AN EXPLORATORY STUDY OF THE POSSIBLE ALIGNMENT BETWEEN THE
BELIEFS AND TEACHING PRACTICES OF SECONDARY MATHEMATICS
PRE-SERVICE TEACHERS AND THEIR COOPERATING TEACHERS AND
ITS EFFECTS ON THE PRE-SERVICE TEACHERS? GROWTH TOWARDS
BECOMING REFORM BASED MATHEMATICS TEACHERS
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 contain any proprietary information.
______________________________
April C. Parker
Certificate of Approval:
______________________________ ______________________________
W. Gary Martin Marilyn E. Strutchens, Chair
Professor Professor
Curriculum and Teaching Curriculum and Teaching
______________________________ ______________________________
Kimberly L. King-Jupiter Stephen Stuckwisch
Associate Professor Assistant Professor
Education Foundations, Leadership & Mathematics and Statistics
Technology
______________________________
Joe F. Pittman
Interim Dean
Graduate School
AN EXPLORATORY STUDY OF THE POSSIBLE ALIGNMENT BETWEEN THE
BELIEFS AND TEACHING PRACTICES OF SECONDARY MATHEMATICS
PRE-SERVICE TEACHERS AND THEIR COOPERATING TEACHERS AND
ITS EFFECTS ON THE PRE-SERVICE TEACHERS? GROWTH TOWARDS
BECOMING REFORM BASED MATHEMATICS TEACHERS
April C. Parker
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
December 17, 2007
iii
AN EXPLORATORY STUDY OF THE POSSIBLE ALIGNMENT BETWEEN THE
BELIEFS AND TEACHING PRACTICES OF SECONDARY MATHEMATICS
PRE-SERVICE TEACHERS AND THEIR COOPERATING TEACHERS AND
ITS EFFECTS ON THE PRE-SERVICE TEACHERS? GROWTH TOWARDS
BECOMING REFORM BASED MATHEMATICS TEACHERS
April C. Parker
Permission is granted to Auburn University to make copies of this dissertation at its
discretion, upon requests of individuals or institutions and at their expense.
The author reserves all publication rights.
Signature of Author
Date of Graduation
iv
VITA
April C. Parker, daughter of Bobby and Penny Cook, was born January 20, 1975, in
Phenix City, Alabama. She graduated from Central High School in 1993. She entered Judson
College in 1993 and graduated with a Bachelor of Science degree in Secondary Mathematics
Education in December, 1996. While working as a mathematics teacher at Central High
School, she entered Graduate School, Troy State University, in January 1998, and graduated
with a Master of Science degree in Secondary Mathematics Education in June, 1999. In July
2002, she became employed by Troy University teaching mathematics. While working at
Troy University, she entered Graduate School, Auburn University, in August, 2002, and
graduated with a Master of Science degree in Applied Mathematics in August 2006. She
married Michael Shannon Parker on June 13, 1998 and they have one child, Michael
Alexander Parker.
v
DISSERTATION ABSTRACT
AN EXPLORATORY STUDY OF THE POSSIBLE ALIGNMENT BETWEEN THE
BELIEFS AND TEACHING PRACTICES OF SECONDARY MATHEMATICS
PRE-SERVICE TEACHERS AND THEIR COOPERATING TEACHERS AND
ITS EFFECTS ON THE PRE-SERVICE TEACHERS? GROWTH TOWARDS
BECOMING REFORM BASED MATHEMATICS TEACHERS
April C. Parker
Doctor of Philosophy, December 17, 2007
(M.S., Auburn University, 2006)
(M.S., Troy State University, 1999)
(B.S., Judson College, 1996)
280 Typed Pages
Directed by Marilyn E. Strutchens
For the mathematics reform movement to continue, cooperating teachers as well
as pre-service teachers must be well equipped to carry out the Standards set forth by The
National Council of Teachers of Mathematics (NCTM). It becomes necessary to explore
the impact of the alignment or misalignment of the cooperating teachers? practices and
the pre-service teachers? approach to teach based on their preparation. Particularly, what
beliefs and practices do cooperating teachers have that support or hinder the growth of a
pre-service teacher immersed into reform-based teaching? What happens when there is a
vi
misalignment of the beliefs and practices held by the cooperating teacher and the
educational background of the pre-service teacher?
Case studies of four different cooperating teacher/pre-service teacher pairs were
used. The cooperating teachers were all teachers that were currently involved in the
university?s mathematics reform initiative program. The pre-service teachers were all
students that were completing requirements in a mathematics education program that
immersed them in mathematics reform techniques. Throughout the study, the researcher
used and collected various types of data to better understand the pairs. The forms of data
included: a beliefs survey; classroom observations; interviews; and completed Reformed
Teaching Observation Protocols (RTOPs) for each classroom observation.
One pre-service teacher was very much reform-minded as was her cooperating
teacher. Because of the support she received from her cooperating teacher, the pre-service
teacher was able to flourish in her internship. Another pre-service teacher was reform-
minded and her cooperating teacher was not. Even so, the pre-service teacher was able to
successfully implement the techniques she had learned in her methods courses. The other
two pre-service teachers ended up imitating the more traditional practices that were
carried out by their cooperating teachers. It is believed that the cooperating teachers?
degree of belief in reform mathematics approaches impacted the actions of the pre-
service teachers. All cooperating teachers were comfortable allowing the pre-service
teachers to try the reform approaches; however, the more traditional cooperating teachers
were not able to mentor the pre-service teachers in ways that would help the pre-service
teachers. As a result, the traditional cooperating teachers? respective pre-service teachers
succumbed to the teaching methods used by them.
vii
ACKNOWLEDGEMENTS
I would first and foremost like to acknowledge my Lord and Savior, Jesus Christ.
He has guided each and every step I have taken along this journey. When times got
tough, He continually reminded me of his promise in Matthew 19:26 which states, ?With
man this is impossible, but with God all things are possible.? None of this would have
been possible without God.
Secondly, I would like to acknowledge my family and friends. Without their love,
help, and support, this whole process would have been unbearable. To my husband,
Shannon, thank you for believing in me and cheering me on when times were rough.
Also, thank you for stepping in and doing the things I didn?t have time to do. To my son,
Alex, thank you for being my inspiration. To my parents, thank you for instilling in me
the importance of a good education. Also, thank you for being there for Shannon and
Alex when I couldn?t be. Finally, to my friend, Paige, I would like to say thank you for
listening and understanding. You?ll soon be writing your own acknowledgements!
Last but not least, I would like to acknowledge my committee. Your support and
guidance is truly appreciated. To Dr. Strutchens, thank you for being so patient with me
and helping me grow through this experience. None of this would have been possible
without you.
viii
Style manual or journal used: Publication Manual of the American Psychological
Association, Fifth Edition
Computer software used: Microsoft Excel, Version 11; Atlas.ti, Version 5.0;
Microsoft Word 2000
ix
TABLE OF CONTENTS
LIST OF TABLES..................................................................................................... xvii
I. INTRODUCTION ......................................................................................... 1
Background Information................................................................................ 3
The Issue ........................................................................................................ 8
II. REVIEW OF LITERATURE ........................................................................ 10
The Internship Experience ............................................................................. 10
Beliefs About Mathematics Instruction ......................................................... 15
Mentoring....................................................................................................... 19
Teacher Efficacy ............................................................................................ 22
Alignment of Standards-based Academic Preparation
with Student Teacher Preparation...................................................... 25
Implications for the Student Teaching Experience........................................ 28
III. DESIGN OF THE STUDY............................................................................ 30
Overview........................................................................................................ 30
Theoretical Basis for the Study...................................................................... 30
Methodology.................................................................................................. 32
Researcher Biases .......................................................................................... 33
Population ...................................................................................................... 34
Math Plus ....................................................................................................... 36
The Mathematics Education Program............................................................ 39
School Demographics .................................................................................... 41
The Cooperating Teachers ............................................................................. 42
The Preservice Teachers ................................................................................ 43
The University Supervisors............................................................................ 44
The Pairs ........................................................................................................ 45
Case 1: Mrs. Smith and Mrs. Franklin............................................... 45
Mrs. Smith.............................................................................. 45
Mrs. Franklin.......................................................................... 46
x
Case 2: Mrs. Johnson and Ms. Walters.............................................. 46
Mrs. Johnson.......................................................................... 46
Ms. Walters............................................................................ 46
Case 3: Mrs. York and Mrs. Windsor................................................ 46
Mrs. York............................................................................... 46
Mrs. Windsor ......................................................................... 47
Case 4: Mrs. Brown and Mrs. Robinson............................................ 47
Mrs. Brown ............................................................................ 47
Mrs. Robinson........................................................................ 47
Instrumentation .............................................................................................. 48
Interviews........................................................................................... 48
Reformed Teaching Observation Protocol (RTOP)........................... 52
Beliefs Survey Used by Math Plus .................................................... 53
Procedure ....................................................................................................... 54
Analysis of Data............................................................................................. 56
IV. RESULTS OF THE STUDY......................................................................... 58
Chapter Organization..................................................................................... 59
Analysis of Data............................................................................................. 59
Case 1: Mrs. Smith and Mrs. Franklin............................................... 63
Mrs. Smith.............................................................................. 65
Spring 2006 and Fall 2006 Classroom Observations. 69
Lesson Design and Implementation........................... 75
Communicative Interactions ...................................... 76
Student-Led Discussions................................ 76
Teacher-Led Discussions............................... 76
Procedural Knowledge............................................... 76
Propositional Knowledge........................................... 77
Student-Teacher Relationships .................................. 77
xi
Mrs. Franklin.......................................................................... 78
Fall 2006 Classroom Observations ............................ 82
Lesson Design and Implementation........................... 88
Communicative Interactions ...................................... 89
Student-Led Discussions................................ 89
Teacher-Led Discussions............................... 89
Procedural Knowledge............................................... 90
Propositional Knowledge........................................... 90
Student-Teacher Relationships .................................. 91
Similarities and Differences between Mrs. Smith
and Mrs. Franklin....................................................... 91
Outcome of the Internship Experience .................................. 92
Case 2: Mrs. Johnson and Ms. Walters.............................................. 94
Mrs. Johnson.......................................................................... 95
Spring 2006 and Fall 2006 Classroom Observations. 99
Lesson Design and Implementation........................... 103
Communicative Interactions ...................................... 104
Student-Led Discussions................................ 104
Teacher-Led Discussions............................... 104
Procedural Knowledge............................................... 105
Propositional Knowledge........................................... 105
Student-Teacher Relationships .................................. 106
Mrs. Walters........................................................................... 106
Fall 2006 Classroom Observations ............................ 110
Lesson Design and Implementation........................... 112
Communicative Interactions ...................................... 114
Student-Led Discussions................................ 114
Teacher-Led Discussions............................... 114
Procedural Knowledge............................................... 115
Propositional Knowledge........................................... 115
Student-Teacher Relationships .................................. 116
xii
Similarities and Differences Between
Mrs. Johnson and Ms. Walters................................... 116
Outcome of the Internship Experience .................................. 117
Case 3: Mrs. York and Mrs. Windsor................................................ 119
Mrs. York............................................................................... 120
Spring 2006 and Fall 2006 Classroom Observations. 125
Lesson Design and Implementation........................... 131
Communicative Interactions ...................................... 133
Student-Led Discussions................................ 133
Teacher-Led Discussions............................... 134
Procedural Knowledge............................................... 134
Propositional Knowledge........................................... 134
Student-Teacher Relationships .................................. 135
Mrs. Windsor ......................................................................... 135
Fall 2006 Classroom Observations ............................ 139
Lesson Design and Implementation........................... 142
Communicative Interactions ...................................... 143
Student-Led Discussions................................ 143
Teacher-Led Discussions............................... 144
Procedural Knowledge............................................... 144
Propositional Knowledge........................................... 144
Student-Teacher Relationships .................................. 145
Similarities and Differences Between
Mrs. York and Mrs. Windsor..................................... 145
Outcome of the Internship Experience .................................. 146
Case 4: Mrs. Brown and Ms. Robinson ............................................. 148
Mrs. Brown ............................................................................ 150
Spring 2006 and Fall 2006 Classroom Observations. 153
Lesson Design and Implementation........................... 160
Communicative Interactions ...................................... 162
Student-Led Discussions................................ 162
xiii
Teacher-Led Discussions............................... 162
Procedural Knowledge............................................... 162
Propositional Knowledge........................................... 163
Student-Teacher Relationships .................................. 163
Mrs. Robinson........................................................................ 164
Fall 2006 Classroom Observations ............................ 167
Lesson Design and Implementation........................... 170
Communicative Interactions ...................................... 171
Student-Led Discussions................................ 171
Teacher-Led Discussions............................... 172
Procedural Knowledge............................................... 172
Propositional Knowledge........................................... 173
Student-Teacher Relationships .................................. 173
Similarities and Differences Between
Mrs. Brown and Ms. Robinson.................................. 174
Outcome of the Internship Experience .................................. 175
Comparison of the Cases ............................................................................... 175
Lesson Design and Implementation................................................... 176
Student-Led Discussions.................................................................... 178
Teacher-Led Discussions................................................................... 178
Procedural Knowledge....................................................................... 179
Propositional Knowledge................................................................... 180
Student/Teacher Relationships........................................................... 181
V. SUMMARY AND RECOMMENDATIONS................................................ 183
Limitations ..................................................................................................... 183
Conclusions.................................................................................................... 184
Implications for Teacher Education Programs .............................................. 186
Implications for Selecting School Leaders and Mentor Teachers ................. 189
Possibilities for Future Research ................................................................... 189
REFERENCES .......................................................................................................... 191
xiv
APPENDICES ........................................................................................................... 197
Appendix A: Interview Questions ............................................................... 198
Appendix B: Information Letters and Consent Forms ................................ 203
Appendix C: Reformed Teaching Observation Protocol (RTOP)............... 210
Appendix D: Mathematics Beliefs Survey .................................................. 216
xv
LIST OF TABLES
Table 1 School Demographics ........................................................................ 42
Table 2 Summary of Cooperating Teacher/Preservice Teacher Pairs ............ 48
Table 3 Summary of Instrumentation Used for Study .................................... 56
Table 4 Utilized Code Words and Frequency................................................. 61
Table 5 Demographic Summary of Riverdale High School ........................... 64
Table 6 RTOP Averages and Median for Mrs. Smith?s
Spring 2006 Classroom Observations................................................ 72
Table 7 RTOP Averages for Mrs. Smith?s Fall 2006
Classroom Observation...................................................................... 75
Table 8 RTOP Averages and Median for Mrs. Franklin?s
Fall 2006 Classroom Observations .................................................... 87
Table 9 Demographic Summary of Riverdale High School ........................... 95
Table 10 RTOP Averages and Median for Mrs. Johnson?s
Spring 2006 Classroom Observations................................................ 102
Table 11 RTOP Averages for Mrs. Johnson?s Fall 2006
Classroom Observation...................................................................... 103
Table 12 RTOP Averages and Median for Ms. Walters?
