils? images of the concept of
function, the flexibility of teacher knowledge, and conceptions about how students learn
mathematics. The teacher expressed the desire for her students to learn conceptually
about functions but felt that students had to learn through the memorization of certain
procedures, practice on examples, and that the conceptual understanding would come
later (Llinares, 2000).
Nathan and Koedinger (2000) found teacher?s perceptions about the algebraic
reasoning of their students could have an effect on the tasks teachers chose for students.
Nathan and Koedinger (2000) asked teachers to make predictions of the level of difficulty
40
that students would have when given different types of problems to solve. Participants in
the study spanned the United States and taught in various school settings. Teachers
ranked twelve problems, six of which were based on arithmetic and the other six were
based on algebra. The six problems in each group were basically the same problem , but
written in different contexts. For example, an arithmetic (result unknown) problem
follows: ?Starting with 81.9, if I subtract 66 and then divide by 6, I get a number. What is
it?? (Nathan & Koedinger, 2000, p. 170). An example of an algebra (start-unknown)
problem follows: ?Solve for x: 6x + 66 = 81.90? (Nathan & Koedinger, 2000, p. 170).
Ninth grade Algebra I students were asked to solve the problems and their difficulties
were compared to the teacher predictions.
Surprisingly, teachers felt that students could more easily solve an algebraic
equation than a word problem, but in reality students were more able to get a correct
answer for a word problem than solve the algebraic equation. Nathan and Koedinger
(2000) noted that for most teachers, textbooks were the primary resource and in many
cases, they were the only resource used by the teachers. Evidence in the teacher
knowledge about their students and their proposed difficulty level of problems may have
stemmed from the sequence of how students were introduced to these types of problems
in their own classrooms (Nathan & Koedinger, 2000). For example, if story problems
were introduced first, students they would not be as difficult for the students.
Furthermore, Nathan and Koedinger (2000) suggested the order that the problems are
presented in the textbook might have been an influencing factor in the teachers?
perceptions about the difficulties that students would have, rather than the reality their
study portrayed.
41
Thus, much like the studies conducted with elementary teachers, weaknesses in
content knowledge were found in the studies with secondary teachers. Teachers
experienced difficulty in understanding slope, function, and difficulty in considering
where students might have difficulties in mathematical tasks.
Measuring Mathematical Knowledge for Teaching
To better understand the content knowledge and pedagogical content knowledge
of teachers, we need to be able to measure these types of knowledge. Researchers
involved in the Study of Instructional Improvement (Hill, Schilling, & Ball, 2004) began
developing mathematics items related to the K-6 curriculum that would be used in
surveying teachers to help in answering the question ?What mathematical knowledge is
needed to help students learn mathematics?? Three test forms were developed and piloted
with participants from the California?s Mathematics Professional Development Institutes,
a project which was aimed at boosting the teachers? subject matter knowledge of
mathematics.
The researchers set out to explore what subject matter knowledge is needed for
teaching and how it is used. Mathematics items were formulated in two different
contexts. Some of the questions required participants to rely on their content knowledge
of mathematics, such as finding decimals that were halfway between two given decimal
numbers. Other questions, referred to as the mathematics-as-used-in-teaching items,
required participants to rely on specialized knowledge of mathematics. This specialized
knowledge included evaluating unconventional methods used by students for solving
mathematics problems, using or examining alternative representations, and providing
explanations why the methods or representations work. One such question, related to this
42
last category of questions, involved participants evaluating three different methods for
multiplying two two-digit numbers. Teachers needed to inspect each step in each of the
processes to understand what was done at each step and decide whether the method
would be mathematically correct for each situation. This type of mathematical situation
arises regularly during classroom instruction, although it may not necessarily occur for
other adults who regularly use mathematics.
Initial findings of this study suggest ?that teachers? knowledge of mathematics for
teaching is at least partly domain specific rather than simply related to a general factor
such as overall intelligence, mathematical ability, or teaching ability? (Hill, Schilling, &
Ball, 2004, p. 26). Evidence from this study supports the conjecture that teachers need
more mathematical content knowledge than the well-educated adult. Teachers need more
depth in their mathematical knowledge beyond the knowledge related to the content in a
textbook. Having strong content knowledge does matter, but teachers also need to know
why mathematical statements are true, be able to represent mathematical ideas using
multiple representations to model phenomena and select what method is best, what
constitutes an appropriate definition for a concept or term, and ways for teachers to
evaluate other mathematical procedures, representations, and solutions (Hill, Schilling, &
Ball, 2004).
In a later study, Ball, Hill, and Bass (2005) relating student achievement to the
Study of Instructional Improvement which was engaged in study school reform. Student
scores were collected on the mathematics portion of the Terra Nova as well as teachers?
performance scores on the mathematical knowledge for teaching instruments (Ball et al.
43
2005). Initial findings found the teachers? scores predicted the size of the gain in scores
of students on their respective test (Ball et al. 2005).
Changes in Teacher Knowledge
Studies have also been conducted to explore changes in teacher knowledge.
Sanders and Morris (2000) studied the mathematical skills of pre-service teachers
whereas Sherin (2002) and Arbaugh and Brown (2002) used teacher study groups as a
basis for improving teacher knowledge.
Even when pre-service teachers are confronted with results that indicate weak
conceptual knowledge in their own mathematical abilities, they sometimes fail to do
anything to correct the problem. Sanders and Morris (2000) studied how the exposure of
the depth of content knowledge empowered or debilitated pre-service teachers. Students
involved in the study were administered a scale to determine their confidence in tackling
certain types of questions. Those students who had poor results were advised to attend
class. Students were expected to take responsibility for making improvements in their
own performances. The researchers included informal conversations with the students as
well as more-formal interviews to clarify their observations. Some of the pre-service
teachers in this study failed to admit that they had mathematics inadequacies, while
others stayed away from tasks they knew would be problematic, and still others admitted
to their inadequacies but did little to correct the problem. Just a few pre-service teachers
admitted they were having mathematical difficulties and worked to make improvements
in their mathematical understanding. Sanders and Morris (2000) felt a critical component
of their study was the need to encourage pre-service teachers to admit that they had gaps
in their mathematical skills and knowledge.
44
Sherin (2002) found teachers developed deeper content knowledge through
reflection on the development of student understanding in a lesson, looking at student
learning, and knowledge of mathematics. Along with videotaping the group reflection
sessions, participants were interviewed to develop an understanding of their knowledge
about linear functions. The content knowledge of the participants was examined within
three classes of interaction involving the teachers? content knowledge and the
implementation of the curriculum (Sherin, 2002). First, some teachers changed the design
of the mathematics lesson but the content knowledge of the teacher did not change.
Second, some teachers used the lesson as it was designed and in doing so they learned
new content knowledge. Third, some teachers developed new content knowledge as well
as made changes in the lesson as it developed during classroom instruction. This type of
interaction involves instructional strategies that are new to both the teacher and lesson.
Arbaugh and Brown (2002) conducted a study in which the researcher was
included in the study group and was considered the ?expert? other. The responsibilities of
the expert other included providing articles that were of interest to the study group,
asking probing questions, challenging the teachers to reflect on their knowledge and
teaching. Both pre- and post-interviews were conducted with the participants. Task
sorting was included as part of the interview process with follow up questions to give the
teachers an opportunity to explain their process in the activity. Task sorting was placed
into four categories: lower-level demands (memorization), lower-level demands
(procedures without connecting to meaning), higher-level demands (procedures with
connections to meaning), and higher-level demands (doing mathematics) (Arbaugh &
Brown, 2002). Through classroom observations, data was collected on the types of tasks
45
teachers in the study were using in their classroom instruction. A portion of the study
group time was devoted to a discussion of the tapes from classroom observations.
Arbaugh and Brown (2002) found that working with the teachers helped them learn about
the levels of cognitive demands. Progress was made in how the teachers were thinking
about the tasks they were choosing for their students, giving consideration to the thinking
and reasoning skills of their students. Through these tasks, teachers began to select and
incorporate more high-level tasks into their teaching practices.
Thus, when pre-service teachers realize they lack content knowledge, they don?t
always do anything to correct the problem. Teachers who reflected on their own teaching
tasks developed deeper content knowledge and changes were made in instructional
practices.
However, much more research is needed. Very little of this research has linked
teacher knowledge with actual classroom practice. Results from these studies seem to
indicate the content knowledge and pedagogical content knowledge are limited and
inadequate in providing the types of instruction necessary for teaching mathematics and
particularly algebra to all students.
Professional Development
Results from the studies done of the content knowledge of teachers suggest a lack
and depth in the understanding of the topics they are expected to teach, as well as
weaknesses in their pedagogical content knowledge. Knowing what to teach and how to
teach it are crucial for student learning. A need for change is evident from the success of
projects such as the Algebra project and the QUASAR project. However, teachers cannot
46
be expected to change their instructional practices just because they are told to do so
(Mewborn, 2003).
Professional development must be provided that affords opportunities for teachers
to experience growth in these types of knowledge that can be transferred to classroom
practice. Guskey (2002) noted that ?professional development programs are systemic
efforts to bring about change in the classroom practices of teachers, in their attitudes, and
beliefs, and in the learning outcomes of students.? Teacher thinking needs to be at the
center of professional development (Mewborn, 2003). Lee (2004/2005) proposes that
requirements for successful professional development need to include participants as
decision makers and consumers, should recruit teachers from the same context, and build
a partnership between university, public, schools, and local education agencies.
Borasi and Fonzi noted these components necessary for successful professional
development. First professional development should be sustained and intensive (Borasi
& Fonzi). Professional development should be long-term and should provide a safe
environment for teachers to come together and discuss their successes and frustrations
(Mewborn, 2003)
Second, professional development should be informed by how people learn best
(Borasi & Fonzi). Three cycles of learning should be considered (Borasi & Fonzi)
1) engaging actively in situations that provoke cognitive dissonance, thus
initiating new constructions of meaning;
2) sharing and discussing these constructions with a group to arrive at consensus
and generalizations;
3) applying these generalizations to new situations. (p. 30)
47
These learning cycles should focus on knowledge of and about mathematics, theories of
mathematics learning, how students develop particular mathematical ideas, ability to plan
instruction, and the ability to effectively interact with students (Borasi & Fonzi).
Third, professional development should focus on activities that are critical to
learning and teaching and should focus on offering a rich set of diverse experiences
(Borasi & Fonzi). In addition, teachers need to learn mathematics the way they are
expected to teach it (Schifter, 1998). Teachers need help to develop their knowledge of
mathematics beyond what nonprofessional functioning adults need (Hill, Schilling, &
Ball, 2004). Teachers need to revisit mathematical concepts to help them gain conceptual
understanding of the mathematical ideas and help them make connections to other
mathematical topics (Mewborn, 2003). Teachers need to be involved in experiencing a
broader version of mathematics, so they can break away from the way they have
traditionally thought about mathematics (Acquarelli & Mumme, 1996). Teachers have to
learn mathematics differently in order to teach it differently (Lee, 2004/2005).
Professional development needs to help prepare teachers for the tasks that they will
encounter on the job (Hill, Schilling, & Ball, 2004). Professional development should
offer a rich set of diverse experiences (Borasi & Fonzi).
Fourth, professional development should focus on developing collaboration
within a community of learners (Borasi & Fonzi). The professional environment needs to
be designed to develop a supportive community among the teachers (Heaton, 2000;
Romagnano, 1994). Professional development should involve multiple grade levels so
that teachers can feel a part of the K-12 mathematics education system rather than feeling
like they are at an isolated grade level (Ruopp, Cuoco, Rasala, & Kelemanik, 1999).
48
Teachers should attend professional development as a collective group from the same
school, rather than as individual teachers, in order to build a stronger teacher community
(Lee, 2004/2005). Furthermore, by working as a community it is easier for teachers to
address issues that might promote or inhibit mathematics teacher reform (Acquarelli &
Mumme, 1996). Teacher leaders need to grapple with the same types of issues that the
teachers that they work with will experience (Acquarelli & Mumme, 1996).
Sowder, Philipp, Armstrong, and Schappelle (1998) found that teachers?
instructional practice changes as their mathematical knowledge changes. Teachers
became more independent and depended less on the prescribed curricula. These teachers
saw their students as being capable mathematics students and expected more conceptual
understanding of mathematics by their students. This resulted in students showing a
deeper mathematical understanding of the mathematics they were studying.
The Renaissance program was a large scale professional development program to
empower mathematics teachers to transform students into mathematical thinkers
(Acquarelli & Mumme, 1996). After the third year of the Renaissance professional
development program teachers were seen using cooperative learning, asking students to
explain their mathematical thinking through writing, encouraging students to use concrete
materials in mathematical tasks, and providing calculators for students to use (Acquarelli
& Mumme, 1996). Since the majority of teachers has learned mathematics in traditional
ways, they need ample time to make changes in their instructional practices. Mathematics
teachers need opportunities to construct new understandings of mathematics and teaching
so they can in turn provide similar opportunities for their students (Acquarelli & Mumme,
1996).
49
Conclusion
Professional development can be successful in increasing content knowledge and
pedagogical content knowledge of teachers. Teachers need to be active participants in
learning, both in the role of a student experiencing mathematics and in the role of a
facilitator learning to understanding how to use this knowledge in the successful teaching
practices. Teachers need to revisit the ?big? ideas of the mathematics they are teaching
and be able to develop both a depth in understanding as well as how topics relate to other
areas of mathematics. Professional development needs to be long term and provide
opportunities for teachers to build a rapport with other teachers in discussing their
failures and successes. However, even if professional development does increase content
knowledge and pedagogical content knowledge of teachers, there is no assurance that
changes in this knowledge will be evident in their instructional practices.
Research Questions
In this section, a synthesis of the literature review is provided, from which the
research questions for the study emerged. Students need to be successful in mathematics,
particularly in algebra, which is the gatekeeper course to future opportunities for success
(RAND, 2003). To increase student learning, teachers need to possess both a deep
knowledge and understanding of the mathematics content as well as pedagogical content
knowledge. However, numerous studies suggest that teachers lack both the content
knowledge and pedagogical content knowledge to be effective in successfully providing
the types of opportunities students need to learn mathematics. Most of this research has
been done at the elementary level, but one might argue that these same conclusions are
50
true about secondary teachers. Further study is needed to verify whether secondary
teachers have the necessary content knowledge and pedagogical content knowledge to be
effective.
Opportunities to develop greater content knowledge and pedagogical content
knowledge and how to effectively use them should be provided through professional
development (RAND, 2003). How to effectively develop content knowledge and
pedagogical content knowledge for teaching remains an unsolved problem for the
improvement of mathematics teaching and learning (Ball et al., 2001). While research
suggests that professional development can increase content knowledge and pedagogical
content knowledge of teachers, we do not know if and how these changes will be
reflected in instructional practices within the mathematics classroom.
To address these issues, the following research questions were proposed to guide
the study.
1. What content knowledge and pedagogical content knowledge of algebra do
high school algebra teachers possess?
2. How are high school algebra teachers? content knowledge and pedagogical
content knowledge reflected in their teaching practices?
3. What growth in content knowledge and pedagogical content knowledge can be
seen from participating in professional development that includes attention to
increasing both content knowledge and pedagogical content knowledge?
4. How are changes in content knowledge and pedagogical content knowledge
reflected in teachers? instructional practices?
51
III. THEORETICAL UNDERPINNINGS AND METHODOLOGIES
The purpose of this study is to better understand the content knowledge and
pedagogical content knowledge of mathematics teachers, how these types of knowledge
are reflected in instructional practices, how intensive professional development promotes
changes in these types of knowledge, and how changes in both types of knowledge are
reflected in instructional practice. In order to answer these questions, four areas of
research design have to be addressed. First, theoretical underpinnings for the study will
be considered. Next, the general design of the study will be described. This study
consisted of two parts, survey research and multi-individual case studies. Then the
context of the study will be addressed. Finally, I will describe the methodologies and
methods used in the two parts of the study.
Theoretical Underpinnings
This study is qualitative in nature, therefore the analysis is both the creation and
interpretation of the researcher (Denzin & Lincoln, 2000). Interpretation of all research is
guided by the set of beliefs and feelings the research holds about how the world should be
studied and understood (Denzin & Lincoln, 1994). However, qualitative researchers
52
attempt to study phenomenon in ?natural settings while attempting to make sense of
phenomena in terms of meanings people bring to them? (Denzin & Lincoln, 1994, p. 2).
Intrepretations from qualitative research are constructed to provide a conduit through
which the voices of those being researched can be heard (Denzin & Lincoln, 2000).
Crotty (1998) denotes four elements that drive qualitative research, as follows:
? Epistemology: the theory of knowledge embedded in the theoretical
perspective and thereby in the methodology.
? Theoretical perspective: the philosophical stance informing the
methodology and thus providing a context for the process and grounding
its logic and criteria.
? Methodology: the strategy, plan of action, process or design lying behind
the choice and use of particular methods and linking the choice and use of
methods to the desired outcomes.
? Methods: the techniques or procedures used to gather and analyze data
related to some research question or hypothesis. (p. 3)
We begin first with the epistemology underlying this research study, which gives
us a way to understand what we gain from a research project and helps us to explain what
we know (Crotty, 1998). Epistemology determines how we go about our research:
?Epistemology is concerned with providing a philosophical grounding for deciding what
kinds of knowledge are possible and how we can ensure that they are both adequate and
legitimate? (Crotty, 1998, p. 8). The epistemological stance taken in this research study is
constructivism. Crotty (1998) defined constructivism as ?the meaning-making activity of
the individual mind? or the individual engaging with research and making sense of it.
53
Constructivism takes into consideration our own individual experiences and suggests that
how we make sense of the world is as worthy and valid as how any other person makes
sense of it (Crotty, 1998). Meaning is constructed through our engagement with the
realities of the world (Crotty, 1998), and the meaning of a phenomenon must be
interpreted to be fully understood (Denzin & Lincoln, 1994).
Constructivism suggests that knowledge is not simply imprinted onto the mind,
but rather concepts and abstractions are formed by the active mind working with the
impressions it receives of phenomena (Denzin & Lincoln, 1994). Different people may
construct different meaning from the same event (Crotty, 1998). Thus, the constructivist
may tentatively come to interpret the language and actions of others through his/her own
constructs, but he/she has to acknowledge that others have realities that are different from
their own interpretations (Ernest, 1998). Ernest (1998) pointed out two principles of
constructivism. First, knowledge cannot be directly transferred from the world or persons
to the mind of the learner or knower. New knowledge is actively constructed from pre-
images within the mind of the learner based on stimuli from the environment. Therefore,
individual learners construct unique interpretations even when exposed to the same
stimuli (Ernest, 1998). Second, knowledge is constructed and cannot reveal anything
definitive about the world or any other domain (Ernest, 1998). For example, observers of
the same phenomenon may offer different interpretations related to the person?s
background and experiences.
The theoretical perspective for this study can be identified as phenomenology.
The actual meaning of phenomenology can be confusing and diluted since its use is so
popular and widespread. As Patton (2002) suggests, ?Phenomenology can refer to a
54
philosophy, an inquiry paradigm, an interpretive theory, a social science analytical
perspective or orientation, a major qualitative tradition, or a research methods
framework? (p. 104). Phenomenology is reflected in most qualitative research and aides
the researcher in attempting ?to understand the meaning of events and interactions to
ordinary people in particular situations? (Bogdan & Biklen, 1998, p. 23). Phenomenology
?requires us to engage with phenomena in our world and make sense of them directly and
immediately? (Crotty, 1998). In addition, phenomenology suggests that we attempt to lay
aside the previous understandings we have about the phenomena, look at our immediate
experiences with it, and possibly develop new meanings from it through a fresh set of
eyes (Crotty, 1998). As Gall et al. (2003) state,
Phenomenology is the study of the world as it appears to individuals when they
place themselves in a state of consciousness that reflects an effort to be free of
everyday biases and beliefs. As such, phenomenology shares the goal of other
qualitative research traditions to understand how individuals construct, and are
constructed by, social reality. (p. 481)
Patton (2002) suggests, ?All our understanding comes from sensory experience of
phenomena, but that experience must be described, explicated, and interpreted? (p. 106).
However, interpretations and descriptions of experiences can become so intertwined they
often become one (Patton, 2002). To understand an experience, interpretation is essential
and experience is included in the interpretation (Patton, 2002). Phenomenology focuses
on how we put the phenomena together in a way to make sense of the world (Patton,
2002).
55
The last of the two elements driving qualitative research are methodology and
methods (Crotty, 1998). These elements will be addressed in the following sections.
Study Design
Understanding how the content knowledge and pedagogical content knowledge of
mathematics teachers relate to each other is not simple. Since this relationship is
complex, one cannot develop a full understanding by looking at these types of knowledge
through one method of data collection. While some understanding can be gained from
data collection from any given source, a deeper and better understanding of the content
knowledge and pedagogical content knowledge of teachers and how they relate to each
can be developed by studying these types of knowledge from multiple data sources
(Crotty, 1998).
This study consisted of two complementary investigations. First, survey research
was conducted with a large pool of teachers in order to understand both their content
knowledge and pedagogical content knowledge. In addition, this phase explored the
attitudes teachers have about how students should learn mathematics. Second, a multi-
case study was used to provide an in-depth examination of these same types of
knowledge, as well as to probe how teachers use their knowledge in mathematical
instruction. The cases also offered opportunities to understand how content knowledge
and pedagogical content knowledge changed as a result professional development and to
see what how these changes are reflected in mathematics instructional practices. More
detail on these two investigations is provided in later sections.
56
Context for the Study
Conducting research in order to answer proposed research questions requires an
available and willing source of teachers. The Multi-District Mathematics Systematic
Improvement Program (MDMSIP) provided a context within which the research
questions could be addressed. The project activities include measuring content
knowledge and pedagogical content knowledge of teachers in grades K-12, as well as
providing professional development which included attention to increasing both of these
types of knowledge.
Multi-District Mathematics Systematic Improvement Program (MDMSIP)
Multi-District Mathematics Systematic Improvement Program (MDMSIP) is a
partnership between two universities and twelve school districts in East Alabama.
Members of the faculty of both the College of Education and the Mathematics
Department at East University and the Mathematics Department of South University are
included in the partnership. The goal of MDMSIP is to improve mathematics education
within the partnership districts, with a focus on increasing overall student achievement
while addressing gaps between demographic groups. These improvements will be met by
aligning the K-12 curriculum, enhancing professional knowledge of practicing
mathematics teachers, ensuing consistency in teaching, developing a cadre of
knowledgeable teachers, and improving the preparation at the university level for future
mathematics teachers (MDMSIP, 2003b). MDMSIP began work with an initial $100,000
grant from East University while submitting a proposal for funding from the National
Science Foundation (NSF). MDMSIP subsequently received a five-year $9,000,000 grant
57
from NSF beginning in October 2003 and continuing through 2008. The following
mission statement was adopted for the MDMSIP project.
To enable all students to understand, utilize, communicate, and appreciate
mathematics as a tool in everyday situations in order to become life-long learners
and productive citizens by (MDMSIP). The mission will be met by:
? Aligning the curriculum K-12
? Ensuring consistency in teaching
? Providing professional development
? Redesigning preparation of new teachers. (MDMSIP, 2003b, p. 1)
Schools from the twelve school districts of MDMSIP were given the opportunity
to apply for two weeks of professional development during the summer of 2004. The
Cohort Selection Committee of MDMSIP was comprised of university mathematics
education professors, university mathematics professors, and persons affiliated with
MDMSIP who had background in K-12 education. The selection of the schools to
participate in the first summer of professional development was based on the following
three criteria: eighty percent of the mathematics teachers were going to attend for the
two weeks, administrative support was evident, and a school-based strategic plan had
been developed for implementing instructional change.
From this selection process, a total of 24 schools, including nine schools that
included grades 9-12, were selected to join the first cohort (Cohort I) of professional
development, meaning they would be included in the baseline data collection for the
project. All other schools in the MDMSIP partnership were asked to participate in
58
baseline data collection for the first year of this project, and one additional secondary
school volunteered to be included.
Once the selection process had been completed and schools were notified of their
acceptance into Cohort I, principals were asked to give preferred dates for data collection
at their schools. Initial data collection at the secondary schools included a grade-
appropriate measurement of teacher knowledge -- the Algebra Content Knowledge
Instrument (ALCKIN) for grades 9-12 -- and the Teacher Attitude Survey (TAS), which
will be discussed later in this chapter. These instruments were administered at all the
cohort schools and the volunteer non-cohort schools. They were also subsequently
administered the following years to measure progress. Classroom observations of
selected teachers were conducted throughout the year and following years using the
Research Teaching Observation Protocol, an instrument developed by SOURCE
(YEAR). Finally, an instrument designed to measure the attitudes of students towards
mathematics and mathematics teaching and learning was administered at selected grades.
Professional development provided by MDMSIP focused on engaging each of the
teachers in the project in 160 contact hours over the span of the five-year project.
Activities provided through the professional development included a two-week summer
institute, with a one-week follow-up summer institute the following summer. Professional
development provided by the MDSMIP during the two week of summer training involved
several key components. The content at each level focused on the ?big picture? of the
content at that level, instructional strategies to help all students be successful, assessment
that aligned with instructional practices, methods for developing support within the
59
community and with parents, and other issues that arise from the effort to implement
institutional changes related to reform mathematics education (MDMSIP, 2003b).
Teachers from Cohort schools were also expected to attend quarterly follow-up
meetings which met one Saturday morning before the beginning of each academic
quarter. These meetings were designed to help teachers implement the ideas from the
summer institute while focusing on the content areas to be covered in the next nine weeks
of student instruction.
Teachers could also select from a variety of optional activities. Planned activities
included the following:
? A course designed to increase the content knowledge of teachers
? An online support course designed to help teachers who are beginning
implementation
? Specially designed course for new teachers
? Mini courses to address special interest or ideas that arise
? A mini course to address parental and community involvement
? A mini course on problem solving and critical thinking for K-grade 8 (MDMSIP,
2003b)
Throughout all activities, focus was placed on a curriculum guide adopted by the
MDMSIP partnership, and attention was consistently given to the research base for the
project (MDMSIP, 2003b). Broader curricular and pedagogical issues were also focuses
of the project (MDMSIP, 2003b). Emphasis was placed on increasing the participants?
content knowledge beyond what they are required to teach (MDMSIP, 2003b).
60
School-based Teacher Leaders (STLs) were selected at each school within the
partnership. STLs were given the responsibilities of meeting with teachers, planning and
conducting school-based planning and inquiry groups, and working with individual
teachers to improve their skills. In addition, School-based Teacher Leaders across the
partnership were encouraged to attend the Cohort I training.
Thus, the MDMSIP project provided an available source of high school teachers
who would be involved in professional development that provided attention to increasing
content knowledge and pedagogical content knowledge. For these reasons, MDMSIP was
selected as the setting for this study of teachers? content knowledge and pedagogical
content knowledge before and after professional development. The project also provided
an opportunity to study the teachers? content knowledge and pedagogical content
knowledge through classroom observations in order to better understand how these same
types of knowledge are reflected in classroom instruction.
The following sections address the methodologies and methods used in the two
investigations included in this study.
Survey Research
Survey research provides the opportunity to study a large population of teachers in order
to develop a broad picture of the phenomenon being studied. In survey research, a sample
population is studied and inferences can then be made to a larger population (Ary, Jacobs,
& Razavieh, 2002). Survey research can be used to measure intangibles such as attitudes,
opinions, or other values and can more easily be used with a larger sample (Ary et al.
2002). Surveys and questionnaires can be reduced to quantifiable data if they are used
61
alone within the research process. Survey research can entail two different types of
questions which require differing types of analysis (Ary, Jacobs, & Razavieh, 2002).
Close-ended questions are more difficult to construct but can be more easily coded and
scanned for analysis; moreover, they lead to easily-quantifiable results (Ary, Jacobs, &
Razavieh, 2002). In contrast, open-ended questions are more easily constructed but are
more difficult to analyze (Ary, Jacobs & Razavieh, 2002). In open-ended types of
questions, participants have the freedom of responding in any manner they choose, which
requires more time to analyze (Ary, Jacobs & Razavieh, 2002) since the responses cannot
be easily quantified. A mixture of open- and closed-ended questions was used used in this
investigation to provide both quantifiable results and deeper analysis of the subjects.
Subject Selection
Teachers from the nine secondary schools who were accepted to be included in
the summer professional development training as part of Cohort I of MDMSIP, along
with the additional school that volunteered to be a part of the baseline data collection of
MDMSIP, were used as the subjects for the survey research portion of this study.
Participants in the survey research at these participating schools included all teachers who
taught mathematics. Each teacher at the participating schools gave consent to be included
in the data collection of MDMSIP, and this study was embedded within that larger data
collection effort.
Instrumentation
Two instruments were used in developing an overall picture of the content
knowledge and pedagogical content knowledge of subjects, the Algebra Content
62
Knowledge Instrument (ALCKIN) and the Teacher Attitude Survey (TAS). Further
description of both of these surveys will be addressed in the following sections.
Algebra Content Knowledge Instrument (ALCKIN)
At the K-8 level, instruments to measure the mathematical knowledge of
participants in MSMSIP were available from other projects. However, the MDMSIP was
unable to identify an instrument for measuring the content knowledge and pedagogical
content knowledge of secondary school teachers. Thus, I volunteered to develop an
instrument that would be used by the project and for this study, and the Algebra Content
Knowledge Instrument (ALCKIN) resulted. The following sections describe the content
knowledge and pedagogical content knowledge that was included on the ALCKIN,
development of the test items, field testing of the ALCKIN.
Content knowledge and pedagogical content knowledge. In order to develop the
test items for this instrument, consideration was given to the types of knowledge that
teachers should possess in order to teach algebra to high school students along with the
major algebraic topics teachers are expected to teach their students. I used five key
sources to make decisions about what algebra content knowledge should be included. The
CBMS report (2000) gives recommended topic areas for the preparation of teachers. The
RAND report (2003) and the PSSM (2000) both contain big ideas in relationship to the
types of algebraic knowledge in which students should be proficient. The last two
documents, the Alabama Course of Study (2003) and the Curriculum Guide from the
MDMSIP project (2003a), provide more specific objectives to be covered in the algebra
courses at the secondary level.
63
The recommendations from the CBMS report (2000), the RAND report (2003),
and the PSSM (2000) have already been discussed in detail in the literature review. Based
on the big ideas from these documents, the following content areas of algebra were given
consideration: families of functions, using algebraic structures in relationship to
expressions, equations, and inequalities, analyzing change in various contexts, using
algebraic reasoning in relationship to other mathematical fields, and properties of number
systems. Attention to conceptual and procedural knowledge (SOURCE) was included, as
were the processes of problem solving, reasoning and proof, making connections,
communicating mathematically, and representing mathematics in a number of ways
(NCTM, 2000).
Teachers? pedagogical content knowledge includes the ability of the teacher to
use his/her content knowledge in different representations as well as different methods
for solving mathematics that may arise in classroom instruction. Therefore, the
instrument included items that asked the participants to assess students? work, including
the use of technology, communication, reasoning and proof, real-world applications, and
relating algebra to other areas of mathematics.
Test format. The process for developing the format for the instrument began by
considering items from other assessments. The items in content instruments developed by
the Learning Mathematics for Teaching (LMT) project (Hill, Schilling, & Ball, 2003) had
formats that would provide the kinds of information desired. These instruments were
multiple-choice in format, and were written so that researchers could get an overall
picture of how teachers could solve mathematics problems that arise in the classroom
(Hill, Schilling, & Ball, 2003). However, I felt that additional understanding of how their
64
mathematical knowledge would be gained by asking them to provide explanations. Thus,
the quantitative summaries possible with close-ended items were merged with the deeper
insights provided by open-ended responses.
Item development. A pool of thirty-five algebra items covering the identified
content areas was developed for consideration to be used on the ALCKIN. Some of the
mathematics problems were drawn from National Assessment of Educational Practices
[NAEP] (NCES, 2003b), RAND (2003), PSSM (NCTM, 2000), Stump (1997), Llinares
(2000), and the LMT project (Hill, Schilling, & Ball, 2003). Additional items were
developed to address areas of algebra content that were not covered by these items.
Field testing. The initial pool of items was field tested in ensure the items were
not confusing and to assure the instrument could be completed in a thirty-minute
timeframe. The ALCKIN was field tested with two undergraduate secondary
mathematics education students, one person with an undergraduate degree in secondary
mathematics education, and four mathematics education doctoral students. All of the
mathematics education doctoral students had previously taught mathematics at the
secondary level. Each participant was asked to complete all items and to record the
amount of time required to complete each test item. They were also asked to note
anything they found confusing in the way the test items were written or in the
accompanying diagrams. Successive revisions were made until the instrument was of an
appropriate length, with well-designed tasks addressing the identified areas.
The final ALCKIN, designed for teachers to complete in a thirty-minute session,
contained 10 questions. Table 1 describes the format and content of the items, and
Appendix A contains the actual instrument.
65
Table 1
Item Format of the Algebra Content Knowledge Instrument (ALCKIN)
Question Algebraic Content Item Format Processes
1 Functions Agree, disagree or
I?m not sure with
teacher explanation
Representations including
multiple methods
Communication
Reasoning and Proof
Connections
2 Simplifying
algebraic
expressions
Multiple-choice with
teacher explanation
Representations including
multiple methods
Communication
Reasoning and Proof
Problem Solving
3
General algebraic
knowledge
Always true, not
always true, or I?m
not sure
Reasoning and Proof
Connections
4 Understanding of
number systems
Agree, disagree, or
I?m not sure
Connections
Reasoning and Proof
5 Solving an
equation
Short Answer with
teacher explanation
Representations including
multiple methods and the use
of technology
Communication
Reasoning and Proof
6 Simplifying
algebraic
expressions
Teacher explanation Problem Solving
Representations
Communications
7 Using algebraic
reasoning in
relationship to
geometry
Multiple choice with
teacher explanation
Real-world applications
Representations including
multiple methods
Reasoning and Proof
Connections
Communications
8 Rate of change Multiple choice with
teacher explanation
Real-world applications
Reasoning and Proof
Communication
Connections
Representations
66
Question Algebraic Content Item Format Processes
9 Families of
Functions
Multiple choice with
teacher explanation
Reasoning and Proof
Communication
Representations
Connections
10 Algebraic
representations
within an
equation
Teacher explanation Representations
Communication
Connections
Reasoning and Proof
Teacher Attitude Survey
An attitude survey was developed by the Evaluation Planning Team of the
MDMSIP partnership to assess teachers? attitudes and beliefs in several areas, as follow.
The attitude survey was used to access their attitudes toward mathematics and to measure
the teachers? beliefs about teaching and learning. It was also designed to find out about
the teachers? instructional and assessment practices used in their classrooms and their
involvement in professional development. In addition, the instrument was designed to
address their impressions of parental involvement and overall school effectiveness.
Furthermore, its design measured the expectations of change and to measure the amount
of involvement with the MDMSIP project and to gather demographic information.
Sources used by the evaluation committee included the RAND report (Ball, 2003a), the
National Assessment of Educational Practices (NAEP, 2003), and several other federal
projects.
67
A preliminary pool of items was selected. The committee did further revisions of
the items selected and narrowed the list until teachers could complete the teacher attitude
survey in twenty to thirty minutes. A final selection was made and the attitude survey was
finalized. Data from the TAS provided information about how teachers viewed their
classroom practices and how they felt students best learn mathematics in their
classrooms. See Appendix B for the complete Teacher Attitude Survey (TAS). In
particular, in this investigation I will focus on items pertaining to classroom practice and
how each teacher felt students best learn mathematics in their classroom.
Procedure for Survey Study
As a member of the Evaluation Planning Team of MDMSIP, I chose to take the
leadership role in collecting the baseline data for grades 9?12 for several reasons. First,
since part of the data that was to be collected would be included in this study, it ensured
that all of the data was collected in a systematic and consistent manner. Second, since I
would be collecting all the data from the teachers, I would have first-hand knowledge of
what happened and could document any similarities or differences in the methods used as
the data collection process proceeded. The collection of data from mathematics teachers
of students in grades 9?12 served as part of the overall baseline data for the MDMSIP
project as well as the basis for this study.
Administration of Surveys
Ten secondary schools were included in the first year?s baseline data collection
for the MDMSIP partnership and furthermore served the basis for this study. Within these
schools, all secondary teachers who taught a mathematics class were to complete the
attitude survey and content instrument part of the data collection process. Eight of the
68
schools included grades 9-12, and the other two schools contained grades kindergarten
through the twelfth grade. The MDMSIP Evaluation Planning Team determined that all
the secondary teachers in K-12 schools who taught mathematics classes from the seventh
to twelfth grade were to complete the surveys. In addition, student surveys were
administered at the same schools. MDMSIP wanted the data collection process to be as
unobtrusive as possible in the participating schools. For this reason, principals at each of
the participating schools were allowed to make the decision on how best to go about
administering the student and teacher surveys.
The TAS and ALCKIN were administered at each of the ten schools. Initial
contact was made at each school via a telephone conversation with the principal. During
the conversation, a meeting was set with the principal along with a request for the number
of students to be surveyed and how the permission letters should be bundled for ease in
distribution. At each of the initial meetings permission letters were delivered and return
dates were set for the administration of the student surveys and teacher TAS and
ALCKIN. A second visit was made to each school at least two days prior to the student
survey administration to deliver the surveys. This was done so that the person or persons
administering the surveys would have an opportunity to see them before administering
them and to relate any questions they had to the MDMSIP office. Each principal made
his/her own decision as to the best way to do the data collection at their respective
schools. Different methods were employed at each school with both the student and
teacher surveys, but my discussion will focus primarily on the teacher surveys.
Surveys from additional participants were collected during the two weeks of
professional development during the summer of 2004. This collection of additional data
69
included 12 cohort I teachers who were absent during the original data collection or who
had recently been hired for the 2004-05 school year. Two additional teachers were School
Teacher Leaders (STLs) at non-Cohort I schools; all STLs were invited to attend the
Summer Institute.
Of the seventy teachers involved in the data collection, sixty-seven teachers
completed the TAS, and sixty-five completed the ALCKIN. Teachers were asked to
provide demographic information on the TAS. The participants of the 67 TAS contained
20 males and 47 females. The 67 participants were also categorized into two racial
groups: 20 were ?Black? and 47 were ?White?. Table 2 contains information regarding the
number of years of teaching experience of the participants; note that a mix of levels of
experience is evident.
Table 2
Years of Teaching Experience
No. of Years No. of Participants
0 ? 5 22
6 ? 10 13
11 ? 15 18
16 ? 20 2
Over 20 12
Data Analysis
70
Multiple choice responses were tabulated from the ALCKIN and from the TAS.
Written explanations from the ALCKIN surveys were entered in Atlas.ti (Muhr, 1991) for
coding. Analysis and conclusions drawn from these instruments are addressed in full
detail in chapter four. Results from the analysis of these documents provide a general
understanding of the content knowledge and pedagogical content knowledge that high
school algebra teachers possess.
Multi-Individual Case Studies
Case studies offer an opportunity for detailed, in-depth data collection over a
period of time, drawing on multiple sources of information (Cresswell, 1998). Having
multiple cases provides the opportunity to compare and contrast the cases in order to
show generalizability or diversity between the cases (Bogdan & Biklen, 1998).
Case studies were incorporated to develop a deeper understanding of the content
knowledge and pedagogical content knowledge of the teachers involved in the cases.
Beyond the information gained from written instruments, additional sources of data
collection allowed for a deeper probe into these same types of knowledge, as well as
observing how these same types of knowledge related to the teachers? instructional
practices. Furthermore, the cases incorporated attention to how these types of knowledge
changed as a result of professional development that included focus on increasing teacher
knowledge. In addition, the cases provided insight into how changes in both types of
knowledge are reflected within instructional practices. Data was analyzed and used to
develop a descriptive model that encompassed all factors related to the cases (Bogdan &
Biklen, 1998).
71
Subject Selection
Teachers selected for the multi individual case studies were chosen from high
school mathematics teachers involved in MDMSIP in east Alabama who were planning
to participate in summer professional development provided by MDMSIP. Given their
involvement in the MDMSIP, these teachers were both a convenient data source and were
willing to participate in the data collection for this research.
When selecting subjects for individual case studies, Cresswell (1998) advocates
selecting a range of subjects. Not having any prior knowledge of the teachers involved in
MDMSIP, the high schools in the project joining Cohort I were used as a means of
selecting a varied set of subjects. Although the school was not the unit of analysis for this
study, it was used to begin the process of selecting teachers. The available teachers in the
selection process taught diverse student populations in relationship to student
achievement on the Alabama High School Graduation Examination [AHSGE] (ASDE,
2003). Racial and socioeconomic factors of each school were taken into consideration.
Table 3 contains information about the demographics, the number of students tested, the
passing percentages of those students tested, the percent of ?Black? students tested, and
the percent of students tested qualifying for the lunch program as obtained from the
Alabama State Department of Education website for the AHSGE in 2003.
72
Table 3
School Information
1
Regarding Demographics and Results of the Alabama High School
Graduation Examination
School
1
No.
Students
Tested
% Students Passing
Mathematics Portion of the
AHSGE
% Black
Students
% Student on
Lunch
Program
*Abbott 183 95.08% 32.24% 24.04%
Youngs 37 75.68% 10.81% 24.32%
*Dover 317 72.24% 17.98% 19.24%
Capstone 181 71.27% 25.41% 24.86%
*Sandsfield 151 64.24% 35.10% 41.72%
Newsville 64 62.50% 65.63% 51.56%
Porter 113 61.06% 24.78% 33.63%
Frazier 31 45.16% 87.10% 61.29%
*Clarion 201 41.29% 100% 61.69%
1
Pseudonyms are used
*Source of subjects for case studies.
From the nine high schools available, teachers were selected from a school with
the highest achieving students, Abbott High School, and from a school with the lowest
achieving students, Clarion High School. In addition, two schools falling between the
73
schools with the highest and lowest achieving students were selected, Dover High School
and Sandsfield High School.
The majority of the school systems within the MDMSIP serve rural schools and
have small mathematics departments, thus limiting the possible pool of teachers from
which to select. With the importance of Algebra I for all students, the pool of possible
teachers from each of the mathematics departments was narrowed to include only
teachers who taught Algebra I, Algebra IA, or Algebra IB.
A further narrowing occurred by the type of algebra course taught. In Alabama,
Algebra I is taught in 140 hours as one credit; alternatively, students can complete the
Algebra I course in two courses, Algebra IA and Algebra IB, which counts 280 hours and
two credits. With more time allowed to teach the Algebra I content in the two courses,
mathematics instruction could possibly be different in the single-course scenario. Thus,
preference was given to teachers who teach Algebra I as well as Algebra IA and/or
Algebra IB. Two of the chosen schools had only two teachers that taught one or more
classes of Algebra I, Algebra IA, or Algebra IB. At the remaining two schools, the
principals were asked to help select two Algebra I, Algebra IA, or Algebra IB teachers
who employed different instructional methods in classroom instruction. This process
yielded eight teachers. Initial classroom observations were made of all the eight teachers.
However, the pool subsequently further contracted. At two of the schools, one of
the two teachers observed was not going to be rehired for the upcoming academic year,
meaning that continuing data collection would not be possible. At the third school, one of
the two teachers did not plan to participate in the summer professional development,
meaning that it was less likely that there would be any changes in their knowledge.
74
Finally, at the highest achieving school, one of the teachers was only teaching Algebra I,
not Algebra IA or IB, which meant that variations in mathematics instruction would less
likely to be observed. Thus, there were four teachers in the final pool. These subjects
gave their assent to participate in the study as an extension of the MDMSIP. The
demographics for the four teachers are given in Table 4.
Table 4
Demographics of Case Study Participants
Name
1
School Years of
Experience
Race Gender Degrees Courses
Taught
Mrs.
Cotney
Abbott High
School
28 W F BS ?
Mathematics
Education
MS ? Language
Arts Education
Algebra I
Algebra IA
Mrs.
Willoughby
Sandsfield
High School
5 W F BS ?
Mathematics
Education
Algebra I
Calculus
Informal
geometry
Mrs.
Pitchford
Clarion
High School
4 B F BS & MS ?
Mathematics
Education
Algebra I
Mrs. Colley Dover High
School
6 W F BS & MS ?
Mathematics
Education
Algebra IB
1
Pseudonyms are used
Data Sources and Procedure
Triangulation of data sources was used to reduce the chance of misinterpretation
of the data. All my data sources provided a useful but complementary perspective in
75
understanding the teachers? content knowledge and pedagogical content knowledge. In
addition, these data sources helped me understand how content knowledge and
pedagogical content knowledge are reflected in instructional practice and how these types
of knowledge change as a result of professional development. Finally, these data sources
provided the opportunity to see how changes in content knowledge and pedagogical
content knowledge are reflected in instructional practices.
Data was collected during three timeframes. The pre-professional development
timeframe refers to the spring of 2004 before the participants had participated in
professional development. The professional development timeframe includes the two
week Cohort I summer professional development as well as the follow up quarterly
meetings held in August and September. The post-professional development timeframe
refers to the spring of 2005, after the participants had participated in initial professional
development. Table 5 summarizes the data collection sources for each of these three
timeframes. The following sections provide further details.
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Table 5
Case Study Time Frames and Data Collection Points
Time Frame
Data Collection Points
Pre-Professional Development Algebra Content Knowledge Instrument (ALCKIN)
Teacher Attitude Survey (TAS)
Classroom Observations
-Reformed Teaching Observation Protocol (RTOP)
-Field Notes
Teacher Interviews
Journal Entries
Professional Development Document Analysis
Professional Development Presenter Interviews
Quarterly Meeting Observations
-Field Notes
Journal Entries
Post Professional Development Classroom Observations
-Reformed Teaching Observation Protocol (RTOP)
-Field Notes
Teacher Interviews
Journal Entries
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Pre-Professional Development
This timeframe of data collection refers to the spring of 2004 prior to involvement
of the subjects in the two-week summer professional development. Data sources included
the ALCKIN and TAS, which they had completed as part of the first investigation, as
well as classroom observations and interviews. Each of these data sources will be
addressed in the following section, along with the procedures and/or protocols related to
each of these sources.
Algebra Content Knowledge Instrument and Teacher Attitude Survey. The
teachers were previously involved in the survey research conducted as the first part of
this study. Thus, their responses to the Algebra Content Knowledge Instrument
(ALCKIN) and the Teacher Attitude Survey (TAS) were included. Descriptions of these
instruments are included in the previous section.
Classroom observations. Classroom observations of each participant were made
during the spring of 2004. The classroom observations were conducted to further develop
an understanding of the case study teachers? content knowledge and pedagogical content
knowledge. Observations focused on evidence of content knowledge and pedagogical
content knowledge and how it was used in their mathematics instruction. These
observations were audiotaped and field notes were kept; both of these were transcribed in
order to be included in the data analysis. Finally, an observation instrument was used,
which is described in the next section. These sources of data are described in the
following paragraphs.
Field notes were used to record those items which would not be recorded on audio
tape. For example, when teachers and/or students worked mathematics problems on the
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whiteboard or used the overhead projector, the work was written in the field notes.
Sketches of classroom settings, notations of dates and times, and other non-verbal
communications were also recorded. For example, if a student nodded his head in
agreement to an answer or if a teacher worked the problem without any verbal
explanation, these were recorded in the field notes taken from the observation.
The Reformed Teaching Observation Protocol (RTOP) was used as an
observation instrument to collect data about the case study teachers? classroom practices
relative to reform practice. The Evaluation Planning Team of MDSMIP made the
decision to use the Reformed Teaching Observation Protocol (RTOP) as part of the
project?s teacher observation process. This instrument was developed by the Arizona
Collaborative for Excellence in the Preparation of Teachers (ACEPT) at Arizona State
University (Sawade et al., 2000). The purpose of the instrument is to assess the degree to
which the instructional practice of observed mathematics teachers is ?reformed?. The
RTOP instrument contains twenty-five items which are scored on a scale rating of zero to
four for a possible total of one hundred points. Five areas are included on the RTOP:
lesson design and implementation, propositional knowledge, procedural knowledge,
communicative interactions, and student/teacher relationships (Sawade et al., 2000). The
instrument also contained space for a drawing of the classroom setting as well as blank
pages for comments to be recorded related to aspects of the classroom instruction. The
comments were used to rate the twenty-five items. The higher the score indicated the
higher the degree of reform. Training was provided by the MDSMIP project on use of the
RTOP before classroom observations began. See Appendix C for the complete RTOP
instrument.
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Initial classroom observations were included for each of the eight teachers in the
original pool. An addition observation was made of the final pool of four subjects.
Classroom observations at Abbott High School, Sandsfield High School and Dover High
School lasted ninety minutes, while the classroom observations at Clarion High School
lasted fifty minutes. Artifacts related to classroom observations such as worksheets
and/or review sheets were collected at the time of the observations. RTOP forms related
to each classroom observations were processed and average scores from these forms were
assigned to each participant for each particular phase of data collection.
Teacher interviews. Interviews of the case study participants were used to further
clarify the depth of content knowledge of algebra they possess as well as the self
perception they had of their own knowledge. These interviews were also used to probe
how teachers used their pedagogical content knowledge to assess how students worked
mathematical problems using non-procedural methods including technology, as well as
their attitudes about whether their students could learn algebra well. In deciding what
questions and/or problems to include in the interviews, a preliminary analysis was
conducted of both the Teacher Attitude Survey (TAS) and the Algebra Content
Knowledge Instrument (ALCKIN). Five mathematical tasks and three questions related
to these instruments were used in the teacher interview.
Two tasks on the ALCKIN seemed to present the greatest variation of responses
from the teachers who had completed this instrument in the field testing. Question five
(See Appendix A) asked teachers to use a spreadsheet in solving a quadratic equation.
Since the equation did not have integer answers, teachers were asked to use the
spreadsheet to give approximate solutions to the problem and to explain why they
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selected their particular answers. Question six (See Appendix A) required the teachers to
provide multiple methods for showing and/or explaining how two expressions are equal
without using the collection of like terms or the distributive property. These two
questions were included in the interview to see if teachers could use different methods
and representations in solving problems.
A third question was taken from the original pool of items on the ALCKIN. The
question required teachers to work a word problem using three different methods. While
it elicited interesting responses, it was not included in the final instrument due to the time
required to complete the task. Teachers were asked to solve the task, as shown in Figure
3. After solving the problem, they were asked to provide a second method for solving the
same problem, and, if one was provided, a third method of solving was requested.
Jameel looked out in the parking lot and decided to count cars and motorcycles by
the number of their wheels, excluding spares. He saw that there were 17 vehicles
with a total of 56 wheels. How many motorcycles did he see?
Figure 3: First Mathematics Task in the Teacher Interview
Teachers were also asked how they would approach teaching this type of problem
to their algebra students and how they felt their students would solve the problem.
Finally, two non-procedural student solutions to the problem were presented to the
teachers, and they were asked to comment on how the student had worked the problem
and if they would accept this solution from their own students. They were also asked to
explain why they would or why they would not accept the solution. This question was
used because 65% of the teachers in the survey investigation had either strongly agreed or
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agreed with statement six on the Teacher Attitude Survey (TAS) which stated, ?In the
mathematics class, each student?s solution process should be accepted and valued.? The
entire task is given in Appendix D.
A fourth mathematical task was posed pertained to solving a quadratic equation.
Given the difficulty teachers had in solving the quadratic equation using the spreadsheet
in the ALCKIN, this question was designed to further probe other ways of solving
quadratic equations. Teachers were asked to respond to two different possible student
solutions. The first student graphed the two sides of the equation and found the solution
from the resulting intersecting graphs. The second student contained incomplete work but
did contain a correct answer. Teachers were asked if they would accept the work and
solutions presented in these two cases and to explain why they would or why they would
not accept them. (See Appendix D.)
Finally, two additional questions from the TAS pertaining to classroom practice
and how each teacher felt students best learn mathematics in their classroom were
included. The two statements follow:
2. It is important for students to figure out how to solve mathematics
problems for themselves.
16. Teachers should model and demonstrate mathematical procedures and
then, ideally, time should be allowed for the students to have the
opportunity to practice those procedures.
Teachers in the survey study generally either strongly agreed or agreed with both of these
statements. Since these two statements are in direct contrast to each other, additional
probing was designed to see how teachers felt both of these types of instructional
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practices could coexist within mathematics instruction and formed the basis for two of
the interview questions. (See Appendix D.)
Finally, given the importance of algebra for all students, teachers were asked what
percentage of the students in all of their Algebra I classes they felt could learn algebra
well. Teachers were asked to explain why they felt this way. For a complete interview
protocol, problems and student solutions see Appendix D.
Each of the case study teachers was interviewed during the spring of 2004. Each
interview lasted about thirty minutes. All interviews were conducted during the planning
period of the teacher either before or following a classroom observation that was
scheduled for the same day. Interviews were audio taped and transcribed to be used in the
data analysis. Teacher work pertaining to the mathematical tasks was also collected.
Professional Development
The focus of this time frame was to understand the professional development in
which the case study teachers were involved and to seek understanding of how presenters
viewed the mathematical knowledge of the case study teachers and the changes in the
content knowledge they may have observed during professional development. Data
sources included document analysis of the presenters guides for Algebra I, Geometry, and
Algebra II in order to better understand the training they received, interviews with their
professional development presenters, and observations of their participation in quarterly
meetings. Each of these data sources are described in the following sections.
Document analysis. Presenters guides from the Cohort I summer professional
development were analyzed. General sessions which related to all three groups were
noted. For example, sessions on assessment and cooperative learning were attended by all
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of the secondary teachers. Each presenter guide was then reviewed for the types of
algebra content that was addressed.
Professional development presenter interviews. Presenters of the MDMSIP
summer professional development training were interviewed to better understand the case
study participants. Presenters were asked questions related to how well the participants
were involved in the activities during the professional development, as well as the
attitudes they exhibited regarding the professional development training they received.
Presenters were also asked about the type of algebra content knowledge they felt the case
study participant possessed, and if they observed any changes in the teacher?s content
knowledge. Presenters were also asked their opinions as to whether they felt the
participants would return to the classroom and implement the types of changes promoted
by MDMSIP. For a complete presenter interview protocol and questions see Appendix F.
Interviews with the presenters of the summer professional development were
conducted during the fall of 2005. Presenters who were teaching in the K-12 system were
interviewed either before school, after school, or during their planning period. Two of the
summer professional development presenters were working at East University and a third
was employed at South University. These presenters were interviewed on the campus of
East University when a convenient time and place could be arranged between the
presenter and researcher. All interviews with the presenters were audiotaped and
transcribed. Each of the interviews was about twenty minutes in length.
Quarterly meetings. I attended three quarterly meetings held in September and
October of 2004 and March of 2005. The first meeting was held at a local junior high
school and the last two meetings were held on the campus of East University. Field notes
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were taken at all three meetings, focusing on content knowledge, pedagogical content
knowledge, and hints that these particular teachers were making changes in their
instructional practices. Since audiotaping was not used at quarterly meetings, field notes
served as the primary source of recorded data.
Post-Professional Development
The last time frame considered changes in the content knowledge and pedagogical
content knowledge of the case study teachers as they returned to the classroom following
the professional development. Data sources included classroom observations and
interviews. Details of each of these follow.
Classroom observations. One classroom observation of the four case study
teachers was completed during the spring of 2005. These classroom observations focused
on changes in the content knowledge and pedagogical content knowledge of the case
study teachers as well as how those changes were reflected in instructional practices. The
lengths of the classroom observations and how they were conducted were the same as
those done during pre-professional development. Audio taping, field notes, RTOP
protocols were again used.
Interviews. A follow-up interview with the teachers took place during the spring
of 2005. The interview addressed the following areas. First, teachers were asked the
questions from the pre-professional development interview related to how well they felt
their students for this academic school year could learn algebra during this school year.
Second, participants were asked questions related to the professional development they
received from MDMSIP, or if they did not participate in the summer professional
development of MDMSIP, questions were asked related to coursework taken toward an
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advanced degree. They were asked about what type of algebra content knowledge they
felt they possessed and how they felt this knowledge had changed as a result of
professional development.
Third, participants were asked to solve a quadratic equation from the pre-
professional development interview (see Appendix D). The participants were asked to
solve this problem using a second method and if one was provided, a third method was
requested. This request for different methods could be used to further develop an
understanding of the depth of the teachers? content knowledge and how it might have
changed. For a complete interview protocol, see Appendix E.
Each interview lasted approximately thirty minutes. Interviews were conducted
during the planning period of the teacher and were audiotaped. The audio tapes were
transcribed. Teacher work pertaining to mathematical tasks was also collected.
Journaling
A journal was kept related to all aspects of the study. My own personal thoughts
about the research, the data collection process, the teachers I was observing, as well as
my insights, my anxieties, and my doubts. This written record of feelings would allow me
to see how my subjective impressions were affecting the conclusions I was forming about
the research. It was a way to record my feelings at the time of data collection, those
things that you may feel differently about as time as passed from the actual events. This
notes were transcribed and included in the data analysis.
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Data Analysis
The data analysis was organized around the three phases of data collection
involved in the multi-individual case studies. Analysis of the pre-professional
development focused on assessing content knowledge and pedagogical content
knowledge before professional development. The professional development phase
focused on understanding the professional development received by participants and to
further understand any growth in content knowledge and pedagogical content knowledge
that had occurred. The post-professional development phase was used to consider the
changes in content knowledge and pedagogical content knowledge of the case study
teachers.
Pre-Professional Development
Coding from the ALCKIN documents was used as the initial basis for coding of
the classroom observations and teacher interviews conducted during the pre-professional
development timeframe. Additional codes were added as needed, especially to describe
classroom situations that might not have arisen in written instruments.
Professional Development
In considering the professional development timeframe, presenter interviews and
field notes related to the quarterly meetings were entered into Atlas.ti (Muhr, 1991) and
coded. The code list developed in the pre-professional phase was used, and additional
codes were added as warranted from classroom observations and teacher interviews.
Post-Professional Development
During the post-professional development timeframe, classroom observations and
teacher interviews were again entered into Atlas.ti (Muhr, 1991) for analysis. The code
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lists used to analyze these documents began with the same as the codes used in the spring
of 2004 for classroom observations. Additional codes related to changes in content
knowledge as well as pedagogical content knowledge were added as needed. This
analysis focused on changes in the teacher?s content knowledge and pedagogical content
knowledge.
Data from all sources was used to develop a picture of each individual teacher
including their content knowledge and pedagogical content knowledge and how the two
types of knowledge relate to each other during classroom instruction. Furthermore, the
pictures of the teachers included the changes they made in both of these types of
knowledge as a result of participation in professional development as well as changes that
occurred in mathematics instruction within their classrooms. Triangulation of data
sources was used to reduce the chance of misinterpretation of the data. This triangulation
of data was done as a process of clarifying meanings through identifying the different
ways in which the phenomenon can be viewed along with increasing the repeatability of
an observation or interpretation (Denzin & Lincoln, 2000). Cases will be considered
individually and will be considered collective addressing similarities and differences
between them.
Conclusion
Interpretation of analysis related to the survey research and synthesis across the
cases provided the basis for answering the research questions related to this study. The
content knowledge and pedagogical content knowledge of teachers will be analyzed by
looking at the combination of information gained from both of the studies. Cases will be
viewed looking for depth of both of these types of knowledge. Cases will also provide an
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insight into the how content knowledge and pedagogical content knowledge are reflected
in instructional practice. Changes in both of these types of knowledge will be found by
looking at the cases. Finally, case will provide the answer to how any changes in both of
these types of knowledge are reflected in instructional practice.
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IV. RESULTS OF THE SURVEY RESEARCH
This chapter reports results from the first part of the study, related to the analysis
of the Algebra Content Knowledge Instrument [ALCKIN] and the Teacher Attitude
Survey (TAS) for 54 of subjects. Each question of the ALCKIN was analyzed separately
to address the correctness of the participants? responses, as well as the types of
explanations they used in justifying their answer choices. Results from the individual
qustions were then compiled to draw overall conclusions relating to the teachers? content
knowledge and pedagogical content knowledge. Analysis of the TAS will involve only
the five questions related to this study. The following sections begin with an overview of
the process of coding, then present the results from item-level analyses, and conclude
with general results drawn across these analyses.
Coding
The analysis of data began with coding of the written explanations provided on
the ALCKIN. Written explanations were entered into Atlas.ti (Muhr, 1991), a program
used to analyze qualitative data and codes iteratively developed to capture new responses.
Many codes were item-specific. For example, the use of the term ?slope? pertained
primarily to question eight, and ?vertical line test? was only used when referring to
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question one. Other codes applied to more than one of the questions. For example,
teachers may have stated that drawings could be used, or they may have actually used a
drawing in clarifying their explanation, resulting in the code of Drawing.
The recoding of documents was an ongoing process as new codes were entered
into the code list from subsequent documents. After all the documents had been coded,
the documents were all revisited to ensure that any additional codes were used when
appropriate. Table 6 contains a list of the codes as well as the frequencies for each code;
detailed definitions are provided in Appendix G.
Table 6
Codes for the Algebra Content Knowledge Instrument
Code Frequency
______________________________________________________
Counterexample 39
Definition 76
Denotes Asymptote 3
Denotes Student Error 5
Discontinuous Function 1
Drawing/Modeling 12
Explanation of Expression 15
Understanding of Expression 35
Formula Manipulation 6
(table continues)
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Table 6 (continued)
Code Frequency
______________________________________________________
Function 45
Graph 11
I Don?t Know 23
Incorrect Reasoning/Answer 74
Uses Manipulatives 13
Multiple Reasons 85
No Explanation Given 176
Procedural Knowledge 28
Provides Example 4
Slope Reference 33
Spreadsheet Analysis 30
Subject Matter Taught 8
Synthetic Division 4
Table 5
Uses Division 18
Uses Factoring 33
Uses Quadratic Formula 11
Value Substitution 22
Vertical Line Test 48
The following sections present the analysis of responses for each task. For each, a
summary of the participants? responses will be provided, followed by an analysis of
correct, incorrect, and ambiguous responses.
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Question One - Functions
In developing the Algebra Content Knowledge Instrument (ALCKIN), functions
were one of the content areas to be included and were the basis for the first question. This
task presented three scenarios in which a fictitious student stated that a graph or equation
was not a function. The student then provided a true statement about the problem which
had nothing to do with whether the graph or equation was a function. The participants
were then teacher was asked to analyze the student?s thinking. Regardless of the answer
selected, participants were requested to provide an explanation about why they chose
their particular answer. Figure 4 contains question one from the ALCKIN.
Analysis of the Results from Question One
Each part of question one represents a function since each element in the domain
is assigned to exactly one element in the range. Therefore, participants should have
selected ?I disagree? for each part of the question. In explaining their responses, it was
hoped that participants would address statements made by the students, identifying
misunderstandings students may have about what defines a function.
1. Mr. Farrow asked his algebra students to determine which of the following
expressions/graphs are functions. For each of the following decide if you
agree, disagree, or you?re not sure about the student?s assessment.
A.
x
y
Noah said A was not a function because
every x value corresponds to the same y
value.
I agree I disagree I?m not sure
1 2 3
Explain your answer.
B.
x
y
Bart said B was not a function because
it had a ?strange shape? and you could
not find an equation for it.
I agree I disagree I?m not sure
1 2 3
Explain your answer.
C.
y =
1
x
Sanchez said C was not a function
because it was undefined when x = 0.
I agree I disagree I?m not sure
1 2 3
Explain your answer.
Figure 4. Question One from the Algebra Content Knowledge Instrument.
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Table 7 shows how the sixty-five participants selected their answers and the
number of participants from each particular group who did not include an explanation for
each of the three parts.
Table 7
Responses to Question One on Functions
Answer Part A Part B Part C
Number
of
responses
Number
with no
explanation
Number of
responses
Number
with no
explanation
Number
of
responses
Number
with no
explanation
Agree 8 7 6 4 21 13
Disagree *52 5 *56 11 *34 7
Not sure 5 2 3 3 7 3
No
answer
0 0 0 0 3 2
*Correct response.
Both parts A and part B contained graphs, and the explanations given by the
participants were quite similar. Thus, the analyses of these two parts were combined. The
majority of their explanations fell within three distinct categories: definition of a
function, vertical line test, and restatement of the fact that it is a function. See a summary
of responses in table 8. Note that some of the participants used two or more explanations
to support their responses. Thus, one participant?s explanation could have been counted
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more than once in the table. For example, a participant who explained by stating the
graph was a function and the graph passed the vertical line test would be counted in the
category stated ?It is a Function? as well as the category Vertical Line Test. Each
explanation will be elaborated in the coming paragraphs.
Table 8
Explanations for Question 1A and Question 1B
Reason Number Part A Number Part B
Definition of a Function 21 16
Vertical Line Test 13 20
Stated ?It is a function.? 13 10
We will first consider acceptable responses to these parts. The teachers who used
the definition of a function for their explanation either wrote ?definition of a function? or
made a statement referring to the definition of a function. The following are examples of
these types of statements. One teacher wrote, ?A function assigns each x value with
exactly 1 y (i.e. not 2). It?s ok for 2 x?s to have the same y.? A second teacher went even
further with her definition, ?Every x value can correspond to the same y but every y value
cannot correspond to the same x value.?
The vertical line test was given as an explanation by several of the teachers.
However, one participant provided an explanation for the vertical line test in writing, ?B
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is a function because there is no place where a vertical line may be drawn to intersect at
more than one point?, although he did not clarify with what it was intersecting.
The following show how some of the participants used a combination of more
than one of the aforementioned reasons in their explanations. One teacher wrote, ?It is a
function. It passes the vertical line test.? Another teacher explained, ?All horizontal lines
are functions because they pass the vertical line test. Each x-value is only being used
once.? One participant related to domain and range as well as the x and y values and
wrote, ?It is a function because each y is paired with a unique x value; it passes the
vertical line test. For each element of the domain there is only one corresponding range
value.?
Several other explanations were provided by one or more teachers. One teacher
stated for part A, ?Line should be vertical to not be a function? as a counterexample to
the horizontal line graphs that would be function. From part B, still another stated ?Every
function is not a straight line, some functions are curves.? One participant noted, ?This is
a piece-wise function.? There were explanations which could not be classified. On both
parts A and B participants used what they called ?the pencil test?, ?straight line test?, and
?parallel test? for their explanations. While they may have been referring to the vertical
line test, no other explanation was provided.
In other cases, errors appeared in the thinking of the participants, suggesting that
some of the participants did not have a deep understanding of functions. Examples of
these errors from part A include the statement by a teacher who wrote, ?Fails ?pencil
test?, (Unique 1-1 correspondence) test.? Although one teacher wasn?t sure whether she
agreed with the student or not, she wrote, ?I?m not sure I understand the graph, is this a
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plane or what?? She went on to state, ?If this is a line, yes it is a function.? One teacher
agreed that the graph in part B was not a function but was unsure of the explanation and
wrote, ?Function is a relationship, I think.? Even though two teachers both disagreed with
the student, their explanations contained errors in their thinking about what a function is.
One teacher explained, ?A function is a one to one corresponds (sic). The range and
domain have that one to one correspondt (sic).? Another teacher stated, ?This is a
function. You can find a shape.?
A little over half of the sixty-five participants selected the correct answer for part
C. The explanations for part C could not be grouped into specific categories as in parts A
and B, and more errors were found in the thinking of the participants than in the previous
two parts. The majority of the participants had idiosyncratic ways of expressing why they
choose a particular answer. Along with using the definition of a function or stating ?It is a
function,? some participants also included sketches of the graph and/or a table of values.
A decision could not be made if the sketches were included to support their explanation
or to help the participants in selecting their answers. The following are examples of
explanations that did not contain errors in the thinking of the participant. One participant
wrote, ?It?s not a continuous function but it is a function,? while another participant
stated, ?Since each x-value would correspond to exactly one y-value this is a function. It
is just not a continuous function, x = 0 is the exception.?
As in parts A and B, the explanations written by some participants contained
errors or information that made it difficult to determine whether the participant truly
understood functions. One of the misconceptions involved thinking that a function had to
be continuous and if an equation could represent a function if it has a restriction on x.
Other misconceptions included if y can undefined at some x value and still be a function
and if functions have to have a one to one correspondence. Still others felt that is y was
undefined this represented a value for y. One teacher?s explanation contained the
following information ?A function assigns exactly 2 y to each x; therefore, each x gives
only 1 answer. Since the ?answer? for x = 0 is that y is undefined, & this is the only
answer, then this is a function.? Other teachers who disagreed provided the following.
One teacher wrote ?C is a function when x ? 0.? Another teacher stated ?The function
still has a one to one correspondce (sic)? as well as the sketch in Figure 5.
Figure 5. Teacher?s incorrect sketch for the graph y =
x
1
.
Some teachers agreed that the equation did not represent a function, but provided
explanations based on incorrect reasoning. For example, one teacher stated, ?A function
cannot be graphed with an ?undefined?.? Two participants felt the student was correct in
his thinking, because ?It is also not a function because the variable has a negative
exponent.? Another participant stated, ?Yes, if the denominator is zero the function is
undefined.? Still another participant wrote ?Sanchez understands that C would not be a
continuous function because he realized that the graph would be undefined when x = 0.?
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Finally, one teacher provided this explanation, ?Functions cannot have an unknown
without restrictions. You cannot divide by zero.?
Others participants were not sure whether to agree or disagree with the student but
included the following types of explanations. One teacher questioned her own thinking as
she wrote, ?It is undefined when x = 0, that I agree with. I believe that the graph would
pass the vertical line test. I?m not sure,? while a second participant suggested, ?I think
this is a function but maybe it needs a limit or has an asymptote.? One participant stated,
?Function is unclear to me.?
Still other participants elected to support their explanations in different ways. For
example, one teacher chose to point out that this topic is not introduced in Algebra I but
did give suggestions for an approach in helping students understand what happens with
this particular equation. He wrote ?In Alg I they have not been introduced to vert.
isomtopes so they may see it as non-function but when get them build table or graph
should make more clear.? Although one participant did not select an answer he wrote in
the explanation space, ?C is a function with domain: {x?x ? 0}.?
Only a few of the teachers responded to the parts of question one by referring to
what the students stated. Those that did responded with correct mathematical thining. For
example, one teacher stated, ?Yes, every x does correspond to the same y value, but that
doesn?t make it not a function. Non-functions pair diff y-values with the same x & it
passes the vertical line test,? and provided an accompanying sketch shown in figure 6.
Figure 6. Example of a graph not passing the vertical line test.
Still another participant explained what part of the student statement in part B he
agreed with as well as what part he was in disagreement with by stating, ?He is right
about not finding an equation, but wrong about it not being a function.? A teacher used
the fact that the graph would have an asymptote within his explanation: ?True, but you
have an asymptote at x = 0. Still a function.? Another teacher explained her answer with
the following statement, ?It is a function, simply a rational function. Yes it is undefined at
x = 0 but this only yields an asymptote. While it is not a continuous function, it is still a
function.?
Conclusion
The results from question one of the ALCKIN suggest that the participants do not
have a deep understanding of functions. When presented with a graphical representation,
the participants were more likely to select the correct answer. Explanations were
generally given as statements such as repeating the definition of a function, stating that a
given situation was a function, or using procedural knowledge such as the vertical line
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test. Some of the participants did use other forms of representations such as drawing
pictures.
The reasoning presented by some participants further suggests that they did not
have a thorough understanding of functions. Errors were observed in whether functions
includes one-to-one correspondence, continuity, can equations with undefined solutions
be a function, and graphing. Some of the participants had difficulty relating that the
statement y =
x
1
was undefined when x = 0 to what this fact might have to do with
whether or not it is a function relationship.
Question Two - Simplifying Algebraic Expressions
Question two addressed simplifying algebraic expressions, in particular the
simplification of the rational expression (x
2
? 4)/(x + 2). See Figure 7 for question two of
the Algebra Content Knowledge Instrument.
101
102
2. Mrs. Jones asked her algebra students to divide x
2
? 4 by x + 2. Seth said, ?I
have an easy method, Mrs. Jones. I just divide the x
2
by x and the 4 by the 2. I
get x ? 2, which is correct.? Mrs. Jones is not surprised by this as she had seen
students do this before. What did she know? (Mark one answer.)
a. She knew that Seth?s method was wrong, even though he happened to get
the right answer for this problem.
b. She knew that Seth?s answer was actually wrong.
c. She knew that Seth?s method was right, but that for many algebraic
fraction division problems this would produce a messy answer.
d. She knew that Seth?s method only works for some algebraic fractions.
e. I?m not sure.
Explain your answer.
Figure 7: Question two from the Algebra Content Knowledge Instrument.
Analysis of Results of Question Two
Table 9 shows a summary of the answers given. There are two responses that
could be considered correct to this particular question. If the method was analyzed for
correctness in all situations, only answer choice ?A? would be considered correct.
However, the wording of answer choice ?D? indicates that there are certain situations in
which Seth?s method would work. This particular method works only in a very few
special cases. Examples can be provided for which the method would work for Seth, but
there are many counterexamples for which the method would not work.
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Table 9
Question Two Responses to Simplifying Algebraic Expressions
Answer Number of Responses
A. Method was wrong, although answer was correct *45
B. Answer was wrong 0
C. Method was correct, but other problems would produce 2
messy answers
D. Method only works for some algebraic expressions **14
E. I?m not sure 3
No answer selected 1
Selected two answers 1
*Correct Response
** Works in limited cases
For those participants who selected answer choice A, the majority of the
explanations fell into three major procedural categories. These explanations related to
factorization of the numerator and cancellation of common factors (n = 19), long
polynomial division (n = 11), and synthetic division (n = 4). The teachers emphasized
these procedural methods for showing how the student should approach simplifying the
expression instead of addressing his invented method. This was true for the majority of
participants whether they recognized the method was correct or incorrect. Most
participants emphasized that the student should remember the rules or procedures for
simplifying these types of algebraic expressions. This was further demonstrated by
participants simplifying the expression using one or more of the procedural methods
and/or writing an explanation for why they should use these procedures. None of the
teachers used any type of conceptual explanation for why the method proposed in the task
is not valid when simplifying any type of algebraic or numerical fraction. The following
are examples of the types of explanations given. One teacher stated:
That is the right answer (x ? 2), but division by common terms is incorrect. It
needs to be factored and then divided. An x
2
can only be divided by an x if it is by
itself or being multiplied (not added or subtracted).