Fall 2006 Classroom Observations .................................................... 112
Table 13 Demographic Summary of Murphy High School.............................. 120
Table 14 RTOP Averages and Median for Mrs. York?s
Spring 2006 Classroom Observations................................................ 130
Table 15 RTOP Averages for Mrs. York?s Fall 2006
Classroom Observation...................................................................... 131
xvi
Table 16 RTOP Averages and Median for Mrs. Windsor?s
Fall 2006 Classroom Observations .................................................... 141
Table 17 Demographic Summary of Yorkshire High School........................... 149
Table 18 RTOP Averages and Median for Mrs. Brown?s
Spring 2006 Classroom Observations................................................ 156
Table 19 RTOP Averages for Mrs. Brown?s Fall 2006
Classroom Observation...................................................................... 160
Table 20 RTOP Averages and Median for Ms. Robinson?s
Fall 2006 Classroom Observations .................................................... 170
Table 21 Overall Outcomes of the Internship Experience................................ 190
1
I. INTRODUCTION
In February 2005, the joint councils of the National Academy of Sciences and the
National Academy of Engineering met to discuss how the United States was fairing in the
global economy at that time (Committee on Science, Engineering, and Public Policy
(CSEPP), 2006). Their conclusion was somewhat bleak. The participants agreed that a
weakening of science and technology in the United States would ultimately lead to a
degradation of the present social and economic conditions which in turn would inevitably
mean that citizens of the United States would not be able to effectively participate in
society or compete for high quality jobs (CSEPP, 2006). Additionally, Schoenfeld (2004)
felt that a lack of access to mathematics was a barrier ? a barrier that left people socially
and economically disenfranchised. Schoenfeld (2004) also stated, ?We are at risk of
becoming a divided nation in which knowledge of mathematics supports a productive,
technologically powerful elite while a dependent, semiliterate majority,
disproportionately Hispanic and Black, find economic and political power beyond reach.
Unless corrected, innumeracy and illiteracy will drive America apart? (Schoenfeld, 2004,
p. 265). In response to all of this information, the joint councils determined that in order
to counteract this existing decline, the United States workforce must be literate in
mathematics and science as well as many other subjects (CSEPP, 2006).
2
The idea of a mathematically literate workforce was not a new one. In fact, it had
been something that had eluded mathematicians and educators for more than a century.
All parties agreed that a literate workforce was the desired outcome. The National
Council of Teachers of Mathematics had even gone so far as to declare ?math for all?
(NCTM, 1989, 2000). The controversy had been over how to produce this outcome. The
two opposing parties were the traditionalists and the reformists (Schoenfeld, 2004). The
traditionalists claimed that the curriculum proposed by the reformists undermined
classical mathematical values such as mastery of basic facts for all four operations,
knowing and using formulas, counting to 100, etc. (Schoenfeld, 2004). On the other hand,
the reformists claimed that their curriculum reflected a deeper and richer view of
mathematics than that of the traditionalists (Schoenfeld, 2004). Schoenfeld (2004) further
described the traditionalists as being content oriented while the reformists were seen as
being more process oriented.
The traditionalists and reformists always argued that the issue was about what was
best for the children. In essence, the argument always comes down to which is more
important, content or process. The traditionalists argue that in order for our students to be
successful in mathematics, they must first understand the skills involved in solving
problems before they can ever employ these skills to solve problems (Van de Walle,
2005). Many people today are comfortable with this method of instruction because this is
the way mathematics was taught back in the good old days. According to Reys (2002),
however, performances over the past thirty years on the National Assessment of
Education Progress and the International Mathematics and Science studies show that the
methods of the good old days have not been effective. Unlike the traditionalists, the
3
reformists argue that in order for our students to be successful in mathematics as well as
today?s growing technological society, they must be able to solve problems, work
cooperatively on mathematics and communicate mathematically, and make sense
mathematically of the world around them. Van de Walle (2005) stated in his article that
one of the only ways for this to be accomplished is by allowing students to construct their
knowledge. In other words, the students must be allowed to build upon what they already
know in order to understand the new concepts. By allowing this, the students make
connections between ideas and concepts. This in turn leads to meaningful networks of
ideas which means there are fewer details to remember, it is easier to recall ideas after
extended periods of time, there is better application of ideas to newer problems, and there
is a feeling that mathematics makes sense (Van de Walle, 2005). Reyes (2002) stated in
his article that research is beginning to emerge that proves reform mathematics is indeed
increasing student learning and producing the type of mathematical citizen that today?s
society demands.
Background Information
Throughout the decades, mathematics education has been and continues to be a
topic of deep debate and controversy for many parties including but not limited to
politicians, the general public, and mathematics educators (Hart & Keller, 2001). In
essence, the big debate is centered on how to best educate our nation?s children.
During the 1950s, the emphasis of mathematics education was on the learner. It
was decided that it was more important to teach practical skills rather than technical
content and theoretical mathematics. The justification was that the United States needed
4
more informed citizens (Hart & Keller, 2001). It was during this time span that
enrollment in advanced high school mathematics courses decreased (Klein, 2003).
Several things ultimately led to the demise of the way of thinking of the 1950s. The most
important was the launch of Sputnik in the fall of 1957. Because Russia beat the United
States into space, the Sputnik launch was perceived as a major humiliation for the United
States (Klein, 2003). It was determined that the reason for this defeat was brought about
by the large number of mathematically illiterate new recruits that were entering the
military (Hart & Keller, 2001).
New Math was born in reaction to the dissatisfaction with the 1950s methods of
teaching mathematics (Klein, 2003). According to Hart and Keller (2001), this approach
to mathematics education was characterized by its emphasis on abstraction and formality.
Klein (2003) also stated that during this period, there was very little attention given to
basic skills or applications of mathematics. Instead, instruction emphasized topics such as
number bases other than base ten, set theory, and various other exotic topics. During this
period, teachers were expected to ask ?the perfect questions? so that students could
investigate and discover the various mathematical topics involved in calculating an
answer (Herrera & Owens, 2001). Along with posing the questions, the teacher more or
less facilitated the investigation process. This was different from the previous role, which
had been telling the students the relevant concepts and then allowing them to practice the
new skill of the day (Herrera & Owens, 2001). In the end, many parents ended up feeling
confused and alienated because they did not know how to help their children. Another
problem was that many teachers were not properly trained in working with this type of
curriculum. As a result of parents feeling confused and alienated, teacher ineptness, and
5
less than satisfactory student performance, public criticism grew. This criticism
ultimately led to the death of New Math (Klein, 2003).
In the 1970s, another pendulum swing occurred. This time, there was an outright
rebellion against New Math. Instead of the abstractness and formality, the public wanted
to go back to teaching students the basic skills of mathematics. This period in
mathematics education was known as the Back to Basics movement. During this time, the
major goal of the mathematics curricula was to train students to be proficient in
computational procedures in the areas of algebra and arithmetic (Schoen, Fey, Hirsch, &
Coxford, 1999). In order to accomplish this goal, much attention was given to the
classroom instructional routine. This routine generally involved teachers explaining and
illustrating the mathematical procedures. Then, the students mimicked the teacher by
practicing the new skills on a plethora of similar exercises (Schoen et al., 1999).
According to Herrera and Owens (2001), a class during this period was typified as
follows: the teacher began class by going over the answer to the previous night?s
homework assignment; the more difficult problems were worked by the teacher or other
students; a brief explanation, if one was given at all, of the new material; and finally, the
students were assigned problems to work on until the end of class. Even though the
emphasis on basic skills was extreme, national tests given at the time showed that student
performance in basic skills either declined or stayed the same. Also, these tests showed
that performance in the area of problem solving was very poor (Hart & Keller, 2001).
According to Klein (2003), during the 1980s the public began to realize that the
quality of mathematics education had been deteriorating. In spite of the efforts of the
Back to Basics movement, many students were not successful problem solvers (Hart &
6
Keller, 2001). Two important works were produced during the 1980s that greatly
influence mathematics education. These two works were An Agenda for Action and A
Nation at Risk (Klein, 2003). It was decided that students must have a certain level of
proficiency in basic skills as well as the ability to understand more abstract mathematical
concepts. Most important of all, the students needed to be able to apply their
mathematical skills and conceptual understanding in order to become proficient problem
solvers. The 1980s ended with the publication of The National Council of Teachers of
Mathematics (NCTM) Curriculum and Evaluation Standards for School Mathematics
(Hart & Keller, 2001).
The mathematics reform of the 1980?s continued throughout the 1990s (Hart &
Keller, 2001). During this time, NCTM published two more standards documents.
Professional Standards for Teaching Mathematics was published in 1991, and
Assessment Standards for School Mathematics was published in 1995.
Hart and Keller (2001) stated that it is believed that the three NCTM documents
fueled the ?standards-based? reform. The pivotal work of NCTM?s standards came in
2000 with the publication of Principles and Standards for School Mathematics (PSSM)
(Hart & Keller, 2001). The ultimate purpose of this work was to help communicate and
implement the new vision for school mathematics (Hart & Keller, 2001). This new vision
proposed that the classroom teacher should be more of a stimulant, sounding board, and
guide throughout the student problem solving process (Schoen et al., 1999). As described
by Herrera and Owens (2001), the teacher should be a facilitator of learning and an
orchestrator of classroom discourse. Overall, the role of the teacher must change from
one who is the transmitter of knowledge to one who orchestrates classroom discourse,
7
creates a learning environment that is mathematically empowering, and engages the
students in mathematical investigation (Herrera & Owens, 2001; Manouchehri &
Goodman, 2000). All of these characteristics typify reform based teaching.
Even after all of this, the debate over what mathematics should be taught
throughout the school curricula is still ongoing (Van de Walle, 2006). Do we teach the
?basics? or do we teach ?reform? mathematics? In order to determine this, it is necessary
to know what each method looks like. According to Van de Walle (2006), the ?basics?
approach consists primarily of arithmetic or computation. This method would include
things such as the following: counting accurately to numbers higher than one hundred;
solving problems involving formulas; mastery of basic computational skills such as
addition, subtraction, division, and multiplication; pencil and paper computation skills;
etc (Van de Walle, 2006). Ultimately, the ?basics? approach suggests that children
mindlessly mimic the things their teachers do. This, however, doesn?t necessarily
guarantee that they will understand what they are being taught (Van de Walle, 2006). On
the other hand, ?reform? mathematics focuses more on how students think and learn.