Another teacher, closest to addressing the issue involved in the problem, stated,
?Seth needs to understand that x + 2 represents one factor and x and 2 should not be
considered separately.? A third teacher responded, ?The correct method is to use long
division, synthetic division, or factor the numerator [(x
2
? 4)/(x + 2)] and cancel out
common factors.? Sample work from a participant can be found in Figure 8.
Figure 8. Participant?s simplification of the algebraic expression by factorization and
division.
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She followed her work with ?Not all binomials are the difference of two squares. Neither
are all quadratics. Factoring or synthetic division would be correct methods to solve the
problem.? Yet another participant wrote:
In order to divide a polynomial by another polynomial, you must factor the
dividend first. (If it is not factorable, you can use long division or possibly
synthetic division.) Only if a factor of the dividend is identical to the divisor can
you cancel those factors out.
The participants (n = 14) who selected answer choice D thought Seth?s method
would work but in only certain situations. Those that selected this answer generally
provided examples that would work as well as counterexamples. One of these teachers
noted that students can make errors but arrive at correct solutions: ?Sometimes students
get the right answer, but the methods are not true for all questions.? Still other
participants gave examples showing this method would only work for the difference of
two squares. One teacher wrote the problem in division form, then explained, ?If there
had been a middle term, his method would not have worked.? Another teacher provided
examples where Seth?s method would work along with counterexamples where the
method would not work. She wrote ?Works on these types: [(x
2
? 4)/(x + 2)]; [(x
2
? 9)/(x
+ 3)]; [(x
2
? 16)/(x + 4)]; Counterexample: [(x
3
? 9)/(x + 3)] ? x
2
? 3.? Yet another
teacher reiterated this same thinking in her response: ?It may have worked for [(x
2
? 4)/(x
+ 2)] but does not work for [(x
2
? 8)/(x + 2)]. This method only works when the
numerator is a difference of perfect squares & the denominator is a factor of the
numerator.? Note that providing examples work and/or counterexamples does not address
the issue for why the method will or will not work.
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Conclusion
The majority of participants could simplify the algebraic fraction using at least
one procedural method. Furthermore, teachers emphasized the students knowing a correct
procedure for the simplification of the given algebraic expression. Participants who
selected the option that Seth?s method would work in some situations often provided
examples that worked and counterexamples where his method would not work. However,
none of these participants suggested that they would encourage Seth to look for
counterexamples so that he could see that his method was not valid in all situations. None
of the participants offered any type of explanation that would help students conceptually
understand why the method he employed would or would not work.
Question Three - General Algebraic Knowledge
Question three involved three of the four areas that were used to select or develop
items for the ALCKIN: using algebraic structures in relationship to expressions,
equations, and inequalities; families of functions; and properties of number systems. The
purpose of this particular question was to see if teachers could use their mathematical
knowledge in evaluating statements indicated as ?rules of thumb.? In the course of
classroom instruction teachers often offer students ?rules of thumb? to help them
remember particular mathematical ideas or procedures. Sometimes, however, these handy
memory devices are not actually true, or they are not true in all situations. The
participants were not asked to explain any of their answer choices. Figure 9 contains
question three from the Algebra Content Knowledge Instrument
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3. Teachers often offer students ?rules of thumb? to help them remember
particular mathematical ideas or procedures. Sometimes, however, these
handy memory devices are not actually true, or they are not true in all
situations. For each of the following, decide whether it is true all of the time
or not. (Mark TRUE FOR ALL SITUATIONS, NOT ALWAYS TRUE, or
I?M NOT SURE.)
True for
ALL
Situations
________
NOT
Always
True
_______
I?m
Not
Sure
______
A. A binomial made up of two perfect squares
cannot be factored into two binomials unless the
two terms have a subtraction sign between them.
1 2 3
B. An asymptote is a line that a graph approaches
but never crosses.
1 2 3
C. Any number to the zero power is equal to one.
1 2 3
D. When graphing linear inequalities, if the
inequality sign is ?? you shade above
the line.
1 2 3
Figure 9. Question three from the Algebra Content Knowledge Instrument.
Table 10 shows how the teachers responded to each answer choice. In each
scenario, participants should have selected ?NOT Always True?. Counterexamples can be
provided for each of the four statements. A counterexample for statement A would be
-16 + x
2
, since this is a difference of squares but not written with a subtraction sign. For
statement B, consider a function with a horizontal asymptote that can be crossed such as
y =
1
2
?x
x
. For statement C consider that 0
0
? 1, and for statement D consider the graph
for 3 > x + y, where the shading of the graph would be below the line even though a ?>?
inequality is used.
Table 10
Question Three Responses to General Algebraic Knowledge
Answer Number of Participants
A B C D
True for All Situations 41 41 47 16
NOT always true *17 *11 *16 *44
I?m not sure 6 11 1 3
No answer selected 2 2 1 2
*Correct Answer
Conclusion
Participants had a difficult time in recognizing that the ?rules of thumb? are not
valid in all situations. Approximately one fourth of the participants were able to decide
that the first three situations were not true all of the time while about sixty-seven percent
answered that the last situation was not true for all situations. These numbers seem to
suggest that the participants did not realize the exceptions to these ?rules of thumbs.?
Although the participants were not asked to respond to any of the above statements or
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explain their reasoning, some of the participants did provide the types of
counterexamples that were mentioned above.
Question Four - Understanding of Number Systems
One of the four areas used in selecting and developing test items for the ALCKIN
was properties of number systems. To develop a better understanding on how well
participants understand the number systems, question four made specific statements
involving numbers belonging to certain sets. Figure 10 contains question four from the
Algebra Content Knowledge Instrument.
4. Consider the following statements related to different sets of numbers. After reading
each statement decide if you agree, disagree, or if you are not sure.
Statement I agree I disagree I?m not
sure
A. -3 is a rational number 1 2 3
B. 2/3 is a real number
1 2 3
C. .010010001? is a rational number. 1 2 3
D. ?5 is a complex number.
1 2 3
Figure 10. Question four from the Algebra Content Knowledge Instrument.
Participants should have agreed with statements A, B, and D, disagreed with
statement C. Negative three is a rational number since it can be written as a fraction
1
3?
.
For part B, two-thirds is a real number because the set of real numbers is the union of all
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110
rational numbers and irrational numbers. Since two-thirds is a rational number, it is also a
real number. In part C, 0.010010001? is not a rational number because a fraction cannot
be found to represent this decimal number. Finally, for part D, since all real numbers are
complex numbers which can be written in the form a + bi where b is equal to zero, ?5 is a
complex number. Table 11 shows the results of how participants responded to each
statement.
Table 11
Question Four Responses to Understanding of Number Systems
Answer Choice Number of Participants
A B C D
I agree *56 *54 19 *20
I disagree 8 10 *42 41
I?m not sure 1 1 4 4
* Correct Answer
Conclusion
The majority of the participants were able to correctly identify the rational
number and the real number, but only about two-thirds of them knew that the decimal in
part C was not a rational number and little more than thirty percent knew that the square
root of five was a complex number. The results suggest the participants do not
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necessarily have a complete understanding of the number systems and how each system
is built from the other.
Question Five - Solving an Equation
One area of consideration in the ALCKIN was using algebraic structures in
relationship to expressions, equations, and inequalities. The emphasis for the fifth
problem on the ALCKIN was not on testing the participants? ability to solve algebraic
equations but rather to give the participants an opportunity to find the solutions for an
equation in a format which may be different from the usual method to which teachers and
students may be accustomed. The participants were told that a teacher had asked his
students to solve the quadratic equation 3x
2
= 4 ? 2x using a spreadsheet. This particular
problem was selected with the intent purpose that the resulting polynomial not be
factorable and that the solution would involve a radical. A spreadsheet table was
presented in which a range of values for x had been substituted in the expressions 3x
2
and
4?2x. This table shows the approximate solution for the quadratic equation should be
between -1.5 and -1.6. Note that there is a second solution between 0.8 and 0.9, which is
not shown in the table. Figure 11 contains question five from the Algebra Content
Knowledge Instrument.
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5. Mr. Casteel is using spreadsheets in his Algebra class to find solutions for quadratic
equations. What approximate solution(s) for the equation 3x
2
= 4-2x should Mr.
Casteel?s students give using the following spreadsheet?
X 3x
2
4-2x
-1.8 9.72 7.6
-1.7 8.67 7.4
-1.6 7.68 7.2
-1.5 6.75 7
-1.4 5.88 6.8
-1.3 5.07 6.6
-1.2 4.32 6.4
-1.1 3.63 6.2
-1 3 6
-0.9 2.43 5.8
-0.8 1.92 5.6
-0.7 1.47 5.4
-0.6 1.08 5.2
-0.5 0.75 5
-0.4 0.48 4.8
Solution(s):
Explain your answer.
Figure 11. Question five from the Algebra Content Knowledge Instrument.
Analysis of the Results From Question Five
The solution for the quadratic equation includes the values for x which make 3x
2
equal to 4-2x. The participants should have looked to see where the second and third
columns are closest to being equal. Twenty-seven of the sixty-five participants (41%)
gave answers that fell within the correct interval of x values, ?between -1.5 and -1.6?, or
reported in different ways that conveyed this thought.
The following are examples of participants who used the spreadsheet to find the
approximate solutions without relying on another procedural method in their justification.
One teacher stated ?When x = -1.6, 3x
2
is greater than 4 ? 2x. When x = -1.5, 3x
2
is less
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than 4 ? 2x. Somewhere between these two x-values 3x
2
should equal 4 ? 2x.? A second
teacher explained:
Look for a row or rows in which the entries in the 2
nd
and 3
rd
columns are equal.
If there are not any, look for a pair of rows where the larger value ?switches?
from column 2 to column 3. The root will be between the x-values for those rows.
Even though these participants relied on the spreadsheet to find and support their
solutions, the order in which work was recorded on their instrument suggested six of
them may have began the process by trying to use an algebra procedure similar to one of
those discussed in the next section.
While we cannot completely infer what the participants did from the written
work, the order in which they presented their work suggests some of the participants
began by setting the equation equal to zero and factoring. For example, one participant
factored 3x
2
+ 2x ? 4 = 0 into (3x ? 2) and (x + 2) and wrote x = 2/3 and x = -2 as her
solutions and used her work as the explanation for her answer. Another teacher also made
errors in his factorization when he wrote (3x + 4) and (x ? 1) as his factors and he did not
provide any further answer or explanation for what he was doing. One participant
exhibited more than one error in his thinking as he indicated an attempt to factor the
quadratic expression, wrote x = -2/5, defended his answer by stating ?both sides of the
equation match?, and drew an arrow pointing to the row that contains the values x = -0.4,
3x
2
= .48 and 4 ? 2x = 4.8. One teacher began by showing 3x-2 and x+2 as factors for the
quadratic expression. He proceeded to use the quadratic formula but made a
mathematical error and ended up with a two in the denominator instead of three and his
solutions of ?1.25, -2.25? were not in the chart thus prompting him to provide ????? as
his explanation. Another teacher also tried factoring but marked through his work and
wrote ?x = -1.5, x = -1.6? and did not provide any explanation for why he picked those
particular solutions.
The order of the work of one teacher suggests that two procedural processes were
tried before using the spreadsheet provided to solve the problem. She attempted to factor
and had written the following parentheses, ?(3x 4) and (x 1)?. She also began to
substitute into the quadratic formula with ?-2? ? and ended up simply writing ?-1.5 <
x < -1.6?. She indicated that when x = -1.5, 3x
2
< 4 ? 2x and when x = -1.6, 3x
2
> 4 ? 2x.
Even though this participant began by using other methods to find the solution for the
equation, it appeared that she was successful in using the spreadsheet in finding the
approximate solution for the equation. It should be further noted that she made an error in
writing the inequality she used for her solution.
Nine participants tried to use the quadratic formula to find the solutions for the
equation. These solutions were left in varying forms of simplification. Participants using
the quadratic formula instead of the spreadsheet also exhibited errors either in
substituting in the formula or in the simplification of the expression. After her
unsuccessful attempt with the quadratic formula, one teacher wrote as her explanation, ?I
guess I do not understand exactly what this is asking for. *You would plug each value
into & check to see which one is correct.? Another teacher actually found the solutions to
be 0.87 and -1.54 but did not provide any explanation other than the work used when
substituting into the quadratic formula and did not refer back to the table.
Some of the respondents either did not attempt to answer the problem or wrote
such responses as ?I don?t know?, ?not sure?, ???, ?can?t do this? or simply rewrote the
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115
problem. Other participants wrote incorrect solutions for the equation. One teacher, who
gave ?-1, 3, and 6? as her solution wrote, ?Because they are all whole numbers, and
decimals throw students for a loop the majority of the time. They have the misconception
that an answer is usually not a decimal.?
Conclusion
The participants had a difficult time finding an approximate solution to the
quadratic equation using the spreadsheet provided. While about one-half of the
participants were able to give a correct approximate solution, only one-third found the
approximate solution without displaying some algebraic procedure such as factorization
or the quadratic formula. Many procedural errors were made by the participants including
incorrect factorizations, incorrect substitution in the quadratic formula, incorrect writing
of an inequality solution, and incorrect simplification of an algebraic expression
involving a radical. Even though some of the participants actually found two solutions for
the equation when using other methods other than looking at the spreadsheet, none of the
sixty-five participants mentioned that there were two solutions for the equation and that
one of these solutions was not contained in the spreadsheet.
Question Six - Simplifying Algebraic Expressions
Question six addresses the content area of using algebraic structures in
relationship to expressions, equations, and inequalities. Figure 12 contains question six
from the Algebra Content Knowledge Instrument.
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6. Without using algebraic manipulation such as collecting like terms or using
the distributive property, what other methods could you as a teacher use to
justify to your students that the expressions 3x + 5 + 5x -3 and 4 (2x + 1/2) are
equivalent to each other?
Figure 12. Question six from the Algebra Content Knowledge Instrument.
Analysis of the Results from Question Six
Responses from the participants fell within three general categories; see table 12.
Ten of the participants used a combination of two of the three categories. Examples of
each of these categories follow. Note that only half were able to provide a satisfactory
response.
Table 12
Question Six Responses
Response Number of Participants
*Substitute in value(s) for x 18
*Use Manipulatives 13
*Drawing/Modeling 12
Didn?t indicate another way 24
No Response 11
*Correct Answer
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The first category includes participants who suggested that a value or values be
substituted in the place of x to show that equal values resulted. For example, one teacher
who responded with, ?Substitute the same value for x,? and a second teacher wrote, ?Try
plugging in values for x. If when simplified the answers are the same, then the
expressions can be equivalent. When x = 2, then both exp = 18.? Another teacher realized
showing the expressions equal did not actually prove them equal, and explaining,
?Substitute values in place of x and simplify using the order of operations, although you
would have to point out that you cannot prove they are equivalent that way because you
cannot check all real numbers.? Along with value substitution, one participant offered an
additional method for showing the two expressions equivalent: ?I would use a graphing
calculator to graph both equations to see if we get the same line, or we could simply pick
a number for x, plug it in and see if we get the same solution.?
Participants also suggested using some type of manipulative to show students that
the two expressions are equivalent. For example, one teacher replied, ?Use an object to
represent x and let them discover the relationship?. Other participants were more specific
in the type of manipulatives they would use to show equivalence such as one participant
who suggested, ?Use physical manipulatives such as algebra tiles?.
Participants also either suggested using drawing or modeling. All but one of these
participants drew a representation to show how the two expressions are equivalent. An
example is given in Figure 13.
Use manipulatives.
x ?
1 ? ?
-1 ? ?
? ?
-1 + 1
(? + ?) = ?
empty set or nothing
3x + 5 + 5x ? 3
? ? ? ? ?
? ? ?
??
? ?
4(2x + ?)
??
? ?
Figure 13. Participant?s pictorial representation to show two algebraic expressions are
equivalent.
However, more than half of the participants (n = 35) were unable to answer this
particular item as requested. Twenty-four of these participants either did not respond,
wrote ?I don?t know?, ?I?m not sure?, or ?Can?t think of anything?, while ten other
participants simplified the two expressions by using the distributive property and the
collection of like terms.
Conclusion
Over half of the participants had difficulty offering any way for teachers to help
students understand that two algebraic expressions were equivalent or simply relied on
procedures that in the instructions they were asked not to use. The majority of the
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119
participants who attempted to offer a way to show that the expressions were equivalent
used a procedure where students would substitute a value or values into the place of x.
Even fewer of the participants offered concrete examples such as various manipulatives
or making a drawing or sketch of the two expressions.
Question Seven - Using Algebraic Reasoning in Relationship to Geometry
Question 7 asked the participants to use algebraic reasoning in other mathematical
fields such as geometry. The focus of question seven related to forming an algebraic
expression to represent a geometric model. Participants were asked to respond to
solutions of two students who had looked at the problem in different ways and had
approached working the fictitious problem differently. Note that the first three answer
choices contained equivalent algebraic expressions. Figure 14 contains question seven
from the Algebra Content Knowledge Instrument.
7. Students who are in Mrs. Simpson?s algebra class were working on the following
problem: You are going to build a square garden and surround its border with square
tiles. Each tile is 1 foot by 1 foot. For example, if the dimensions of the garden are 10
feet by 10 feet, then you will need 44 tiles for the border.
10
10
How many tiles would you need for a garden that is n feet by n feet? Two
students gave the following diagrams, representations, and reasons for their
answers.
James stated you could find the number of tiles needed for the
border by finding the area of the entire garden and border and
subtract out the area of the garden.
Which representation best reflects his method?
A. 4n + 4
B. 4(n + 1)
C. (n + 2)
2
? n
2
D. All are equally correct
Ann said you could take the entire length of each side of the
border which is n and multiply by four, then add in each of the
corners.
Which representation best reflects her method?
A. 4n + 4
B. 4(n + 1)
C. (n + 2)
2
? n
2
D. All are equally correct
Which student has the best understanding of the tile problem?
A. James
B. Ann
C. Both show equal understanding.
Explain your answer:
Figure 14. Question seven from the Algebra Content Knowledge Instrument.
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121
Analysis of the Results from Question Seven
Responses to the first two parts of question seven are summarized in Table 13.
The correct algebraic expression for James? interpretation should have been answer
choice ?C?. Since the border adds two to the length and width of the garden, the area of
the garden and border would be (n + 2)
2
and the area of the garden would be n
2
. Since the
first area also includes the area of the garden, James removed this and is left with the area
of the border. The correct algebraic expression for Ann?s interpretation should have been
answer choice ?A?. Since the border of the garden is n tiles on each side, four n would
represent the total tiles on the sides of the garden. Ann then includes the tiles from the
corners. Note that while all three expressions are equivalent, only one correctly represents
the interpretation given by each student.
Table 13
Question Seven Responses on Using Algebraic Reasoning in Relationship to Geometry
Number of Participants
Answer Choice James Ann
A. 4n + 4 1 *50
B. 4(n + 1) 2 4
C. (n + 2)
2
? n
2
*51 2
D. All are equally correct 7 5
E. No response 4 4
*Correct Answer
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Almost eighty percent of the participants were able to select the correct
expression for James? interpretation, and about seventy-eight percent were able to select
Ann?s algebraic expression that fit her interpretation. Twelve of the participants noted
that all three of the algebraic expressions are equivalent.
The participants were then asked to choose which of the two students had a better
understanding of the problem and to explain why they chose that particular student.
Participants should have stated that both students had an equal understanding of the
problem, since both correctly represented a solution. Table 14 summarizes the responses
and those with no explanation of their choice.
Table 14
Question Seven, Part C Responses
Answer Choices Number of Participants Contained No Explanation
James 12 8
Ann 11 5
Both show equal understanding *37 11
No response 5 5
*Correct Answer
Participants who felt both students understood the problem equally well explained
their answers using some way of stating that there are various ways of approaching and
solving problems in mathematics and were more likely to point out that both students
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arrived at the same solution. For example, one teacher stated: ?Answer A best represents
Ann?s view while choice C is representative of James? view. Since the 2 are equal one
might assume their understanding is equal.? A second teacher explained, ?Either method
produces the same correct result, and each student shows critical thinking skills.? Another
teacher actually showed that the two expressions (n + 2)
2
? n
2
and 4n + 4 were equivalent
to each other, while another stated: ?Although they approached it from different
viewpoints, they both are correct in their view of the problem. Furthermore, they get the
same result when you simplify.?
After selecting James as having the best understanding of the problem, one
teacher wrote, ?This is the easiest way to explain it to a class.? A second teacher
explained:
I feel that James seems to better understand, because he seems to know it is more
logical to get a more correct answer by using what you really know. With Ann?s
method, how do you really know the ?measure? of the corners.
The following are examples from participants who selected Ann as having the
better understanding. One participant wrote ?Ann is finding perimeter. James is finding
area of tiles.? A second participant explained with ?James? answer involves area; Ann?s
involves perimeter, which is what a ?border? is.?
Still other participants focused on the geometric aspects, perimeter and area, as
factors in their explanations. For example, one participant wrote, ?Both ways are
acceptable. Ann is more focused on perimeter whereas James is focused on area, but both
are ok.? Another participant explained:
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James? approach involves area, while Ann?s involves perimeter. ? As long as
only 1x1 tiles are used in one row around garden, each method works, although
Ann?s seems to be simpler. However, if the tiles are in two or more rows, problem
becomes more difficult for Ann.
Other participants focused on the practicality of the two methods. For example, one
teacher felt ?Ann?s method is more practical than James? and another teacher explained
?James? answer is more complicated, but also works.?
Errors can be noted in some of the explanations of the participants who selected
Ann as having the best understanding. For example, one teacher failed to realize that
since the area of the tiles was one that the area of the border would also be equal to the
number of tiles in the border. She stated, ?Even though there eq are the same, Ann?s
method will give you the # of 1x1 tiles, James? way will give you the area of the border
(not the # of tiles).? A second teacher felt James was missing something as she stated,
?James is close to understanding, but is missing the 4 corner pieces.? Another teacher
seemed to think the numbers used in the problem promoted which method was better
when he wrote, ?As long as the dimensions of the garden are both even numbers, then
James? method will do nicely. Otherwise, Ann has the better approach.?
Conclusion
More than three-fourths of the participants were able to correctly identify the
algebraic expressions representing the geometric interpretations of the two students.
However, only about half of the participants felt that both students had an equal
understanding of the problem. The data suggest that these teachers felt there is more than
one method for solving mathematical problems, some even pointing out procedurally that
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the two algebraic expressions representing the two students? choices are equivalent.
Errors in the thinking of the participants could be seen by the indication of some of the
participants who felt that if you talk about the border of a patio you have to relate it to
perimeter and not to area.
Question Eight ? Rate of Change
Question eight addressed the content area of rate of change was written within a
context of steepness of steps. The participants were asked to determine which group or
groups had a correct Figure 15 contains question eight from the Algebra Content
Knowledge Instrument.
8. Mrs. Ledbetter gave her algebra students the following two diagrams of steps and asked them
to determine which set of steps were the steepest. The students could not agree on an answer and
gave the following answers and reasons for their answers.
Steps BSteps A
6
4
6
4
6
4
6
4
3
2
3
2
3
2
3
2
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A. Group 1 said that both sets of steps have
the same steepness. ?First connect the
points at the edges of the steps with a line
and measure the angle formed between
this line and the ground. The measure of
this angle is the same in both set of steps.
Therefore they have the same steepness.?
B. Group 2 said Steps B were steeper
because the steps in Steps B are taller than
in Steps A.
C. Group 3 thought both sets of steps had
the same steepness since doubling both
the length and height of the steps does
not affect the steepness.
D. Group 4 said both sets of steps had the
same steepness. ?If you sketch in a line
that contains the edges of the steps and
determine the slope of the line in each set
of steps, you will find they have equal
slope. Steps A have a slope of 2/3 and
Steps B have a slope of 4/6 which is
equivalent to 2/3. Therefore, they have the
same steepness.?
Which group(s) have a correct understanding of slope? (Circle all that apply).
A. Group 1
B. Group 2
C. Group 3
D. Group 4
Of the four groups which group has the best understanding of slope?
______________________
Explain why you chose this group.
Figure 15. Question eight from the Algebra Content Knowledge Instrument.
126
127
Analysis of the Result from Question Eight
Table 15 summarizes whether the participants felt the students did or did not
understand slope from the explanation given. Six of the participants chose not to respond
to the question.
Table 15
Question Eight Responses on Rate of Change
Answer Choices
Number of Participants
Understand Do Not Understand No Response
A. Group 1 *40 19 6
B. Group 2 5 *54 6
C. Group 3 *38 21 6
D. Group 4 *54 5 6
*Correct Answer
Group 1 in part A described a trigonometric representation for slope. This group
understood that the steps had the same slope because the tangent of the angle formed with
the ground would be the same in both sets of steps. Group B did not understand
steepness relates to both vertical change as well as horizontal change. Steepness is not
based on ?tallness.? Group C understood that doubling the ratio of the change in the
height of the rows as related to the change in the length of the steps, results in equivalent
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ratios. Group D understood slope and steepness and defended their answer in a traditional
view of finding slope of a line.
For Group A, the trigonometric representation 61% of the participants correctly
selected that the participants understood slope. For Group B, 82% were correct in
selecting that the group did not understand slope. For Group C, 58% of the participants
selected that the group understood slope. For Group D, 82% of the participants said the
group understood slope.
After the participants chose the group or groups they felt had a good
understanding of slope, they were asked to select which of the four groups had the best
understanding of slope and to explain why they felt this group had the best
understanding. Table 16 shows how the participants selected which group or in some
cases groups had the best understanding of slope.
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Table 16
Group with the Best Understanding of Slope
Group Number of Participants
A 1
B 1
C 5
D 45
A and D 1
A, C, and D *1
No Response 11
*Correct Answer
The following are examples from the participants explaining why they selected
each group or groups. One teacher selected Group A and explained his answer by stating,
?Group A brought in the idea of angle w/ the ground which is a key thought for slope
although group 4 could do slope group 1 had a better working knowledge.? A second
teacher felt that Group B had a better understanding and wrote, ?The steps are higher.?
Another teacher selected group C and stated, ?It?s the simplest answer,? and yet another
explained, ?Because they understand how the slope will not change if you double both
numbers.?
Of the participants who chose Group D as having the best understanding and
provided an explanation, most fell into one of two distinct categories. Their statements of
explanation either pointed out that the students used terminology related to slope or the
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students actually calculated the slope. The following are examples from some of the
participants who explained their choice with terminology. One teacher wrote, ?Slope is
discussed as rise/run, which they thoroughly explained,? while a second teacher
explained, ?I feel all students understand well, but group D uses the terminology in
context and more fully explains problems.? One participant stated ?Group C understood
the concept of change in slope but Group D understood exactly what a slope and how to
determine how to get the slope to compare the steepness.? A second participant wrote
?Group 1 & 3 showed an understanding, but group 4 actually related the problem back to
slope and used slope in their explanation.?
The other category of participants used the idea that the students actually
calculated the value for the slope and thus had a better understanding of slope than the
other groups. The following are examples of these types of explanations. One teacher
explained, ?They calculated slope instead of measuring angles.? A second teacher goes
even further than just stating the students know how to compute slope when she wrote,
?Group D has a better understanding because they know how to find slope, and they
understand what slope represents. They also can recognize equivalent fractions. Group 1
understands that grade and slope go together, but they did not actually find it.? Another
teacher offered, ?They have shown they understand the concept of the slope graphically
(drawing the lines) and algebraically (calculating the slopes and showing they are
equivalent).?
There were other explanations that did not fall into these two categories. For
example, one participant wrote ?Their reasoning best meats (sic) my expectations of
understanding slope.? A second participant stated, ?The easiest way to measure steepness
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is slope,? while another teacher wrote, ?Understands slope as rise/run. Same slope means
same steepness.?
Two of the participants opted to select more than one group as having the best
understanding of slope. One teacher selected both groups A and D and stated, ?The 2
groups I circled both show understanding of what it means to have steep stairs? as his
explanation. A second teacher selected groups A, C, and D and explained, ?All three have
an equal understanding. Each deals w/ a different concept of slope & its app.?
Conclusion
Participants were just as likely to select the incorrect group (Group B) as they
were to select the group that found the numerical slope (Group D). A little more than half
of them selected the group using the trigonometric representation as understanding. Even
fewer of the participants realized that doubling the rise and run (Group C) was a valid
understanding of slope More participants (n = 54) selected group D as understanding
slope than either group A (n = 40) and group C (n = 38). Furthermore, their
overwhelming selection of group D as having the best understanding suggests that
participants felt students had to find the numerical value of slope and/or use terminology
such as slope or rise/run in order to have an understanding of slope. Only one participant
felt that all three groups had an equal understanding of slope.
Question Nine - Family of Functions
Families of functions were the focus of the selected content areas of the ALCKIN
for question nine. To further understand what the participants know about a family of
functions, this question was represented by a parabola and its translation in the coordinate
plane. Figure 16 contains question nine from the Algebra Content Knowledge Instrument.
9. Mr. Seng?s algebra class is studying the graph of y = ax
2
+ bx + c and how changing
the parameters a, b, and c will cause different translations of the original graph.
Translated graph
y= ax
2
+ bx + c
How do you think Mr. Seng will explain the translation of the original graph
y = ax
2
+ bx + c to the translated graph?
A. Only the a value changed
B. Only the c value changed
C. Only the b value changed
D. At least two of the parameters changed.
E. You cannot generate the translated graph by changing any of the parameters.
Explain your answer choice:
Figure 16. Question nine from the Algebra Content Knowledge Instrument.
132
133
Analysis of Results of Question Nine
After selecting an answer, participants were asked to explain why they selected
their particular answer. Answer choice ?D? which states that at least two parameters
changed is the more general answer for generating any translation of a graph. However,
changing both the b and c parameters would result in graphs which do not share a
common y-intercept. The problem did not directly specify that the two graphs shared the
same y-intercept, so participants would have to infer this from looking at the graph to
select ?C? as the correct response. Thus, answer choice ?D? would be accepted as the
general response, which might reflect some knowledge of how changing both the
parameters translates graphs on the coordinate plane. On the other hand, if participants
inferred from the graphs that they shared the same y-intercept, then answer choice ?C?
might also be considered correct. Table 17 showed how the distribution of responses to
the question.
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Table 17
Question Nine Responses to Families of Functions
Answer Choice Number of Participants
Only the a value changed 6
Only the b value changed **15
Only the c value changed 8
At least two of the parameters changed *17
You cannot generate the translated graph by
changing any of the parameters 6
No response 13
* Correct response ? General Case.
**Correct response ? Special Case.
This question appeared to cause the participants a lot of difficulty in both their
answer selection, as well as in the explanations provided for those answers. Note that 13
participants did not even attempt to answer the problem, and many more did not explain
their answers. No matter which answer choice was selected by the participants, there was
not a lot of commonality in the explanations provided. Some explanations contained
statements about what changes the parameters would cause in the original equation, in
attempting to explain what parameters must have changed. One teacher noted ?c ? ? and
a makes it skinnier & fatter? while another teacher acknowledged ?a = changes the width
and direction of opening, c = shifts the parabola up & down ?? A somewhat more
complete explanation follows: ?The steepness of the two graphs did not change. The
vertex is all that changed & the b-value is what switched the vertex from one side of the
y-axis to the other.? Another participant suggested, ?The axis of symmetry x = -b/2a is
what changed. b was originally positive and changed to a negative. a & c stayed the
same.?
Still others used the analysis of the general form for the equation for a parabola.
One teacher had a question mark and ?maybe? written by her answer choice with the
following statements as her reasons:
{
2
22
2
2
2
2
2
2
2
4
4
)
2
(
4
)
4
(
)(
a
bca
a
b
xay
a
b
c
a
b
x
a
b
xay
cx
a
b
xay
a
?
++=
?+++=
++=
negative shifts it to the right. You would have to change b/2a to shift the graph
left or right, but I can?t decide if you can do that without shifting the graph up or
down simultaneously! I would need more time to play with some numbers.
There were errors in the thinking of some of the participants. For example, two
participants, who stated that the b value changed, stated ??the b value moves the graph
left and right?? Two other teachers, who indicated that only the c value changed, used
the following as their explanation: ?Changing the c value shifts the graph left or right.?
Another teacher chose to simply reiterate the answer he selected and stated, ?You must
change two parameters in order to translate the particular graph.? One participant wrote
?Changing the coefficient of the x
2
term changes the width of the curve. These are not
from the same family of graphs.? Additional errors came from participants who said the
graph could not be translated by changing any of the parameters. For example, a second
participant felt ?? horz by adding or sub from quadratic as a whole not changing
135
136
quadratic by coefficient value? and another participant wrote, ?Translation is a shift of
axis here, not a change in graph?s parameters.?
It was difficult to determine what the participants were thinking in some of the
explanations. For example, one teacher declared ?Axis of symmetry is ?b/2a and a is
positive ? neg/pos = neg and a is negative ? neg/neg = positive.? Another teacher
stated, ?The vertex changed, involving 2 points (h, k)? and one teacher stated ?a? affects
the orientation and steepness of curve along with ?Another format for quadratic: (x ? h)
2
+ k, vertex at (h, k); x
2
? 2hx + h
2
+ k; -2k = b; (h
2
+ k) = c; in this format, h alone has
changed, from + to -.?
Conclusion
The participants had a difficult time with this particular problem, suggesting that
they do not have a full understanding of how families of functions work and the effects
that are caused by changes in parameters. Participants were more likely to list any and all
information they thought they knew about the problem, and very few of the participants
were able to provide an adequate explanation for which parameter or parameters caused
the change in the graph.
Question 10 - Algebraic Representations within an Equation
This particular question on the ALCKIN comes from the content area of using
algebraic structures in relationship to expressions, equations, and inequalities. The item
first gave background information in the format of a word problem related the amount of
time it took to bake chocolate chip cookies and plain cookies, then asked them to
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interpret one part of an equation representing that relationship. Figure 17 contains
question ten from the Algebra Content Knowledge Instrument.
10. Mrs. Westbrook?s algebra students were working on writing algebraic equations for
problems similar to the following:
It takes 0.2 hour to bake a dozen chocolate chip cookies and 0.15 hour to
bake a dozen plain cookies, how many dozen cookies can be baked in
fifteen hours?
Mrs. Westbrook?s students came up with the following equation:
0.2x + 0.15y = 15
where x = number of dozens of chocolate chip cookies and y = number of dozens
of plain cookies.
What does the 0.2x represent in the equation?
Figure 17. Question ten from the Algebra Content Knowledge Instrument.
Analysis of Results of Question Ten
Participants were asked to give an explanation as to what the 0.2x represented in
this particular equation. A correct response should have indicated that 0.2x represented
the time to bake the chocolate chip cookies and/or the part of the fifteen hours need to
bake the chocolate chip cookies. The majority of the responses to this question fell into
the categories listed in Table 18.
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Table 18
Question Ten Responses to Algebraic Representations within an Equation
Response Category Number of Participants
Translated the expression into Words 11
Analyzed the Expression Correctly *31
Total Number of Chocolate Chip Cookies 11
Did Not Answer 3
Other 9
*Answered Correctly
Less than half of the participants were able to analyze what the expression
represented in the equation. The following are examples of analyses given by some of
these participants. The majority were very much like one teacher who wrote, ?The total
amount of time it would take to bake x dozen chocolate chip cookies.? A second teacher
explained with ?The amount of time of the 15 hours used to bake the chocolate chip
cookies depending on the # of dozens of chocolate chip cookies.? Another teacher simply
stated, ?Time used in baking chocolate chip cookies!?
Errors were also found in the explanations of this expression. For example, one
participant wrote ?# of hours needed to bake a dozen choc. chip cookies,? and a second
participant stated ?The time it takes to make x number of chocolate chip cookies.?
Another participant used ?Rate = time in hours per one dozen choc. chip cookies.? One
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teacher explained ?The number of hours that it takes to bake cookies? not specifying
chocolate chip cookies.
Examples of how the participants tried translating the expression into words are as
follows. One wrote ?.2 hours times the number of chocolate chip cookies.? Another
explained with ?The number of minutes x the number of dozens of choc chip cookies.?
Other examples were written similar to one teacher who stated ?0.2 hours per 1 dozen
chocolate chip cookies? and still other explanations were similar to a second teacher who
wrote ?0.2x represents the fact that it takes 2/10 hr. to bake one dozen chocolate ch.?
Eleven of the participants elected to explain the representation by saying it either
represented the number of chocolate chip cookies or number of dozens of chocolate chip
cookies that were baked. An example of these types of explanations came from one
teacher who wrote ?The total number of dozens of chocolate chip cookies that can be
baked in 15 hours.?
Still other participants used explanations that were difficult to categorize. For
example, one teacher wrote ?2/10 of the total number of cookies? and a second teacher
stated ?.2 hour and x is how many dozen chocolate chip cookies.? Another teacher used
?The percentage of chocolate chip cookies in 15 hours.?
Conclusion
Less than half of the participants gave a plausible explanation for the algebraic
expression. This suggests that participants had a difficult time in determining a correct
explanation for an algebraic expression that is part of an algebraic equation. It appears
that teachers do not have a full understanding of how the expressions contribute to the
overall equation used to solve problems.
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Teacher Attitude Survey
There were five questions on the TAS that directly pertained to classroom practice
and how each teacher felt students best learn mathematics in their classroom. The
following five statements were considered:
2. It is important for students to figure out how to solve mathematics
problems for themselves.
6. In a mathematics class, each student?s solution process should be accepted
and valued.
7. Students learn mathematics best from their teacher?s demonstrations and
explanations.
16. Teachers should model and demonstrate mathematical procedures and
then, ideally, time should be allowed for the students to have the
opportunity to practice those procedures.
17. Rather than demonstrating how to solve a problem, a teacher should allow
students to figure out and explain their own ways of solving mathematics
problems, including word problems.
Teachers were asked to respond to the above five statements which appeared on
the Teacher Attitude Survey (TAS) with ?strongly agree?, ?agree?, ?neutral?, ?disagree?,
and ?strongly disagree?. Table 19 summarizes how the 54 participants responded to each
part.
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Table 19
Teacher Attitude Survey Summary
Statement
Number of Participants
Strongly
Agree
Agree
Neutral
Disagree
Strongly
Disagree
No
Answer
Statement 2
20 26 6 0 1 1
Statement 6 14 22 12 6 0 0
Statement 7 3 16 18 17 0 0
Statement 16 14 33 5 2 0 0
Statement 17 6 21 16 9 1 1
The participants were generally in agreement with the statement that students
should figure out how to solve mathematics problems for themselves. About two-thirds
agreed that each student?s solution process should be accepted and valued, while only
about 35% thought that students learn best by teacher?s demonstrations and explanations.
Note that this is in direct contrast to the results for statement 16, where almost 90%
thought teachers should model and demonstrate mathematical procedures and allow time
for students to practice these procedures. Half of the participants agreed that students
should figure out and explain their own ways of solving mathematics, which is in contrast
to their agreement with statement 2.
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Conclusion
Had the intention of this instrument only been to see if the participants had the
necessary content knowledge to answer questions, there is little doubt that the
participants would have done relatively well on a majority of the questions. However,
when using a format that required them to look at mathematics in the context of a variety
of lenses such as student work and solving problems using technology, a different picture
emerged. The purpose of the ALCKIN was to gain understanding of the content
knowledge and pedagogical content knowledge of the participants.
Content Knowledge
The analysis of their content knowledge includes both the procedural knowledge
and conceptual knowledge along with the processes of: reasoning and proof,
communication, connections, problem solving, use of technology (NCTM, 2000), and
real world applications (CBMS, 2000).
In only seven of the twenty-five tasks where teachers were asked select an answer
were more than 80% of the participants able to select the correct answer. The content
knowledge exhibited tended to be procedural in nature, since the majority of the
reasoning given, when any at all was given, was algebraic manipulations. For example,
participants used or suggested the procedures of ?vertical line test?, ?factoring
binomials?, ?synthetic division?, and ?quadratic formula.? On every item of the ALCKIN
instances can be noted where teachers made errors in mathematical computations or in
the reasoning they provided for their answer choices or where they exhibited a significant
lack of understanding of the mathematical topic in question. Even though participants
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could generally write algebraic expressions, many had difficulty in expressing what
algebraic expressions meant in equations.
The majority of the participants exhibited a limited ability to view mathematics in
multiple ways or in ways that demonstrated the conceptual basis for the problem.
Multiple methods of solving problems suggested involved different procedures related to
the same topic such as factorization or long division. A few conceptual examples were
used, such as drawing pictures. While some participants did suggest the use of
manipulatives, a conclusion cannot be drawn about whether this would be specifically to
develop mathematics conceptually or used as another procedural method.
Pedagogical Content Knowledge
There was very little evidence of pedagogical content knowledge in on the
responses to the items of the ALCKIN. The value teachers placed on doing mathematics
procedurally was very evident when the majority of participants selected the group that
found the numerical value for slope, and only one of the participants recognized that
three of the four groups had equal understanding of their representations of slope. The
majority of the participants seemed to think that a student had to find the value of slope to
really understand it.
Participants did not respond to the errors in student reasoning given on the
ALCKIN. Since they did not respond to these errors, alternative explanations or models
with not offered by the participants.
In conclusion, participants in this part of the study had a procedural knowledge of
mathematics but displayed little conceptual knowledge of the same mathematical topics.
Participants made mathematical errors, gave incorrect reasoning, and exhibited a lack of
144
understanding in some of the content areas. The pedagogical content knowledge of the
participants was very limited in these participants. There were few alternative
representations or methods used, except those that were procedural in nature.
Connections were not made between mathematical topics. Student errors were generally
ignored and not responded to. They emphasized the use of procedural knowledge and
suggested how students should have worked problems using set procedures.
In the next chapter, results from four case study teachers will be presented in an
effort to gain a better understanding of content knowledge and pedagogical content
knowledge. Examination of the instructional practices of the case study teachers also
provides further insight into how both of these types of knowledge are used in
mathematics instruction.
V. CASES
This chapter contains case studies of four secondary mathematics teachers who
taught Algebra I at the high school level. Multiple data sources were used to develop a
deep understanding of their content knowledge and pedagogical content knowledge. In
addition, the cases help us understand how content knowledge and pedagogical content
knowledge of the cases are reflected in instructional practices. Furthermore, the cases will
further provide information about the changes in their content knowledge and
pedagogical content knowledge resulting from professional development. In addition, the
145
cases will further provide information on how changes in bother of these types of
knowledge are reflected in instructional practices. Data collection sources were surveys,
classroom observations, interviews, and quarterly meeting observations.
The structure of each case consists of three main sections of analysis: pre-
professional development, professional development, and post-professional development.
First, each subject?s content knowledge and pedagogical content knowledge was
analyzed prior to professional development, referred to as pre-professional development.
The supporting data for this analysis was collected in the spring of 2004 and included
surveys, written instruments, classroom observations, and interviews.
The second analysis, referred to as professional development, involved the
teachers? participation in professional development, which included attention to
increasing content knowledge and pedagogical content knowledge. Data for this analysis
was collected during the summer of 2004, fall of 2004, and the spring of 2005 and
included both teacher and presenter interviews, analysis of professional development
presenter documents, and quarterly meeting observations.
The third section of analysis, referred to as post-professional development,
focused on changes in both content knowledge and pedagogical content knowledge of the
case teachers after professional development, as well as how changes in these types of
knowledge are seen in instructional practice. Data for this section of analysis was
collected during the spring of 2005 and included both classroom observations and
interviews.
The following sections begin by describing the professional development and the
coding of the documents related to the case study teachers. Each case will be described
146
organized around the timeframes of pre-professional development, professional
development, and post-professional development. Finally, conclusions will be drawn on
each case.
Description of Professional Development
Professional development at the MDMSIP summer institute was provided for
Algebra I, Geometry, and Algebra II, addressing content related to specific areas. In
addition, there were general sessions that all secondary teachers attended during the
summer professional development that did not directly relate to content knowledge of
these content areas. These sessions dealt with pedagogical issues, which included
assessment, working with diverse students, cooperative learning, incorporating
technology into classroom instruction, questioning, teaching via problem solving, and
developing mathematical understanding. Presenters for the Multi-District Mathematics
Systemic Improvement Project [MDMISP] summer professional developmental training
modeled the types of instructional strategies they were encouraging the teachers to
implement into their own classrooms.
Participants were expected to serve in the roles of both students and teachers
while actively participating in the activities, as well as reflecting on how each of these
pedagogical issues might appear within their own classroom. Issues related to how they
might facilitate changes to improve the learning of mathematics for their own students
were also addressed. Presenters for the MDMSIP summer institute also worked with
participating teachers on transforming procedural lessons into investigative lessons, as
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well as facilitating mathematical lessons from Interactive Mathematics Program (IMP)
units (Fendel et al., 2000).
During quarterly meetings, case study teachers met to topics that would be
covered in the next grading period. Opportunities were also provided further develop
content knowledge and pedagogical content knowledge.
Coding
Documents related to the case studies were entered into Atlas.ti (Muhr, 1991).
The data from the case study participants was already included from the first part of this
study. Additional documents included field notes from classroom observations, and
transcriptions of interviews of teachers and professional development presenters. The
codes developed in the survey study were used as a beginning point for coding the case
study documents. New codes were added to this list of codes as the text from the
documents was analyzed. The frequencies for some of the codes became so large it
became obvious that additional codes needed to be added to further clarify the existing
code. For example, Teacher Question was one of the initial codes. Different types of
teacher questions began to evolve as additional documents were analyzed, and the code
Teacher Question was broken down into the types of questions that a teacher asked such
as ?Teacher questions involves asking students to name the next step or procedure? or
Teacher Questions ?why? or ?how??.
Additional codes were also added to capture events that may normally be
observed in classroom observations but may not be a part of written instruments. For
example, student conversation was a part of classroom observations and codes such as
148
?Student Gives Correct Answer? or ?Student Indicates They Don?t Understand? had to be
included.
Once all of the documents had been coded, each document was revisited to ensure
consistency of coding throughout all of the documents. Table 20 contains a list of the
codes that evolved as well as the frequency of each code. Appendix H provides more
detailed explanations of how each code was used.
Table 20
Codes for Case Studies
Code Frequency
Checks Work 18
Conceptual Knowledge 24
Definition 52
Denotes Using Inverse Operations 2
Formula Manipulation 7
Gives Praise to Students 30
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Multiple Reasons 35
Problem Instructions 2
Provides Example 19
Step by Step Procedures 225
Student Agrees with Teacher 137
Student Answers ?Why? or ?How? 41
Student Gives Correct Answer 177
Student Gives Correct Procedure 86
Student Gives Incorrect Answer 31
Student Gives Incorrect Procedure or Next Step 15
Student Indicates They Don?t Understand 17
Student Practice, Practice Problems or Quiz 12
(table continues)
Table 20 (continued)
Code Frequency
Student Question 96
Student Recognition of an Error 16
Student Recognizes Teacher Error 3
Student Short Answers to Arithmetic or Algebraic Question 236
Teacher Answers Own Question 76
Teacher Answers Student Question 25
Teacher Can?t Recognize Student Error 3
150
Teacher Asks Clarifying Question 174
Teacher Comment or Question Indicates a Task is Easy 23
Teacher Gives Hint 117
Teacher Gives Non Mathematical Instructions 121
Teacher Ignores Incorrect Student Procedure 18
Teacher Ignores Student?s Question 6
Teacher Gives Incorrect Information 6
Teacher Question Involves Asking for an Answer to an Arithmetic
Problem or Simplifying an Algebraic Expression 134
Teacher Question Involves Asking if Student has
Other Questions or Understands 168
Teacher Questions ?How? or ?Why? 133
(table continues)
Table 20 (continued)
Code Frequency
Teacher Questions Involves Asking Students to
Name the Next Step or Procedure 220
Teacher Recognizes Correct Student Procedure 35
Teacher Recognizes Incorrect Student Answer 30
Teacher Recognizes Incorrect Student Procedure 37
Teacher Reiterates Students? Reply 210
Teacher Reminds Students of ?Steps? or ?Rules? 90
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Uses Manipulatives 20
Uses Mnemonics 17
Each case will be considered in the following sections. Analysis of each case
study teacher will be considered in the three phases of analysis: pre-professional
development, professional development, and post-professional development. Each phase
of analysis will consider the content knowledge and pedagogical content knowledge of
the cases as well as looking for changes in these types of knowledge resulting from their
involvement in professional development. The cases will be summarized in an effort to
understand how their content knowledge and pedagogical content knowledge are used in
instructional practices both before and after professional development.
Mrs. Cotney
Mrs. Cotney is a White female and when this research study began she was
completing her twenty-eighth year as a classroom teacher. Mrs. Cotney was short in
stature and soft spoken. Her educational background included a Bachelor of Science
degree in mathematics education and a Master?s degree in language arts education. She
taught at Abbott High School, a ninth through twelfth grade school, where she was
primarily responsible for teaching Algebra I and Algebra IA. Abbott High School had
95% of students passing the Alabama High School Graduation Exam (ASHGE), 32% of
its students were Black, and 24% of the students were on the free or reduced lunch
program.
Pre-Professional Development
152
In this section, we will consider Mrs. Cotney?s classroom environment, how she
viewed her students? ability to learn algebra and responses to statements contained on the
Teacher Attitude Survey. Next, Mrs. Cotney?s content knowledge and pedagogical
content knowledge will be addressed from the ALCKIN and interview. Finally, a further
look at her content knowledge and pedagogical content knowledge will be drawn from
classroom observations, which includes how these types of knowledge were reflected in
her instructional practices.
Classroom Environment
Mrs. Cotney had a smile on her face for the students that she was teaching and
gave encouragement to all of her students. For example, comments such as ?Good for
you? or ?You?re trying so hard? and ?You had it (the answer), you were saying it, take
some credit? were often heard as she observed pairs or small groups of students working
on their practice problems. Mrs. Cotney frequently moved among her students to observe
how her students were working problems, offered encouragement, and answered their
questions.
Her classroom consisted of student desks in straight rows facing the front of the
room where the overhead projector, pull-down screen, and whiteboard were. This
particular classroom setting suggested that Mrs. Cotney used a traditional approach to
teaching mathematics. However, students were observed forming pairs or small groups
when time was allowed for them to practice the mathematical instruction Mrs. Cotney
had provided. The daily homework assignments for each of her classes were written on a
portion of the whiteboard. Looking from the back of the classroom, the right wall was
153
covered with bookcases containing extra textbooks. Mrs. Cotney?s teacher desk sat at the
back of the classroom along with an additional table with chairs.
Mrs. Cotney had four function calculators available for students to use in the
classroom, which could be found in a basket at the back of the room. The left-hand wall
had an additional whiteboard. Spaces above the whiteboard were covered with
mathematical posters and what Mrs. Cotney called ?helping devices? for her students.
One such poster contained a table of counting numbers along with their squares. Still
another poster consisted of the mnemonic, ?Please Excuse My Dear Aunt Sally,? meant
to help students remember the order of operations of parentheses, exponents,
multiplication, division, addition and subtraction.
Student Ability
Mrs. Cotney had different thoughts about the abilities her students possess to
learn mathematics based on whether they were in Algebra I or Algebra IA. For those
students who are in the Algebra I classes, she felt that a large majority of them could
learn algebra well because they have:
Enough math background that I can build on. Those who don?t when they are
sitting in a class with kids who seem to understand push themselves a little harder
to get it, and the other ones will come in before school for help because they don?t
want to look bad with the other ones.
In contrast, Mrs. Cotney felt that only half of her students in her Algebra IA and
Algebra IB classes could learn algebra well. For the half from this group that could learn
algebra, she commented,
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They are the ones that have gotten through, you know they?ve learned enough. I
don?t know that any of them this past semester could have passed regular Algebra
I. They couldn?t have done it in eighteen weeks what we did in eighteen weeks.
Ah, but going slowly, giving the repetition, we could build on what they did have.
On the other hand, she explained why she felt the others could not learn algebra well as
follows:
The fifty percent that can?t when I looked back at their school history most of
them have not passed a math class since maybe second grade. So they are really,
really, really lacking in skills. They are so weak and they have no concept of
number. They don?t understand. I mean you can take half of twelve and they are
perplexed. They don?t understand number period. They are just the sad ones.
Thus, Mrs. Cotney?s comments suggest she felt if student have been successful
with mathematics in their past courses, they can learn algebra. In addition, students need
to have this background in order for her to build on it.
Teacher Attitude Survey
Mrs. Cotney was questioned about the two conflicting statements from the TAS
related to related to instructional practices. When asked, ?It is important for students to
figure out how to solve mathematics problems for themselves?? she replied:
Oh, I agree that is one of my, I agree absolutely. (Laughs). It?s hard. When we
are learning something new, systems of equations. And I have given them some
tools you know I have shown them ways to solve different kinds, maybe not all
the different methods. I?ll try to give them some that don?t look like we?ve been
doing and ask them in groups sometimes because some of them are insecure, you
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know. Ah, ask them to figure out how to solve this. You know, use what you
know about how systems work together, use what you know about equations,
numbers in general, and come up with a strategy.
Mrs. Cotney was also asked to reply to the second statement, ?Teachers should
model and demonstrate mathematical procedures and then ideally time should be allowed
for the students to have the opportunity to practice those procedures.? In addition, she
was asked how she thought they could coexist in mathematics instruction. She replied:
Oh they can. The second, the second part you read, I think is important when you
are introducing new skills. You know they need to see it modeled. And then let
them go with it after they have an understanding of the process.
Thus, Mrs. Cotney?s agreement with the statements suggest she felt teachers
should model and demonstrate for students and that by allowing time for student to
practice those procedures she was allowing them to develop an understanding of the
mathematics.
Content Knowledge
Mrs. Cotney?s content knowledge will be viewed following the definition of
content knowledge in the literature review, considering in turn procedural knowledge,
conceptual knowledge, and mathematical processes.
Procedural knowledge. Mrs. Cotney exhibited some procedural knowledge on
the ALCKIN. She correctly answered 14 of the 25 mathematical tasks. On question five
on the ALCKIN, she was unable to provide any approximate solution to the quadratic
problem involving the spreadsheet. However, she exhibited strong procedural content
knowledge during her interview. She was able to correctly solve the wheels and vehicles
problem by using the set procedure of solving a system of linear equations by
elimination. The problem follows:
Jameel looked out in the parking lot and decided to count cars and motorcycles by
the number of their wheels, excluding spares. He saw that there were 17 vehicles
with a total of 56 wheels. How many motorcycles did he see?
She provided the written work in Figure18 and commented, ?Because that is the kind of
thing I would give them when we are working with a system of equations. I think we do
one with chickens and cows.?
Figure18. Mrs. Cotney?s work for the wheels and vehicles problem.
Conceptual knowledge. Conceptual knowledge is the ability to know why and
how algorithms work. Mrs. Cotney exhibited weak conceptual content knowledge during
this same timeframe. She was able to give examples for showing the two algebraic
expressions were equivalent from question six on the Algebra Content Knowledge
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157
Instrument [ALCKIN] (see Appendix A), which may suggest she may have had some
conceptual understanding of this topic. Mrs. Cotney wrote, ?Manipulatives (algebra tiles,
money, for instance), draw pictures.? Other than this one particular incident, there were
no other examples where she demonstrated conceptual understanding of the mathematics
being addressed.
Mathematical processes. When asked to solve the wheels and vehicles problem
(see Appendix D) using a different method than the procedural method she had already
used, Mrs. Cotney was unable to provide one, and stated, ?I?m sure there are like I tell
my kids fifty different ways to do it but right this minute, no I can?t.? Mrs. Cotney stated
that she believed there are many different ways to solve mathematics problems and
students should be allowed to solve problems using methods of their own choosing as
long as they were mathematically sound. She stated that students did not have to work
problems in her classroom using her particular method, ?I?m just not big on you have to
follow my twelve steps.?
Thus, Mrs. Cotney?s content knowledge consisted of strong procedural
knowledge and limited conceptual knowledge. Observations of her use of mathematical
processes were limited to problem solving in which a set procedure was used. There was
no evidence of her mathematical reasoning, use of connections, or ability to use multiple
representations or methods.
Pedagogical Content Knowledge
Pedagogical content knowledge will be examined through the lens of the
definition provided in Chapter 2. Pedagogical content knowledge is the ability to
?unwrap? and present mathematical topics so that students can be successful in learning
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mathematics. Pedagogical content knowledge includes the ability to access different
representations as well as methods for solving mathematics problems. Pedagogical
content knowledge is the ability to recognize student errors and be able to respond to
them with alternative models and explanations. Pedagogical content knowledge includes
the ability to responds to questions, and to pose questions and problems that are
productive to students learning mathematics.
Mrs. Cotney had difficulty in recognizing errors made by students on the
ALCKIN. For example, on question one from the ALCKIN (see Appendix A), she
disagreed with the student in part B and agreed with the student in part C without
providing any explanation for what she thought about what the students had stated. In
addition from question seven on the ALCKIN (see Appendix A), Mrs. Cotney was unable
to recognize that both students had presented viable methods for finding the number of
tiles in the border of the patio. She stated, ?James is close to understanding but missing
the 4 corner pieces.? In general, she had a hard time generating alternative solution
methods.
On question eight from the ALCKIN, Mrs. Cotney selected only two of the three
correct groups, Group 1 and Group 4, as having a good understanding of slope.
Furthermore, she felt Group 4 had the best understanding of slope because they were the
only group that found the slope by using rise over run and stated, ?They understand the
relationship of rise to run.?
However, Mrs. Cotney did not have any difficulty in assessing the two elementary
non-procedural approaches used by students to solve the wheels and vehicle problem in
the interview (see Appendix D). She felt both were viable and would be methods she
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would accept in her own classroom. Mrs. Cotney showed hesitancy in accepting the
correctness of a non-procedural method (see figure 19) used to solve the quadratic
problem x
2
= 2x + 8.
To find the solution I graphed each side of the equation.
-15 -10 -5 5 10 15
18
16
14
12
10
8
6
4
2
-2
hx() = 2?x+8
fx() = x
2
The two graphs intersected at two points (-2,4) and (4, 16). From these I can find the two
solutions for the equation x
2
= 2x + 8. x would be equal to -2 and 4.
Figure 20. Student?s non-procedural method for solving x
2
= 2x + 8.
Mrs. Cotney admitted that she would have to further investigate the graphing method to
convince herself that the process would always work. Her thoughts on the graphing
approach were:
Graph each side of the equation. (Pauses) Oh, Okay. (Pauses) It?s working out.
We?ve never done one like this before. I can?t figure out what (Pause) exactly she
is doing, what she?s doing. (Pauses) I?m going to try and work it in my head and
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make sure it?s working out some way. (Laughs) Yea, (Pauses) Looks like it
works. (Pauses) After you leave I?m going to pull some out and play with them.
She ended up by checking the answer to the problem by solving the problem by
factorization and stated factorization would be an easier way to solve the problem. This
indicated that Mrs. Cotney relied on procedures and place a value on procedures.
Thus, Mrs. Cotney was able to correctly evaluate different representations, as well
as different methods of working problems presented by fictitious students. However, she
did have difficulty in accessing errors made by students on the ALCKIN.
Further insights Mrs. Cotney?s pedagogical content knowledge will be provided
in the following section describing her classroom instruction, as along with insights into
her ability to apply this knowledge in her classroom.
Classroom instruction
Mrs. Cotney?s used limited pedagogical content knowledge in presenting
mathematics, which was evident in her typical method of providing step-by-step
instructions to her students on how to solve a problem. Mrs. Cotney was competent with
the procedural knowledge of the topics she taught during classroom observations. She
was not observed making any mathematical errors. The following is an example of a
classroom dialogue when Mrs. Cotney?s Algebra I class was simplifying radicals:
Mrs. Cotney: Look at this one. These are simple. Take the square root of five over
eight. One of the ways you are going to see this you can write this, okay. We can?t
keep a radical in our denominator can we?
Student: No.
Mrs. Cotney: Isn?t that one of our steps for simplifying radicals.
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Student: What do we do?
Mrs. Cotney: Let me show you what to do. This is easy. This is very easy. What
you do to get rid of the fraction, ah the radical in the denominator is multiply by
1. That doesn?t change the problem does it?
Student: No.
Mrs. Cotney: But instead of writing one over one but we?re going to write it
(writes square root of eight over the square root of eight) Isn?t this one, isn?t this
one by itself? So I?m not changing the problem.
Student: Why don?t you use the five?
Mrs. Cotney: Because we don?t care if we have a fraction, a radical in our
numerator. Now think back to the very beginning before Jimmy started falling
asleep. We can multiply a radical times a radical. What is the square root of five
times the square root of eight?
Student: Square root of forty.
Mrs. Cotney: Very good. We?re getting silly. Square root of eight times square
root of eight is? Don?t you tell me the square root of sixty-four. Think.
Student: Eight.
Mrs. Cotney: Eight. Remember when you square a square root you get the
number. We?re not through.
Student: If you have a perfect square in the radical do you still have the radical?
Mrs. Cotney: It?s cool on the bottom, right? Remember this is what we want.
What perfect square will divide into forty?
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Student: Five.
Mrs. Cotney: Five?s not a perfect square is it?
Student: Four.
Mrs. Cotney: Okay. Four. What is the square root of four?
Student: Two.
Mrs. Cotney: Think of radicals just like you do variables. If we had two x over
eight we would simplify it, right? To one x over four, ignore the radical. Two into
itself goes?
Students: One.
Mrs. Cotney: And two goes into eight?
Student: Four.
Mrs. Cotney: You got it? So that?s the square root of ten over four. Simplified.
Okay. Make sure you?ve got it. The main thing, the main thing you?ve got to
remember is it?s the denominator we?re trying to get rid of, alright. Not the
denominator but the radical in the denominator we?re trying to get rid of so we
multiply by that radical over itself so that gives us the one which won?t change
the problem but will let us work with it.
Student: What if the denominator is already a perfect square?
Mrs. Cotney: If it?s already a perfect square, well at least you?re done.
After providing the step-by-step instruction, Mrs. Cotney allowed time for
students to work on a problem very similar to the one that she had just demonstrated.
After working through some examples with the entire class, she assigned additional
problems for students to work. Students were able to work with a partner or within a
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small group on problems which were provided from their textbook or as part of a
worksheet to be completed during class.
Mrs. Cotney was not observed giving her students opportunities to develop an
understanding of the mathematics they were taking. Even when opportunities arose in her
classroom, she did not give her students the chance to fully explore the purpose of why
they were doing particular procedures. For example, when solving two simultaneous
equations with two unknowns she pointed out,
Mrs. Cotney: What is it you are looking for when you are solving a system of
equations?
Student: The ordered pair.
Mrs. Cotney: The ordered pair when the lines cross.
Mrs. Cotney had an average score of 32 on the RTOP instrument from classroom
observations during the spring of 2004. This score suggests that she did not use reformed
teaching practices as a strong point in her mathematics instruction. In general, classroom
instruction reflected a very limited use of pedagogical content knowledge.
Student questioning. Mrs. Cotney used questioning that appeared to reinforce the
procedural knowledge of her students. The majority of her questions involved asking
students to give answers to mathematical problems and/or provide what the next step
would be in a series of steps to simplify expressions or solve algebraic equations. The
following was a teacher and student dialogue which involved learning how to simplify
radicals.
Mrs. Cotney: There you go. I would like for you to try one all on your own. Let?s
see what you could do. The square root of eighty. Think what will divide this
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number? Mentally, you know, you don?t need to list it. That?s kind of a crutch but
it might make you feel a little more comfortable. What is the biggest number on
the left side that will divide into eighty? (As the student tells the Mrs. Cotney
what to do, she works the problem on the overhead projector.)
Student: Sixteen. Five stays in. Sixteen squared is four. Four goes on the outside.
Mrs. Cotney: So you?re saying sixteen times five and that gives you
Student: Eighty.
Mrs. Cotney: What is the square root of sixteen, don?t you say eighty.
Student: Four.
Thus, Mrs. Cotney posed questions during her classroom instruction that required
students to provide answers to mathematical problems, simplifying algebraic expressions,
or solving algebraic problems. She was able to respond to her students? questions but the
questions were similar to the same types of questions she asked in class. Mrs. Cotney?s
questions were not the types of questions or problems that would produce success in
student learning.
Analyzing student errors. While Mrs. Cotney had difficulty in recognizing student
errors on the ALCKIN, she could recognize and react to errors made by her students in
executing mathematical procedures. In the following excerpt, students were solving for
the lengths of unknown sides in a right triangle using the Pythagorean Theorem.
Mrs. Cotney: No calculators! Oh, my? The a is six, lets say centimeters and the b
is eight centimeters and we want to find out what c is, it?s coming back to you
now.
Student: Um humph.
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Mrs. Cotney: Pretty much?
Student: Um humph.
Mrs. Cotney: Yes.
Student: thirty-six plus sixty-four equals a hundred and then you
Another Student: c equals a hundred.
Mrs. Cotney: No, use common sense. If this thing is six centimeters and this is
eight centimeters, do you really think this is gonna be a hundred? (Referring to
the hypotenuse of a right triangle.)
Student: No.
Another Student: Do the radical thing.
Mrs. Cotney: That?s c squared. We don?t want c squared, we want c.
Thus, this shows that Mrs. Cotney was able to recognize and respond to the errors
her students made in carrying out procedures, although she did try to get her students to
think about the reasonableness of their answer. However, her response to student errors
did not provide them with alternative explanations or other models that might help them
better understand their errors.
Multiple methods of solving problems. In the interview, Mrs. Cotney stated that
students did not have to follow her step-by-step instructions and she valued other
methods used by her students in solving problems. In her classroom, she was able to
assess and accept different ways of solving problems. However, she was often quick to
point out which method was the easiest for students to use. For example, when solving
systems of linear equations involving two variables using substitution she stated, ?I
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promise if you hang in there with me we?re going to find an easier way to do this, but we
have to go through this.? However, the majority of different methods observed from the
work exhibited by students consisted of putting extra steps into procedures as opposed to
omitting steps.
On one occasion when students were finding the area of irregular shaped
polygons Mrs. Cotney told students a particular way to solve a problem but agreed that
the way some of the students had worked the problem was also correct, as in the
following:
Mrs. Cotney: If we get it into two rectangles, we?ll be in good shape. (Sketched in
a dotted line to separate the irregular shaped polygon into two rectangles.)
Number 8. First I would draw it on paper, I sure would. I?m at an advantage
because I can mark on these (referring to the overhead sheet she was working on)
and you can?t mark in the book. Actually we?ve got one big area then we have to
subtract this. That?s how David is doing his right now and it?s worked perfectly.
Student: That?s what I was doing.
Mrs. Cotney: Does that mean that I?m wrong and you?re right? Maybe? It just
means there are different ways you can work the very same problem which is kind
of good and kind of bad too. Sometimes it easier if you?ve got one. The way I was
going to do it was just to find this part and this part. (Indicates the two small
rectangles formed by her dotted line.) Find the area of this skinny rectangle and
the area of this box and then add them together. Do you know what I am saying?
The way you two guys did it was to get the area of the great big rectangle and
then subtract this one. What did ya?ll get?
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Also while solving for unknown sides of a right triangle using the Pythagorean Theorem,
she stated
Okay where are you? You know what c is. You can set it up in an equation a
2
+ b
2
= c
2
and solve it but it?s easier to do c
2
? a
2
= b
2
. You subtract the a squared from
this c squared. Good.
Even though Mrs. Cotney indicated she valued methods students used in solving
problems, she often pointed out the easiest method to use. The methods she termed as
?different? tended to very similar ways to the procedures demonstrated for solving
problems, where students simply used slightly different steps. For example, although
students may have divided a polygon in slightly different ways to find its area, methods
to arrive at the answer were the same.
Conclusion
Mrs. Cotney felt that the students who had a good mathematics background could
learn algebra well, while the students that did not have a mathematics background could
not learn algebra well. Mrs. Cotney exhibited strong procedural knowledge but limited
conceptual knowledge of the algebra content. There was little evidence of use of
mathematical processes. Her classroom instruction primarily consisted of teaching
students step-by-step procedures for simplifying algebraic expressions or for solving
algebraic equations. Explanations on the ALCKIN were procedural in nature and the
only exception related to the problem where she suggested students use manipulatives in
showing the two algebraic expressions were equivalent. She did not provide opportunities
for her students to develop a conceptual understanding of the mathematics they were
taught.
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Mrs. Cotney was able to recognize errors that her students were making.
However, it appeared she had difficulty in recognizing and assessing errors made in
student statements on the ALCKIN, since she generally did not make any response to
them, possibly these statements extended beyond merely following set procedures. Her
questions to students generally called on their procedural knowledge. She recognized
students in her class who worked problems in what she termed different ways, although
these differences generally related to minor variations of the procedural methods other
students were using. Her content knowledge was reflected in her instructional practices
but her pedagogical content knowledge was limited. Although she used questioning,
recognized and responded to student errors, and noted ?different? methods used by
students, they were rather superficial representations of pedagogical content knowledge.
Professional Development
Mrs. Cotney participated in the Algebra I section of the summer of 2004
professional development training of Cohort I. Mathematics content addressed within the
Algebra I group involved working in the Interactive Mathematics Program (IMP) Baker?s
Choice (Fendel et al., 2000) and Solve It units (Fendel et al., 2000). Teachers were
expected to be familiar with the concepts of how to graph an equation, and how to use
graphing and at least one other method for solving a system of linear equations involving
two variables. Participants were involved in investigating inequalities and developing the
subsequent rules that apply to inequalities. From the Solve It unit (Fendel et al., 2000),
teachers used different approaches to solving equations.
Mrs. Cotney appeared to be enthusiastic about the training she received through
MDSMIP. She indicated that she had taught algebra for several years and was looking for
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different ways to teach: ?I thought it was wonderful. I really did. We all came back just
charged up. We were so excited. I don?t know what to say, it was just great.? One of the
presenters for the summer institute, stated:
Out of all the participants that I had, she was definitely one that stood out as
having a very positive attitude about the training. A lot of people were there just
because they had to be there and she did not have that attitude at all. She seemed
to really enjoy the sessions and really wanted to know more about it. She asked a
lot of questions and she was kind of a quiet person but she, you could just tell the
way that she was acting that she was enthusiastic about the training. Definitely.
The presenter also expected that Mrs. Cotney would try the things she had learned from
the institute in her classroom. On the other hand, another of Mrs. Cotney?s presenters was
hesitant in thinking that Mrs. Cotney would make any changes in her instructional
practices:
I don?t know if I would go that far in how comfortable she was. Ah, I think she
would try them. I think that she ah made a connection. I think she could see how
it would work. I don?t know if she went as far as to, I don?t know if she would
implement it in her classroom or not.
The presenters of the Algebra I sessions of the MDSMIP Cohort I summer
professional development training agreed that Mrs. Cotney was knowledgeable about the
Algebra I content. One presenter stated, ?I don?t think she was lacking in content
knowledge at all?I think she was strong.?
In addition to the MDSMIP summer development, Mrs. Cotney took Internet
classes offered through Harvard University. She had taken a class on differentiating
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instruction and was presently taking a class on fostering algebraic thinking in high school
classes. When Mrs. Cotney was questioned about the impact these courses may have had
on her classroom instruction, she replied:
I was telling Brittany Johnson (another teacher at her school). Monday I was
sitting here doing my homework, cause it was due Tuesday and I was showing her
what was on there and I said Dr. Allen (project director for MDSMIP) and
MDSMIP could have just written this course because it is MDSMIP. I mean it is
everything, everything we read, talked about, philosophy, it?s just reiterated. I
mean they are talking about Core Plus, CMP (Connected Mathematics Program,
IMP and modeling, internal models and I?m going I didn?t need to take this but
it?s reaffirming. (Note that pseudonyms were used.)
While participating in a small group discussion at a quarterly meeting, Mrs.
Cotney indicated that she may have been experiencing some difficulties with
implementing the different methods of representation. Mrs. Cotney was overhead telling
her group:
I?ve never used them (algebra tiles) until this year so I?m not real comfortable
with them. I have two Algebra IB classes, one of them I don?t enjoy. They cannot
read it (IMP unit). Alice (IMP unit) does get into rules but?. I want them to
know five and the square root of five are not the same.
Thus, the professional development presenters thought Mrs. Cotney was
knowledgeable of the Algebra I content. She was an active participant in the professional
development, but the presenters were split on whether she would implement any
instructional changes. Growth in pedagogical content knowledge was noted in her
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conversation about algebra tiles since she had not been familiar with them prior to
professional development.
Post-Professional Development
During post-professional development, the topics of changes in classroom
environment, changes in student ability, changes in content knowledge, changes in
pedagogical content knowledge, and how the changes of content knowledge and
pedagogical content knowledge are reflected in instructional practices will be addressed.
Each of these will be addressed in the following sections:
Classroom Environment
The layout of the classroom during post-professional development remained
unchanged. Mrs. Cotney continued to encourage all of her students as she had in the pre-
professional development timeframe.
Student Ability
In the follow-up interview, Mrs. Cotney indicated she felt a large percentage of
her students could learn algebra well. However, she made the following comments about
why some of them may not learn algebra well.
Mrs. Cotney: I think ninety-five percent of them can learn algebra well. They?re
not. But they can.
Interviewer: So why do you think they can even though they?re not?
Mrs. Cotney: OK. A few things. Major immaturity for ninth graders. I mean it?s
just this group, not just mine, the whole ninth grade, we?re seeing it across the
board. They have been spoon fed and I?m not pointing fingers, I spoon feed too.
Not as much now but you know and if I can?t show them every possible variation
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on a process they don?t understand because they don?t know how to think
(laughs). I mean they don?t know how to think it through and, not they don?t
know how, they haven?t been required too much and so they don?t want to and if
you don?t give them all the steps they go home and you?re not teaching them. So
they can learn, they really can. I don?t think I have a slow one in the algebra one
class but they are not being successful.