According to Van de Walle (2006), reformers have five goals for students: value
mathematics; be confident in the ability to do mathematics; become mathematical
problem solvers; learn to communicate mathematically; and learn to reason
mathematically (Van de Walle, 2006). As a result of these goals, manipulatives,
cooperative group work, calculators, etc. have become the hallmarks of reform
mathematics (Van de Walle, 2006).
8
The Issue
According to Curcio and Artzt (2005), one of the most difficult and challenging
jobs for teacher educators is to prepare future teachers to support the reform efforts that
ultimately lead to high-quality teaching; however, in order for the mathematics reform
movement to continue, the existing teachers as well as pre-service teachers must be well
equipped to carry out the standards set forth by NCTM (1989, 2000). As stated by
Graham and Fennell (2001), over the years, change has been made in teacher education
programs. From the early to mid-1900s, many of the teachers were produced by two-year
normal schools. In the mid-1900?s, these normal schools grew into four-year institutions.
As the change to the four-year institutions was made, so were the course requirements for
prospective teachers. These improvements were pertinent during the mid-1900s;
however, recent studies have shown that not much has changed in teacher education
programs since the mid-1900s (Graham & Fennell, 2001). Fortunately, the National
Council of Teachers of Mathematics (NCTM) has initiated standards for changing what
mathematics should be taught, how it should be taught, and how it should be assessed
(Taylor, 2002).
One of the biggest challenges to NCTM?s proposed change(s) has been changing
teachers? views of mathematics. Up to this point, math has always been associated with
following the teacher?s rules and finally getting the ?one right answer? (Taylor, 2002).
Now, teacher educators and mathematics supervisors must ?move teachers away from
mathematics as teachers have most likely experienced it as students for over a decade and
guide teachers toward a view of mathematics that is more consistent with the NCTM
standards? (Taylor, 2002, p. 138). Ultimately, teachers must build a new image of
9
teaching and learning (Taylor, 2002). In light of this building process, it is necessary to
explore the impact of the alignment or misalignment of the cooperating teachers?
practices and the pre-service teachers? approach to teaching based on their preparation. In
particular, what beliefs and practices do cooperating teachers have that support or hinder
the growth of an intern indoctrinated into reform-based teaching? Also, what happens
when there is a misalignment of the beliefs and practices held by the cooperating teacher
and the educational background of the intern?
10
II. REVIEW OF THE LITERATURE
In order to fully understand the issue at hand, there are several areas that must be
explored. First, the general internship experience is discussed. It is important to
understand who the participants are during the internship experience as well as how each
participant influences the other. Next, is a discussion concerning the beliefs about
mathematics instruction. This section explores different ideas about what mathematics
should be taught in the classroom. It also examines how mathematics should be taught.
Then, it is necessary to explore the various views on the alignment of standards-based
academic preparation with internship experiences. Here, the reader is exposed to various
ideas about how what the pre-service teachers learn through their university classes either
is or is not reinforced throughout their internship experience. Finally, the impact of the
above mentioned concepts on the overall internship experience is discussed.
The Internship Experience
Borko and Mayfield (1995) stated that learning to teach is a complex process,
especially in the field of mathematics education; however, despite its complexity,
learning to teach is also considered to be one of the most important aspects of any
educational program. There are several issues that come into existence whenever any
educational program begins to try to place student teachers within school systems. The
following are some of those issues: what student teaching model will be utilized; what
11
school system(s) will participate; who the cooperating teachers will be; what role will the
cooperating teacher play; what role will the university supervisor play; what will be the
responsibilities of the student teacher.
Over the past several years, much research has been conducted on internships in
education (Mtetwa & Thompson, 2000). The consistent problem, however, is that a
majority of the research that has been done up to this point is very generic. Hardly any of
the research that has been conducted thus far has been subject-specific (Mtetwa &
Thompson, 2000).
Frykholm (1998), reported that the student teaching experience is generally
thought of as the most formative and significant element of the entire educational
program. McIntyre, Byrd, and Foxx (1996) agreed, but also added that in the past few
years, this practice has come under increased scrutiny. The reason for this is due in part to
an increased desire by the educational community to produce new teachers who are
capable of analyzing and reflecting on teaching practices (McIntyre et al., 1996).
McIntyre et al. (1996) inferred that a possible remedy for this problem is to modify the
current student teaching experience.
Under the current model, the tripartite model, there are three key players: the pre-
service teacher, the cooperating teacher, and the university supervisor (Tsui, Lopez-Real,
Law, Tang, & Shum, 2001). In this model, it is quite obvious that the role of the pre-
service teacher is to learn how to teach (Borko & Mayfield, 1995). The cooperating
teacher, in general, is thought of as the person that helps build and foster self-confidence
rather than to give constructive criticism on instruction (Borko & Mayfield, 1995).
12
Giving constructive criticism on instruction is generally viewed to be the role of the
university supervisor (Borko & Mayfield, 1995).
According to several sources, cooperating teachers are the most important
influences within the student teaching experience (Beck & Kosnik, 2000; Drafall &
Grant, 1994; Fueyo, 1991). One of the main jobs of the cooperating teacher is to try to
help the pre-service teacher through various phases of thought development.
Unfortunately, many cooperating teachers do not feel like they have been adequately
trained to carry out this role (Drafall & Grant, 1994). According to Beck and Kosnick
(2000), the reason for this is a lack of clarity and agreement about the role of the
cooperating teacher. There appears to be two separate models for the role of the
cooperating teacher. The first is the practical initiation model. In this model, described by
Beck and Kosnick (2000), the role of the cooperating teacher is to initiate the pre-service
teacher into the field of teaching. In other words, the internship is viewed more like an
apprenticeship. According to Beck and Kosnick (2000), there are two approaches to this
model. The cooperating teacher either takes the sympathetic approach or the sink or swim
approach. The other model that Beck and Kosnick (2000) reported on is the critical
intervention model. In this model, the role of the cooperating teacher is to encourage the
pre-service teacher to become more reflective and analytical of the implemented teaching
practices. This role seems to be one way to ward off some of the scrutiny that was
previously mentioned by McIntyre et al. (1996).
Borko and Mayfield (1995) reported on a longitudinal study named Learning to
Teach Mathematics (LTTM). In this study, four pre-service teachers were observed
throughout their internship experience. All four pre-service teachers were interested in
13
teaching mathematics in grades six through eight. According to the authors, the
cooperating teachers that were assigned to the four pre-service teachers had a varied
range of teaching experience and mathematical knowledge (Borko & Mayfield, 1995).
The university supervisors who participated in the study were three graduate students.
Like the participating cooperating teachers, the university supervisors all had various
amounts of mathematical knowledge and teaching experience (Borko & Mayfield, 1995).
Throughout the study, various forms of data collection techniques were used. These
techniques included the following: interviews with the cooperating teachers, pre-service
teachers, and university supervisors; observations of pre-service teachers? mathematical
instruction taking place in the classrooms; observations between pre-service teachers and
cooperating teachers; and observations between pre-service teachers and the university
supervisors (Borko & Mayfield, 1995). It was discovered that many of the conversations
held between cooperating teachers and the pre-service teachers rarely included in-depth
exploration of issues of teaching and learning mathematics. Likewise, conversations
between the university supervisors and the pre-service teachers were frequently too
rushed and based on insufficient data concerning the pre-service teachers? teaching.
Borko and Mayfield (1995) ultimately concluded that the pre-service teachers involved in
LTTM learned not to expect much out of their relationships with the cooperating teachers
and university supervisors.
Based on their research, Borko and Mayfield (1995) proposed several reasons for
the limitations and potential solutions for changing the situation involving student
teaching experiences in the area of mathematics education. One reason involved the
belief systems of all parties involved in the student teaching experience. All three parties
14
involved, the cooperating teachers, the pre-service teachers, and the university
supervisors all reported that a person learns to teach by teaching. In other words, learning
to teach is accomplished through experience, practice, and making mistakes (Borko &
Mayfield, 1995). Based on this, the authors further concluded that it becomes too easy for
the cooperating teachers and university supervisors to offer too few suggestions or
challenges to the pre-service teachers. Also for the same reason, the pre-service teachers
pay very little attention to the feedback that is given by the cooperating teachers and
university supervisors. In essence, the status quo is maintained (Borko & Mayfield,
1995). Borko and Mayfield (1995) concluded that the student teaching experience should
be considered as a beginning point rather than a culminating point of the pre-service
teacher?s learning instead of the other way around. Another factor that appeared to hinder
the influence of the cooperating teacher and the university supervisor was the shared
desire to maximize comfort and minimize risks. Borko and Mayfield (1995) suggested
that the cooperating teachers and university supervisors should be supportive of the pre-
service teachers, but they should also allow them to take the risks that are necessary for in
depth learning. Finally, Borko and Mayfield (1995) reported that both the cooperating
teachers and the university supervisors needed to have a more active role in the student
teaching experience. In order to function in this more active role, tasks such as modeling
new forms of pedagogy and challenging pre-service teachers? beliefs and practices
through more frequent and more extensive conversations are expected of both university
supervisors and cooperating teachers. In order to accomplish this, however, cooperating
teachers and university supervisors alike needed to develop a sense of efficacy as teacher
educators. For the cooperating teachers, this involves learning how to engage the pre-
15
service teachers in more in-depth discussions that focus on teaching and learning as well
as how to be more reflective about their practice. For the university supervisors, this
process entails shifting from a role of critiquing specific lessons to a role of enabling the
cooperating teachers to become teacher educators. This means that the university
supervisors would use their time helping the cooperating teachers learn ways to observe
pre-service teachers as well as conduct meaningful conversations with the pre-service
teachers that ultimately lead to self reflection on teaching practices (Borko & Mayfield,
1995).
Beliefs About Mathematics Instruction
According to Thompson (1992), there are four dominant and distinct views of
how mathematics should be taught. One is the learner-focused or constructivist view.
From this viewpoint, mathematics teaching focuses on the learner and the knowledge that
he can construct. Here, the teacher is a facilitator and stimulator of student learning. Her
job is to ask intriguing thought-provoking questions, pose situations for investigation, and
challenge students to think (Thompson, 1992). Another of the four dominant views is
content-focused with an emphasis on conceptual understanding also known as the
Platonist view (Thompson, 1992). Here, mathematics teaching is driven by the
mathematical content itself but emphasis is placed on conceptual understanding. This
view emphasizes students? understanding of the logical relations among various
mathematical topics and the logic underlying the mathematical procedures. The role of
the teacher is very similar to that of the previous view (Thompson, 1992). The next view
is content-focused with an emphasis on performance also known as the instrumentalist
16
view (Thompson, 1992). Here, the emphasis is on student performance particularly on the
mastery of mathematical rules and procedures. The role of the teacher here is to
demonstrate, explain, and define the mathematics the students need to know in an
expository style. In this case, the role of the student is to listen, answer questions that
have been asked by the teacher, and then complete exercises or problems using the
procedures that have been previously demonstrated by the teacher (Thompson, 1992). It
is important to note that this type of instruction does not actively engage the students in
the process of exploring and investigating various mathematical ideas. Thus, mathematics
is many times misrepresented to students when this view is utilized (Thompson, 1992).
The final view is the classroom-focused view. Here, mathematics teaching is based on
knowledge about effective classrooms. The teacher is portrayed as directing all classroom
activities, clearly presenting the mathematical material to the whole class, and providing
opportunities for the students to work individually. From this perspective, teachers who
are effective can skillfully explain, assign tasks, monitor student work, provide feedback
to students as well as manage the overall classroom environment while at the same time
eliminate or prevent disruptions that might interfere with the flow of the planned activity.
Playing off the role of the teacher, the students? job is to listen, answer questions when
asked, follow directions, and complete tasks assigned by the teacher (Thompson, 1992).
Vacc and Bright (1999) reported on a study of pre-service teacher education
programs at three sites that was carried out by the University of Wisconsin. In their
report, Vacc and Bright focused on the site located at the University of North Carolina.
At that site, the researchers explored changes in pre-service elementary school teachers?
beliefs concerning teaching and learning mathematics along with their abilities to offer
17
mathematics instruction that was structured around the way children think. Throughout
the study, the thirty-four participants were exposed to Cognitively Guided Instruction
(CGI) as part of their mathematics methods courses. A CGI Belief Scale was used to help
determine if significant changes in their beliefs and perceptions about mathematics
instruction took place throughout the duration of their methods courses and internship
experience. Observations were also used to explore how two participants in particular
used their knowledge about their students? mathematical thinking in instruction
throughout their internship. The two participants, Helen and Andrea, were chosen
because their cooperating teachers both taught at the same school and both taught the
same grade level. The difference was that one of the cooperating teachers had extensive
CGI training while the other had only been briefly exposed (Vacc & Bright, 1999).
Andrea was placed with the cooperating teacher that had only been exposed to CGI in a
two-hour workshop (Vacc & Bright, 1999).
Helen was placed with the cooperating teacher that had extensive CGI training.
Because of her placement, Helen was able to observe her teacher incorporating CGI
principles into her mathematics instruction prior to taking over full instruction of the
classes. Also, throughout her internship experience, Helen was constantly encouraged by
her teacher to gather information about her students? thinking in order to adapt her
instruction for the students (Vacc & Bright, 1999).
At the beginning of her program, Helen wrote she believed that the teacher?s role
was to model problem solutions for the students. She also stated that a teacher should
question students to find out what they were thinking as they were solving problems.
Throughout her internship experience, Helen consistently planned and implemented
18
instruction that was based on problem solving. Additionally, she facilitated critical
thinking skills and student understanding by using high level questioning that extended
beyond basic arithmetic problem types (Vacc & Bright, 1999). The authors commented
that the instruction she provided her students appeared to be consistent with her beliefs
about teaching and learning mathematics (Vacc & Bright, 1999).
Andrea was placed with the cooperating teacher that had only been briefly
exposed to CGI principles. She reported that she received very little support from her
teacher. Andrea also stated that most of the time, she taught straight from the textbook
unless she knew her lesson was going to be video-taped, then, she taught a ?CGI-type?
lesson. From the way Andrea commented, her teacher encouraged her choice of when to
incorporate the CGI principles (Vacc & Bright, 1999).
At the beginning of her program, Andrea stated that the framework of learning
mathematics was the memorization of facts; however, by the conclusion of her internship
experience, Andrea stated that children should be provided opportunities to explore and
discover various mathematical concepts. She also commented that asking questions was
more important than telling students what they needed to know. Vacc and Bright (1999)
further commented that the questioning was important to Andrea as long as the students?
responses to her questions matched up with what she expected them to say. In essence,
her focus ultimately became more directed toward procedure building with the teacher
being the ultimate authority on the concept being learned (Vacc & Bright, 1999).
As a result of the study, Vacc and Bright (1999) reported that teachers? beliefs
about teaching and learning mathematics greatly influenced the form and type of
mathematical instruction that was delivered. In particular, Vacc and Bright (1999) stated
19
that if teachers? beliefs were compatible with the underlying philosophy and materials
comprising the mathematics curriculum they were utilizing, they were more likely to
fully implement the curriculum. On the other hand, the same could not be said if the
beliefs were not in alignment with the existing curriculum.
According to Vacc and Bright (1999), pre-service teachers are somewhat set in
their ways when it comes to their beliefs about teaching and learning mathematics. In
particular, many of these beliefs are derived from their own experiences as students. As a
matter of fact, it was reported that because of these vivid personal experiences, ?learning
new theories and concepts may have little effect in changing pre-service teachers? general
beliefs about teaching practices? (Vacc & Bright, 1999, p. 91). Cooney, Shealy, and
Arvold (1998) also stated that these beliefs seldom change dramatically without
significant intervention. In light of this information, it was suggested that in order for
existing beliefs to be replaced or restructured, new beliefs must be intelligible and appear
plausible (Vacc & Bright, 1999).
Mentoring
An internship experience can also be thought of as a mentoring relationship that
exists between the cooperating teacher and the pre-service teacher. Nolder, Smith, and
Melrose (1994) stated that the perceived success or failure of the internship experience
hinges on the quality of the relationship formed and the expectations of both parties with
regard to the roles to be played by the pre-service teacher and the cooperating teacher.
These roles can be viewed as: supportive fellow professional; listening friend; supportive
critic; gatekeeper and guide; and link agent (Nolder, Smith & Melrose, 1994).
20
When playing the role of the supportive fellow professional, the mentor treats the
pre-service teacher not as a student or teacher?s aide but as a novice professional. In this
situation, both parties professionally contribute to the relationship. Everything the mentor
does is perceived by the pre-service teacher as a model of professional practice (Nolder,
Smith & Melrose, 1994).
During the course of the internship experience, the pre-service teacher just needs
someone to listen. This is where the role of the listening friend comes into play. In this
role, the mentor is there when the pre-service teacher needs to confide his/her fears,
his/her joys, or his/her successes or failures in the classroom. Nolder, Smith and Melrose
(1994) stated that in order to build this facet of the relationship, it is essential that regular
times be set aside for meetings between the cooperating teacher and the pre-service
teacher where privacy and confidentiality are respected. Nolder, Smith and Melrose
(1994) also commented that availability and approachability seemed to be key features in
encouraging the pre-service teachers to relate to their cooperating teacher.
When acting in the role of the supportive critic, the mentor is in essence acting
like a critical friend. Being a supportive critic involves many tasks. One is observing pre-
service teachers? lessons. Another is offering praise. Giving constructive criticism is yet
another. Finally, a supportive critic is available to support the pre-service teacher in
follow-up activities (Nolder, Smith & Melrose, 1994).
The role of gatekeeper and guide is another important aspect of a mentor. Here,
the mentor is the one who assists the pre-service teacher in getting acquainted with the
school and its functions. Some other responsibilities include: provide knowledge about
the backgrounds and abilities of the children and what to expect from them; explain the
21
system within the school such as knowing about discipline, sanctions, and rewards; and
know what is likely to work in mathematics classrooms within the school (Nolder, Smith
& Melrose, 1994).
The final role of a mentor as discussed by Nolder, Smith and Melrose (1994) is
the link agent. In this capacity, the cooperating teacher serves as a liaison. She provides
opportunities that ensure the pre-service teacher is familiar with the school, the staff, the
students, and other teachers such as English teachers or Special Needs teachers (Nolder,
Smith & Melrose, 1994).
From the above descriptions, the roles of supportive fellow professional and
supportive critic can be viewed as somewhat comparable roles. Both roles are viewed as
ones that assist the pre-service teacher in improving teaching practices. The supportive
fellow professional provides guidance for the pre-service teacher by modeling acceptable
professional practice. The supportive critic provides guidance for the pre-service teacher
by observing lessons, giving constructive criticism, supporting the pre-service teacher in
follow up activities, and giving positive recognition when it is due. All of these tasks help
the pre-service teacher grow inside the classroom.
Two other roles that can be viewed as comparable roles are those of gatekeeper
and guide and the link agent. Both of these roles assist the pre-service teacher with things
not directly associated with teaching a lesson. As mentioned above, the primary role of
the gatekeeper and guide is to ensure that the pre-service teacher is familiar with the
school and its functions. The link agent is similar in that he/she ensures that the pre-
service teacher has opportunities to reach out and meet other support agents of the school
22
such as Special Needs teachers. All of these tasks help the pre-service teacher grow
outside the classroom.
All of the above mentioned roles have the potential of being a friend; however,
they are not the same as a listening friend. The role of the listening friend can be viewed
as simply a sounding board. He/She is there when the pre-service teacher just needs to
talk to someone. The things discussed may or may not always be directly related to
teaching practices or the internship experience at all. Unlike the other mentoring roles,
the listening friend is there for the emotional well-being of the pre-service teacher.
Philippou and Charambous (2005) reiterated that a mentor?s role encapsulates a
wide spectrum of responsibilities such as being considered as teaching models and
critical friends who assist newcomers with planning, teaching, and evaluating students to
simply being there to provide assistance to pre-service teachers only when requested. In
particular, it has been determined that mentors affect pre-service teachers? teaching image
by their teaching style, the feedback they provide to the pre-service teachers, and the
underlying messages that their behavior and body language conveys (Philippou &
Charambous, 2005). It was also stated that even though mentors are in a position to guide
pre-service teachers? participation in practices of teaching and various pedagogical
responses, they seldom take advantage of this position (Philippou & Charambous, 2005).
Teacher Efficacy
As defined by Smith (1996), a teacher?s sense of efficacy is his/her belief in their
ability to have a positive effect on student learning. This sense of efficacy can be viewed
as one of two types. The first is teaching efficacy (Smith, 1996). Teaching efficacy refers
23
to general beliefs about teachers? ability to produce student learning in spite of various
external challenges such as low motivation levels in the students, weak student ability,
and etc. The other type of efficacy is personal teaching efficacy (Smith, 1996). This type
is an individual teacher?s own sense of his/her ability to take effective action in teaching.
As a natural result, teachers with a strong sense of efficacy generally attribute the success
of their students to things that they as teachers did to bring about the success. They
disregard other factors that may have also influenced student success. On the other hand,
teachers with a weaker sense of efficacy believe that other factors besides their teaching
influence student success (Smith, 1996).
According to Smith (1996), a sense of efficacy is a self-attribution. In other
words, a person must construct his/her beliefs about the connection between his/her
actions and the consequences of those actions. This connection involves two things:
beliefs about himself and herself and beliefs about the world. In general, a person must
believe that he/she has the ability to have an effect on things along with the belief that the
world will respond in a positive manner (Smith, 1996). There are various sources of these
beliefs, however, as stated by Smith (1996), a history of perceived past successes plays
the most important role. Based on this information, a strong sense of teaching efficacy
requires the teacher to: conceptualize what is efficacious about his/her actions and find
the positive results of those actions in student learning; reflect on, maintain, and draw
upon a personal history of past teaching successes; and recognize that his/her
effectiveness will vary from student to student and context to context (Smith, 1996).
Smith (1996) stated that teachers? sense of efficacy is an important influence on
their practice as well as their students? learning. As reported by Smith (1996), teachers
24
who had a higher sense of efficacy: produced higher measures of student achievement;
maintained learning environments that were responsive to students; persisted longer with
struggling students; and orchestrated more productive small-group work. In general, these
teachers knew that their authority in the classroom was a direct result of their competence
and not their social position, were more committed to teaching, and were usually more
willing to attempt new and innovative practices in their classrooms (Smith, 1996).
Charambous, Philippou, and Kyriakides (2004) also stated that teachers with a strong
sense of efficacy have more positive attitudes toward innovation and are more likely to
implement it and regard the innovation as important and compatible with their usual way
of working. Additionally, Charambous, Philippou, and Kyriakides (2004) commented
that teachers with a strong sense of efficacy are generally more willing to experiment
with new teaching approaches and materials and are usually less anxious about the
reform and the possible limitations or complications deriving from it. On the other hand,
teachers who have a lower sense of efficacy: attributed student failure to things that were
beyond their control such as students? lack of ability, lack of student motivation, flaws
within a student?s character, or poor home environment; intentionally overlooked
students who incorrectly answered questions; and maintained classrooms that were more
rigid and controlling (Smith, 1996).
It is believed that efficacy beliefs are important for the success of any reform
program. Charambous, Philippou, and Kyriakides (2004) addressed three levels of
concerns that teachers have when it comes to reform. They are self concerns, task
concerns, and impact concerns. Self concerns typically relate to the teacher?s anxiety
about his/her ability to take over new demands and responsibilities in the school
25
environment. Task concerns refer to the every day jobs associated with teaching,
especially in relation to numerous limitations such as time constraints, teaching larger
numbers of students, having fewer resources, etc. Impact concerns focus on teachers?
anxiety concerning students? outcomes (Charambous, Philippou, & Kyriakides, 2004). In
order to defuse some of these concerns and maintain a strong sense of efficacy,
Charambous, Philippou, and Kyriakides (2004) stated that it was imperative that teachers
receive ample information about the philosophy and aims of the reform.
Alignment of Standards-based Academic Preparation with Student
Teaching Experiences
One of the biggest challenges to NCTM?s proposed change(s) has been changing
teachers? views of mathematics. Up to this point, math has always been associated with
following the teacher?s rules and finally getting the ?one right answer? (Taylor, 2002).
Now, teacher educators and mathematics supervisors must ?move teachers away from
mathematics as they have most likely experienced it as students for over a decade and
guide them toward a view of mathematics that is more consistent with the standards?
(Taylor, 2002, p. 138). Ultimately, teachers must build a new image of teaching and
learning (Taylor, 2002).