Thus, while Mrs. Cotney previously felt her students could not learn algebra well
if they were not mathematically prepared, during the post-professional development
phase, her views had changed somewhat. She now suggested that her students are
immature and can?t learn algebra well unless she ?spoon feeds? them, meaning she needs
to give them step-by-step instructions.
Changes in Content Knowledge
Mrs. Cotney was confident that she knew the content knowledge of Algebra I well
and did not feel she learned any additional algebra content as a result of MDSMIP
summer professional development training. However, she did feel that she had developed
a better understanding of how the content fits together. No changes in her procedural
knowledge or conceptual knowledge were observed during the interview and classroom
observation.
During an interview in the spring of 2005, Mrs. Cotney was asked to solve the
quadratic problem x
2
= 2x + 8, which she was able to do by setting the equation equal to
zero and using factorization. Before continuing she checked her answer for correctness
by using substitution. Previously during an interview in the spring of 2004 when asked to
provide another method for solving an algebraic problem, she was unable to do so.
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However, on this occasion she provided other procedural methods for solving the
quadratic problem, such as graphing or setting the equation equal to zero and using the
quadratic formula. Mrs. Cotney also suggested a non-procedural scheme of using the trial
and error method where students could substitute in values for x until they found which
number or numbers made the statement true.
Thus, Mrs. Cotney did exhibit some changes in the mathematical processes
related to her content knowledge. Previously she had been unable to offer different ways
of solving problems, but now she offered more than one procedural way, as well as a
non-procedural way for solving the same problem. This may also reflect a deeper
conceptual understanding of the mathematics.
Changes in Pedagogical Content Knowledge
No changes in her pedagogical content knowledge were evident in observations of
her classroom. She used the same types of strategies when recognizing student errors or
the types of questions asked of students after attending professional development. There
was no additional evidence, such as displayed student work or projects that indicated
students were involved in IMP units (Fendel et al., 2000).
She still pointed out student errors instead of taking advantage of these
opportunities to challenge her students to look at the work of other students to decide if
the answers, methods, and/or procedures used were correct. For example, during a review
of homework problems, a student wrote the following work on the whiteboard
corresponding to homework problem twenty-six:
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(5x ? 6)(5x ? 6).
26. (5m ? 6)(5m ? 6)
5m(5m ? 6) ? 6(5m ? 6)
25m
2
? 30m ? 30m + 36
+30m +30m
25m
2
+ 36
Instead of asking the students if they agreed with what the student had done, Mrs. Cotney
erased the two 30m terms which had been added, and the following classroom discussion
took place:
Mrs. Cotney: Why don?t we do this?
Student: There?s no equal sign.
Mrs. Cotney: There?s no equal sign. So you didn?t swap them from one side of
the equal sign to the other did you and you know not to change signs. So what
should it be?
(Student went back to the whiteboard and wrote 25m
2
? 60m + 36.)
Mrs. Cotney?s questions still focused on having students provide answers to
mathematical problems or to provide the next step in a procedure. She did not always
allow adequate wait time for students to answer questions. In some cases questions were
rephrased to give students hints for the answer, and in other cases Mrs. Cotney merely
answered her own question. For example, Mrs. Cotney and her students were working
with the following diagram to represent the area of the rectangle. She said:
So if I wanted to I could go in and calculate the area right now. And how would I
do that? (No pause.) I would just say that times that right? (Indicates the x and x +
3.) We could calculate area right now. (Writes l = x + 3 and w = x on the
overhead sheet.)
x
x+ 3
So we agree that our length right now is x plus three and the width is x?
Although Mrs. Cotney continued to state that her students could work problems in
any manner they chose, I continued to frequently observe her telling students a particular
way to proceed with problems, rather than allowing them to access different
representations and different methods for solving problems. For instance in responding to
a student answer, Mrs. Cotney stated, ?You didn?t sketch it,? indicating the student
should have made a geometric sketch of the problem in order to be able to solve it. In one
particular discussion, Mrs. Cotney questioned the students about a different way of
writing (2x-3)(2x-3). She asked:
Mrs. Cotney: How else could I write this? It?s a really tricky question; you?re
going to have to think. What?s another way I could have written this problem
other than doing it that way?
Student: (3-2x)(3-2x).
Mrs. Cotney: Without changing signs? Okay so let?s fill out the variable part.
What?s another easy way I can write this problem?
Students: (Talking at once)
Mrs. Cotney: So if I multiply something by itself, I?m squaring it correct?
(Wrote (2x - 3)
2
next to the problem.)
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177
Thus, Mrs. Cotney?s pedagogical content seemed largely unchanged from the
period before professional development, as reflected both in the interview and classroom
observations.
Reflection of Changes in Knowledge in Instructional Practices
Mrs. Cotney indicated that she was using different instructional practices and
when she expressed an excitement about what was going on with her Algebra IB class.
She said, ?I?m having a good time. You know last year I cried over my Algebra IB class.
These are my favorite classes. The students are getting so excited about learning.? She
was further asked if she felt her students were retaining the material better, and she
answered, ?Oh, yes! I?m having to reteach some of the Algebra IA material but they are
remembering the new material better.? However, changes in instructional practices, much
less changes in content knowledge and pedagogical content knowledge, were not
observed during the spring of 2005.
Mrs. Cotney was questioned concerning the types of changes she had made in her
classroom practices as a result of professional development:
I am doing more, oh gosh (laughs) I?m regressing this semester but until recently
a lot more I call them open ended activities in class. I?ll always do team work and
team projects and things but I?m not giving step by step here?s how you do it. I
mean I?m having them work toward coming up with processes.
When asked to provide an example, Mrs. Cotney said:
I?m going to pull in one of the IMP things. With my Algebra B students rather
than talk about here?s how you make your in out table, here?s what you do with it,
you know because some of that should be review for them, I presented you know
178
each days unit or each days activities, they got with their teams and together they
would come up with how?s the best way to get this information. How can we and
what they were doing was learning functions. I mean that?s what they were
learning but it wasn?t until way, way down the road that we used the words
functions, domains, and ranges but they understood what it meant when we got to
them. You know so it more was indirect teaching as opposed to what I?m doing
right now standing in front of the room, datta, datta, da.
Mrs. Cotney further commented on her difficulty in making changes in her
instructional practices, contradicting the successes she reported she was having with her
class, when she stated:
You know it was really, really hard first semester to try to implement things and
to get kids at this age to buy into a new way of doing things. I mean we were in
tears half the time, not the kids but the teachers. (Laughs) Going (Sighs) It?s not
working, they?re not doing anything and so you start letting it go and dropping
back into your old ways. You don?t need this, I do.
The RTOP score of 35 was recorded from the classroom observation of Mrs.
Cotney during the spring of 2005. This reflects a slight increase from the RTOP score of
32 received during the pre-professional development timeframe.
Mrs. Cotney indicated she had used ?in-out tables? during classroom instruction
suggesting growth in pedagogical content knowledge had provided other ways to present
mathematics instruction to her students. Thus, some changes in instructional practices
resulted from growth of pedagogical content knowledge.
Conclusion
179
While Mrs. Cotney thought the summer professional development training was
good, the presenters were split in their thoughts on whether Mrs. Cotney would
implement any instructional changes into her classroom instruction. Mrs. Cotney did
exhibit some changes in her content knowledge and pedagogical content knowledge,
when she discussed using ?in-out tables?, algebra tiles, and when she offered multiple
procedural as well as non-procedural methods students could use in solving problems.
However, the changes in her content knowledge and pedagogical content knowledge
were not observed in her instructional practices. Mrs. Cotney reported that she had tried
implementing the IMP units (Fendel et al., 2000) during the fall semester of 2004 but fell
back, in the spring semester of 2005, into the same type of instructional practices she had
used in the 2003-2004 academic school year. She indicated that the students and their
parents felt she was not teaching her students mathematics if she did not show them all of
the steps. RTOP scores from the spring of 2004 and the spring of 2005 were very similar.
Conclusion of Case
Mrs. Cotney felt her Algebra IA students could learn algebra if they had the
mathematical background on which to build, but considered all of her Algebra I student
capable of learning. Mrs. Cotney stated she demonstrated and explained mathematics to
her students and allowed time for them to practice, in turn helping them to understand
mathematics. Mrs. Cotney had strong content knowledge but somewhat limited
pedagogical content knowledge. Growth in both her content knowledge and pedagogical
content knowledge was observed as a result of professional development. Content
knowledge and limited pedagogical content knowledge were observed during her
instructional practices during the post-professional development timeframe, but they were
180
generally unchanged from the pre-professional development timeframe. However, she
did indicate that observations growth in these types of knowledge could have been
observed in classroom instructional practices during the previous semester.
Mrs. Willoughby
Mrs. Willoughby is an older White female who was teaching high school
mathematics as a second career. Mrs. Willoughby stands at an average height and talks
with a normal level of voice. At the beginning of this study, Mrs. Willoughby was in her
fifth year of teaching at Sandsfield High School. She holds a Bachelor of Science degree
in mathematics education and was working on her Master?s degree in mathematics
education. At Sandsfield High School, a ninth through twelfth grade school, her primary
teaching responsibilities were Algebra I, Calculus and informal geometry. Passing rate on
the Alabama High School Graduation Exam at Sandsfield High School was 64%, the
student population was 35% Black, and 41% of the students were on the free or reduced
lunch program.
Pre-Professional Development
In this section, we will consider Willoughby?s classroom environment, how she
viewed her students? ability to learn algebra, and her responses to statements contained
on the Teacher Attitude Survey. Next, Mrs. Willoughby?s content knowledge and
pedagogical content knowledge will be addressed from the ALCKIN and interview.
Finally, a further look at her content knowledge and pedagogical content knowledge will
be considered from classroom observations, which included how these types of
knowledge were reflected in her instructional practices.
181
Classroom Environment
Her classroom appeared to be a student-friendly environment and her students
seemed at ease, although she dominated the classroom conversation. Student participation
was minimal and generally consisted of Mrs. Willoughby asking her students questions
that could be answered quickly and with short answers.
The student desks in Mrs. Willoughby?s classroom formed straight lines which
faced the whiteboard at the front of her classroom. This type of classroom setting might
suggest that Mrs. Willoughby used a traditional approach to teaching mathematics and
nothing from classroom observations contradicted this assessment. The teacher?s desk
was at the front of the classroom. From the back of the classroom, the left wall was
covered with windows while the back wall contained storage cabinets. The right hand
wall held both a bulletin board and a whiteboard. During classroom visits work done by
students such as projects from her geometry classes were displayed. Although calculators
were not easily accessible to students in Mrs. Willoughby?s algebra classes, they were
able to easily borrow one from her when the need arose.
Student Ability
Mrs. Willoughby felt almost all of her algebra I students could learn algebra well.
She said, ?I teach advanced algebra. And so in the advanced algebra I would say ninety-
seven, eight percent, learn it well. They?re dedicated. They?ll study. They?ll retake test.
They ask questions. Those are the kids that succeed.? Thus, Mrs. Willoughby felt the
majority of her students could learn algebra well because they would put in the necessary
effort; they are the students that will generally succeed.
182
Teacher Attitude Survey
Mrs. Willoughby was asked to respond to the two conflicting statements from the
TAS with which a majority of teachers had agreed. Her response to, ?It is important for
students to figure out how to solve mathematics problems for themselves? was, ?First I
demonstrate a concept and then I give them I call guided practice and I give them work to
do and let them work at those problems.? When asked to respond to ?Teachers should
model and demonstrate mathematical procedures and then, ideally, time should be
allowed for students to have the opportunity to practice those procedures? and how this
statement and the first statement could coexist in mathematics instruction, she replied,
?Ah, now I don?t know how well they will coexist because if one is letting them figure
out how to let them work it themselves and another is modeling it, I model but I don?t
know how it would coexist unless some concepts you might have to teach. I?m not sure.?
Thus, Mrs. Willoughby responded that she actually taught a concept and let her
students practice it which is what the second statement implied. Since the second
response contradicted what she had previously stated, it suggests that she does not think
you can allow students solve mathematics problems for themselves and model for them
in instructional practices.
Content Knowledge
Mrs. Willoughby?s content knowledge will be viewed in terms of procedural
knowledge, conceptual knowledge, and mathematical processes. These will be addressed
in the following sections.
Procedural knowledge. Mrs. Willoughby correctly answered 15 of the 25
mathematical tasks on the ALCKIN. However, Mrs. Willoughby exhibited strong
procedural knowledge of the algebra one content during interviews. When asked to work
the wheels and vehicles task during an interview in the spring of 2004 (see Appendix D),
she was able to take the information from the problem, put it in a table representing the
number of wheels per vehicle, number of vehicles with each type of wheel, and the total
number of wheels from each type of vehicle (see Figure 20).
Figure 20. Mrs. Willoughby?s setup to solve the wheels and vehicles problem.
She used only one variable to represent the number of cars, then represented the other
quantities in the problem in terms of that variable. She set up an equation by adding the
resulting expressions to total fifty-six, the number of wheels for the two types of vehicles.
From classroom observations, Mrs. Willoughby possessed procedural knowledge of the
mathematical topics she taught and no observations were made of Mrs. Willoughby
making mathematical errors.
One significant error can be pointed out in Mrs. Willoughby?s content knowledge.
On question three of the ALCKIN (see appendix A), Mrs. Willoughby noted that the
statement, ?Any number to the zero power was equal to one? was true for all situations.
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184
Conceptual knowledge. Mrs. Willoughby exhibited some gaps in conceptual
knowledge in the data collected during the spring of 2004. On question six of the
ALCKIN (see Appendix A), she could not offer any other method to show the two
algebraic expressions were equivalent except the procedural methods indicated in the
instructions. On question ten from the ALCKIN (see Appendix A), she explained the 0.2x
in the equation as ?the time (0.2 hour) times the number of dozens of chocolate chip
cookies?, indicating a verbal description of the expression, instead of what the expression
represents in the equation. Thus, Mrs. Willoughby appeared to have some limits in her
conceptual understanding of the mathemaics. Showing two algebraic expressions
equivalent was limited to a set procedure, and she did not exhibit understanding of the
algebraic expression in an equation.
Mathematical processes. Use of mathematical processes was not evident in the
ALCKIN and interview. From question five on the ALCKIN (see Appendix A), Mrs.
Willoughby was able to find a correct solution to the quadratic equation by using the
spreadsheet and justified her answer with ?at that point answers are closest.? In question
eight (see Appendix A), she was able to correctly identify only two of the three groups
that correctly described slope. The group that she did not select as understanding slope
was group one, which related slope to the tangent of the angle. Furthermore, she selected
the group that found the value for slope as having the best understanding, explaining that
?(group) D is actually calculating the actual slope.? Thus, Mrs. Willoughby displayed
limited use of the mathematical processes. Some different representations she could
understand such as the spreadsheet, while she did not recognize all representations of
slope.
185
Pedagogical Content Knowledge
Her pedagogical content knowledge will be examined by in the definition from
the literature review. Pedagogical content knowledge is the ability to ?unwrap? and
present mathematical topics so that students can be successful in learning mathematics.
Pedagogical content knowledge includes the ability to access different representations as
well as methods for solving mathematics problems. Pedagogical content knowledge is the
ability to recognize student errors and be able to respond to them with alternative models
and explanations. Pedagogical content knowledge includes the ability to responds to
questions, and to pose questions and problems that are productive to students learning
mathematics.
Mrs. Willoughby?s pedagogical content knowledge as exhibited on the ALCKIN,
seemed to be based on knowledge that was primarily procedural in nature. She tended to
ignore errors in reasoning made by students. During an interview Mrs. Willoughby was
asked to respond to student work on a quadratic equation problem where the student had
graphed both sides of the equation x
2
= 2x + 8 (see Appendix D). Her analysis follows:
(Long pause) I would have never thought of that. I mean because I teach them
what the graph actually looks like instead of what they are doing. Ah, does it
work every time? I would have to check. (Laughs.) Pretty good. I would have
never thought of that. But I see. Ah, Yea because that would be like a system of
equations and where they, yea because where they cross would be, I think so.
Mrs. Willoughby believed students should be allowed to work problems using
methods or procedural steps different from her own. She also suggested that students
could use the guess and check method in finding the solution to the wheels and vehicles
186
problem (see Appendix D). Mrs. Willoughby noted that Seth?s method in question two on
the ALCKIN (see Appendix A) would work for some algebraic fractions. She explained
with ?works only on difference of squares.? Thus, Mrs. Willoughby did exhibit limited
pedagogical content knowledge. She was able to access different forms and
representations of problems by fictitious students both in the interview and on the
ALCKIN. However, she did not respond to the errors made by students on the ALCKIN.
Further clarification of Mrs. Willoughby?s pedagogical content knowledge will be
focused in the following section describing her classroom instruction, as well as her
ability to apply this knowledge in her classroom.
Classroom instruction
Mrs. Willoughby?s content knowledge was largely reflected in mathematics
instruction where students were given step-by-step instructions for working problems.
For example when instructing her students on how to find the distance between ordered
pairs such as (9, -2) and (3, -4), she told her students:
Label these x sub one, y sub one, x sub two, y sub two. (Above nine she placed x
sub one, above negative two she put y sub one and above three she placed x sub
two and followed y sub two above the negative four.) So I?ve got three minus nine
squared plus negative four minus negative two squared. When I put that minus
negative, what do you when you?re subtracting? Add the inverse. ? Every time
she sees subtraction, she automatically thinks add the inverse. So I?ve got the
square root of negative six squared plus negative two squared. That?s the square
root of thirty-six plus four which is the square root of forty. You?ve got to factor
now, right? Forty factors into two times two times two times five. I?ve got a
perfect square so I can factor out two square roots of ten.
After Mrs. Willoughby gave students the ?steps? for simplifying expressions or for
solving equations, she proceeded to work more examples using the same general pattern
described above. Students worked the examples as she worked them at the whiteboard.
After working a few examples she questioned the students as she worked the problems.
Instead of accessing different methods or representations, she stressed that
students should know the steps in each procedure and was often heard reminding students
of what rules they should follow or steps they should take next. For example when
simplifying problems such as )3102)(5325( ?+ using the First, Outer, Inner and
Last (FOIL) method she remarked, ?You can?t multiply a number under the radical
because it is an irrational number not a?. Be careful never, never multiply inside and
outside. Let?s do it in order.?
Step-by-step procedures were also stressed in the following conversation
concerning the Pythagorean Theorem:
Okay. Ah first thing, the first thing we did in this chapter was where we used the
Pythagorean Theorem?Okay a is square root of five, b is six, and we?re going to
find c. Now remember on the Pythagorean Theorem what is c? (Students were
trying to answer but Mrs. Willoughby never slowed up in her talking.) c is what?
It?s the hypotenuse. It?s the one that is always by itself. Okay? So if I?m not
wrong, features you need to keep in mind, unless I specify different, c is the
hypotenuse. So the Pythagorean Theorem is a squared plus b squared is equal to c
squared.
187
188
Mrs. Willoughby?s classroom instruction did not offer students the opportunities
to develop a conceptual understanding of the mathematical procedures they were
learning. Mrs. Willoughby had an average RTOP score of 26.5 from the classroom
observations done during this timeframe of the research study, suggesting her
instructional practices were not aligned with reform mathematical instructional practices.
Thus, Mrs. Willoughby?s content knowledge was reflected in instructional
practices, but limited pedagogical content knowledge was evident. She emphasized
students doing step-by-step procedures instead of accessing different representations and
procedures that students could use.
Student questioning. Mrs. Willoughby did not ask questions or pose problems
productive in helping students learn mathematics. Questions generally prompted students
to remember procedures. For example, the following classroom dialogue took place when
students were solving equations involving radicals.
Mrs. Willoughby: What about the last one? It?s the only one that is a little bit
difficult. We?ve got the radical isolated so we?ve got square root of x plus two is
equal to x minus four. What do we need to do?
Student: Square both sides.
Mrs. Willoughby: So I?ve got x plus two. Is that equal to x squared plus eight? x
squared plus sixteen?
Student: x squared minus
Mrs. Willoughby: x squared minus what?
Student: You have to foil it out. Eight x.
Mrs. Willoughby: Eight x plus?.Okay. Now what do we do?
Student: Subtract
Mrs. Willoughby: Subtract
Another S: Set it equal to zero.
Mrs. Willoughby: Okay. (Pause) So I can do that at the same time so zero is equal
to x squared minus nine x plus fourteen. Okay now we?re factoring. What are two
factors of fourteen, what can we tell about the factor, positive, negative, one of
each?
Mrs. Willoughby also often asked her students to provide answers to problems.
For example when solving for a in the equation 25=a , the following classroom
dialogue took place.
Mrs. Willoughby: Okay. What do we do about a radical, what should we do?
(Pause some mumbling of answers) How do we solve number one? What do we
do? (No pause for students to respond.) We?re going to square both sides because
if we have two radicals but they are both by themselves, right? They are both
isolated. One on one side, one on the other. So if I square this side I?m going to
get?
Student: a
Mrs. Willoughby: a. Okay if I square this side what do I get? (Students can be
heard trying to answer but Mrs. Willoughby continues.) What?s five squared?
Student: Twenty-five
Mrs. Willoughby: Twenty-five times the square root of two squared.
Student: Two
189
Mrs. Willoughby: Okay?
As in previous examples of classroom dialogue, note that Mrs. Willoughby repeated
statements made by her students.
Thus, these example show Mrs. Willoughby questioned her students largely to
remember steps in procedures or to provide answers to problems. These types of
questions are not likely to be productive in increasing students? conceptual
understanding.
Analyzing student errors. Mrs. Willoughby was able to respond to her students
who made mathematical errors or used incorrect procedures. However, she did not
respond with alternative explanations or models. For example, when solving the equation
1310 =? y the class had manipulated the equation to the point of having - 93 ?=y .
The following classroom discussion subsequently took place:
Mrs. Willoughby: Square both sides will give you three y is equal to eighty-one.
Now this is what I do because I look at things a little bit different. The first thing I
do is get rid of my negative and I multiply by negative one. But what happens? I
get the same answer. Okay so you could square both sides. I can see that negative
there and ooh I?m going to get rid of it because I don?t like it. One way is no
better than the other. Then what do we do?
Student: Shouldn?t it be negative three y?
Mrs. Willoughby: No and that?s fine, you get rid of the negative sometimes is
because when you negative square root of three y times negative square root of
three y. What?s a negative times negative? [Shows ( )( )yy 33 ?? =]
190
Student: Positive
Mrs. Willoughby: It?s a positive?
As another example, she recognized incorrect methods suggested by students
when asking for the next step in solving the equation 10736
2
=+? xa .
Mrs. Willoughby: ?And now what do I do?
Student: Square the radical
Mrs. Willoughby: No, I square both sides?
Thus, Mrs. Willougbhy was able to recognize and respond to her students? errors.
However, her responses generally consisted of correcting students, rather than offering
them an alternative explanation or model that might help them to see the source of their
error.
Multiple methods of solving problems. Mrs. Willoughby was able to recognize
when students were using other procedural methods for solving problems which were
different from her own. Two examples follow. First, the following classroom discussion
took place involving solving the equation 12
5
4
=
x
.
Student: I have no idea what I did but somehow I had twenty and twenty-five in
there and still got one hundred and eighty. (One hundred and eighty was the
correct answer to the problem.)
Mrs. Willoughby: Somehow you got twenty and twenty-five and then you still got
one eighty. Let?s see if we can name a new math concept after you. (Mrs.
Willoughby moved to the student, looked at her work and walked to whiteboard
and proceeded to show others what the student had done.) Okay let me show you.
191
I think what you did, I don?t, you forgot what you did but I see exactly what you
did. Okay let?s look at it another way. We?ve got, we?re back to this point. Okay?
Now this is what Augustine, I had Augustine and I?m thinking of somebody
Amber, I?ve got Amber that sits over there. This is what she did. Okay. She did,
same as Jarrett, okay and then she said okay, times square root of five over the
square root of five. She going to rationalize this denominator is equal to twelve.
So she?s got the square root of twenty x over five is equal to twelve. Okay? Then
she squared both sides and got twenty x over twenty-five is equal to one forty-
four. Okay? Is that valid?
Student: Um humph.
Mrs. Willoughby: Of course it is. If we go on and multiply by twenty-five and
divide by twenty we will still get one eighty.
()
144
25
20
12
5
20
12
5
5
5
4
12
5
4
2
2
=
=
?
?
?
?
?
?
?
?
=?
=
x
x
x
x
Mrs. Willoughby: Which is the best?
Student: Shortest one
(Student laughs)
Mrs. Willoughby: The best is what works for you. Okay. Just because I do it
simple and Jarrett went this way and Amber did this way. What happens? We all
192
get the same answer. Okay? So there is no one way where you stop and say okay
that?s it, I can?t do anything else?
Mrs. Willoughby noted when procedures suggested by students were probably not
the easiest ones, such as the one above for simplifying expressions. She stated when they
should not need to use a suggested procedure, even though she did not offer any
explanation for why it should not be used, as in the following:
Mrs. Willoughby: What about this one? What are we going to do? What do I need
to do on that one?
39
5
4
=?
x
Student: Separate (Indicates separating
5
4x
into
5
4x
) the four x and five.
Mrs. Willoughby: Separate. We?re not going to do that now. How am I going to
get this nine out of my way?
Another student says: Square both sides.
Mrs. Willoughby: Okay we?re going to square this but do we want to square it
first? (Student mumbling) I?m going to add nine. I knew folks were thinking
about it and I wanted you to say it for me. Okay. So we?ve got the square root of
four x over five is equal to twelve. Now we?ve got a radical is equal to something.
We square both sides. If I square this side what do I have?
Although the following classroom discussion shows three different procedural
approaches to simplifying the expression given to the class, it demonstrates that Mrs.
193
194
Willoughby recognized and took instructional time to show students each of the methods.
The work shown on the whiteboard during the discussion is interspersed:
Mrs. Willoughby: There are quite a few ways to approach this problem. (Refers to
the following problem.) Okay. Let?s start with the first time, let?s separate the
radical. Okay? We?re going to do it that way. I?m going to do it a couple ways
and let you see what you like. So I?ve got the square root of five x to the fourth
divided by the square root of four n to the fifth. Now can I take the square root of
four?
Student: Two
Mrs. Willoughby: Yea, it?s two but I can do this I can take the square root of five
x to the fourth divided by two n square roots of n, right? Is that okay? Does
everyone agree with that?
Student: n squared
Mrs. Willoughby: n squared instead of n. Ya?ll okay with that? Then what would
we have to do?
Student: Multiply by the square root of n
Mrs. Willoughby: Multiply by the square root of n. We do not want a radical on
the bottom. So we?ve got the square root of five x to the fourth n divided by two n
cubed. n squared times n.
3
4
2
4
5
4
5
4
2
5
2
5
4
5
4
5
n
nx
n
n
nn
x
n
x
n
x
?
Mrs. Willoughby: Is it simplified?
Student: No
Mrs. Willoughby: What do I need to do? What?s wrong with this?
Student: Too many x?s
Mrs. Willoughby: There?s too many x?s up there. I can take the square root of that
and what do you?x squared times the square root of five n over two n cubed.
That?s one way to do it. Okay.
3
2
3
4
2
5
2
5
n
nx
n
nx
=
(Goes back to the same problem for other suggested ways of simplifying the
problem.)
Mrs. Willoughby: Alright Jarrett you don?t like my work.
Jarrett: That?s not the way I worked it.
Mrs. Willoughby: Okay
Jarrett: I got the wrong answer.
Mrs. Willoughby: What?s another way to do it?
Jarrett: Separate the radical
195
196
Mrs. Willoughby: You want to separate what?
Jarrett: Like the square root five x, the square root of five x to the fourth over four
n to the fifth.
Mrs. Willoughby: Now what do you want to do Jarrett?
Jarrett: Then multiply square root of four n by
Mrs. Willoughby: Okay so we?ve got the square root of twenty x to the fourth n to
the fifth over four n to the fifth, right? We?re in good shape. That?s right Jarrett.
But where did you go from there because we?ve got to get an answer. Okay let?s
do a prime factorization of twenty, how would you factor. Two goes twenty ten
times, and two goes in five times. So I?ve got two times two times five times x
four times and y, I mean n five times. Okay so I?m going to circle my pairs. So
I?ve got two x square n squared and I?ve got five n under that radical over four n
to the fifth. Did you get that? (Could not understand what the student said.) Okay,
now look at it, is it simplified? The simplest it can be is simplified. What else can
I do to this? (Pause) Can I simplify that? Okay what do I do?
Jarrett: the n
Mrs. Willoughby: the n, I?ve got two n?s up here and five down here, right? So
the rule says I would have x squared n squared minus five over two so I?ve x
squared n to the negative three square roots of five n over two, right? So x
squared square root of five n over two n cubed. So you were on the right track you
just didn?t take, you didn?t.
Jarrett: simplify.
Mrs. Willoughby: You didn?t go on and get it in simplest form. (Student laughs)
Okay Jarrett remember if you got a n in two places, something is wrong unless
one?s under the radical, if they?re both outside the radical then you?ve got to do
something. You don?t want an n in two places, top and bottom unless one is under
the radical. Okay?
3
2
522
5
22
5
54
5
5
5
4
5
4
2
5
2
5
4
52
4
20
4
4
4
5
4
5
n
nx
nnx
n
nnx
n
nx
n
n
n
x
n
x
?
?
Mrs. Willoughby: So yea you were doing it right. You just started out and stopped
way to early. Okay?
Thus, Mrs. Willoughby did acknowledge different methods used by her students
even though they were variations of the same general procedure. She also took
instructional time to demonstrate these methods to students. However, this is a limited
way of accessing different representations or different models.
Conclusion
Mrs. Willoughby felt that all of her students could learn algebra. She exhibited
strong procedural knowledge but limited conceptual knowledge of the algebra content.
She used procedural explanations on a majority of the items on the ALCKIN and felt
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198
students who computed slope had a better understanding. She showed two algebraic
expressions were equivalent using procedures (which she was asked not to use in the
instructions) and agreed that Seth?s method worked only on certain algebraic expressions.
Mrs. Willoughby was able to work the quadratic problem on the ALCKIN using the
spreadsheet.
Her classroom instruction exhibited her content knowledge as she provided step-
by-step instruction of procedures. However, the student methods recognized by Mrs.
Willoughby as ?different? from her own but were variations of the same procedure and
were not accessing truly different methods. Mrs. Willoughby recognized and responded
to errors of her own students but did not respond with alternative explanations and
models. Students were asked questions related to answers for arithmetic procedures,
simplification of algebraic expressions, or telling her the steps in algebraic procedures.
These types of questions were not the type that would be productive to successful
learning of mathematics by her students. Therefore, Mrs. Willoughby exhibited limited
pedagogical content knowledge in her instruction.
Professional Development
Mrs. Willoughby attended the geometry sessions during the two weeks of summer
of 2004 professional development provided by MDMSIP. The general topics covered in
the geometry sessions were the same as in the Algebra I sessions. Mathematics related to
algebra covered within the IMP units (Fendel et al., 2000) used during the geometry
sessions included proportional reasoning and constructing ?in-out? tables, as well as
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discussions on how to introduce students to the terminology related to the IMP units
(Fendel et all, 2000).
When Mrs. Willoughby was questioned about the professional development she
received during the summer professional development training, she stated,
Mrs. Willoughby: Overall I was very impressed with it because to start I did not
think it would work and I had some ?., now I?ve not used some of the IMP units
like they did but I have used the concepts of those and used the discovery and
have definitely found out that it is for the lower students as well.
Interviewer: What was your attitude toward attending the summer professional
development training?
Mrs. Willoughby: It was very lousy. I just thought, I just thought I can teach and I
didn?t think that that changing was, I didn?t see anything wrong with what I did.
And so I thought that it was just a waste of my time and that I was just going for
two weeks and in my journal I started out letting them know what I thought and
then you know after the first week I thought you know this is pretty good and Dr.
Thompson (MDMSIP presenter) even said he could tell the difference in what I
wrote in that two weeks, how I changed from thinking you know they?re wasting
my time to, you know, I can see where I can use this.
One of Mrs. Willoughby?s presenters of professional development commented on
her attitude as well as her engagement in the institute during the two week period.
Ah, to begin with Veronica was very resistant about any of the possible methods
that we were discussing. She was openly resistant. She said I?m here but I don?t
know that I?m going to get anything out of it. As the days progressed and as the
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activities progressed she began to see how this could work in her classroom and
by the end of institute she was very excited about going back to the classroom and
trying some of these things. She was very engaged. She got involved in all of the
activities. She asked questions about well, how would this work in a regular
classroom and what kind of modifications could be made, and she made
suggestions about how she might be able to use things herself, rather than asking
?What can I do?? It was a drastic change. She came in very negative, not wanting
to learn anything, not wanting to get anything out of the workshop, but she left
with a lot of new techniques, a lot of new things to try and she was very positive
at the end of the workshop.
This same presenter also felt that Mrs. Willoughby would implement the instructional
strategies learned during the two week professional development in her own classroom
practices.
During the follow-up quarterly meeting Mrs. Willoughby participated in the
sessions and shared strategies that she found successful during classroom instruction. For
example, she shared the following after one presenter began a lesson plan with algebra
tiles.
The tiles are absolutely wonderful. The algebra tiles are absolutely wonderful! I
even have students now who will sketch out on an algebra test the problem (Mrs.
Willoughby does an algebra tile drawing on the board) who don?t have the rules
down but they get the right answer. On their quiz, I tell them I don?t want them to
just factor but I want them to draw it. You go from the tiles to factoring.
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Mrs. Willoughby expressed that before the professional development she did not
know how to use algebra tiles.
At another quarterly meeting, Mrs. Willoughby used algebra tiles to represent the
following expressions along with different colors to represent the positive and negative
tiles. She drew the tiles and stated:
What is that? (Referring to her drawing.) The students will come up with
2x
2
? 2x + 2
-x
2
+ 3x + 1
(They can zero out a pair of similar tiles that have different colors.) Then they can
zero it and come up with x
2
+ x + 3. Different shapes and different colors. I
discovered algebra tiles this year. I did not know how to use them before this
year. I solve equations with them. I started them off with positive and negative
numbers. They came up with the rules, but if they forget the rules they can go
back and draw the circles with + and ? in them.
Mrs. Willoughby expressed that she felt the methods learned at the summer
institute were more helpful for some of her students than others. She stated:
You know I think that the things I learned during the summer at institute helped
me more with my ah (Algebra) A/B classes and my remediation class. I mean my
remediation class I used hands on with them so much to introduce topics and, you
know, I had kids tell me after, you know, after we did algebra tiles, I had a little
boy come and tell me that that was the first time he understood what it meant to
multiply binomials.
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One of the presenters gave this assessment of Mrs. Willoughby?s algebraic
knowledge:
I think she has a very good content knowledge of algebra one. I know she?s taught
algebra one also and she could make the connections between the geometry and
the algebra one. I?m having trouble remembering exactly what we did that
actually would have centered on algebra since this was a geometry workshop ?
But we did do some, we looked at patterns, we looked at in out tables and how to
introduce students to the terminology of the IMP where they used in out tables,
which is something the kids probably had not called that before, and so we did do
that in the workshop but we basically added the content.
Thus, professional development presenters felt Mrs. Willoughby possessed good
content knowledge of algebra and could make connections between the algebra content
and geometry content. Presenters also felt she would implement changes into her
instructional practices. Changes in content knowledge and pedagogical knowledge were
demonstrated through her ability to use algebra tiles to model algebraic manipulations
and willingness to use them in her instruction.
Post-Professional Development
During the post-professional development timeframe, changes in several areas
will be addressed, including classroom environment, her views of student ability, content
knowledge, and pedagogical content knowledge. How the changes in content knowledge
and pedagogical content knowledge are reflected in instructional practices will be
addressed. Each of these areas will be addressed in the following sections.
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Classroom Environment
The classroom layout remained unchanged, and no changes not noted in Mrs.
Willoughby?s demeanor. However, student work from the IMP units (Fendel et al., 2000)
was displayed on the classroom walls, evidence that some changes in instructional
practices may have occurred.
Student Ability
When Mrs. Willoughby was questioned about how well her students could learn
algebra, she commented:
I went to the counselor first, after the first test and said some of the students did
not need to be in the advanced (Algebra I) class because that?s a fact they needed
to be in the more hands on Algebra A/B and they struggled?
She also felt that the students who are in the Algebra IA and Algebra IB classes could
also learn algebra but maybe not at the same pace as those students in the Algebra I
classes. She stated, ?You know I think it all, I think it more depends on the individual
student and the background they have. Whether they can learn at the pace that we set in
the advanced classes (Algebra I).?
Thus, Mrs. Willoughby feels all of her students can learn algebra, no matter which
section of algebra they are taking. She did note that her students in Algebra IA and
Algebra IB would need to learn algebra at a different pace. Moreover, she seemed to
think that use of ?hands-on? methods was more important for the lower level courses than
for Algebra I.