Taylor (2002) reported that teachers can be categorized into one of two states of
being. The first is the teacher in motion. These are the teachers that see themselves as
learners. Because they view themselves as learners, they are more likely to evolve and
grow in their teaching (Taylor, 2002). The second is the teacher that is at rest. These are
the teachers who see themselves as having completed their fundamental learning upon
26
receiving their certification. These teachers tend to make only superficial changes to their
teaching if they make any changes at all (Taylor, 2002). According to Taylor (2002), in
order for any kind of significant change to occur, teachers must continually reflect on
their teaching, reflect on how their teaching affects their students, seek professional
development, and be willing to make changes based on the new understanding(s) they
gain from the whole process.
Taylor (2002) recommended two strategies to help overcome the above
mentioned challenges. He also noted that these strategies work especially well for pre-
service teachers. The first strategy is immersion, and the second strategy is instillation.
The immersion strategy is designed to encourage pre-service mathematics
teachers to implement standards-based teaching upon entering the field as a certified
teacher. According to Taylor (2002), there are three key factors related to immersion.
First, the teacher educator must have standards-based materials readily available for the
pre-service teachers and use them on a regular basis with the pre-service teachers. Some
materials that Taylor (2000) suggested are standards-based curricula, videotapes of
standards-based teaching, and narrative cases of standards-based teaching. All of these
are very effective for challenging the pre-service teachers? beliefs about mathematics
education. The second key to immersion is to immerse the pre-service teacher in both
theory and practice (Taylor, 2002). Bristor et al. (2002), stated that many times, ?teacher
preparation programs fail to link theory with practice, leave content area knowledge
disconnected from methods, and do a poor job of relating instructional practices to
learning and development? (p. 689). Taylor (2002) suggested that one way to immerse
the pre-service teachers into theory and practice is to engage them as mathematical
27
learners with an inquiry approach. The third key of immersion is to transition the pre-
service mathematics teacher into the real classroom. This can sometimes be an issue if
there is inconsistency between the kind of teaching the pre-service teacher has been
prepared for and the experience they have through their field experiences and student
teaching. Taylor (2002) stated that if this type of inconsistency occurs, the teacher
educator would then have to find a way to bring the two worlds closer together.
According to Taylor (2002), the best way to do this is to make sure the pre-service
teacher gets placed with a teacher whose teaching is in line with the standards. Taylor
(2002) recommended that if no such teachers exist, professional development should be
done to train the needed cooperating teachers. Peterson and Williams (1998) warn that if
this does not occur, the pre-service teachers will be less inclined to utilize the standards-
based strategies they have been taught throughout their teacher preparation program.
The instillation strategy is designed to instill some of the professional habits
necessary to keep mathematics teachers and their students actively engaged (Taylor,
2002). According to Taylor (2002), there are three key factors involved in the instillation
process. The first is to read and discuss practice articles as well as theoretical-research
articles. The reasoning behind this is to form habits early and to reinforce the idea that
this is a practice that needs to be continued even after they have been teaching for thirty
years. Another factor for the instillation strategy is to unite pre-service teachers with
other pre-service teachers. This process gets them used to the idea of acting
professionally with other people. The purpose for uniting with other pre-service teachers
is to get them to associate with other people who have similar experiences up to that point
in their career (Taylor, 2002). The third factor is to network the pre-service teachers with
28
in-service teachers. Taylor (2002) cautioned here that pre-service teachers need to learn
to interact with in-service teachers so that they don?t get used to only associating with
colleagues their own age.
Implications for the Student Teaching Experience
As stated by Pourdavood 1999), existing classroom norms and the cooperating
teachers? methods of instructions have profound impact on pre-service teachers? beliefs
and practices. According to the research, it seems that if pre-service teachers are to
internalize coherent applications to teaching and learning mathematics, the environment
in which they complete their internship and the support they receive need to be consistent
with the principles being advocated in their professional preparation program (Vacc &
Bright, 1999). As quoted by Vacc and Bright:
Although we believe that providing pre-service teachers with a robust research-
based model of children?s thinking during a mathematics methods course changes
their beliefs about teaching and learning mathematics, their abilities to incorporate
these beliefs during student teaching may depend on the support pre-service
teachers receive from the classroom teacher who supervises their student-teaching
experiences. (1999, p. 109)
It seems that extensive field experiences and linkages between theory and practice are
essential elements for changing pre-service teachers? beliefs (Vacc & Bright, 1999). The
problem is finding field placements that support the philosophy of reform-based teacher
preparation programs. According to the research, recent evidence suggests that
incongruent field placements may be counterproductive and damaging in developing
29
open-minded attitudes toward reform among pre-service teachers (Curcio & Artzt, 2005).
Curcio and Artzt (2005) further stated that in order for fieldwork to be most effective, it
needs to take place in an environment in which the philosophy is aligned with that of the
teacher preparation program. The bottom line is that the framework underlying the
content presented in mathematics methods courses needs to be consistent with the
framework of the mathematics education program that pre-service teachers observe and
implement during field experiences. If the two frameworks are not in sync, the theories
and concepts presented during the mathematics methods course may not seem plausible
and may ultimately be rejected by the pre-service teacher (Vacc & Bright, 1999).
30
III. DESIGN OF THE STUDY
Overview
This study incorporated the input from cooperating teacher/pre-service teacher
pairs. The purpose of the study was to explore the impact of the alignment or
misalignment of the cooperating teachers? practices and the pre-service teachers?
approach to teaching based on their preparation. The specific questions of research that
were investigated using qualitative methods were:
1. What beliefs and practices do cooperating teachers have that support or
hinder the growth of a pre-service teacher immersed in reform based teaching?
2. What happens when there is a misalignment of the beliefs and practices
held by the cooperating teacher and the educational background of the pre-service
teacher?
Theoretical Basis for the Study
Constructivism is a learning theory where people construct their own
understanding of the world (Ishii, 2003; Telese, 1999); hence, the construction of their
own knowledge (Ishii, 2003). In turn, constructivism is thought of as a lens with which to
know or understand the world (Ishii, 2003).
31
Ishii (2003) pointed out that professional literature describes constructivism in
several different forms. These forms include, but are not necessarily limited to, the
following adjectives: contextual, dialectical, empirical, humanistic, information-
processing, methodological, moderate, Piagetian, post-epistemological, pragmatic,
radical, rational, realistic, social, and socio-historical (Ishii, 2003). Regardless, every
form of constructivism incorporates the idea of individually constructed knowledge
(Ishii, 2003).
Using constructivism as the lens, the classroom is viewed as a mini society or a
community of learners, in particular, a group of learners that are engaged in activity,
discourse, and reflection (Telese, 1999). In these classrooms, the teacher is responsible
for providing concrete and contextually meaningful experiences in which the students
feel comfortable asking questions as well as constructing models, concepts, and strategies
(Telese, 1999). In essence, the students and teacher must know and be at ease with the
community?s language, customs, typical problems, and tools (Greenes, 1995).
Constructivism suggests certain classroom practices and social norms (Wheatley,
Blumsack, & Jakubowski, 1995). Some of these social norms include the following: a
task that requires time and investigation; students explaining their reasoning to their
classmates; and collaboration among peers (Wheatley, Blumsack, & Jakubowski, 1995).
These norms imply certain classroom practices associated with constructivist teaching.
These practices are: the mathematics studied must be analyzed to determine the major
concepts and relationships; it is important to build models of students thinking; tasks are
designed that have potential learning opportunities; all activities must have the potential
of being meaningful to the students; meaning must be negotiated; and a major
32
responsibility of the teacher is to facilitate classroom discourse (Wheatley, Blumsack, &
Jakubowski, 1995).
In this study, all of the participants can be viewed as ?students? who were
engaged in utilizing the constructivist theory in some shape, form, or fashion. The
cooperating teachers were all involved in the university?s reform initiative program. This
program was driven by the constructivist view. It was geared toward teaching teachers
how to help their students build their own knowledge base in mathematics. By doing so,
many of the teachers involved in the program had to take a serious look at the way
mathematics instruction was being implemented in their own classrooms. At the same
time, the pre-service teachers involved in this study were in the process of constructing
their personal teaching style. In their methods courses, the pre-service teachers were
exposed to multiple ways to help their students become engaged in meaningful
mathematics. Throughout the internship experience, the pre-service teachers were also
exposed to other ways to teach their students. Sometimes these methods coincided with
what the pre-service teachers had learned in their college courses, and sometimes the
methods were contradictory to what the pre-service teachers had been taught. Then, the
pre-service teachers also had to contend with the methods they were exposed to in grade
school. Which method or combination of methods would work best for them? This was
the question that the pre-service teachers had to battle with throughout this study.
Methodology
In general, interpretative research practices were utilized to collect data for this
study. According to Gubrium and Holstein (2003), these practices are defined as the
33
?constellation of procedures, conditions, and resources through which reality is
apprehended, understood, organized, and conveyed in everyday life? (Gubrium &
Holstein, 2003, p. 215). More importantly, interpretative research practices ?engage both
the hows and whats of social reality? (Gubrium & Holstein, 2003, p. 215). Furthermore,
these practices focus on how people construct their worlds and experiences (Gubrium &
Holstein, 2003).
More specifically, the researcher utilized case studies throughout the data
collection process. According to Schwandt (2001), a case study is simply a strategy for
doing social inquiry. By definition, a case study is preferred under the following
conditions: when the researcher wants answers to how or why questions; when the
researcher has little control over events being studied; when the object of study is a
contemporary phenomenon in a real-life context; when boundaries between the
phenomenon and the context are not clear; and when it is desirable to use multiple
sources of evidence (Schwandt, 2001). More importantly, case studies seek to discern and
pursue understanding of issues intrinsic to the case (Schwandt, 2001).
Researcher Biases
It is important to note here that the researcher is a graduate student that was
exposed to many of the same teachings, philosophies, and techniques as the pre-service
teachers. As a result, the researcher may have tended to pay more attention to certain
details as opposed to others. For example, when completing the classroom observations,
due to prior training, the researcher could have inadvertently dismissed important
information for paying attention to other details. Also, by exploring the topic before ever
34
studying the subjects, the researcher formulated ideas about the outcome of the study.
Being human, this means that the researcher naturally looked for things that would
support those preconceived notions. Nonetheless, the researcher made a valiant effort to
take all information into account when analyzing the data.
Population
Pseudonyms have been used for all students, teachers, and schools included in this
study. The population for this study was comprised of cooperating teachers, pre-service
teachers, and university supervisors. More specifically, the study focused on the
cooperating teacher/pre-service teacher pairs.
Initially, there were six cooperating teachers and seven pre-service teachers that
were chosen as potential candidates to participate in this study. During the Spring 2006
semester, when making initial contact with the principals and the cooperating teachers, it
was discovered that three of the teachers that all taught for the same system could not be
cooperating teachers because they didn?t meet the system?s criteria for doing so. It was
made clear that in that particular school system, a teacher had to have taught in that
system for at least three years before he/she could be considered as a cooperating teacher.
The three teachers that were chosen by the university did not meet that qualification;
hence, without consulting the university, the school system replaced them with three
other teachers. One of those teachers chose not to participate in this study. In another
situation, one of the teachers originally contacted reported back to the researcher that she
would not be teaching mathematics at her present school the next school year because she
was relocating as a school librarian in another school. Upon being asked, she did
35
recommend another teacher to act as her replacement. So, after all of that, there were five
cooperating teachers that decided to participate in this study. Once all of the teachers had
agreed to participate, the researcher conducted the initial three classroom observations.
At the last observation with one of the teachers, he informed the researcher that he was
relocating to a different school the next year. Due to various circumstances surrounding
this situation, that particular teacher could no longer participate in the study. So, there
were four cooperating teachers left to participate in this study.
As mentioned previously, there were initially seven pre-service teachers. They all
agreed to participate, so the researcher commenced to observing them in their methods
course in which they were all enrolled. At the completion of that course, there were two
pre-service teachers whose participation in the upcoming internship was questionable due
to their current grade point average. These two pre-service teachers had to take a class
during the summer semester to help raise their grade point average so that it would be
high enough to participate in the internship experience. Unfortunately, these two were not
allowed to participate; therefore, there were five pre-service teachers left to participate in
this study.
During the Spring 2006 semester the coordinator of the secondary mathematics
education program carefully paired the cooperating teachers with the pre-service
teachers. At that time, the pre-service teachers were strategically placed with cooperating
teachers that were currently participating in the mathematics reform initiative program,
Math Plus. When all of the above described changes began to occur, the cooperating
teacher/pre-service teacher pairs changed as well. As an end result of the above described
changes, there were four cooperating teachers and four pre-service teachers that were
36
utilized in this study. The four cooperating teachers all functioned in various capacities in
the Math Plus program. One of the cooperating teachers was a Math Plus presenter,
district teacher leader, and school teacher leader. Two of the teachers were school teacher
leaders. The final cooperating teacher just attended professional development sessions
that were provided by Math Plus.
Math Plus
Math Plus, the mathematics reform initiative program, is a partnership between
Valley University?s College of Education and College of Sciences and Mathematics,
Cartersville University, and fifteen school districts located in the Eastern portion of the
state. The purpose of the program is to improve mathematics throughout the Eastern
portion of the state. Eventually, the program would like to accomplish the following:
increase overall student achievement; address gaps in mathematical performance that can
be found among the various demographic groups; improve professional development that
is offered to practicing mathematics teachers; foster a group of knowledgeable teacher
leaders; and enhance the preparation of pre-service teachers at the university level. The
mission statement of the program involves enabling all students to understand, utilize,
and communicate mathematics as a tool in everyday situations. The final goal is for all
students to become life-long learners of mathematics as well as productive citizens. Math
Plus hopes to accomplish this agenda by implementing the following: align the K-12
mathematics curriculum; ensure consistency in teaching mathematics throughout the
state; provide quality professional development designed for practicing mathematics
teachers; and improve the preparation of the pre-service teachers.
37
Math Plus would not succeed without the participation of the state?s mathematics
teachers. Besides the various partnerships that are incorporated within the program, the
beauty of the program is that there are various levels in which the teachers can choose to
participate. A teacher might be selected to become a presenter, a district teacher leader, or
a school teacher leader. Other teachers involved in the program receive professional
development specifically for their grade level and/or grade band.
A Math Plus presenter facilitates various professional development sessions that
help demonstrate ways to achieve the above mentioned goals in today?s mathematics
classrooms. Not just anyone can be a Math Plus presenter. In order to be considered as a
presenter, the participant must be active in Math Plus. This means that he/she regularly
attends and actively participates in the various programs sponsored by Math Plus. Then,
the teacher is nominated as a potential presenter. Next, he/she is asked to attend a
workshop or meeting where his/her participation can be observed by the Math Plus staff.
From that point, if the Math Plus staff believes the teacher would be a good presenter;
he/she is invited to a presenter planning meeting. Finally, the participant is allowed to co-
present at one of the next Math Plus meetings.
The principal of a school selects who will represent the school as a school teacher
leader for Math Plus. School teacher leaders have the opportunity to acquire several hours
of professional development. The amount that is acquired varies dependant upon the
amount of participation by the school teacher leader. At a yearly minimum, a school
teacher leader should have sixteen hours from attending the quarterly meetings; however,
a school teacher leader could have many more hours than that if he/she attends all of the
workshops and meetings that are sponsored by Math Plus. A school teacher leader has the
38
following responsibilities: coordinate activities at the school; act as a change agent for
individual teachers; act as a change agent for groups of teachers; and act as a change
agent for reform. In order to coordinate activities at the school, the school teacher leader
should: work with individual teachers to improve their skills; plan and conduct school-
based planning and inquiry groups; and develop a learning community at the school. In
order to act as a change agent for individual teachers, the school teacher leader should
incorporate some if not all of the following activities into his/her schedule: peer coach;
co-teach; demonstrate lessons for other teachers; plan; advise; and debrief after classroom
observations. In order to act as a change agent for groups of teachers, the school teacher
leader should: design and/or deliver workshops; lead study groups; and facilitate
meetings among mathematics departments at various grade levels. Finally, in order to act
as a change agent for reform, the school teacher leader should: create an awareness of the
Math Plus agenda; provide proof of reform work; engage teachers in discussions about
mathematics reform; and demonstrate lessons that have been used with actual students.
The district teacher leaders are recommended by a representative from their
school district. District teacher leaders have the opportunity to acquire several hours of
professional development. The amount that is acquired varies dependant upon the amount
of participation by the district teacher leader. At a yearly minimum, a district teacher
leader should have twelve hours from attending the quarterly meetings; however, a
district teacher leader could have many more hours than that if he/she attends all of the
workshops and meetings that are sponsored by Math Plus. The responsibilities of a
district teacher leader are the same as those of the school teacher leader except they are
performed at the district level instead of at the school level.
39
The first year that a school participates in Math Plus, all teachers who teach
mathematics are expected to participate in a two-week professional development program
called the Summer Institute. The second year, the teachers are expected to participate in a
one week follow up of Summer Institute. During the Summer Institute, the teachers are
oriented to the goals and objectives of Math Plus. Additionally, they are provided
opportunities to learn and practice many of the reform mathematics techniques. The
teachers are also given a curriculum guide that will help them implement reform
strategies in their classrooms throughout the academic year. Also at the Summer Institute,
there are designated meetings for the school teacher leaders as well as the district teacher
leaders. In addition to Summer Institute, the teachers of participating schools are
encouraged to attend quarterly meetings. These meetings are designed to provide
additional professional development for the teachers as well as opportunities for
networking with other teachers who are implementing reform mathematics techniques.
The Mathematics Education Program
The teacher education programs in the university?s College of Education are
designed to ensure that program graduates have the knowledge, skills and dispositions to
help all students learn. These programs maintain selective admission, retention and
graduation requirements and are in compliance with the Alabama Teacher Certification
Code. In addition, the university offers an assurance of competence that articulates its
guarantee with regards to graduates of the teacher education programs.
All students desiring an undergraduate degree in education must meet certain
eligibility requirements in order to enter any internship experience. First, the student must
40
complete and submit his/her internship application one year prior to participating in the
internship. Next, the student must have satisfactorily completed all courses that are
designated as prerequisites for internship. Also, the student must have a minimum 2.5
GPA on all college coursework that was attempted as well as all coursework attempted at
the university, in the program, in professional studies, and in the teaching field.
Additionally, the student must have a grade of ?C? or better in all professional studies
courses. In addition to regular general studies courses, the following are those that
students in the mathematics education program complete: Calculus I; Calculus II;
Calculus III; Differential Equations; Linear Algebra; Discrete Math; Applied Probability
and Statistics I; Foundations of Math; History of Math; Analysis I; Abstract Algebra;
Geometry I; Geometry II; Cryptography; Teaching Mathematics in the Middle School;
Mathematics Curriculum and Teaching; and Technology in Teaching Secondary
Mathematics. The student must also have a passing score on the state?s prospective
teacher subject assessment. Also, the student must have a clear background check.
Finally, the student must demonstrate a potential for teaching and obtain departmental
approval.
All students desiring a degree via the fifth year certification program must also
meet certain eligibility requirements in order to enter an internship experience. First, the
student must complete and submit his/her internship application form two semesters prior
to participating in the internship. Also, the student must have a 3.0 GPA on all
coursework carrying graduate credit. Additionally, the student must maintain a grade of
?C? or better on all coursework carrying graduate credit. See above for a listing of the
courses that are completed by students in the mathematics education program. The
41
student must also have a passing score on the basic skills assessments as well as the
subject matter assessment. Also, the student must provide documentation of a clear
background check. Finally, the student must demonstrate a potential for teaching and
obtain departmental approval.
School Demographics
The cooperating teacher/pre-service teacher pairs were assigned to three different
schools: Riverdale High School, Murphy High School, and Yorkshire High School. Two
of the pairs were assigned to Riverdale High School while the other two were assigned to
Murphy High School and Yorkshire High School respectively. Riverdale was housed
within a city school system while Murphy and Yorkshire were housed within separate
county school systems. Riverdale was the largest of the three schools and housed grades
9-12. Murphy was the next largest. Like Riverdale, it also housed grades 9-12. Yorkshire
was the smallest of the three and housed grades 7-12. It was noted that Yorkshire and
Murphy had similar racial background breakdowns with the student body being
predominately White. On the other hand, Riverdale had a predominately African
American student body. Also, unlike the other two schools, Riverdale had more of a
racially diverse student population. Another difference in the three was seen in the
socioeconomic background of the students. Yorkshire had the highest percentage of
students that were eligible for the free or reduced-price lunch program. Riverdale had the
next highest percentage. Murphy had the smallest percentage of students that were
eligible for the free or reduced-price lunch program. A comparison of these schools can
be found in Table 1.
42
Table 1
School Demographics
System Grades
Serviced
Total
Population
Student/
Teacher
Ratio
Students
Eligible for
Free or
Reduced-
Price Lunch
Racial
Background
Murphy
High
School
County
9-12
1003
20
39%
White ? 61%
African American ? 36%
Hispanic ? 1%
Asian ? <1%
American Indian ? <1%
Riverdale
High
School
City
9-12
1312
16
49%
African American ? 59%
White ? 38%
Asian ? 2%
Hispanic ? 1%
American Indian ? <1%
Yorkshire
High
School
County
7-12
703
18
58%
White ? 60%
African American ? 40%
Hispanic ? <1%
American Indian ? <1%
Asian ? <1%
The Cooperating Teachers
As stated above, the cooperating teachers that were chosen for this project are
currently involved in the mathematics reform initiative program, Math Plus. This,
however, does not imply that all participants are performing at the same level of change.
In fact, almost every cooperating teacher was functioning at a different level. As
previously discussed in the Literature Review, there are four dominant and distinct views
of how mathematics should be taught. The views are as follows: constructivist view
which is thought of as learner-focused; Platonist view which is thought of as content-
43
focused with an emphasis on conceptual understanding; instrumentalist view which is
thought of as content-focused with an emphasis on performance; and the classroom-
focused view (Thompson, 1992). Refer to pages fifteen and sixteen of the Literature
Review for further explanations of these four views on the teaching of mathematics.
Because the cooperating teachers in this study were chosen from schools participating in
Math Plus, the desire was that most of the teachers fell under the views of learner-focused
instruction or content-focused instruction with an emphasis on conceptual understanding;
however, some of the cooperating teachers fell under the views of content-focused
instruction with an emphasis on performance. The desire was also that none of the
cooperating teachers involved in the study fell under the view of classroom-focused
instruction.
The Pre-Service Teachers
The pre-service teachers that participated in this study were all enrolled in the
college of education. Specifically, they are all completing requirements in a mathematics
education program that is focused on mathematics reform. Again, these pre-service
teachers have been taught how to make mathematics education student-centered instead
of teacher-centered. This is evident in the objectives of the internship experience, which
is the culmination of their program. According to the internship syllabus, there are
several objectives the pre-service teacher must accomplish by the end of the internship
experience. The objectives are as follows: to use fundamental mathematical operations,
algorithms, and measurements essential to teaching the full range of secondary
mathematics; to use language and symbols of mathematics accurately in communications;
44
to use a variety of manipulative and visual materials to help students explore and develop
mathematical concepts; to conduct and lead students in inquiry mathematics activities;
and to use technology and other resources to enhance the teaching of mathematics and to
promote students? understanding of mathematical concepts.
The University Supervisors
There were three university supervisors that were involved throughout this study.
As mentioned previously, the major emphasis of this study was on the cooperating
teacher/pre-service teacher pairs so the university supervisors were not formally included
in the data collection process. It is important to note, however, that the university
supervisors cannot be ignored altogether. University Supervisor A, a graduate teaching
assistant, was responsible for Mrs. Franklin and Ms. Walters. University Supervisor B, a
professor in the secondary mathematics teaching program, was responsible for Mrs.
Windsor. Finally, University Supervisor C, a professor and program coordinator of the
secondary mathematics teaching program, was responsible for Ms. Robinson.
Throughout the internship experience, all three university supervisors had the
same responsibilities. First, they conducted an orientation meeting with all of the pre-
service teachers. At this meeting, the pre-service teachers were familiarized with the
course syllabus as well as assignments and other expectations for the internship
experience. Second, the university supervisors met with the cooperating teacher/pre-
service teacher pair to discuss the course syllabus and internship expectations with the
cooperating teacher. Then, each pre-service teacher was observed a minimum of three
times. After each observation, each university supervisor held a debriefing session with
45
his/her respective pre-service teacher. At these meetings, the university supervisor
discussed things such as parts of the lesson that went well, parts of the lesson that needed
improvement, suggestions for new techniques to try, etc. It should be noted that if the
university supervisor felt that additional observations were needed, he/she could visit
his/her pre-service teacher as many times as he/she saw fit. At the midpoint of the
internship experience, all university supervisors and all pre-service teachers attended a
debriefing meeting where progress of the pre-service teachers was discussed as a group.
This same type of debriefing was held at the end of the internship experience as well.
Also at the conclusion of the internship experience, the university supervisor met with the
cooperating teacher/pre-service teacher pair one last time to finalize paperwork and
discuss any other concerns any of the involved parties might have.