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Changes in Content Knowledge
Mrs. Willoughby felt that she had good algebra content knowledge, stating, ?I?m
real comfortable with algebra. I enjoy algebra and kids don?t seem to find it as exciting as
I do. I enjoy it. I enjoy algebra.? Mrs. Willoughby did not feel she learned any new
algebra content as a result of the professional development, pointing out:
Well, they did the Pythagorean Theorem this summer in and so I did not see, I
was not able to teach it this last nine weeks, so I didn?t get to use but I have
introduced the Pythagorean theorem by one of the games that we used in the
geometry.
When asked to solve a quadratic equation during her final interview, Mrs.
Willoughby was not only able to solve the equation by setting the equation equal to zero,
factoring the quadratic expression, and setting each factor equal to zero but she was able
to offer other methods that could be used on the same problem. For example, she stated,
?Let?s see you could do it where, now I could do it with x squared minus two x minus
eight is equal to zero and x squared and (Pauses while she is working) and these are, I?ve
got two negative x?s, a negative?.I could do it with algebra tiles? and provided a
drawing for her method. Other techniques were also suggested:
Mrs. Willoughby: Well you could use the quadratic formula, you could graph it.
Interviewer: Okay so what do you mean by graphing it?
Mrs. Willoughby: To me this is more algebra two. It?s just choose points and for x
and solve and if you know it?s a parabola and you know what a parabola looks
like and I for one note that they would have to randomly choose points and they
could get it.
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Thus, Mrs. Willoughby demonstrated different ways of representing and solving
mathematical tasks using a variety of techniques, both procedural and conceptual. This
indicated a growth in her content knowledge from the pre-professional development
phase.
Changes in Pedagogical Content Knowledge
Mrs. Willoughby praised students when they used methods that she herself had
not thought of using. For example, during a problem in which students were given the
ordered pairs for two points and asked to find the equation of a line that contained these
two points, Mrs. Willoughby commented:
You know but if I see something new like Marquette, I saw how he got that
problem today. I never thought of about. All he had was two points and I, but one
of the two points was the y intercept and I never thought about that?s a way to get
the slope and he?s real bright and you know I enjoy, but I thought you know
something different, that?s something that because I like to, you know, and I don?t
if what I do is right or wrong because I like to show different techniques for the
same, to do the same thing because different people think different ways and so I
like to know different ideas and that?s what I try to. That?s what I want to get
from MDSMIP is some different ways to teach the same things we?ve taught.
Thus, Mrs. Willoughby was able to understand a different representation her student had
for finding the equation of the line.
Mrs. Willoughby indicated she used algebra tiles in developing an understanding
for understanding why algebraic rules she used. She said:
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I picked it up (algebra tiles) somewhere else because I saw that the that the hands
on worked ? it?s because I tried to show them what I was doing before I put,
actually put it on the board and I would give them a test, a quiz on multiply this,
drawing out the tiles because I would tell them you can?t use these on the exit
exam. So we did a lot and they would come in, are we going to be able to do
algebra tiles.
Thus, Mrs. Willoughby demonstrated access to different representations as well as
different methods for developing an understanding for algebraic rules. This suggests
growth in her pedagogical content knowledge
One error that was pointed out in the mathematical knowledge of Mrs.
Willoughby on the ALCKIN also was manifested in the knowledge constructed by her
students. A group of students had concluded that any number to the zero power was equal
to one. This was written by students on their displayed work from the IMP Alice unit
(Fendel et al., 2000). Thus, her lack of understanding may be reflected in the mathematics
her students learn. Thus, Mrs. Willoughby did exhibit growth in pedagogical content
knowledge as evidenced through the demonstration of how to use algebra tiles and build
the bridge between the physical models, to drawings, and to developing an understanding
for the rules.
Reflection of Changes in Knowledge in Instructional Practices
Mrs. Willoughby had reported using hands on techniques with her classes;
however, this was not observed. When asked if she was had used these same types of
strategies with the particular Algebra I class that was observed, she responded:
207
With this class in eighteen weeks to cover the course of study, I did very little. I
did probably maybe four or five activities. I mean you know but not near as much
as with the geometry class where with the geometry B that I?ve got where you
have eighteen weeks with half the book. In eighteen weeks to cover everything in
the algebra one content you don?t have time to do a lot of activities. I did Alice
with them, that was one I used with them, and I used probably about four or five
different activities in that.
Her follow-up RTOP score during the spring of 2005 was 41, which was
somewhat higher than her RTOP score from the previous spring and suggests that some
instructional change may have occurred. The same comments made about how her
content knowledge and pedagogical content knowledge were reflected in her instructional
practices prior to professional development could be said about these types of knowledge
post-professional development.
Thus, no changes in her content and pedagogical content knowledge were actually
observed in her instructional practices. Although she indicated she had used algebra tiles
with some of her classes, there was no evidence in the follow up observations that Mrs.
Willoughby had made changes within the algebra one class.
Conclusion
Mrs. Willoughby possessed content knowledge which includes strong procedural
knowledge and limited conceptual knowledge. Mrs. Willoughby had limited pedagogical
content knowledge Growth in mathematical processes was observed through problem
solving in different ways and using different representations such as algebra tiles. It was
expected that Mrs. Willoughby would make instructional changes as a result of
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professional development. An error made by Mrs. Willoughby was manifested in the
work constructed by her students. Growth in both her content knowledge and pedagogical
content knowledge resulted from professional development. Although growth in content
knowledge and pedagogical content knowledge was demonstrated in professional
development activities, it was not reflected in her instructional practices.
Conclusion of Case
Mrs. Willoughby felt that all of her students could learn algebra but maybe at
different paces. She felt she had good algebra content knowledge, and while she
exhibited strong content knowledge, evidence of pedagogical content knowledge was
limited. Evidence of growth in both of these types of knowledge was seen in the post-
professional development timeframe. Mrs. Willoughby demonstrated the use of algebra
tiles during the quarterly meetings and discussed how to use them in instruction.
Thetudent work displayed also suggested she was implementing instructional changes.
However, how these changes in her content knowledge and pedagogical content
knowledge were actually reflected in her instructional practices was not clear, because it
was not evident in the classroom observations I did.
Mrs. Pitchford
Mrs. Pitchford is a Black female, and she was in her fourth year of teaching
mathematics when this study began. She was shorter than average and had a normal
speaking voice that was easily heard in her classroom. Mrs. Pitchford held a Bachelor of
Science degree, as well as a Master?s degree in mathematics education. During this
research study, Mrs. Pitchford was working on her Education Specialist?s degree in
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mathematics education. At Clarion High School, a ninth through twelfth grade school,
her teaching responsibilities included Algebra I. Clarion High School had a passing rate
of 41% on the Alabama High School Graduation Exam, the student population was 100%
Black, and 61% of its students were on the free or reduced lunch program.
Pre-Professional Development
In this section, we will consider Mrs. Pitchford?s classroom environment, how she
viewed her students? ability to learn algebra and responses to statements contained on the
Teacher Attitude Survey. Next, Mrs. Pitchford?s content knowledge and pedagogical
content knowledge will be addressed from the ALCKIN and interview. Finally, a further
look at her content knowledge and pedagogical content knowledge will be considered
from classroom observations, which includes how these types of knowledge were
reflected in her instructional practices.
Classroom Environment
She generally sat at her desk while her students worked in pairs or small groups.
She instructed her students that one person from each pair or group could come and ask
her for help when they needed it. Only once during any classroom observations was she
observed moving between the pairs and small groups to observe what the students were
doing and to answer their questions.
The student desks in Mrs. Pitchford?s classroom formed two sets of straight rows,
each set facing toward the center of the classroom. This type of setting might suggest that
Mrs. Pitchford used a traditional approach to teaching mathematics. However, since there
was very little mathematics instruction, classroom observations did not contradict nor
confirm this belief. The back wall contained a whiteboard with the teacher desk at the far
210
corner of this particular wall. From this reference point, the right hand wall had a
whiteboard while the left hand wall contained windows. The front of the room contained
a television, computer, overhead projector and pull down screen. Posters depicting
famous Black Americans and pertinent information about each adorned the remaining
wall space in the classroom.
Mrs. Pitchford encouraged her students to borrow calculators for classroom use.
During one classroom observation she made the following statements.
?It?s not three times two, it?s three, it?s negative three squared. That?s negative
three times negative three. That?s why I told you?re going to need a calculator
tomorrow. You?re going to need it because when you do the square root, you?ve
got to take the square root. It?s not always a perfect square root for the number.
That means it doesn?t work out. That?s what I?m going to be looking for
tomorrow. So how many of you have found you a calculator, a scientific
calculator? What are ya?ll going to do tomorrow? Huh?
Student Ability
Mrs. Pitchford felt only a little more than half of her students could learn algebra
well because ,?I was looking at them as (having) more general math skills.? In regards to
the other students who she did not feel could learn algebra well, she indicated that they
were lacking in the mathematical skills they should have already learned in previous
mathematics classes. Thus, she largely saw their success as depending on their prior
knowledge.
211
Teacher Attitude Survey
Mrs. Pitchford was also asked to reflect on two conflicting statements from the
TAS. She was first asked to respond to, ?It is important for students to figure out how to
solve mathematics problems for themselves.? She indicated that this was a part of her
instructional practice and stated, ?By letting them work individually especially with
practice problems and really trying to get them to use math, mental math rather than
depending on their calculators.? In response to ?Teachers should model and demonstrate
mathematical procedures then ideally time should be allowed for students to have the
opportunity to practice these procedures? and how the two statements can coexist in the
mathematics classroom, she replied, ?That?s how, that?s how I really, how I teach my
class. By modeling you know the examples and showing them how to work them and go
through the steps and then after I?ve done this, I let them, you know, practice on working
the problems themselves.?
Thus, Mrs. Pitchford felt she was allowing students to figure out how to solve
mathematics for themselves by modeling and demonstrating examples. She then allowed
students to practice these same types of examples.
Content Knowledge
Mrs. Pitchford?s content knowledge will be viewed through the lenses of
procedural knowledge, conceptual knowledge, and mathematical processes. Each of these
will be addressed in the following sections.
Procedural knowledge. Mrs. Pitchford correctly answered 9 of the 25
mathematical tasks on the ALCKIN, suggesting weak overall knowledge of algebra. She
exhibited significant weaknesses in her procedural knowledge. On parts B and C of
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question one (see Appendix A), she agreed with the students that neither of the situations
represented a function and offered no explanations to defend her incorrect answers
suggesting she may not understand functions.
When presented with the wheels and vehicles problem (see Appendix D), she was
unable to find any method to correctly solve the problem. She responded with the
following:
Mrs. Pitchford: (Pauses) I think I would, seventeen vehicles, fifty-six wheels, and
you know that cars have four wheels and you would divide four into the fifty-six
and what is left should be the two wheels on the motorcycle. That?s how I think I
would teach.
Interviewer: Okay so will you work it for me. Just work it on there (indicated the
sheet of paper that the problem was written on) will be fine.
Mrs. Pitchford: (Pauses) Humph. Still not going to balance out. It going to come
out to be fourteen. (Pauses) Humph. That?s the only logical reason I could see
how to, how to figure it out. You know a car has four wheels.
Thus, Mrs. Pitchford exhibited weak procedural knowledge in selecting answers
on the ALCKIN. She was also unable to solve the wheels and vehicles problem using any
procedure.
Conceptual knowledge. Mrs. Pitchford also exhibited limited conceptual
knowledge. On question six on the ALCKIN she was unable to provide a method for
showing that the two algebraic expressions were equivalent and left the question blank
(see Appendix A). From question eight on the same instrument, she selected the group
using the tangent of the angle as the only group which had an understanding of slope.
213
Since she elected not to support her answer with an explanation, it is not clear why she
selected this particular group. From question ten on the ALCKIN, she explained the 0.2x
with ?the # of chocolate chip cookies that can be baked in 15 hrs? which is not what the
expression represents within in the equation. These examples suggest that Mrs. Pitchford
had very limited conceptual understanding.
Mathematical processes. In question five on the ALCKIN (see Appendix A) Mrs.
Pitchford was unable to correctly provide a solution to the quadratic problem using part
of a spreadsheet. She substituted -1.8 into the expression 4-2x and got 6.7. She explained
her answer as follows: ?Student?s should use the value of x in each place in x in column?s
2 & 3.? This example suggests that Mrs. Pitchford had limited ability to use mathematical
processes.
Pedagogical Content Knowledge
Very little could be determined about Mrs. Pitchford?s pedagogical content
knowledge. Since few of her answers on the ALCKIN were supported with written
explanations, it was difficult to assess her pedagogical content knowledge.
In the interview, Mrs. Pitchford was asked to respond to the student work
involving graphing both sides of the quadratic problem x
2
= 2x + 8 to obtain its solution
(see Appendix D). It wasn?t clear whether she understood what the student had done and
she responded with the following:
Interviewer: ?Here?s the first one and tell me what you think about what this
student did.
Mrs. Pitchford: (Pauses ? Long Pause) Humph.
Interviewer: What do you think of that?
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Mrs. Pitchford: If I presented it to my class, it would be difficult for them.
Interviewer: So do you think it is an okay way to work it?
Mrs. Pitchford: It?s an okay to work it but I think more or less they, I believe they
let the computer do it. (Laughs) That?s what I am thinking, I mean you know.
She was unable to provide a method of solving the wheels and vehicles problem
during the interview. However, she recognized that both of the elementary methods she
was asked to analyze were suitable ways of solving the same problem (see Appendix D).
Thus, Mrs. Pitchford exhibited limited pedagogical content knowledge in the
ALCKIN and interview. Further assessment of her pedagogical content knowledge was
done through her classroom instruction which will be detailed in the following section.
Classroom Instruction
There was little evidence of pedagogical content knowledge in her classroom
instruction. During the majority of the time students put homework problems on the
whiteboard, worked in pairs or small groups with little instruction provided by Mrs.
Pitchford. When procedural instructions were given, it was apparent that the students had
already previously worked on these same types of problems. For example:
Mrs. Pitchford: Does anyone remember what we are doing?
Students: (Many different answers are given.)
Mrs. Pitchford: Add exponents. (She wrote a
2
(a
3
)(a
6
) on the whiteboard.) Multiply
ones, add exponents. (She wrote a
2+3+6
and a
11
. )
(Without any additional comments, she moved to another problem and wrote
(l
2
k
2
)(l
3
k) on the whiteboard,
Mrs. Pitchford: Separate them. (She wrote l
2+3
k
2+1
= l
5
k
3
.).
Mrs. Pitchform: Next problem (10
2
)
3
(Even though she asked the students for the
next step she wrote 10
6
and continued.) You?re just going to multiply your
exponents.
Mrs. Pitchford typically gave her students verbal instructions on what steps to use,
then simply demonstrated solutions to sample problems on the whiteboard. For example:
Mrs. Pitchford: These are multiple step equations so you use opposite operations.
(She then proceeded to work the problem on the whiteboard for the students.)
2
3
12
3
3
612663
43
4
63
4
3
4
63
=
=
?=?+
?=
+
?
=
+
x
x
x
x
x
Thus, Mrs. Pitchford was able to use her content knowledge to work these particular
problems. However, she was not posing problems productive to learning or presenting
them in ways so students could successfully learn.
When a student requested that a particular problem be worked, Mrs. Pitchford
frequently exhibited weak content knowledge. Furthermore, she offered no explanations
for the procedures she used. For example, when asking students if they had any questions
about the problems on a worksheet, this exchange followed:
Mrs. Pitchford: What about the quadratics?
Students: Quadratics?
Mrs. Pitchford: Starts on the second page.
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216
Student: Seventeen, no number eighteen.
Mrs. Pitchford: (Writes 4x
2
? 11x ? 3 = 0 on the whiteboard.) You?re going to get
it in the foil method. You?re going to factor it out. (Writes on the whiteboard:
(4x 3)(x 3) = 0. She then erased the 3 in the first parentheses and wrote (4x 1)(x 3)
= 0, Followed by 4x
2
12x 1x 3.)
Mrs. Pitchford: We want a -11x so 12x has to be negative. (She wrote, 4x
2
? 12x
+ 1x ? 3.) You come back to the original 4x
2
? 11x ? 3. What signs go in the
parentheses?
Students: (Guess both positive and negative.)
Mrs. Pitchford: (Puts a subtraction sign in front of the three and an addition sign
in front of the one. Fills in original parentheses and writes (4x + 1)(x ? 3) = 0.)
Thus, even though Mrs. Pitchford?s exhibited correct content knowledge in her
instructional practices, she exhibited weaknesses in this knowledge.
In addition, Mrs. Pitchford indicated to the students when a procedure should be
used on all the assigned problems. For example, in a lesson where students were
simplifying expressions involving the power rule for exponents, the following dialogue
took place:
Student: Hey don?t you multiply the whole numbers? (Called Mrs. Pitchford over
to her desk.) Do you multiply or add?
Mrs. Pitchford: Multiply. You multiply on all of these. (To another student.) The
only time you multiply is when you have a number on the outside.
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Simply telling students what to do is not presenting mathematics so that students can
successfully learn it.
Mrs. Pitchford had an average RTOP score of 9 1/3 based on classroom
observations during the spring of 2004 suggesting that her instructional practices were
not at all conducive to helping students develop a conceptual understanding of the
mathematics that was taught during classroom instruction.
Student questioning. Very little dialogue took place in Mrs. Pitchford?s classroom.
Thus, it was difficult to determine evidence of her pedagogical content knowledge in her
questioning. The typical types of questions she generally asked did not require much
mathematical thought and could be answered with short answers. For example:
Mrs. Pitchford: Factor it out, you know two times three and one times six is six.
We want to get seventeen in the middle. We know what numbers multiply to
seven?
Student: one and seven
In another classroom observation Mrs. Pitchford asked, ?What is the square root of thirty-
six?? and the student answered ?six.? That was the extent of the interaction.
Thus, there was very little opportunity to observe how Mrs. Pitchford responded
to questions of her students. In addition, Mrs. Pitchford was not observed posing
problems and questions that would be productive to student learning.
Analyzing student errors. There was little evidence that Mrs. Pitchford could use
her mathematical knowledge to recognize the errors of her students. She had difficulty in
responding to students who had made errors. For example, a student went to the
whiteboard and showed the following work:
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Student: -4(cs
2
t)
2
-4(c
2
s
4
t
2
)
-4c
2
-4s
4
-4t
2
Mrs. Pitchford: Who did this? The answer should be negative four c squared s to
the fourth t squared. I don?t know where you got all those extra negative fours.
Thus, Mrs. Pitchford had difficulty in recognizing where and why her students
made errors. When she did recognize an incorrect answer, she did not understand why the
student made the error nor did she respond with alternative explanations and models.
Multiple methods of solving problems. Mrs. Pitchford was not observed working
mathematical problems using different methods nor did she encourage her students to use
alternative ways of solving problems during classroom instruction.
Conclusion
Mrs. Pitchford expressed that not all of her students could learn algebra well
because they lacked the necessary general mathematics skills needed for learning algebra.
Mrs. Pitchford showed limited procedural knowledge and limited conceptual knowledge.
Mrs. Pitchford was observed on more than one occasion making mathematical errors or
showing a lack of understanding for the algebraic topics she was teaching. Errors were
also exhibited in the few explanations that she provided on the ALCKIN and neither did
she assess any of the errors in the statements made by students on the same instrument.
She exhibited difficulty in recognizing why students made the types of mathematical
errors they were making. Her classroom instruction consisted of showing students
procedures to work problems while giving very little verbal instruction, displaying very
little evidence of pedagogical content knowledge. Mrs. Pitchford asked her students
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questions that required primarily required them to use arithmetic. It appeared that Mrs.
Pitchford lacked the ability to solve problems using a variety of methods and there was
no evidence that working problems in different ways was valued in classroom instruction.
Professional Development
Mrs. Pitchford did not attend the two-week MDSMIP professional development
training with the other teachers from her school. Instead, she participated in classes at a
local university that were required for her education specialist degree in mathematics
education. In the interview, she was asked to describe the classes she had completed and
how these classes had promoted any change in her classroom instructional practices. She
gave no indication that these classes helped her make any changes. Mrs. Pitchford said,
?From the classes, well the classes I?m taking right now are dealing with curriculum. I
took an advanced curriculum class which I had to design my own curriculum and then the
other class I had was a statistics class.? She further indicated that what she had learned in
her statistics class could not be used in the classroom:
Those higher level classes I?m taking (laughs) higher level math classes have
nothing to do with algebra to be honest with you. I mean I?m taking like I said six
hundred level courses but they?re more or less like this statistics class it?s dealing
with variance and this here kind of stuff that I don?t get to present to them.
Thus, Mrs. Pitchford indicated that the courses she was taking for her advanced degree
were not beneficial to her instructional practices.
Post-Professional Development
We will now consider changes that occurred during the post-professional
development timeframe, including changes in classroom environment, view of student
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ability, content knowledge, and pedagogical content knowledge. We will also consider
how the changes of content knowledge and pedagogical content knowledge were
reflected in changes in instructional practices. Each of areas will be addressed in the
following sections:
Classroom Environment
Mrs. Pitchford exhibited the patterns of behavior during this timeframe that she
did during pre-professional development timeframe. The classroom layout remained
unchanged. There was no other evidence of change in the classroom environment.
Student Ability
Mrs. Pitchford felt a majority of her students could learn not algebra well.
Regarding the ability of her students in her algebra classroom who she did not feel could
learn algebra well, she stated:
To be honest with you they are terrible. You build upon and you have to come in
with something to, you know, to get something. I?m seeing children come in here
without any type of computation skills, division, just basic math skills. They don?t
have, they?re not having it. They?re not having the mental skills that they need to
have to work problems quickly without having their hands on a calculator, but if
you don?t know how to put the information into the calculator, it?s still not going
to come out correctly.
Thus, Mrs. Pitchford felt her students do not have the necessary computational skills to
learn algebra well. She stated that these skills were necessary, so that she could build
upon them to help her students learn algebra.
Changes in Content Knowledge
When questioned during an interview in the spring of 2005, Mrs. Pitchford stated
that she was confident that she knows the algebra content well. She commented with
?Oh, very well.? However, continued weaknesses in her content knowledge were still
observed.
In the interview, Mrs. Pitchford used a set procedure to correctly solve the
quadratic problem x
2
= 2x + 8 she was given (see Figure 21) and provided a second
procedural way for students to solve the problem involving using the quadratic formula.
Figure 21. Mrs. Pitchford?s work on the quadratic problem.
Her third method for solving the problem reflected incorrect mathematical knowledge:
Interviewer: Can you think of a third way?
Mrs. Pitchford: Do ah, take the square of both sides, yea take the square of both
sides and you might be able to do it that way. I think. I?m not sure.
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Thus, while Mrs. Pitchford was able to correctly solve the mathematical task this
time in two different ways, she suggested a third method that did not reflect correct
mathematics.
Changes in Pedagogical Content Knowledge
In one classroom observation in the spring of 2005, students had apparently been
working on using the quadratic formula to solve equations. Mrs. Pitchford began the class
as follows:
Mrs. Pitchford: Now what I?m going to do is I?m going to put the quadratic
formula on the board and also the formula you use to find the determinants. Now
the determinant is easy. That?s why I could not figure out why ya?ll could not do
those problems where you were finding the determinant. Now that homework I
gave you, ya?ll remember that is some of your quiz grade. So did ya?ll do it?
Students: Yes
Mrs. Pitchford: Alright everybody turn it in?
Student: No
(On whiteboard Mrs. Pitchford had written.)
-b ? 4ac
a
acbb
x
2
4
2
???
=
Mrs. Pitchford: When you find the determinant, when you?re trying to find out
how many roots you have this is all you use. (Indicates the first expression
written.) Right here, I believe or is it b squared?
Students: b
2
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Mrs. Pitchford: Are you sure?
Student: No that?s right, that?s right.
Student: b squared
Mrs. Pitchford: (Picks up a textbook, looks it up and changes the b on the
whiteboard to b
2
? 4ac.) b squared minus four a.
Mrs. Pitchford did not appear to know the algebraic expression for the determinant and
the relationship between the determinant and the quadratic formula since she had
correctly written the quadratic formula on the whiteboard.
Thus, no particular changes were evident in Mrs. Pitchford?s pedagogical content
knowledge. However, she did exhibit weakness in making connections between
mathematical topics.
Reflection of Changes in Knowledge in Instructional Practices
Mrs. Pitchford had a RTOP score of eleven on the classroom observation done
during the spring of 2005 which was about the same as the previous spring. This might
not be surprising since no changes were observed in her content knowledge or
pedagogical content knowledge. Mrs. Pitchford generally sat at her desk and gave verbal
instructions when her students had a question. When students gave incorrect solutions for
the problems they were assigned to work on, the following exchange resulted:
Mrs. Pitchford: Are ya?ll finished with number three? What is the answer?
Student: Negative two point sixty-one.
Teacher: No
Students: (Other answers are called out.)
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Mrs. Pitchford: Didn?t you have a negative number under the radical? You?ve got
to add the?. What did you have up under the radical?
Students: (Mumbling)
Mrs. Pitchford: Sixteen and what?
Student: Sixteen and thirty-six? (More student mumbling)
Mrs. Pitchford: Sixteen minus twenty-four is equal to what? Negative what?
Student: No
Mrs. Pitchford: What is sixteen minus twenty-four?
Student: Negative eight.
Mrs. Pitchford: Negative eight! (Student talking) Any time you end up with a
negative number up under the radical it?s no solution. Because you are going to
end up with an imaginary number. So you draw the little circle and put a slash
through it and means no solution to that problem. So any time you come up with a
negative number under the radical it?s no solution, you stop. Any questions about
that? (No questions.) Okay did you guys move on to number four?
Conclusion
Mrs. Pitchford did not participate in the MDSMIP professional development.
Rather, she was involved in taking classes for her education specialist degree in
mathematics education. She did not feel these classes helped her in making changes in
her instructional practices, and no real changes were observed. The comparison of the
RTOP scores from the spring of 2004 and the spring of 2005 were almost the same. In
classroom instruction, Mrs. Pitchford used the same general types of instructions as she
has used the previous spring. She continued to struggle with the algebraic content,
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making significant algebraic errors, and no changes in content knowledge or pedagogical
content knowledge were evident.
Conclusion of Case
Mrs. Pitchford felt that students who had general mathematics skills could
learn algebra well. Mrs. Pitchford exhibited some weaknesses in her content knowledge
and limited pedagogical content knowledge. She was seldom observed giving any type of
mathematics instruction to her classes. She made mathematical errors and errors in her
mathematical reasoning and was unable to solve problems using a variety of methods.
Mrs. Pitchford did not attend the summer professional development training, and no
changes in her knowledge were seen as a result of the coursework she took her education
specialist degree. No changes were seen in her content knowledge and pedagogical
content knowledge, and she continued to make algebraic errors.
Mrs. Colley
Mrs. Colley is a White female who was in in her sixth year of teaching when this
study began. Mrs. Colley was tall in statue and had a booming voice to complement her
height. She holds both a Bachelor of Science degree as well as a Master?s degree in
mathematics education. Mrs. Colley taught at Dover High School, a ninth through twelfth
grade school, where her primary teaching responsibility was Algebra IB. Dover High
School had a 72% passing rate on the Alabama High School Graduation Exam, the
student population was 18% Black, and 19% of the students were on the free or reduced
lunch program.
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Pre-Professional Development
In this section, we will consider Mrs. Colley?s classroom environment, how she
viewed her students? ability to learn algebra and responses to statements contained on the
Teacher Attitude Survey. Next, Mrs. Colley?s content knowledge and pedagogical
content knowledge will be addressed from the ALCKIN and interview. Finally, a further
look at her content knowledge and pedagogical content knowledge will be considered
from classroom observations, focusing on how these types of knowledge were reflected
in her instructional practices.
Classroom Environment
Mrs. Colley exhibited a pleasing personality and set a comfortable classroom
setting where her students appeared to be at ease. She exhibited patience in working with
her students. Mrs. Colley?s classroom consisted of student desks arranged in straight rows
facing the front of the classroom. This setting suggested that Mrs. Colley used a
traditional format for mathematical instruction, and during the spring of 2004, Mrs.
Colley was the dominant character in facilitating mathematics instruction. The front of
the classroom contained a whiteboard, overhead projector and a pull-down screen, the
teacher desk sitting on the left end of this wall. The right-hand wall also had a whiteboard
along with a bulletin board. The left-hand wall contained windows. During one classroom
observation, student dodecahedrons hung from the ceiling. Students had researched a
mathematician and had included facts pertaining to their mathematician on the regular
polygonal faces.
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Student Ability
Mrs. Colley?s felt that only about one fourth of her students could learn algebra
well. She stated during an interview in the spring of 2004,
I guess they?re learning it well the way I?m teaching it. The other ones are
struggling. You know what I?m saying? Ah and that?s what the MDSMIP thing is,
it?s learning it how to teach it so the other half can get it, the other seventy-five
percent but those that it comes, the twenty-five percent is just that it comes easy
to, those that just pick up on it. Those are the ones to me that are going to learn it
and retain it, where the other seventy-five percent may struggle. They?ll learn it
long enough to take the test and then be done.
Thus, Mrs. Colley did not feel many of her students could learn algebra well. She
indicated that they did not learn well from the instructional practices she was employing.
Her students focused on learning material for the test.
Teacher Attitude Survey
Mrs. Colley was asked to respond to ?It is important for students to figure out
how to solve mathematics problems for themselves.? She replied:
Humph. Let?s see. Humph. Well they have to work their own stuff. Ah, I
usually after we?ve done notes we ah, after we?ve done notes I usually give them
classwork. And during their classwork we?re, I don?t let them work in so called
groups but I don?t mind them doing peer tutoring across the row or anything like
that and then I?m always up here and around helping out where you need help but
it?s mostly them having to work the problems. So, yes, then the next day we
usually come in and I give a daily quiz and that?s independent and if you know if
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you?ve done the classwork the day before you should be successful in the daily
quiz.
Mrs. Colley was asked respond to ?Teachers should model and demonstrate
mathematical procedures and then ideally time should be allowed for the students to have
the opportunity to practice those procedures? and how both of these statements could
coexist in the mathematics classroom. Her response follows:
See I don?t know. That?s that to me is what scares me cause I know a lot of the
MDMSIP stuff and a lot of the stuff that I?ve seen, cause I came on board a little
late but a lot of the stuff I?ve seen has been letting them explore and come up their
own way of doing it. I know there was a tower thing that we did over at West
State University where and it?s on video tape and I?ve seen it on public television
and all of that sort of stuff where the kids had to come up with how many
different varieties of two color towers could there be as long as they are five tall
and it?s just seeing the kids work through it. But to me, you know, the objectives
in the course of study say the kids have to be able to do slope, you know or the
kids in the course of study have to be able do this. And then the graduation exam
is going to present it written form, test form, that sort of thing. ? [T]hat?s one
thing I?m you know, I?m looking, I?m hoping that?s clarified for me, because how
can they go from well this and this goes together, but how am I supposed to take
what I?ve figured out here to work this problem? I don?t, I don?t, I don?t get how
they coexist.
Thus, Mrs. Colley allows students to work together in figuring how to do mathematics.
But she contradicts this statement in implying that she does not know how students can
make sense of the mathematics themselves if teaching are modeling and demonstrating
for them.
Content Knowledge
Mrs. Colley?s content knowledge will be examined from the perspectives of
procedural knowledge, conceptual knowledge, and mathematical processes in the
following sections.
Procedural knowledge. Mrs. Colley exhibited strong procedural knowledge. The
majority of her explanations used procedures on the ALCKIN. Mrs. Colley successfully
solved the wheels and vehicles problem procedurally (see Appendix D) by setting up two
equations, solving for one variable in terms of the other, and solving the problem by
substitution. Figure 22 contains Mrs. Colley?s work on this problem.
Figure 22. Mrs. Colley?s work on the wheels and vehicles problem.
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It should be noted, however, that Mrs. Colley made errors on question four on the
ALCKIN that related to sets of numbers (see Appendix A). She disagreed that 2/3 was a
real number, agreed that .010010001? was a rational number, and disagreed that 5was
a complex number. This reflects limited knowledge involving number systems and how
they are related.
Conceptual knowledge. Mrs. Colley exhibited limited conceptual knowledge.
Mrs. Colley offered no additional way for showing the two algebraic expressions were
equivalent on question six on the ALCKIN (see Appendix A). Mrs. Colley was able to
correctly identify the groups of students that understood slope from question eight on the
same instrument. She stated that group four had the best understanding of slope and
wrote the following as her explanation ?Group 1 & 4 both understand slope. It?s just that
Group 4 was able to right the solution down better and convince the reader of the
understanding. They also were able to show the two slopes were equivalent by reducing.?
Mrs. Colley provided a conceptual approach for solving the wheels and vehicles
problem as her second method. She stated:
Mrs. Colley: Hum, I can think of it but it doesn?t come to me as quickly as this
one did. The other way would be to use (pause) see I?m having trouble thinking it
through. It would just trial and error, just basically drawing cars or having a cube
to represent a car and counting up that way.
Interviewer: Ah just trial and error and using manipulatives?
Mrs. Colley: Right.
Interviewer: Are those two different ways?
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231
Mrs. Colley: No, I?m using it together because I?m a visual. Yes, either I?d have
to draw a picture or I?d have to use something. Do you know what I?m saying? To
do trial and error.
So, Mrs. Colley exhibited limited conceptual knowledge. However, she was able
to provide a conceptual example for solving the wheels and vehicles problem.
Pedagogical Content Knowledge
Her pedagogical content knowledge will be examined by how it is defined in the
literature review, looking at the following abilities: to ?unwrap? and present
mathematical topics so that students can be successful in learning mathematics, to access
different representations as well as methods for solving mathematics problems, to
recognize student errors and be able to respond to them with alternative models and
explanations, to respond to questions, and to pose questions and problems that are
productive to students learning mathematics.
Mrs. Colley?s pedagogical content knowledge appeared limited based on her
responses to the ALCKIN. Mrs. Colley?s was able to recognize that Seth was using a
procedure that was not mathematically correct on question two of the ALCKIN (see
Appendix A). She selected the correct answer and followed her selection with the
following explanation, ?Many students do not realize that you cannot cancel #?s when
they are being added/subtracted to other #?s You can only cancel when #?s are be
multiplied to one another.?
Mrs. Colley was unable to find the solution for the quadratic problem using the
spreadsheet which was problem five on the ALCKIN (see Appendix A). She recognized
that the methods used by both students on question seven of the same instrument were
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correct and explained her answer choice by stating, ?They both found correct solutions to
the problem by looking at patterns & how the original problem was worked.?
However, during an interview Mrs. Colley recognized that using graphing to find
the solution for the problem x
2
= 2x + 8 (see Appendix D) was feasible, declaring:
Humph, wait a minute. Now I see what they did. OK. How?d they get two
equations out of this one equation? OK I got it. (Laughs) OK, I?ll get it ? OH. It?s
different. (Pause) I would have to say I?ve never worked it that way before and it
looks to me like it?s valid but I would have never thought of it myself.
In the wheels and vehicles problem in the interview (see Appendix D), Mrs.
Colley recognized that both the elementary methods and solutions were viable and
indicated that she would accept them as ways to work the problem in her classroom.
However, she pointed out that she felt that students needed to know the usual procedures
for working the problem, as follows:
Yea, because to me this is the trial and error. Because like I said I?d have to draw
pictures or use manipulatives. I don?t know that I would thought of it this way I
mean this makes sense but like I said before I was having trouble, I?m so used to
doing it this (points to her work) doing this is a stretch even for me. But it would
still concern me that they need to be able to do it this way (Indicates the way she
worked the problem.) because this is the way the graduation exam, the SAT?s, the
ACT?s are all going to be set up so it concerns me that yes you can rationally
think it, but what if the problem was given to you in this form and you didn?t have
the written thing. Could they still get to the solution?
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Thus, Mrs. Colley exhibited limited pedagogical content knowledge on the
ALCKIN. However, she was able to determine the elementary methods for solving the
wheels and vehicles were acceptable ways of solving problems.
Further clarification of Mrs.Colley?s pedagogical content knowledge will be
evident in the following section describing her classroom instruction, as well as her
ability to apply this knowledge in her classroom.
Classroom instruction
Mrs. Colley?s content knowledge was primarily reflected in her providing
students with step-by-step instructions on how to solve and/or simplify algebraic
expressions. No errors in her content knowledge were observed. Mrs. Colley showed an
understanding of each of a variety of algebraic topics as well as the ability to
procedurally work related problems. For example, the following classroom dialogue took
place when Mrs. Colley was reviewing how to solve quadratic equations by factoring in
preparation for a semester exam:
Mrs. Colley: Okay let?s start off by looking at the factoring. Okay? (She wrote
ex1) x
2
? x ? 12 = 0 on the whiteboard.) The first thing we need to check for when
we?re given the problem is it is in standard form? Who can tell me what is
standard form of a quadratic is?
Student: ax
2
+ bx + c = 0
Mrs. Colley: Close. a x squared plus b x plus c. Okay. And all that needs to be on
one side of the equal sign, right? The whole thing needs to be set equal to zero in
order for us to solve it? Okay. (Writes formula and reminds students it needs to be
set equal to zero.) Now the first problems done with factoring were problems
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where there?s one in front of the x squared. A one for our a value. Right? Those
were the first type we looked at. Okay and for those we took the c value, right,
and in this case the c value is negative twelve and we looked at it?s factors, one
and twelve, two and six, three and four, right? (She writes these off to the side on
the whiteboard.)