The Pairs
The following information is to serve only as a brief introduction of the pairs.
More detailed descriptions of the pairs will follow in Chapter IV.
Case 1: Mrs. Smith and Mrs. Franklin
Mrs. Smith
Mrs. Smith, a teacher at Riverdale High School, has been teaching high school
mathematics for thirty-four years. Her teacher certification is in grades seven through
twelve. During the Spring 2006 semester, Mrs. Smith taught two blocks of pre-calculus
and one block of remedial math. The Fall 2006 semester she taught one block of pre-
calculus, one block of remedial mathematics, and one block of calculus.
46
Mrs. Franklin
Mrs. Franklin was placed at Riverdale High School and was paired with Mrs.
Smith. During the Fall 2006 semester, she taught one block of pre-calculus, one block of
remedial mathematics, and one block of calculus.
Case 2: Mrs. Johnson and Ms. Walters
Mrs. Johnson
Mrs. Johnson, a teacher at Riverdale High School, has been teaching high school
mathematics for six years. Her teacher certification is in grades four through twelve
because she also has a middle school endorsement. During the Spring 2006 semester,
Mrs. Johnson taught two blocks of Algebra I and one block of Algebra II. The Fall 2006
semester, she again taught two blocks of Algebra I and one block of Algebra II.
Ms. Walters
Ms. Walters was placed at Riverdale High School and was paired with Mrs.
Johnson. During the Fall 2006 semester, she taught two blocks of Algebra I and one
block of Algebra II.
Case 3: Mrs. York and Mrs. Windsor
Mrs. York
Mrs. York, a teacher at Murphy High School, has been teaching high school
mathematics for seventeen years. Her teacher certification is in grades seven through
twelve. During the Spring 2006 semester, Mrs. York taught two blocks of advanced
geometry and one block of Algebra 1B. The Fall 2006 semester, she only taught two
blocks of geometry. Mrs. York was given two planning periods the Fall 2006 semester
because she was the head of the school?s SACS review committee.
47
Mrs. Windsor
Mrs. Windsor was placed at Murphy High School and was paired with Mrs. York.
During the Fall 2006 semester, she taught two blocks of geometry. It is important to note
that due to the fact that Mrs. York only taught two classes, Mrs. Windsor was placed with
another teacher at Murphy High School for her third class. This third class was not used
in this study.
Case 4: Mrs. Brown and Ms. Robinson
Mrs. Brown
Mrs. Brown, a teacher at Yorkshire High School, has been teaching high school
mathematics for 18? years. Her teacher certification is in grades seven through twelve.
During the Spring 2006 semester, Mrs. Brown taught one period of Algebra I, three
periods of Algebra II, and two periods of pre-calculus. It is important to note here that
prior to the 2005-2006 school year, Yorkshire High School utilized block scheduling. The
2005-2006 school year was the first year for Yorkshire High School to have the seven
period day. During the Fall 2006 semester, Mrs. Brown taught two periods of Algebra II,
one period of remedial mathematics, two periods of pre-calculus, and one period of
Algebra 1B which was an inclusion class.
Ms. Robinson
Ms. Robinson was placed at Yorkshire High School and was paired with Mrs.
Brown. During the Fall 2006 semester, she taught 2 periods of Algebra II, one period of
remedial mathematics, two periods of pre-calculus, and one period of Algebra 1B which
was an inclusion class.
48
Table 2
Summary of Cooperating Teacher/Pre-Service Teacher Pairs
School Student Population Classes
Taught
Mrs. Smith and Mrs. Franklin Riverdale 11
th
and 12
th
grade Pre-Calculus
Remediation
Calculus
Mrs. Johnson and Ms. Walters Riverdale 9
th
and 10
th
grade Algebra I
Algebra II
Mrs. York and Mrs. Windsor Murphy 9
th
and 10
th
grade Geometry
Mrs. Brown and Ms. Robinson Yorkshire 10
th
, 11
th
,and 12
th
grade Algebra II
Remediation
Pre-Calculus
Algebra 1B
Instrumentation
Interviews
The initial interview was conducted in the Spring 2006 semester. Each
participant, cooperating teachers and pre-service teachers, was asked questions from a
predetermined list of questions. This was to ensure that all topics of interest were
addressed as well as to ensure that all participants were asked the same questions. The
second interview was conducted in the Fall 2006 semester. As before, each participant
was asked questions from a predetermined list of questions. Refer to Appendix A for the
following sets of interview questions: Spring 2006 Teacher Interview Questions; Spring
2006 Pre-Service Teacher Interview Questions; Fall 2006 Teacher Interview Questions;
and Fall 2006 Pre-Service Teacher Interview Questions.
49
The Spring 2006 Teacher Interview Questions were chosen because they
addressed the two topics of interest, the first concerning beliefs about teaching and
learning mathematics, the second concerning the internship experience. Questions were
asked to investigate what the teachers thought mathematics involved. The cooperating
teacher participants were asked to describe the best way for students to learn mathematics
as well as the most effective ways to teach mathematics. They were also asked to portray
how they thought they had an impact on student learning. Also, the teachers were asked
to analyze their teaching by explaining when they knew they had delivered a good
mathematics lesson, depicting a typical lesson in their classroom, discussing the various
tasks that were used in their classroom, and portraying the learning environment of their
classroom. In addition, the teachers were also asked to expound upon what most
influenced their beliefs and practices in the mathematics classroom. In conjunction with
the above questions, the teacher participants were also asked to answer questions
pertaining to the internship experience. They were asked to define the roles of the
cooperating teacher and the university supervisor. Additionally, they were asked to
discuss things they would expect from the pre-service teacher. The researcher also asked
the teachers to talk about who would have the most influence on the pre-service teacher
and why. Finally, the teachers were asked to discuss the problems they thought might
arise if the pre-service teachers had differing beliefs about teaching and learning
mathematics.
The Spring 2006 Pre-service Teacher Interview Questions were chosen because
like the Teacher Interview Questions, they addressed the two topics of interest, the first
concerning beliefs about teaching and learning mathematics, the second concerning the
50
internship experience. Questions were asked to investigate what the pre-service teachers
thought mathematics involved. The pre-service teacher participants were asked to
describe the best way for students to learn mathematics as well as the most effective ways
to teach mathematics. They were also asked to describe how they thought they had an
impact on student learning. Also, the pre-service teachers were asked to analyze their
teaching even though by this point, their actual teaching time in a real classroom had
been somewhat limited. They were asked to explain when they knew they had delivered a
good mathematics lesson, describe what they thought a typical lesson would look like in
their classroom, discuss the various tasks they imagined would be used in their
classroom, and portray the ideal learning environment for their classroom. In addition,
the pre-service teachers were also asked to discuss what most influenced their beliefs and
practices in the mathematics classroom. In conjunction with the above questions, the pre-
service teacher participants were also asked to answer questions pertaining to the
internship experience as a whole. They were asked to define their role as a pre-service
teacher along with the roles of the cooperating teacher and the university supervisor. The
researcher also asked the pre-service teachers to talk about who they thought would have
the most influence on them and why. Finally, the pre-service teachers were asked to
discuss the problems they thought might arise if the cooperating teachers had differing
beliefs about teaching and learning mathematics than they had.
The Fall 2006 Teacher Interview Questions were used to see if any perceptions
had changed at the end of the internship experience. They were first asked to discuss any
changes in the way they felt about teaching and learning mathematics. Following that
discussion, the teacher participants were once again asked to define the roles of the
51
cooperating teacher, the pre-service teacher, and the university supervisor. Additionally,
they were asked to elaborate on who they thought influenced the pre-service teacher the
most. Concerning the internship experience as a whole, the teacher participants were
asked to talk about whether or not they thought the internship experience turned out the
way they initially envisioned it. They were also asked to expound upon the most
beneficial part or parts of the internship experience along with any changes they would
recommend. Finally, the teacher participants were asked to discuss whether or not they
and their pre-service teacher had differing beliefs about mathematics learning and
teaching and then to elaborate on any issues that may have come about because of these
differing beliefs.
The Fall 2006 Pre-service Teacher Interview Questions were used to see if any
perceptions had changed because of the internship experience. The pre-service teachers
were asked to discuss what they thought mathematics entailed, the best way for students
to learn mathematics, and the most effective ways of teaching mathematics. The
researcher asked the pre-service teachers to analyze their teaching by asking them to
describe how they knew they had taught a successful lesson, how they knew they had
taught an unsuccessful lesson, what their typical mathematics lesson looked like, the
types of tasks that the students engaged in during a mathematics lesson, and the overall
learning environment of their classroom. Like the teacher participants, the pre-service
teachers were asked to define the roles of the cooperating teacher, the university
supervisor as well as their role as a pre-service teacher. Additionally, they were asked to
discuss who had the most influence on them throughout the internship experience and
why. Concerning the internship experience as a whole, the pre-service teacher
52
participants were asked to talk about whether or not they thought the internship
experience turned out the way they envisioned it in the beginning. They were also asked
to elaborate on the most beneficial part or parts of the internship experience along with
any changes they would recommend. Finally, the pre-service teacher participants were
asked to talk about whether or not they and their cooperating teacher had differing beliefs
about mathematics learning and teaching and then to elaborate on any issues that may
have come about because of these differing beliefs.
Reformed Teaching Observation Protocol (RTOP)
The Reformed Teaching Observation Protocol (RTOP) is an observational
instrument that was designed by a participant of the Arizona Teacher Excellence
Coalition (AzTEC), the Evaluation Facilitation Group (EFG) of the Arizona
Collaborative for the Excellence in the Preparation of Teachers (ACEPT), to provide a
standardized means of determining to what degree mathematics instruction reform had
taken place (AzTEC, 2002). The instrument is comprised of twenty-five items that focus
on reform. These items are organized into five categories containing five questions each.
The categories are as follows: lesson design and implementation; content: propositional
pedagogic knowledge; content: procedural pedagogic knowledge; classroom culture:
communicative interactions; and classroom culture: student/teacher relationships
(AzTEC, 2002). Each question is scored on a five point Likert scale where 0 represents
?Never Occurred? and 5 represents ?Very Descriptive? (AzTEC, 2002). A copy of the
RTOP can be found in Appendix C.
Prior to utilizing the RTOP as an observational instrument, the researcher along
with several other Math Plus participants engaged in training to learn how to effectively
53
use the instrument. The training took place in two segments. During the first segment, the
participants watched a recording of a teacher teaching a mathematics lesson. Then, the
participants used the RTOP to determine at which level of reform the teacher was
performing. Next, the group discussed the ?scores? that were given. At this point, any
discrepancies in the scores were resolved. Finally, this process was repeated using
another recorded lesson. The purpose of this activity was to eventually get all RTOP
users to score the same lesson similarly. This would ultimately ensure reliability among
observers. The second segment of the training involved the same process. The difference
was that this segment was completed a couple of weeks after the initial training. The
purpose here was to ensure that observers were still utilizing the things that were learned
at the initial training.
Beliefs Survey Used by Math Plus
The beliefs survey consisted of five components. They were as follows: the
teacher?s beliefs about teaching and learning mathematics; information about the
mathematics classes being taught; background information; and involvement in Math
Plus. The components that were the most relevant to this project were: beliefs about
teaching and learning mathematics, and information about the mathematics classes
taught. With the exception of the questions pertaining strictly to demographic
information, each component contained questions that were to be answered in the
following manner: strongly agree, agree, neutral, disagree, or strongly disagree. A copy
of this survey can be found in Appendix D.
54
Procedure
At the beginning of the Spring 2006 semester, principals were contacted and then
provided information letters explaining the intent and design of the study; see Appendix
B for a copy of the information letter that was provided to the principals. Once the
principals had given permission for their teachers to be included in this study, the
participating cooperating teachers were contacted and provided information letters
explaining the intent and design of the study. Additionally, the participating cooperating
teachers were asked to sign a consent form in order to participate. A copy of the
information letter in addition to the consent form can be found in Appendix B.
Simultaneously, the participating pre-service teachers were also given information letters
explaining the intent and design of the study. Just like the cooperating teachers, the
participating pre-service teachers were asked to sign a consent form in order to
participate in the study; refer to Appendix B for a copy of the information letter as well as
the consent form.