Student: Um humph.
Student: Yes.
Mrs. Colley: Alright, then what did we look at when we saw all these factors we
had listed?
Student: The middle number.
Mrs. Colley: Which one of these could equal the b value, the middle number,
right?
Student: negative four and three
Mrs. Colley: Negative four and three. Okay. Multiply together they do give us the
negative twelve, added together they do give us the negative one, right? So our
factors for this problem were what?
Student: x plus three and x minus four.
Mrs. Colley: x plus three and x minus four. So when you?re solving by factoring
you?re not done yet are you?
Student: No
Mrs. Colley: So what do we do? (Students mumbling.) Set them both equal to
zero. Good. And then we solve each of them independently, right? How do you
get x by itself here?
Student: Minus three
Mrs. Colley: You subtract three on both sides. So I get x is equal to negative
three. How do I get x by itself here?
Students: Add four
Mrs. Colley: Add four. So x is equal to four. Those are your two solutions,
alright?
After giving students the ?steps?, Mrs. Colley gave the students a similar problem to
work and walked around the room observing and answering any questions that they
might have.
Classroom instruction did not seem designed to encourage students to develop
conceptual understanding for the algebraic topics they were. The average of the RTOP
scores from the classroom observations during the spring of 2004 was 27.5.
Thus, classroom instruction revealed generally strong content knowledge but
limited pedagogical content knowledge. Mrs. Colley was not observed making errors of
her own.
Student questioning. Mrs. Colley?s questions primarily focused on answers to
arithmetic problems, providing the next step in the procedure, or the simplification to an
algebraic expression. The following conversation is typical:
Mrs. Colley: (Wrote 34)
xx
12
2
1
=+ on the overhead projector.) Alright what?s the
first thing we need to do?
Student: Find a common denominator.
235
Mrs. Colley: Find what the common denominator is right? See what they have in
common. What is the common denominator?
Student: two x
Mrs. Colley: Good. Two x. Two goes into two x how many times.
Student: x
Mrs. Colley: x times. x times one is?
Student: x
Mrs. Colley: x. x goes into two x
Student: two times
Mrs. Colley: two times. Two times two is?
Student: Four
Mrs. Colley: x goes into two x
Student: two
Mrs. Colley: Two times. Two times one is?
Student: two
Mrs. Colley: Does anybody need to see that redone? (Goes back through changing
the fractions to common denominators again.) Okay. (Writes on the overhead)
x
x
2
24 =+
Now we can just look at the numerator. x plus four is equal to two. So
how do I solve for x?
Student: Subtract four.
Mrs. Colley: subtract four. So x is equal to negative two. Alright?
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It should be pointed out, as in the previous teacher/student dialogue, Mrs. Colley
frequently repeated her students? statements.
Thus, Mrs. Colley questioned her students often but her questions did not require
much mathematical thought. They were not the types of questions or problems posed to
be productive in helping students learn mathematics.
Analyzing student errors. Within her own classroom, Mrs. Colley handled the
errors made by her students in two distinct ways. First, Mrs. Colley would give further
comments on questions asked by her students. For example, when using the quadratic
formula to solve quadratic equations students had taken the square root of sixteen and got
four. One student wanted to take the square root again and Mrs. Colley responded with
?No, once you?ve taken the square root you don?t take the square root again.? Secondly,
Mrs. Colley responded with yes/ no responses without any further explanation. For
example, Mrs. Colley had asked her students for three methods to use when solving
quadratic equations, and the following classroom dialogue took place:
Mrs. Colley: ?..Alright. Let?s go back and look at where we started this nine
weeks. We started this nine weeks in chapter ten. Okay. In chapter ten that was
that horrible word factoring. Alright? Ya?ll just thought that was horrible.Okay,
alright so we started off looking at the three ways to solve quadratics. Alright the
first way we learned how to use and that was factoring, right?
Student: Yea.
Mrs. Colley: Does anybody remember the other two ways?
Student: Elimination
Mrs. Colley: No
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Student: Substitution
Mrs. Colley: Those two things would be the same thing as factoring.
Student: The quadratic equation.
Mrs. Colley: The quadratic equation, the quadratic formula and the other one was
completing the square. Okay do ya?ll remember the quadratic formula?
Students: Yes
Thus, Mrs. Colley was able to recognize and respond to the errors her students
were making. However, she did not respond by suggesting alternative explanations and
models.
Multiple methods of solving problems. Mrs. Colley?s classroom instruction did not
include attention to accessing different representations or different methods for solving a
problem. Mrs. Colley controlled the instruction within her own classroom with little input
from the students so it was difficult to determine whether the students were allowed to
work problems using methods other than the ones she stressed in class.
Conclusion
Mrs. Colley felt that all students given time and the proper type of instruction
could learn algebra. Mrs. Colley exhibited strong procedural knowledge but limited
conceptual knowledge. Explanations on the ALCKIN were all procedural in nature, and
Mrs. Colley could not offer another way to show the two algebraic expressions were
equivalent on the same instrument.
Mrs. Colley appeared to know the algebra content she was teaching and did not
appear to make mathematical errors. Her classroom instruction consisted of giving step-
239
by-step procedures. The types of questions Mrs. Colley asked her students were ones that
could be answered with a mathematical solution or the simplification of an algebraic
expression or simply stating what should be the next step in a mathematical process.
Mrs. Colley was limited in her ability to work algebra problems using a variety of
methods. Moreover, on the ALCKIN she was unable to solve the quadratic problem using
the spreadsheet. She appeared to be able to recognize when students were giving
incorrect answers or when students suggested the use of incorrect methods for solving
problems. Responses on the ALCKIN indicate she did not always choose to respond to
the errors in the statements provided by the students.
Professional Development
During the summer of 2004, Mrs. Colley attended the algebra two sessions of the
two week MDSMIP summer professional development training. Mrs. Colley?s content
sessions included covering the IMP Fireworks unit (Fendel et al., 2000) along with parts
of the Pennant Fever unit (Fendel et al., 2000). Content covered within the Fireworks unit
(Fendel et al., 2000) dealt with methods for solving quadratic equations which involved
factoring, root finding, completing the square, and using algebra tiles. The content
covered within the Pennant Fever unit (Fendel et al., 2000) dealt with different types of
probability.
Mrs. Colley?s felt her attitude changed as the professional development
progressed during the summer sessions, as follows:
Mrs. Colley: Ah, I don?t know, I don?t want to give up two weeks of my
summer.
Interviewer: So how did your attitude change over the two weeks you were there?
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Mrs. Colley: I realized that once I got in there I was getting a lot of good stuff.
And that was worth it.
The presenters saw little or no change in Mrs. Colley?s attitude as she continued
to attend the professional development training. One presenter felt that Mrs. Colley was
?a little apprehensive? at the beginning of the professional development. A second
presenter stated:
I don?t know that I saw her attitude change. I think she came in with a good
attitude and she sounded like she wanted to learn something. She did not, she did
not appear resistant to being there and there were occasions that she discussed
how her students were and how they might react to what we were doing, but she
didn?t seem like she was arguing about it. She seemed like she was willing to give
it a go and see if she could do this with her students.
The presenters expressed that they felt Mrs. Colley showed a willingness to make
changes in her classroom instruction as a result of the professional development. One of
the presenters stated:
She seemed to be open to it. There were others in the class that argued about that
their students couldn?t possibly do it, what we were doing but she didn?t come
across with that kind of attitude in the class, like she was willing to do something
different.
Another presenter pointed out ?Yes, yes but she was, did keep questioning the
problems that all of us have. For instance how does she make her students want to do
241
this?? A third presenter further reiterated what the other two presenters had to say about
Mrs. Colley.
Well I think she was very receptive to the MDMSIP model of learning and ah I
see her as a very receptive person overall and would be someone who would be,
you know, someone who would be willing to try new things and try a new style of
teaching and things like that.
Mrs. Colley attended all of the quarterly meetings during the fall of 2004 and the
spring of 2005. She felt like she got more from the quarterly meetings than she did from
the two weeks of summer professional development training; see the following excerpt
from the interview:
Mrs. Colley: I actually got more out of the Saturday meetings than I did from the
others.
Interviewer: Okay, tell me how it was more helpful.
Mrs. Colley: How, well the others teachers were coming back and saying, you
know, I used this, this and this and it worked.
Interviewer: Okay
Mrs. Colley: So I came back and I was immediately able to put into play and saw
that it really did work instead of having to wait until the time came to do it
because we?re all just about in this same area as far as where we are in teaching
the curriculum. So when you go to one of the Saturdays and say hey you know
last week we did such and such and we did it this way in my class and it really
worked and so you?re able to go back and say hey, you know, look I?ve got this
new way to do this. Look at it this way and I really got more, I get more from the
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Saturdays than I do from the other because I?m able to immediately come back
and use it.
Mrs. Colley shared her experiences with other teachers in order to help them
make better decisions regarding their own classrooms based on what had happened with
her. For example,
Algebra B, trying to put IMP units in is overwhelming! We goofed. We did the
way we usually do. But use About Alice (referring to All About Alice, an IMP
unit), use sections eight one and eight two, days one through seven, and section
eight three, days sixteen and seventeen. The rest is longs and fractional exponents.
We (indicating the teachers at her school) are discussing what we are going to do
because they are not getting it!
One presenter noted that Mrs. Colley ?knows her algebra.? When asked about
Mrs. Colley?s content knowledge, another presenter stated, ?I believe all of the basic
knowledge.? A third presenter assessed her knowledge in terms of the types of
mathematics that were covered during the two weeks of summer professional
development training provided by MDSMIP. She stated:
For the quadratic stuff she seemed to be perfectly at ease with all of it. I mean
there wasn?t, there wasn?t any difficulty with that. And she seemed to be at ease
with the, with the other part also. I mean she was I think particularly in that group
she was very good.
Thus, Mrs. Colley was very involved in the professional development training .
Presenters felt her knowledge of the algebra content was good. They also felt she would
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try to implement changes into her instructional practice. She was willing to share success
and failures with the other teachers at the quarterly meetings.
Post-Professional Development
During the post-professional development timeframe, we will consider changes in
her classroom environment, in her views of student ability, in her content knowledge, and
in her pedagogical content knowledge. Moreover, we will consider how changes in
content knowledge and pedagogical content knowledge are reflected in instructional
practices. Each of these areas is addressed in the following sections:
Classroom Environment
The layout of Mrs. Colley remained unchanged from the previous school year.
She continued to be personable and exhibited a patience attitude with her students. Group
work from the Fireworks IMP unit (Fendel et al., 2000) was displayed on the hallway
wall outside of Mrs. Colley?s classroom, suggesting that ?.
Student Ability
In the spring of 2005, she stated that she felt all of her students could learn
algebra well and supported this statement as follows:
The percentage of them that can learn algebra well given the desire to do so. It
would be one hundred percent. They all could do learn to do it. But a lot of them
don?t have the desire to do it. ?Ah, I think a lot of it has to do with how it is
presented and see I?m looking at the kids in my class, I?m not looking overall.
And I really and truly believe that I can get to all of them. But I think at some
point it does fall on the students shoulders. They have to be willing to accept me
helping them to get it.
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Thus, Mrs. Colley felt that all of her students can learn algebra but learning may
depend on how it is presented to students. However, students need to be willing to seek
help.
Changes in Content Knowledge
Mrs. Colley was asked to assess her own content knowledge of algebra. She
responded as follows:
Not very good. Algebra one I?ve got. Algebra one I could battle with just about
anybody else on the planet. Algebra two I?m still struggling?
Mrs. Colley felt that the professional development training did have an affect on her
content knowledge of algebra, and she reaffirmed that she had learned new content when
she stated:
Ah, let?s see. Yea I completely got where the completed square form came from.
And I don?t think I had really gotten that before. I had seen it done but I didn?t
really get it. I could do it, I could manipulate it the way somebody showed it to
me to manipulate it but I didn?t get it. And I got it.
She further stated that the training ??helped in making connections. But it also
gave me a chance to realize that I?m not the only blooming idiot out there. There are
other people who are struggling with this stuff just like I am.?
It should be noted that during an interview in the spring of 2005 that Mrs. Colley
was able to solve the quadratic equation problem x
2
= 2x + 8 (see Appendix E) using
factoring, using the ?magic square? method, and using algebra tiles (see Figure 23).
Figure 23. Mrs. Colley?s work for the quadratic problem.
The three methods all involved factoring but in each method she used a different type of
factoring.
She noted the effect these different methods had on the success of her students
within her own classroom. Her conversation related to this transition went as follows.
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Mrs. Colley: I taught it all three ways. I started off with the algebra tiles. And then
once they got the fact that ah they began to see the pattern of the fact that hey the
sides are multiples of the c value. They started making that transition and so then
they started working it that way and so then I showed them the magic square way.
And some of them have jumped to the first way that I done it just because they
have developed the complete understanding and can skip sections. But I?ve learn
to let them do that on their own. But some of them are still having difficulty
making the jump from the tiles to that.
Interviewer: Last year did you do the algebra tiles?
Mrs. Colley: No
Interviewer: Last year did you do all three ways?
Mrs. Colley: No I did the first one.
Interviewer: Okay, so have you seen a difference in what your students are doing?
Mrs. Colley: Oh yea and I can see where I?m getting to different ones of them
without them having to bang their heads. And them getting frustrated. Octavia in
fact would never get the first way. They got the algebra tiles.
Interviewer: Okay, so this way (points to her drawing of algebra tiles)
Mrs. Colley: That way, yes, and they are still there. They?re one of the ones
having a hard time making it to here (magic square).
Interviewer: But they could solve this? (Points to the quadratic equation problem.)
Mrs. Colley: They could solve that using it. If they had enough algebra tiles to
work the large numbers, that?s the way they would choose to do it.
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Thus, Mrs. Colley demonstrated changes in her content knowledge. While she
may have previously suggested the use of manipulatives, she demonstrated solving a
quadratics problem by using them. In addition, she was bridging from the physical
manipulatives, to drawing pictures, to her ?magic square?, and eventually to applying
rules for factoring. Furthermore, she gave an example of content that she better
understood as a result of professional development.
Changes in Pedagogical Content Knowledge
Growth in her content knowledge directly connected to changes in Mrs. Colley?s
pedagogical content knowledge. This growth in pedagogical content knowledge was
evident from the previous example in her use of algebra tiles and in helping students
make connections between physical objects and the rules for factoring polynomials. From
her conservation, it suggests that her increased pedagogical content knowledge aided in
the success of her students learning mathematics.
Reflection of Changes in Knowledge in Instructional Practices
With growth in both her content knowledge and pedagogical content knowledge,
one might expect that these changes were reflected in her instructional practices. In one
instance, Mrs. Colley had problems for her students to factor written on the whiteboard as
students came into her classroom. One of the students factored 16a
2
? 24a + 5 into factors
(4a + 1) and (4a + 5). The following classroom dialogue took place:
Mrs. Colley: Added together gives you the negative twenty-four. Correct? Sixteen
a squared minus four a, or negative four have a four a in common. Right? Twenty
a and five, have a five. Sixteen a squared, negative twenty a have a four a.
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Negative four a and five have a one. So this is your answer (4a + 1)(4a + 5), whoa
what?s wrong here?
Student: The 4a should be negative.
Another Student: The four is not negative.
Mrs. Colley: Which four?
Student: In your graph, in the box on the side with your first negative. One of the
four a?s has to be negative because in your second block on the top, the four a is
negative. Then make your four a negative, then make your other four a negative
so when you multiply you get negative twenty a
Mrs. Colley: Humph?
(Students think one or both of 4a?s should be negative)
Student: Do you want me to show you?
Mrs. Colley: Show us.
Student: This right here has to be negative so when you multiply these right here
you get negative. Okay make this negative but then you come right here and this
will make this negative so you add this one negative and that works out down
here and gives you negative twenty a so in that way it comes in a negative twenty
a. (Student goes to whiteboard and puts in negative on both 4a?s and comes up
with this.)
5-20a
-4a16a
5
-4a
1-4a
Mrs. Colley: Alright, do you see what she did?
Student: No.
Mrs. Colley tried to explain what the student had done but had to check the
answer by multiplying the two factors together to make sure that they equaled the original
algebraic expression. The factors did in fact give the correct answer and she proceeded to
go over what the student had done. Mrs. Colley then asked if there was another way to
factor the same expression, and a student came up with factoring out negative one and
negative five and writing the factors as (4a ? 5) and (4a ? 1). She agreed that both sets of
factors were correct in the following discussion:
Mrs. Colley: Okay you could have done your four a minus one and your four a
minus five. And still works, right? That?s positive sixteen a squared, that?s
negative four a, that?s negative twenty a, and that?s negative five. And that?s what
threw me off because I?m used to doing it that way instead of the other way.
Student: That?s five
Mrs. Colley: Both, they?re both right. Okay. So in that case your answer would be
four a minus one and four a minus five, right? Okay, Good!
Thus, from the use of different representations and different methods for factoring
polynomials, Mrs. Colley was able to respond to her student?s error. What resulted was a
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different representation of the factors of the trinomial that she did not immediately
recognize as correct. She was able to determine that the student had factored the trinomial
correctly, even though she checked it with procedures.
Further evidence of use of a different kind of mathematical knowledge was
evident in a lesson in which Mrs. Colley used a hands on activity related to the difference
of two squares in her Algebra IB class during the spring of 2005. The paper square
manipulatives had another square marked within the upper right hand corner. The first
square had numerical measurements, while the second square had variable measurements.
The class worked as a whole group and Mrs. Colley took time during instruction to allow
students to make and record observations which were discussed within the class. Mrs.
Colley instructed the students to physically remove the smaller square and find the
lengths of the sides after the smaller square was removed, area of the original large
square, area of the smaller square, and the area of the remaining polygon. The students
took the remaining polygon and divided it along a diagonal so that the two pieces formed
a rectangle. Dimensions of the rectangle were found and the area computed. The students
surmised from the exploration that the area formed by the region from the differences of
a squared and b squared had the same area as the rectangle whose dimensions were a plus
b and a subtract b. The RTOP score during this classroom observation was fifty-two
which was significantly higher than her previous RTOP scores.
Conclusion
Mrs. Colley exhibited a positive attitude during professional development and the
presenters felt she would be willing to make instructional changes based on this
professional development. Growth was seen in both her content knowledge and
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pedagogical content knowledge. This growth was seen in her ability to work problems
incorporating non-standard methods, as well as recognizing student answers she was not
expecting. There was evidence from classroom observations, quarterly meetings, and
work exhibited by her students that she was implementing changes in her instructional
practices. Although she was allowing students to do activities, she directed instruction
and still had students do mathematics in a procedural way. Colley RTOP scores showed
an increase from the classroom observations made during the spring of 2004 to the spring
of 2005.
Conclusion of Case
Mrs. Colley felt that all of her students could algebra given the proper type of
instruction, this attitude did not change. She felt she knew the algebra I content well.
Prior to the professional development, she exhibited strong procedural knowledge and
limited conceptual knowledge. Her pedagogical content knowledge was also limited. She
was not observed making mathematical errors. However, changes in both her content
knowledge and pedagogical content were observed following the professional
development. Using algebra tiles, making drawing, moving to a ?magic square?, and
bridging these methods to formal rules were all evident. Evidence that this type of
instruction had been used was seen in the types of work displayed by students, which was
not observed during pre-professional development.
Conclusion
The content knowledge and pedagogical content knowledge of these cases prior to
professional development will first be addressed. All of the participants in the case
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studies felt they had good algebraic content knowledge. Three of the case study teachers,
Mrs. Cotney, Mrs. Willoughby, and Mrs. Colley, exhibited strong procedural knowledge
while Mrs. Pitchford exhibited weaknesses in her procedural knowledge. However, all of
the subjects exhibited limited conceptual knowledge. None of the case study participants
could work problems using multiple methods. However, all of the cases could recognize
that the elementary approaches provided in the wheel and vehicle problem were viable
ways of working the problem. Little evidence of pedagogical content knowledge was was
seen in either the ALCKIN or interviews.
We next consider how the participants used their content knowledge and
pedagogical content knowledge during classroom instruction prior to the professional
development. Their procedural content knowledge was reflected in the instructional
practices, which largely consisted of presenting step-by-step procedures to their students.
Mrs. Cotney, Mrs. Willoughby, and Mrs. Colley did not make exhibit errors in their
content knowledge during instruction, while Mrs. Pitchford did. All of the case study
participants asked questions involving answers to arithmetic problems, simplifying
algebraic expressions, or naming the next step in a procedure. However, these types of
questions were not of the type most productive in promoting successful learning of
mathematics by students. None of the participants were observed providing classroom
instruction that was conducive to students developing a conceptual understanding of
mathematics.
Three of the participants, Mrs. Cotney, Mrs. Willoughby, and Mrs. Colley,
provided step-by-step mathematical instruction to their students. They were able to
recognize the errors in the thinking of their students and respond to them. However, their
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responses did not include alternative explanations or models for representing problems.
They were able to recognize when students worked problems using a variation of the
procedure they presented, but no solutions significantly different from the presented
solution were given or encouraged. The average RTOP scores during the spring of 2004
for these three case study teachers were similar.
Mrs. Pitchford was different from the previous three participants, since she
provided very little instruction in the classes that were observed. Thus, it was difficult to
determine how her content knowledge and pedagogical content knowledge were used in
instruction. In the limited classroom instruction that was observed, she made
mathematical errors. She also had difficulty in recognizing and responding to why her
students were making errors. The RTOP score on this case study teacher was lower than
the other three cases.
Classroom observations revealed similarities in the pedagogical content
knowledge all four cases used. Their instructional practices involved procedures and
emphasizing procedural knowledge through the questioning they provided.
Three of the participants, Mrs. Cotney, Mrs. Willoughby, and Mrs. Colley,
attended the summer professional development training. All of these case study teachers
agreed that the professional development training was beneficial. None of the participants
felt their content knowledge had changed as a result of professional development, a view
with which the presenters agreed. Mrs. Pitchford did not attend the summer professional
development training but was finishing up coursework for her education specialist
degree.
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The three case study participants who attended the professional development,
Mrs. Cotney, Mrs. Colley, and Mrs. Willoughby, all exhibited changes in content
knowledge and/or pedagogical content knowledge following the professional
development. Mrs. Cotney referred to ?in/out? tables to before graphing linear equations,
and mention that she better understood how the algebra content fit together. Mrs. Colley
and Mrs. Willoughby both emphasized the use of algebra tiles when working with
algebraic expressions. All three could offer more than one method to work the quadratic
equation during the second interview. Mrs. Colley and Mrs. Willoughby also worked the
quadratic problem using conceptual methods such as algebra tiles or drawing pictures.
The other two case study teachers did exhibit changes in mathematics instruction.
Mrs. Willoughby demonstrated the use of algebra tiles at a quarterly meeting. She shared
how she used them in her instruction with the other teachers. Mrs. Colley used algebra
tiles, drawings and her ?magic square? when factoring trinomials with her students. Both
of these participants felt that all students could learn algebra but at different paces. These
two teachers had increased RTOP scores from the spring of 2004 until the spring of 2005.
Presenters of both of these teachers felt they would implement changes in their classroom
instruction, and observations suggested that they did.
However, even though three of the cases exhibited changes in their content
knowledge and pedagogical content knowledge and two of them exhibited some changes
in their instructional practices, the pedagogical content knowledge of all the case study
teachers was still quite limited. No changes in content knowledge and pedagogical
content knowledge of Mrs. Willoughby were observed in her instructional practices. Mrs.
Colley?s students used ?magic squares? when displaying work on the board. However,
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the changes in content knowledge and pedagogical content knowledge were not observed
in classroom observations. It can be said that they had used manipulatives or ?in-out
tables?, which was accessing different representations, and in some instances students
were making the connections from these to the rules. However, it cannot be said that the
manipulatives or other forms of representations were used in a different way from
teaching procedures since their introduction was not observed. In addition, other areas
considered in pedagogical content knowledge were not obvious. How they ?unwrapped?
the mathematical topics and presented them so students could successfully learn
mathematics was not observed. How these teachers made connections between
mathematical topics was not seen. The teachers did not respond to student errors with
alternative explanations and models, although it might be argued that using manipulatives
was using alternative models. Neither was using them in response to student errors
obvious. How the teachers responded to questions still remained the same as before
changes in knowledge. It did not appear that these teachers posed problems and questions
that facilitated the successful learning of mathematics.
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VI. SUMMARY AND RECOMMENDATIONS
To become productive members of society, adults of the twenty-first century need
to be proficient in mathematics (Ball, 2003a), and the need for mathematics in our
everyday lives continues to grow (NCTM, 2000). However, in the United States we are
not preparing our students for the demands to be mathematically proficient. According to
results of the Third International Mathematics and Science Study (TIMSS) (NCES,
2003a) and the National Assessment of Educational Practices (NCTM, 2004), our
students do not achieve at the levels necessary for mathematics success. Mathematics
preparation of students reaches a critical juncture with the Algebra I course. Algebra
serves as a gatekeeper course, offering differential opportunities for entry into advanced
mathematics courses (Ball, 2003a), for preparation for college (Pascopella, 2000,
Lawton, 1997, Chevigny, 1996, Silver, 1997, Olson, 1994), and for preparation to enter
the world of work (Silver, 1997).
Teachers play a key role in ensuring that all students have the opportunities and
experiences needed to learn mathematics (Mewborn, 2003). Therefore, we must consider
the types of knowledge needed to provide all students with equitable opportunities to
learn algebra. Studies have shown that teacher?s content knowledge is often thin and
inadequate to provide the instructional opportunities needed for students to successfully
learn mathematics (Ball, 1998a, 2003b, Ball & Bass, 2000; Fuller, 1996, Ma, 1999,
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Mewborn, 2001, Stacy, et al., 2001). Moreover, teachers need to know how to use their
mathematics knowledge in facilitating the learning of mathematics by their students
(Sherin, 2002), which is often referred to as pedagogical content knowledge (Shulman,
1987).
This study seeks to better understand teacher?s content knowledge and
pedagogical content knowledge. While some research has been conducted in this area at
the elementary level (cf. Ball et al. 2005), far fewer studies have been done with
secondary teachers. We need a better understanding of content-specific knowledge and
pedagogical content knowledge (RAND, 2000), as well as how they are used during
mathematics instruction (Ball, 2003a).
An additional question lies in the extent to which professional development can
change both content knowledge and pedagogical content knowledge of teachers.
Effectively developing content knowledge and pedagogical content knowledge for
teaching remains an unsolved problem for the improvement of mathematics teaching and
learning (Ball et al., 2001). Furthermore, even if professional development can facilitate
changes in both content knowledge and pedagogical, we do not understand how these
types of knowledge might influence changes in teachers? instructional practices. The
possession of content knowledge and pedagogical content knowledge do not
automatically mean successful mathematics instruction (Ball et al. 2001).
This study was design to answer the following specific research questions related
to the broader issues raised above:
1. What content knowledge and pedagogical content knowledge of algebra do
high school algebra teachers possess?
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2. How are high school algebra teachers? content knowledge and pedagogical
content knowledge reflected in their teaching practices?
3. What growth in content knowledge and pedagogical content knowledge can be
seen from participating in professional development that includes attention to
increasing both content knowledge and pedagogical content knowledge?
4. How are changes in content knowledge and pedagogical content knowledge
reflected in teachers? instructional practices?
In the following sections, answers to these questions will be presented, based on
the findings of this study. We will then consider limitations of the research and
implications for the field of practice. Finally, we will explore areas for further research.
Summary
This study was comprised of two major investigations, one involving survey
research and the other multi-case studies. Survey research provided the opportunity to
study the content knowledge, pedagogical content knowledge and attitudes of a large
group of teachers. The multi-case studies provided the opportunity for an in-depth study
of four teachers and included the opportunity to understand how the case study teachers
used their content knowledge and pedagogical content knowledge in their instructional
practices. The case studies also provided the opportunity to probe for depth in content
knowledge and pedagogical content knowledge. The results of each of these
investigations have been summarized in previous chapters, and we will now review what
we have learned about of the research questions as a culmination of both of these
methodologies.
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1. What content knowledge of algebra and pedagogical content knowledge
do algebra teachers possess?
We will focus our attention to the areas that make up content knowledge. They
are procedural knowledge, conceptual knowledge, and the mathematical processes.
Did participants demonstrate the ability to use the rules and algorithms of mathematics,
or procedural knowledge? On the parts of the ALCKIN that required a selection of
answers, only seven of the twenty-five tasks were answered correctly by more than eighty
percent of the participants. This ability to do mathematics procedurally was predominant
in explanations and in the procedures suggested in clarifying answer choices. The
majority of the case study teachers exhibited strong procedural knowledge and performed
additional mathematical tasks using procedures.
However, there were errors made in the procedures used by some of the
participants. This was true for all the items on the ALCKIN. This included computational
errors and errors in mathematical notations. The majority of the case study participants
did not make procedural errors.
Was conceptual knowledge evident in the participants? The conceptual
knowledge of all the participants in both studies was limited. A limited view of
conceptual knowledge was exhibited by using drawings or algebra tiles or through the
suggestion of using manipulatives. Gaps were exhibited in the understanding of topics on
the ALCKIN, such as understanding of algebraic expressions, understanding of functions,
and slope.
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Did participants exhibit use of the mathematical processes? Participants in both
studies had a difficult time in providing different representations for mathematical
situations. This was true with the case study teachers as well. Some of the participants did
not recognize that algebraic expressions, although equivalent, cannot be used to describe
different interpretations by students. Less than half of the participants could explain the
meaning an algebraic expression within the context of an algebraic equation. In addition,
the majority of participants did not provide or use multiple methods for solving
mathematical situations, and they tended to be quite procedural in nature. When the case
study teachers were pushed to provide other ways of solving mathematical tasks, even
alternative procedural methods could not be provided.
It might also be noted that a mathematics education degree does not always
ensure the content knowledge and pedagogical content knowledge to teach mathematics
(Ball et al., 2001). This was observed in the case of the teacher who was completing her
education specialist degree in mathematics education. She exhibited weak content
knowledge, both in procedural and conceptual knowledge. She had difficulty in
recognizing why her students made errors and difficulty herself in working mathematical
tasks. In addition, she exhibited a lack of understanding for the connections between
mathematical topics and difficulty in using and recognizing different representations as
well as methods for solving mathematical problems. One would hope that this teacher is
an isolated case!
In considering the pedagogical content knowledge of teachers, our examination
will be guided by how the definition from the literature review, which briefly follows.
Pedagogical content knowledge is the ability to ?unwrap? and present mathematical
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topics so that students can be successful in learning mathematics. Pedagogical content
knowledge includes the ability to access different representations as well as methods for
solving mathematics problems. Pedagogical content knowledge is the ability to recognize
student errors and be able to respond to them with alternative models and explanations.
Pedagogical content knowledge includes the ability to responds to questions, and to pose
questions and problems that are productive to students learning mathematics.
First, was the ability to ?unwrap? mathematics topics and present them so
students can be successful in learning mathematics evident? Actual mathematical
instruction was not used in answering this question, however if participants lack an
understanding of mathematics topics it could be argued that they do not have the ability
for unwrapping them and presenting them in ways so students can be successful.
Next, do teachers have the ability to access different representations as well as
methods for solving mathematics problems that arise within mathematics instruction?
Participants had a difficult time in solving the spreadsheet problem on the ALCKIN. The
different representations of slope also caused a majority of the participants? problems in
recognizing they were equally valid. The case study teachers were able to use their
pedagogical content knowledge in analyzing non-procedural methods used by
hypothetical students within problem solving context during interviews.
Did the participants have the ability to recognize student errors and be able to
respond to them with alternative models and explanations? Participants had difficulty in
assessing student errors on the ALCKIN. Instead of responding to errors or statements
made by students, participants generally ignored them or responded how the student
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should have worked the problem using a set algebraic manipulation, rather than providing
alternative explanations or models.
Did the participants have the ability to respond to questions, and to pose questions
and problems that are productive to students learning mathematics? While the design of
the ALCKIN did directly address this issue, none of the case study teachers exhibited this
particular part of pedagogical content knowledge in interviews or in classroom
observations.
Overall, it can be stated teachers in this research study had content knowledge.
However, their knowledge was primarily procedural and they had limited conceptual
knowledge. Also lacking was their ability to use various representations as well as use
different methods to solve problems. We can claim that the participants did not have a
deep understanding of the algebra content. Participants also exhibited limited pedagogical
content knowledge. This was evident in all areas addressed related to the definition of
pedagogical content knowledge.
2. How are high school algebra teachers? content knowledge and pedagogical
content knowledge reflected in their teaching practices?
Again we return to what constitutes content knowledge as defined in the literature
refiew. First, procedural knowledge was predominant within instructional practice. Case
study teachers demonstrated use of procedures within a variety of mathematics topics in
almost all of the classroom observations. Three of the cases were not observing making
procedural mistakes, while one case study teacher did. However, when we consider their
conceptual knowledge, it was seen to be very limited. There was not an occasion that
could be identified where the case study teachers exhibited they knew why rules and
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algorithms worked. Were the mathematical processes evident in instructional practice?
Case study teachers were not observed using different methods of working problems.
Conjectures were not investigated, nor was the case study teachers? ability to develop and
evaluate arguments observed either. Teachers did communicate ideas and clarify what
they meant but it was in superficial ways and relating to providing procedures and
explaining what to do next. Case study teachers were not observed making connections
between mathematical topics. Representations of mathematics did not go beyond the
procedures for doing mathematics.
Let us return to the definition of pedagogical content knowledge and see how that
definition is reflected in instructional practice. First, consider the ability to ?unwrap? the
mathematical topics and present the content in ways for students to successfully learn the
mathematics. My conclusion is that there was no unwrapping of mathematical topics or
ideas, only presentation of particular procedures.
Second, did the teachers access different representations as well as different
methods for solving mathematics problems that may arise within the mathematics
instruction? Once again this was not obvious. While teachers may have felt that their
students were using alternative methods, in actuality they were generally the same
procedures used by the teacher, but using a different order of steps or omitting some steps
altogether. Since the teacher was dominant in mathematics instruction, little opportunity
was given for other types of mathematical problems to arise that would necessitate
different representations.
Next, did the teacher use her ability to direct students in making connections
between mathematical topics as well as the connectedness of different representations for
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those same topics? Connections between mathematical topics were not noted and since
the case study teachers did not use, or have students use, different representations for
mathematical topics, this facet of pedagogical content knowledge was not present either.
What was the ability of teachers to understand where and why students make
errors and be prepared with alternative explanations and models? Analysis of student
errors within the context of mathematical instruction generally related to incorrect
mathematical computation errors, the teacher recognizing incorrect answers to algebraic
expressions or equations or, or using incorrect procedures in completely algebraic tasks.
All student errors related to the mathematical topics covered during a particular day?s
instructions. Neither did results show teachers offering alternative explanations or
models.
Did the pedagogical content knowledge of the case study teachers include their
ability to respond productively to students? questions and pose problems and questions
that are productive to student learning? Case study teachers could respond to questions
their students asked during instruction, but the questions were generally about the
correctness of an answer, if a student was doing the work correctly, or what the student
should do next in solving the mathematical task. Moreover, teachers did ask their
students questions productive in successful learning of mathematics, but they generally
related to asking students to give short answers to arithmetic problems, the simplification
of algebraic expressions, or to name the next step in a procedural process. Therefore,
once again the type of questions required of a teachers? pedagogical content knowledge
was limited since these questions were not the type that would be productive to students
learning mathematics in the ways they need to.
265
From the Teacher Attitude Survey, participants felt they had to model and
demonstrate mathematics and then allow time for students to practice these procedures.
This is how mathematical instruction was reflected in a majority of the case study
teachers. Half of the cases felt that by allowing students time to practice procedures, this
was the same as helping them make sense of the mathematics. This view of mathematics
instruction is limited when compared to what pedagogical content knowledge includes.
This prevalent attitude along with the lack of depth in content knowledge may further
suggest why the pedagogical content knowledge was limited in instructional practices.
Content knowledge, but not a deep understanding of it, was reflected in the instructional
practices of the case study teachers. However, limited pedagogical content knowledge
was observed during these classroom settings.
3. What growth in content knowledge and pedagogical content knowledge can be
seen from participating in professional development that includes attention to increasing
both content knowledge and pedagogical content knowledge?
Changes in the procedural knowledge of the case study teachers were not
observed. However, changes in conceptual knowledge were evident, which was
expressed either by demonstration and/or in conversations. Mrs. Willoughby
demonstrated the use of algebra tiles in showing the addition of algebraic expressions.
Mrs. Colley drew pictures of algebra tiles and a ?magic square? in solving a quadratic
problem. Both of these examples show that the case study teachers could solve problems
in different ways and could communicate their understanding clearly. They were also
able to makes connections to the procedures used and it was obvious they used different
representations in demonstrating they understood how to solve the quadratic problem
266
using different methods. All of these case study teachers were able to suggest other non-
procedural ways of solving the quadratic equation. Two of them offered conceptual ways
of solving the problem. In pre-professional development only one of the case was able to
supply more than one way of solving problems.
Growth in pedagogical content knowledge was also evident. Mrs. Willoughby
presented the use of algebra tiles in ways she said she used them in instructional
practices. She provided the connections to using algebra tiles which included: using the
tiles, drawing pictures and the bridging to the algorithms related to their use. Mrs. Colley
also demonstrated that she was using algebra tiles for the multiplication of binomials and
factorization. Although instruction was not observed using algebra tiles, it was evident
that students had learned with them from the work they did on the board.
Conversations with one of the case study teachers included such terminology as
?in-out? tables and how they could be used before introducing terms such as function,
domain, and range. She also indicated that she had used them during instruction.
4. How are changes in content knowledge and pedagogical content knowledge
reflected in teachers? instructional practices?
Strictly limiting the answer to this question to how changes in content knowledge
and pedagogical content knowledge were observed in classroom observations, the answer
would be simple. There was very little difference in how the content knowledge and
pedagogical content knowledge was reflected through during classroom observations
between pre- and post-professional development. It can be noted that one of the cases
was using an activity to help students develop the conceptual understanding for factoring
the differences of two squares, but even in this one instance, she controlled the classroom
267
conversation and limiting the opportunities for her student to really develop the
understanding they needed. However, when we consider the types of observations such as
the demonstration of algebra tiles with the two case study teachers and the conversation
about ?in-out? tables, we might be able to suggest that it was possibly reflected in their
instructional practice just not on the days of observation. Also the display of student work
from the IMP units (Fendel et al., 2000) would suggest that changes in both of these types
of knowledge are reflected in the case study teachers? instructional practices. These are
limited views of pedagogical content knowledge, because they do not get everything the
definitions of these types of knowledge entail. Furthermore, without these observations
within the classroom context, we cannot say that it is used as a tool in helping students
make sense of the mathematics or as just another procedure.
Limitations
Although survey research and multi individual case studies were considered the
best methodologies for this study there were several limitations for each. First, survey
research allows for a sample population to be studied and inferences made about the
general population (Ary, Jacobs, & Razavieh, 2002). However, this pool of participants
may not be representative of the population of the state in which this study was
conducted, much less to all algebra teachers in the nation. Moreover, despite the fact that
some of the data could be reduced to simple numerical analysis such as percentages, the
explanations of the participants were coded using software designed for qualitative
analysis and was subjective to my own interpretations (Bogdan & Biklen, 1998).
268
Second, while the use of multiple case studies allows for a detailed, in-depth data
collection over a long period of time drawing from multiple sources of information
(Cresswell, 1998) with the ability to compare two or more cases in order to show
generalizability or diversity between the cases (Bogdan & Biklen, 1998), use of the case
study limits the number of participants in the study. While conclusions will be drawn
about the four individual cases, these generalizations cannot be presumed true about the
population from which these cases were selected, much less to the general population of
all algebra teachers in the nation (Bogdan & Biklen, 1998).
Third, the longevity of the study was a limitation of this study (Bogdan & Biklen,
1998). Data collection occurred over a span of one and a half years. One phase of the
study involved assessing changes in content knowledge and pedagogical content
knowledge of the cases. It cannot be assumed that sufficient time had elapsed for changes
to occur in either type of knowledge as well as in both of them. Since the reflection of
changes in both of these types of knowledge was considered within instructional practice,
the length of this study may not have allowed enough time for the changes in both types
of knowledge to be reflected in instructional practice. In addition, even though classroom
observations were announced, there is still the possibility that changes in both of these
types of knowledge were more observable when the research was not present.
Finally, while I tried to minimize my initial feelings about the phenomena I was
studying, as a researcher, I brought my own bias to the interpretations of the study
(Bogdan & Biklen, 1998). First, I have my own set of experiences as a mathematics
classroom teacher and beliefs about the mathematical tasks performed on both the
ALCKIN and during teacher interviews. Although I tried to lay aside my preconceived
269
notions about what I expected to see, it is difficult to separate myself from the
phenomena I are studying (Bogdan & Biklen, 1998). I was involved in all parts of the
data collection process. First the instruments were all administered in my presence. There
was personal interaction with all participants, including the case study teachers and the
presenters of professional development. Classroom observations may have also been
influenced by my presence, not only from the teacher?s standpoint but also in the
reactions of the students as well (Crotty, 1998). While interviews had certain protocols to
follow and data from the all sources of collection including classroom observations,
quarterly meeting observations, or interviews were entered into software designed for
qualitative analysis, the coding was created by me and was subject to my own
interpretations (Bogdan & Biklen, 1998). Final conclusions and inferences were also
drawn from my own perceptions of this analysis.
Implications of the Study
In the following section I will address the implications of this research study.
They include implications for teachers, teacher education programs, professional
development, and for policy makers.
Implications for Teachers
Teachers need to recognize that while they may know procedures, they may not
have the deep understanding of content knowledge and pedagogical content knowledge
for teaching mathematics. Without strong content knowledge including a deep
understanding, pedagogical content knowledge will suffer and affect successful
mathematics instruction. Realizing and working toward increasing content knowledge
270
can and should influence the way in which mathematics is presented within the classroom
domain. Recognizing this lack of knowledge should influence teachers to seek
professional development that offers them the opportunity to develop a deep
understanding of the mathematics they are teaching. They should seek professional
development that offers them the opportunity to learn mathematics in a different ways if
they are going to teach it differently. Professional development should include the
revisiting of the big ideas that teachers are expected to learn. Professional development
cannot be considered a quick fix to the problem and should be considered a long-term
investment.
With increases in both content knowledge and pedagogical content knowledge,
teachers really need to use it. Why would you want to learn something new if you did not
intend to put it to use? Both content knowledge and pedagogical content knowledge
should be reflected in the instructional practices of teachers. With strong content
knowledge and pedagogical content knowledge, teachers can move from being providers
of mathematical procedures to facilitators of instruction that provide students with tasks
to build mathematical knowledge along with the types of understanding necessary for
them to be successful in mathematics.
Change for any of us is never easy. Committing to change will be necessary for
effective instructional change to take place. Saying that it will be easy for teachers to
implement different instructional strategies that research has proven to be effective is not
realistic. The majority of us may remember our first year of teaching and how everything
did not always go well, but we persevered. The same will be true for those working
toward creating the seamless connection between knowledge and practice that will make
271
a difference in the success students have at learning mathematics which in turn have a
direct affect on their future opportunities for success (Ball et al., 2001).
Implications for Teacher Education
Preparing pre-service teachers who have strong content knowledge, including a
deep understanding of the content, and strong pedagogical content knowledge should be
the goal of teacher education programs. Coursework should be designed so that they may
revisit the big ideas of the mathematics they will be teaching, using the types pedagogical
practices they will be expected to use. We cannot expect changes to take place in the K-
12 setting if professors at the university level are not willing to provide those teachers
with opportunities to develop both content knowledge and pedagogical content
knowledge through courses in both mathematics and mathematics education.
Pre-service teachers should be paired with in-service teachers who demonstrate
strong content knowledge and strong pedagogical content knowledge that is reflected in
their instructional practices. This pairing will ensure continuity between the coursework
taken at the university and reinforcement in how both of these types of knowledge should
be effectively used. These situations also offered the opportunity for interns tp use and
further develop their content knowledge and pedagogical content knowledge under the
guidance of an experienced in-service teacher. In this way, pre-service teachers?
knowledge will be more likely to reflect in their instructional practices.
Faculty from both the mathematics department and the mathematics education
department must collaborate to ensure that the courses taught within the two departments
align with common goals in preparing pre-service teachers. Pre-service teachers need to
know that content knowledge and pedagogical content knowledge are both valued within
272
both departments. Teamwork will make a stronger program in preparing future teachers
to teach mathematics.
Implications for Professional Development
Teachers need to be provided professional development where they can
experience growth in both content knowledge and pedagogical content knowledge in
ways that are transferable to their instructional practices. Professional development
should be intensive and long-term providing teachers with a safe environment to discuss
their success and failures (Mewborn, 2003). Teachers need to be provided with
opportunities to revisit mathematical concepts to help them build their conceptual
knowledge of those topics. Opportunities should also be provided to help teachers make
connections to other mathematical topics (Mewborn, 2003). Professional development
needs to prepare teachers for the tasks they will encounter on the job (Hill, Schilling, &
Ball, 2004). Moreover, we cannot expect ?quick fixes? to these problems. While some
evidence of change was seen in this study, a much longer timespan might be necessary to
see the profound changes I had hoped to see.
Implications for Policy Makers
Policy can effect what happens in the classroom. Taking a set number of courses
in mathematics does not equate to a pre-service teacher having the strong content
knowledge, with deep understanding, that will be sufficient to be successful in teaching
students mathematics. Nor is students only gaining exposure to the types of pedagogy
sufficient in helping them to use their content knowledge in assuring successful learning
of mathematics for all students. Nor does a score on a test (NCLB, 2001) assure that
mathematics teachers are adequately prepared for the job they will be called to do. Any of
273
these in isolation can undermine the importance of the kinds of knowledge teachers need
to have to be successful in teaching students mathematics. Policy makers need to
consider what knowledge teachers need to be successful and require coursework that will
develop that knowledge. These courses should include opportunities for the intertwining
of the domains of content knowledge and pedagogical content knowledge. Coursework
should be required of students to revisit the topics they will be expected to teach and in
ways that will be useful in the classroom, adding depth to their knowledge of the
mathematical topics.
Implications for Future Research
Conclusions drawn from this study suggest we do not know nearly enough about
the content knowledge and pedagogical content knowledge of high school algebra
teachers. Neither do we now have a full understanding about how changes in these types
of knowledge are reflected in instructional practices. This study suggests results similar
to studies done with elementary teachers (Ball, 1988a, Ma, 1999, Ball & Bass, 2000,
Stacy et al., 2001. Swafford, Jones, & Thornton, 1997, Borko et al., 2000), algebra
teachers have inadequate content knowledge and pedagogical content knowledge for
successful mathematics instruction. Additional research needs to be conducted to further
develop an understanding of the content knowledge and pedagogical content knowledge
that high school teachers possess. Results from this study suggest that both of these types
of knowledge are limited in their use in instructional practice. Therefore, more research
needs to be done in understanding how to create a seamless connection between
mathematical knowledge and instructional practice (Ball et al., 2001). In addition, we
274
need to develop a better understanding of what content knowledge and pedagogical
content knowledge of algebra pre-service teachers have. If the mathematics and
mathematics education faculty members collaborate to ensure that pre-service teachers
develop content knowledge and pedagogical content knowledge, what kind of content
knowledge and pedagogical content knowledge will result? Furthermore, how does the
the mathematical knowledge of pre-service teachers compare to that of pre-service
teachers attending universities where the mathematics department and mathematics
education departments do not collaborate to ensure the acquisition of both of these types
of knowledge?
Conclusion
Do you hear the doors closing on the productive futures of many of students in the
U. S.? Without adequate preparation in mathematics in general and algebra in particular,
those doors to future educational and career opportunities will remain closed. A central
key in opening this door lies in learning algebra and learning it with deep understanding.
But the key to these opportunities lies in the hands of teachers who have the strong
content knowledge and strong pedagogical content knowledge needed to be successful in
opening that door for the adults of the twenty-first century. However, along with many
past studies, this study demonstrates that many teachers do not possess these types of
knowledge. This study also demonstrates that providing teachers with strong content
knowledge and strong pedagogical content knowledge is possible, and it can influence
instructional practices. This process of change, however, is neither simple nor direct.
Nonetheless, we need to maintain our commitment to finding ways of accomplishing this
275
goal, which in turn affects the learning of students and provides them all the opportunities
they all deserve.
276
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research monograph. SUNY Series, Reform in Mathematics Education. Ithaca,
NY: State University of New York Press.
Stacey, K., Helme, S., Steinle, V., Baturo, A., Irwin, K., & Bana, J. (2001). Preservice
teachers? knowledge of difficulties in decimal numeration. Journal of
Mathematics Teacher Education, 4(3), 205-225.
Strutchens, M. E., & Silver, E. A. (2000). NAEP findings regarding race/ethnicity: The
students, their performance, and their classrooms. In E. A. Silver and P. A.
Kennedy (Eds.), Results from the seventh mathematics assessment of the National
Assessment of Educational Progress (pp. 45-72). Reston, VA: National Council
of Teachers of Mathematics.
Stump, S. L. (1997, March 28). Secondary mathematics teachers? knowledge of the
concept of slope. Paper presented at the Annual Meeting of the American
Educational Research Association, Chicago, IL.
Stump, S. L. (2001). High school precalculus students? understanding of slope as
measure. School Science and Mathematics, 101(2), 81-89.
287
Swafford, J. O., Jones, G. A., & Thornton, C. A. (1997). Increased knowledge in
geometry and instructional practice. Journal for Research in Mathematics
Education, 28(4), 467-483.
Thompson, A. G. & Thompson, P. W. (1994). Talking about rates conceptually, Part I:
A teacher?s struggle. Journal for Research in Mathematics Education, 25( ),
279-303.
Thompson, P. W. & Thompson, A. G. (1996). Talking about rates conceptually, Part II:
Mathematical knowledge for teaching. Journal for Research in Mathematics
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knowing: Representations of knowledge in teaching. In J. Calderhead (Ed.),
Exploring teachers? thinking (pp. 104-124). Philadelphia, PA: Cassell
Educational.
288
APPENDICES
289
APPENDIX A
ALGEBRA CONTENT KNOWLEDGE INSTRUMENT (ALCKIN)
Algebra Content Knowledge Instrument (ALCKIN)
11. Mr. Farrow asked his algebra students to determine which of the following
expressions/graphs are functions. For each of the following decide if you agree,
disagree, or you?re not sure about the student?s assessment.
A.
x
y
Noah said A was not a function because
every x value corresponds to the same y
value.
I agree I disagree I?m not sure
1 2 3
Explain your answer.
B.
x
y
Bart said B was not a function because
it had a ?strange shape? and you could
not find an equation for it.
I agree I disagree I?m not sure
1 2 3
Explain your answer.
290
C.
y =
1
x
Sanchez said C was not a function
because it was undefined when x = 0.
I agree I disagree I?m not sure
1 2 3
Explain your answer.
12. Mrs. Jones asked her algebra students to divide x
2
? 4 by x + 2. Seth said, ?I have an
easy method, Mrs. Jones. I just divide the x
2
by x and the 4 by the 2. I get x ? 2,
which is correct.? Mrs. Jones is not surprised by this as she had seen students do this
before. What did she know? (Mark one answer.)
f. She knew that Seth?s method was wrong, even though he happened to get the
right answer for this problem.
g. She knew that Seth?s answer was actually wrong.
h. She knew that Seth?s method was right, but that for many algebraic fraction
division problems this would produce a messy answer.
i. She knew that Seth?s method only works for some algebraic fractions.
j. I?m not sure.
Explain your answer.
291
292
13. Teachers often offer students ?rules of thumb? to help them remember particular
mathematical ideas or procedures. Sometimes, however, these handy memory
devices are not actually true, or they are not true in all situations. For each of the
following, decide whether it is true all of the time or not. (Mark TRUE FOR ALL
SITUATIONS, NOT ALWAYS TRUE, or I?M NOT SURE.)
True for
ALL
Situations
________
NOT
Always
True
_______
I?m
Not
Sure
______
A. A binomial made up of two perfect squares
cannot be factored into two binomials unless the
two terms have a subtraction sign between them.
1 2 3
B. An asymptote is a line that a graph approaches
but never crosses.
1 2 3
C. Any number to the zero power is equal to one.
1 2 3
D. When graphing linear inequalities, if the
inequality sign is ?? you shade above
the line.
1 2 3
14. Consider the following statements related to different sets of numbers. After reading
each statement decide if you agree, disagree, or if you are not sure.
Statement I agree I disagree I?m not sure
A. -3 is a rational number 1 2 3
B. 2/3 is a real number
1 2 3
C. .010010001? is a rational number. 1 2 3
D. ?5 is a complex number.
1 2 3
293
15. Mr. Casteel is using spreadsheets in his Algebra class to find solutions for quadratic
equations. What approximate solution(s) for the equation 3x
2
= 4-2x should Mr.
Casteel?s students give using the following spreadsheet?
X 3x
2
4-2x
-1.8 9.72 7.6
-1.7 8.67 7.4
-1.6 7.68 7.2
-1.5 6.75 7
-1.4 5.88 6.8
-1.3 5.07 6.6
-1.2 4.32 6.4
-1.1 3.63 6.2
-1 3 6
-0.9 2.43 5.8
-0.8 1.92 5.6
-0.7 1.47 5.4
-0.6 1.08 5.2
-0.5 0.75 5
-0.4 0.48 4.8
Solution(s):
Explain your answer.
16. Without using algebraic manipulation such as collecting like terms or using the
distributive property, what other methods could you as a teacher use to justify to your
students that the expressions 3x + 5 + 5x -3 and 4 (2x + 1/2) are equivalent to each
other?
17. Students who are in Mrs. Simpson?s algebra class were working on the following
problem: You are going to build a square garden and surround its border with square
tiles. Each tile is 1 foot by 1 foot. For example, if the dimensions of the garden are
10 feet by 10 feet, then you will need 44 tiles for the border.
10
10
How many tiles would you need for a garden that is n feet by n feet? Two
students gave the following diagrams, representations, and reasons for their
answers.
James stated you could find the number of tiles needed for the
border by finding the area of the entire garden and border and
subtract out the area of the garden.
Which representation best reflects his method?
A. 4n + 4
B. 4(n + 1)
C. (n + 2)
2
? n
2
D. All are equally correct
Ann said you could take the entire length of each side of the
border which is n and multiply by four, then add in each of the
corners.
Which representation best reflects her method?
A. 4n + 4
B. 4(n + 1)
C. (n + 2)
2
? n
2
D. All are equally correct
Which student has the best understanding of the tile problem?
A. James
B. Ann
C. Both show equal understanding.
Explain your answer:
294
18. Mrs. Ledbetter gave her algebra students the following two diagrams of steps and
asked them to determine which set of steps were the steepest. The students could not
agree on an answer and gave the following answers and reasons for their
answers.
Steps BSteps A
6
4
6
4
6
4
6
4
3
2
3
2
3
2
3
2
32
E. Group 1 said that both sets of steps
have the same steepness. ?First
connect the points at the edges of
the steps with a line and measure
the angle formed between this line
and the ground. The measure of
this angle is the same in both set of
steps. Therefore they have the
same steepness.?
F. Group 2 said Steps B were steeper
because the steps in Steps B are
taller than in Steps A.
G. Group 3 thought both sets of steps
had the same steepness since
doubling both the length and height
of the steps does not affect the
steepness.
H. Group 4 said both sets of steps had
the same steepness. ?If you sketch
in a line that contains the edges of
the steps and determine the slope of
the line in each set of steps, you
will find they have equal slope.
Steps A have a slope of 2/3 and
Steps B have a slope of 4/6 which is
equivalent to 2/3. Therefore, they
have the same steepness.?
Which group(s) have a correct understanding of slope? (Circle all that apply).
A. Group 1
B. Group 2
C. Group 3
D. Group 4
Of the four groups which group has the best understanding of slope?
______________________
Explain why you chose this group.
295
19. Mr. Seng?s algebra class is studying the graph of y = ax
2
+ bx + c and how changing
the parameters a, b, and c will cause different translations of the original graph.
Translated graph
y= ax
2
+ bx + c
How do you think Mr. Seng will explain the translation of the original graph
y = ax
2
+ bx + c to the translated graph?
F. Only the a value changed
G. Only the c value changed
H. Only the b value changed
I. At least two of the parameters changed.
J. You cannot generate the translated graph by changing any of the parameters.
Explain your answer choice:
296
297
20. Mrs. Westbrook?s algebra students were working on writing algebraic equations for
problems similar to the following:
It takes 0.2 hour to bake a dozen chocolate chip cookies and 0.15 hour to
bake a dozen plain cookies, how many dozen cookies can be baked in
fifteen hours?
Mrs. Westbrook?s students came up with the following equation:
0.2x + 0.15y = 15
where x = number of dozens of chocolate chip cookies and y = number of dozens
of plain cookies.
What does the 0.2x represent in the equation?
298
APPENDIX B
TEACHER ATTITUDE SURVEY
299
300
301
302
303
304
305
306
307
308
APPENDIX C
RTOP CLASSROOM OBSERVATION SURVEY
309
310
311
312
313
314
APPENDIX D
FIRST TEACHER INTERVIEW PROTOCOL
315
? My name is Joy Black, and I am from Auburn University. I work with the
MDSMIP project which is trying to understand and improvement mathematics
in your school.
? We are interested in finding out what teachers think about math and how they
feel about teaching math.
? During this interview I will ask questions pertaining to your opinions and
beliefs about how you and your students would work mathematics problems
and questions about how you would approach teaching certain algebraic
topics. This is not a test, so there are no right and wrong answers; it?s only
your opinions about methods you use in your classroom that matter.
? Everything you tell me is completely confidential. That means, we never
share your answers with anyone.
? First, I need to get your permission to participate in this interview.
1) Give two copies of the permission form to the teacher and ask them to sign one
and return to you. The second permission form is for the teacher to keep.
2) Say: ?I want you to think about the students you have in your algebra classes.
What percent of them do you think can learn algebra well?
a. If the teacher answers all or almost all ask: ?Why do you think almost all
of the students can learn algebra well??
b. For other answers ask: ?Why do you think this percentage of your
students can learn algebra well?? After they answer ask this: ?Why do
you think the other students cannot learn algebra well??
3) Say: ?88% of the teachers surveyed agreed or strongly agreed with the following
statement: It is important for students to figure out how to solve mathematics
problems for themselves. How are you carrying this out in your own classroom??
4) Say: ?On the other hand 88% of these same teachers also agreed or strongly
agreed with the following statement: Teachers should model and demonstrate
mathematical procedures and then, ideally, time should be allowed for the
students to have the opportunity to practice those procedures. In what ways can
these coexist in the mathematics classroom??
5) Give the teacher the following problem (on a separate sheet of paper) and ask
him/her to work it:
Jameel looked out in the parking lot and decided to count cars and motorcycles by
the number of their wheels, excluding spares. He saw that there were 17 vehicles
with a total of 56 wheels. How many motorcycles did he see?
6) When the teacher finishes working the problem ask him/her to work the problem
using a different method.
316
7) When the teacher finishes working the problem a second way, ask the teacher to
work the problem using a third method. If the first two methods appear to
basically be the same such as setting up two equations ? solving the first by
elimination and the second by substitution, say ?Can you work the problem in any
other way that is really different from the first two ways you have worked the
problem??
8) Give time for the teacher to work the problem a third way. If the teacher can not
quickly come up with a third way move on to the next question.
9) Ask the teacher how he/she would teach this problem to his/her algebra classes.
10) When she is finished say: ?If you gave this problem to your algebra students to
solve, which of the methods that you used do you think your students would
use??
11) After the teacher chooses a method say: ?Why do you think your students would
use this method?? If the teacher says that the students would use none of the three
methods ask: ?What method do you think your students would use to solve the
problem and why do you think they would use this method??
12) If the teacher did not give a picture solution to the problem, give her the picture
solution you have prepared (See Interview Protocol page 4). Say: ?Look at this
student solution and tell me what you think about it.? After the teacher replies
ask: ?Would you accept this as a solution to this problem?? Depending on their
answer ask ?Why or why not?? (If the teacher gave a picture solution, go directly
to the question ?Would you accept this as a solution to the problem?? and follow
it by ?Why or why not??)
13) Give them the second solution which is a reasoned solution (See Interview
Protocol page 5). Say: ?Look at this student solution and tell me what you think
about it.? After the teacher replies ask: ?Would you accept this as a solution to
the problem?? Depending on their answer ask ?Why or why not?? (If the teacher
gave a reasoning type solution, go directly to the question ?Would you accept this
as a solution to the problem?? and follow it by ?Why or why not??)
14) Give the teacher the following problem on a separate sheet of paper:
x
2
= 2x + 8
Give the teacher the first student solution where the student graphed both sides of
the equation (See Interview Protocol page 6). Say ?Explain what you think about
what this student did to solve the problem.? After the teacher explains, ask: ?Do
you think this is an acceptable way for a student to work the problem? Why or
why not??
317
15) Give the teacher the second student solution where the student shows some work
then comes up with the correct solution (See Interview Protocol page 7). Say
?Explain what you think about what this student did to solve the problem.? After
the teacher explains, ask: ?Do you think this is an acceptable way for a student to
work the problem? Why or why not??
16) Thank the teacher for his/her participation.
318
319
1. Solve for x:
x
2
= 2x + 8
To find the solution I graphed each side of the equation.
-15 -10 -5 5 10 15
18
16
14
12
10
8
6
4
2
-2
hx() = 2?x+8
fx() = x
2
The two graphs intersected at two points (-2,4) and (4, 16). From these I can find the two
solutions for the equation x
2
= 2x + 8. x would be equal to -2 and 4.
320
321
322
APPENDIX E
PROTOCOL FOR SECOND TEACHER INTERVIEW
323
? My name is Joy Black, and I am from Auburn University. I work with the
MDSMIP project which is trying to understand and improvement mathematics
in your school.
? We are interested in finding out what teachers think about math and how they
feel about teaching math.
? During this interview I will ask questions pertaining to your opinions and
beliefs about how you and your students would work mathematics problems
and questions about how you would approach teaching certain algebraic
topics. This is not a test, so there are no right and wrong answers; it?s only
your opinions about methods you use in your classroom that matter.
? Everything you tell me is completely confidential. That means, we never
share your answers with anyone.
? First, I need to get your permission to participate in this interview.
? Give two copies of the permission form to the teacher and ask them to sign
one and return to you. The second permission form is for the teacher to keep.
1. Give me your overall impression on the cohort I professional development
training you received this summer through MDSMIP.
As the teacher gives her impressions, I will be looking for answers to the following
questions. If the teacher does not address the questions, these will be used as follow
up questions to the first overall question.
2. Prior to the summer cohort I training, what was your attitude toward attending the
summer professional development training? In what ways do you think it
changed as the two weeks progressed?
3. What changes have you made in your classroom practices as a result of the
professional development training you have received? Give me an example. Do
you think it was successful? Why or why not? Will you use this approach again?
How do you think you will change it? Why?
4. What kind of content knowledge of algebra do you think you possess? Did you
learn anything new about algebra this summer? (If yes) Give me an example. In
what ways do you think the professional development training affected your
content knowledge of algebra?
5. Last spring I asked you about the percentage of students you had in your algebra
classes that could really learn algebra well. Think about the students you have in
your algebra classes this year. What percent of them do you think can learn
algebra well? If the teacher answers all or almost all ask: ?Why do you think
almost all of the students can learn algebra well?? For other answers ask: ?Why
do you think this percentage of your students can learn algebra well?? After they
answer ask this: ?Why do you think the other students cannot learn algebra
well??
324
6. Was there any part of the summer professional development that may have
changed your perception about the percentage of your students who can learn
algebra well? If so, give me an example.
7. Did you participate in any other professional development training besides the
two week cohort I professional development training? If yes, what training did
you participate in? Do you think this professional development training had an
impact on your classroom practices for this academic school year? What impact
has it had? Can you give me an example?
8. You?ve participated in follow-up cohort I meetings, give me your impression of
these meetings. Were they helpful? If yes, in what ways and why do you think it
was helpful. If the answer was no, why do you think it was not helpful?
9. Give the teacher the following problem (on a separate sheet of paper) and ask
him/her to work it:
x
2
= 2x + 8
10. When the teacher finishes working the problem ask him/her to work the problem
using a different method.
11. When the teacher finishes working the problem a second way, ask the teacher to
work the problem using a third method. If the first two methods appear to
basically be the same such as setting up two equations ? solving the first by
elimination and the second by substitution, say ?Can you work the problem in any
other way that is really different from the first two ways you have worked the
problem??
12. Give time for the teacher to work the problem a third way. If the teacher can not
quickly come up with a third way move on to the next question.
13. Ask the teacher how he/she would teach this problem to his/her algebra classes.
14. When she is finished say: ?If you gave this problem to your algebra students to
solve, which of the methods that you used do you think your students would use??
15. After the teacher chooses a method say: ?Why do you think your students would
use this method?? If the teacher says that the students would use none of the three
methods ask:?What method do you think your students would use to solve the
problem and why do you think they would use this method??
325
APPENDIX F
PROFESSIONAL DEVELOPMENT PRESENTERS INTERVIEW PROTOCOL
326
1. Tell me about your overall impression on how (the teacher) participated in the
cohort I summer professional development training.
As the presenter gives the impression on the teacher, I will be looking for answers to
the following questions. If the presenter does not address the questions, these will be
used as follow up questions to the first overall question.
2. How engaged was this teacher during the professional development? Can you
give me an example?
3. How did you see her attitude change during the summer professional development
training? Can you give me an example?
4. Did you think this teacher was open to the ideas presented during the summer
training? Why or why not? Did she seem to have an openness to implement the
types of classroom instruction that were being encouraged through the
professional development? Why or why not?
5. What kind of algebraic content knowledge was addressed during the summer
professional development training? What do you think this is the kind of algebraic
content knowledge this teacher possesses?
327
APPENDIX G
CODE LIST FOR THE ALGEBRA CONTENT KNOWLEDGE INSTRUMENT
328
CODES FOR ALGEBRA CONTENT KNOWLEDGE INSTRUMENT
________________________________________________________________________
Code Frequency Explanation of Code
________________________________________________________________________
Counterexample 39 Provides an example that would make a
statement false such as 0
0
? 1.
Definition 76 Use the definition of a term as an
explantion.
Denotes Asymptote 3 Denotes that the function has an asymptote.
Denotes Student Error 5 ?Maybe they were thinking? type
staements.
Discontinuous Function 1 States the function is not continuous
Drawing/Modeling 12 States or draws a model for the explanation.
Explanation of Expression 15 Put into words such as two tenths times x.
Understanding of Expression 35 Indicates an understanding that 0.2 x means
the number of hours it would take to bake x
dozen chocolate chip cookies.
Formula Manipulation 6 Analyzes general quadratic equation to see
which parameter reflects the graph.
Function 45 Indicates that the graph or problem is a
function.
Graph 11 Teacher sketches a graph in the Cartesian
plane to use in the answer decision or to
clarify the selected answer.
I Don?t Know 23 Teacher writes ?I don?t know. I?m not sure!
Or Not sure! Took a Guess.
Incorrect Reasoning/Answer 74 The reasoning is incorrect; teacher gave an
incorrect answer; teacher gave a partial
answer, teacher doesn?t really explain the
answer; partial explanation; answer has
nothing to do with the question.
329
________________________________________________________________________
Code Frequency Explanation of Code
________________________________________________________________________
Use Manipulatives 13 Teacher suggests some type of manipulative
such as algebra tiles or counters in the
explantion.
Multiple Reasons 85 Teacher uses more than one method in
supporting his/her explanation.
No Explanation Given 176 No explanation was provided or participant
did not choose an answer but provided an
explantion.
Procedural Knowledge 28 Uses procedures for explanation.
Provides Example 4 Gives a number example in the explanation.
Slope Reference 33 Noted that a particular group used slope.
Spreadsheet Analysis 30 Uses the spreadsheet correctly to find the
solution for x in the quadratic equation.
Subject Matter Taught 8 Relates explanations to the fact that their
students do not learn this or the teacher does
not teach this particular concept.
Synthetic Division 4 States or demonstrates synthetic division in
the explanation.
Table 5 Creates a chart or suggests that a chart be
used.
Uses Division 18 States or demonstrates polynomial division
in the xplantion.
Uses Factoring 33 Breaks the expressions down into factors
and cancels out the common factors.
Uses Quadratic Formula 11 Uses the quadratic formula in an attempt to
find the solution instead of using the
spreadshet.
330
________________________________________________________________________
Code Frequency Explanation of Code
________________________________________________________________________
Value Substitution 22 Suggest that students can use the
substitution of a value or values for the
varible to show two algebraic expressions
are quivalent.
Vertical Line Test 48 Denotes or demonstrates that the vertical
line test in testing for functions.
331
APPENDIX H
CODE LIST FOR THE CASE STUDIES
332
CODES FOR CASE STUDIES
________________________________________________________________________
Code Frequency Code Explanation
________________________________________________________________________
Checks Work 18 Teacher demonstrates or encourages
students to check their solutions.
Conceptual Knowledge 24 Teacher makes a statement that
refrs to the conceptual
understanding of the
mathematical topic being discussed
in class such as referring to algebra
tiles.
Definition 52 Defines terms, gives formulas, or teacher
asks the students for the definition of terms
or formula.
Denotes Using Inverse 2 Teacher indicates that the procedural step in
Operations solving an algebraic equation would involve
doing an inverse operation such as taking a
square root of both sides of an equation.
Formula Manipulation 7 Includes correct substitution into formula
and solving for unknowns
Gives Praise to Students 30 Gives praise or encouragement to students.
You can do this! Think! Teacher expresses
that students can do the work because they
are advanced students.
Multiple Reasons 35 Multiple ways of working problems
proceduraly.
Problem Instructions 2 Gives the students instructions on their
asignment.
Provides Example 19 Begins procedural instruction by working an
example.
Step by Step Procedures 225 Teacher goes through step by step
procedures for working a problem.
333
_______________________________________________________________________
Code Frequency Code Explanation
________________________________________________________________________
Student Agrees with Teacher 137 Yes madam, um humph, etc.
Student Answers ?Why? or ?How? 41 Teacher asks a question for why he did
something. Student explains how to do
something procedurally or why he/she chose
a particular procedure.
Student Gives Correct Answer 177 Student gives a correct answer to an
arithmetic problem, correct solution to an
equation, or correct simplification to an
algebraic expression. For example, ?What is
five squared??
Student Gives Correct Procedure 86 ?Find a common denominator? ?Subtract
forty-eight.? ?Just divide everything by
two.?
Student Gives Incorrect Answer 31 Student answers a procedural question
incorectly.
Student Gives Incorrect Procedure 15 ?You added the four sides together?
or Next Step
Student Indicates They Don?t 17 Student tell teacher they don?t understand
Understand a procedure.
Student Practice, Practice Problems 12 Students are given class time to work
or Quiz several practice problems or a quiz over
previously covered material. Students work
problems after procedural instruction from
the teacher. Students put homework
problems on the whiteboard.
Student Question 96 Student questions such as ?Are these bonus
questions??
Student Recognition of an Error 16 Student recognizes own error or the error of
a clasmate.
334
________________________________________________________________________
Code Frequency Code Explanation
________________________________________________________________________
Student Recognizes Teacher Error 3 Student recognizes when the teacher writes
something incorrectly on the whiteboard.
Student Short Answers to 236 Student answers the teacher?s request for an
Arithmetic or answer to an arithmetic procedure or for the
Algebraic Question simplification of an algebraic expression.
Teacher Answers Own Question 76 Teacher asked a question but does not allow
time for students to answer. Teacher
answers her own question.
Teacher Answers Student Question 25 Answers student question or responds to an
answer given by the student.
Teacher Can?t Recognize Student 3 Can?t or does not recognize student error.
Error
Teacher Asks Clarifying Question 174 Teacher may ask ?Do you mean?? What did
you get??
Teacher Comment or Question 23 That was easy. Or Okay a quick way to do
Indicates a Task is Easy this is?
Or Can be done quicker
Teacher Given Hint 117 It?s a multi-step equation.
Teacher Gives Non Mathematical 121 Think before you speak. Your assignment
Instructions is?
Teacher Ignores Incorrect Student 18 Teacher ignores incorrect student suggestion
Procedure for working problem.
Teacher Ignores Student Question 6 Teacher ignores student question of
coment.
Teacher Gives Incorrect Information 6 b ? 4ac (instead of b
2
? 4ac)
335
________________________________________________________________________
Code Frequency Code Explanation
________________________________________________________________________
Teacher Question Involves Asking 134 Procedure, step, names a formula, ask for
for an Answer to an agreement from the students.
Arithmetic Problem or
Simplifying an Algebraic
Expression
Teacher Question Involves Asking 168 For example: Alright? Do you have any
if Students have other questions? Do you understand? Which
Questions or Understands problems do you need worked?
Teacher Questions ?How? or 133 Why you cannot do something, how you
?Why? did something.
Teacher Question Involves Asking 220 As the teacher is working, she asked
To Name the Next Step or students to tell the next step or procedure.
Procedure
Teacher Recognizes Correct Student 35 Correct. Um humph.
Procedure
Teacher Recognizes Incorrect 30 Denotes that student answer is incorrect.
Student Answer
Teacher Recognizes Incorrect 37 Teacher denotes that student procedure is
Student Procedure incorrect.
Teacher Reiterates Students? Reply 210 Teacher repeats what the student just said.
Teacher Reminds Students of 90 Reminds students of steps that should be
?Steps? or ?Rules? taken to solve problem or reminds students
of rules associated with working a problem.
Uses Manipulatives 20 Uses manipulatives, drawings, graphs, etc.
Uses neumonics 17 Uses a procedure considered to be a
neumonic. For example: Please excuse my
dear aunt Sally or Remember these are the
problems we put in a chart.