Once consent had been obtained by all participating parties, the formal data collection
process began. This process began by administering the mathematics beliefs survey to all
participating cooperating teachers and pre-service teachers; refer to Appendix D for a
copy of the beliefs survey. Then, during the last half of the Spring 2006 semester, each
cooperating teacher was observed three times. The Reformed Teaching Observation
Protocol (RTOP) was used in conjunction with these observations; a copy of the RTOP
can be found in Appendix C. Also, during the last half of the Spring 2006 semester, each
cooperating teacher was interviewed; refer to Appendix A for a listing of the Spring 2006
Cooperating Teacher Interview Questions. All interviews were audio-taped and then
55
transcribed at a later date. When the researcher was not observing the cooperating
teachers, she was observing the pre-service teachers in their mathematics methods course.
These observations continued until the end of the Spring 2006 semester. Additionally,
there were at least seven of these types of observations. At the beginning of the Summer
2006 semester, each pre-service teacher was interviewed; see Appendix A for a listing of
the interview questions. As with the cooperating teachers, all interviews were audio-taped
and then transcribed at a later date. Also at the beginning of the Summer 2006 semester,
the researcher observed the cooperating teachers while they attended the two-week long
summer training sessions provided by the university sponsored mathematics reform
initiative program. At the onset of the Fall 2006 semester, the researcher observed the
cooperating teachers one more time. As before, the RTOP was utilized. Next, each pre-
service teacher was observed on three separate occasions throughout the Fall 2006
semester. As with the cooperating teachers, the RTOP was utilized. At the end of the Fall
2006 semester, each participating cooperating teacher and pre-service teacher was
administered the same mathematics beliefs survey they had been given in the Spring
2006 semester. Finally, the participants were asked some follow up interview questions.
A copy of the Fall 2006 interview questions can be found in Appendix A. As before,
these questions were transcribed at a later date.
56
Table 3
Summary of Instrumentation Used for Study
Cooperating Teachers Pre-Service Teachers
Spring 2006 Interview ? ?
Fall 2006 Interview ? ?
Spring 2006 Beliefs Survey ? ?
Fall 2006 Beliefs Survey ? ?
Spring 2006 RTOP Observations ?
Fall 2006 RTOP Observations ? ?
Analysis of Data
The researcher utilized Atlas.ti software to assist in the coding of the data. The
coding process suggested by Strauss and Corbin (1990) was used. In this process, there
are three main types of coding: open coding; axial coding; and selective coding (Strauss
& Corbin, 1990). It is important to note that these are analytic types of coding; therefore,
the researcher may or may not move from open coding to axial coding to selective coding
in a strict consecutive manner. As defined by Strauss and Corbin (1990), open coding is
the portion of data analysis that involves labeling and categorizing the phenomena as
indicated by the data. In open coding, the researcher must ask questions and make
constant comparisons about the data. Initially, the data are broken down by asking simple
questions such as what, where, how, when, how much, etc. Then, data are compared and
57
similar incidents are grouped together and given the same conceptual label. This process
of grouping concepts at a higher, more abstract, level is called categorizing (Strauss &
Corbin, 1990). Axial coding takes the coding process to another level. Unlike open
coding which separates the data into concepts and categories, axial coding puts the data
back together in new ways by making connections between a category and its sub-
categories; therefore, axial coding is the process of developing main categories and their
sub-categories (Strauss & Corbin, 1990). Finally, selective coding involves the
integration of the categories that have been developed to form the initial theoretical
framework (Strauss & Corbin, 1990).
Fine, Weis, Weseen, and Wong (2003) define triangulation as a method of adding
layers to the data source by using multiple items to measure the same construct. In an
effort to achieve triangulation, as defined above, a plethora of data was analyzed for this
study. The researcher utilized the Reformed Teaching Observation Protocol (RTOP) form
to help gather information from classroom observations. The researcher also used the
beliefs survey that is used by the university?s mathematics reform initiative program.
This survey assisted the researcher in analyzing each participant?s beliefs about learning
and teaching mathematics. In addition to these sources of data, the researcher also used
course syllabi as well as interviews to help gather information for this study.
58
IV. RESULTS, ANALYSIS, & INTERPRETATION OF THE DATA
Borko and Mayfield (1995) stated that learning to teach is a complex process,
especially in the field of mathematics education; however, despite its complexity,
learning to teach is also considered to be one of the most important aspects of any
educational program. Frykholm (1998) further elaborated that the student teaching
experience is generally thought of as the most formative and significant element of the
entire educational program. In light of this, Pourdavood (1999) stated that existing
classroom norms and the cooperating teachers? methods of instructions have profound
impact on pre-service teachers? beliefs and practices. According to the research, it seems
that if pre-service teachers are to internalize coherent applications to teaching and
learning mathematics, the environment in which they complete their internship and the
support they receive need to be consistent with the principles being advocated in their
professional preparation program (Vacc & Bright, 1999).
As stated earlier, the purpose of this study was to explore the impact of the
alignment or misalignment of the cooperating teachers? practices and the pre-service
teachers? approach to teaching based on their preparation. Interviews, beliefs surveys, and
classroom observations were all utilized to help explore: what beliefs and practices
cooperating teachers have that support or hinder the growth of a pre-service teacher
indoctrinated in reform based teaching; and what happens when there is a misalignment
59
of the beliefs and practices held by the cooperating teacher and the educational
background of the pre-service teacher.
Chapter Organization
In order to best relay the details of the study in a logical manner, the researcher
determined that it would be best to combine the results of the data, the analysis of the
data, and the interpretation of the data into one chapter. In this chapter the reader can
expect to find the following information about each case: demographic information about
the school; background information about the participants; beliefs about teaching and
learning mathematics; expectations of the internship experience; vignettes from
classroom observations; information about lesson design and implementation;
information about communicative interactions in the classroom; information about
procedural knowledge; information about propositional knowledge; and information
describing the student/teacher relationships.
Analysis of the Data
Analysis and interpretation of observations and interviews with the four
cooperating teachers and four pre-service teachers was completed using a qualitative
computer software program called Atlas.ti (Scientific Software Development GmbH,
2003). At the conclusion of the data collection process, all taped interviews were
transcribed and all observation notes were typed up so that they could be loaded into the
aforementioned software for analysis. As a result of this process, each document was
linked to Atlas.ti and was then accessed through that link. By having the document linked
60
to the program instead of using the program to create the document, the document could
not be changed by the program. Atlas.ti allows the researcher to code text and organize it
into hermeneutic units, which is a way to name a single unit of text documents. When
using the software, a single word, multiple words, sentences, or even paragraphs from
several documents can be coded with a word or phrase which refers back to the meaning
that is associated to the words. For example, cooperating teachers and pre-service
teachers discussed various ways to design and implement their lessons; therefore, the
code ?lesson design and implementation? was utilized. It should be noted here as well
that identical phrases or portions of phrases could be coded with multiple codes. For
example a phrase coded with ?lesson design and implementation? could also be assigned
to the code ?communicative interactions?. Ultimately, any selected code can be used to
sort the text so that the researcher can see all text that is associated with that particular
code along with identifying information to tell the researcher from which document the
phrase or phrases originated.
After reading through each of the documents, various patterns and similarities
emerged. These patterns and similarities along with the headings from the RTOP were
used to generate the list of codes. For example, even though responses were as varied as
the responders, all interviews followed a defined set of questions, so that similar response
categories were more apparent. The list of codes follows in Table 4 with the number of
times each code was used after all of the documents had been coded.
61
Table 4
Utilized Code Words and Frequency
Code Frequency
Lesson Design and Implementation 293
Procedural Knowledge 122
Propositional Knowledge 75
Communicative Interactions 258
Student-Led Discussion 78
Teacher-Led Discussion 145
Student/Teacher Relationships 145
Utilizing Atlas.ti permitted an accurate and efficient method of recording and
tracking selected codes so that major areas could readily be identified. For example, with
?lesson design and implementation? resulting in 293 responses, it is evident that this was
a major category. At the same time, however, it is obvious that with 75 resulting
responses, ?propositional knowledge? was the least of the categories.
As mentioned previously, the researcher used the coding process suggested by
Strauss and Corbin (1990). In the initial phase of coding, the process of open coding as
defined by Strauss and Corbin (1990) was utilized. In order to generate the overall
categories that were used for the open coding, the researcher used the categories from the
RTOP. The categories used for the open coding process were: Lesson Design and
Implementation; Communicative Interactions; and Student/Teacher Relationships. Once
62
the open coding had been completed, the researcher divided the larger groupings into
smaller subsets using the process of axial coding as defined by Strauss and Corbin
(1990). Lesson Design and Implementation was divided into two subsets: Propositional
Knowledge and Procedural Knowledge. Both of these subset categories were derived
from the RTOP. The other large category, Communicative Interactions, was divided into
two subsets as well: Student-Led Discussion and Teacher-Led Discussion. Both of these
subset categories were generated by the researcher. The third category of Student/Teacher
Relationships was not subdivided.
When coding the data, any information that had anything to do with lesson design
or lesson implementation was placed in the category Lesson Design and Implementation
(293 occurrences). Then, that information was further subdivided into the subset
categories of Propositional Knowledge (75 occurrences) and Procedural Knowledge (122
occurrences). Propositional knowledge was considered as anything in the lesson design or
implementation that allowed the students opportunities to explore mathematical concepts
for themselves. Procedural knowledge was considered as anything in the lesson design or
implementation that focused only on the procedures involved in solving mathematical
problems. It should be noted that the occurrences of the subsets Propositional Knowledge
and Procedural Knowledge do not add up to the total number of occurrences of Lesson
Design and Implementation. That is because there were things that were placed in the
Lesson Design and Implementation category that were strictly about the lesson and had
nothing to do with either subset.
When coding the data, any information that had anything to do with
communicative interactions was placed in the category Communicative Interactions (258
63
occurrences). Then, that information was further subdivided into the subset categories of
Student-Led Discussion (78 occurrences) and Teacher-Led Discussion (145 occurrences).
Student-led discussions were considered any type of discussion where the students either
initiated the conversation or were presenting some type of mathematical information.
Teacher-led discussions were considered any type of discussion where the teacher
dominated most or all of the conversation. It should be noted that the number of
occurrences of the subsets Student-Led Discussion and Teacher-Led Discussion do not
add up to the total number of occurrences of Communicative Interactions. That is because
there were instances of conversations that were neither student-led nor teacher-led.
Finally, when coding the data, any information that had to do with student/teacher
relationships was placed in the category Student/Teacher Relationships (145
occurrences). These occurrences were things such as a teacher?s actions toward a student
or group of students. The occurrences were also things such as a student?s or group of
students? actions toward the teacher.
Case 1: Mrs. Smith and Mrs. Franklin
Mrs. Smith is the White female cooperating teacher discussed throughout Case
One. She is a veteran teacher that has been teaching mathematics for thirty-four years.
Mrs. Franklin is a White, devout Muslim, female, pre-service teacher discussed
throughout Case One. Mrs. Franklin is working on her degree in Secondary Mathematics
Education.
The internship experience discussed throughout Case One takes place at Riverdale
High School. Riverdale High School is a city school that services students in grades nine
through twelve. Its total population is 1312 students. The student-teacher ratio for the
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school is sixteen to one. Forty-nine percent of the student body is eligible for the free or
reduced-price lunch program. The student body has the following racial components:
African American, White, Asian/Pacific Islander, Hispanic, and American
Indian/Alaskan Native. Fifty-nine percent of the students are African American. Thirty-
eight percent of the students are White. Two percent of the students are Asian/Pacific
Islander. One percent of the students are Hispanic. Finally, less than one percent of the
students are American Indian/Alaskan Native.
Table 5
Demographic Summary of Riverdale High School
System
Type
Grades
Serviced
Total
Population
Student/
Teacher
Ratio
Students
Eligible for
Free or
Reduced-Price
Lunch
Racial
Background
City 9-12 1312 16 49% African
American ? 59%
White ? 38%
Asian/
Pacific Islander ? 2%
Hispanic ? 1%
American Indian/
Alaskan Native ? <1%
65
Mrs. Smith
Mrs. Smith is a White female in her mid fifties with at least two adult children.
She is very tall and has a very dominant presence in the classroom due to her stature, loud
voice, and no nonsense attitude. Mrs. Smith has been teaching high school mathematics
for thirty-four years. Twenty of those thirty-four years have been spent teaching at
Riverdale High School. She has a bachelor?s degree in mathematics and a master?s
degree in mathematics education.
Mrs. Smith is also the Math Plus School Teacher Leader at her school. Refer to
pages 37 and 38 for more detailed information about the role and responsibilities of a
school teacher leader. Mrs. Smith reported that over the past year, she has spent
approximately forty-one to eighty hours involved in some type of professional
development. Of those hours, she stated that six to ten of those hours were spent in
professional development settings that specifically focused on mathematics. Additionally,
she reported that over twenty of her professional development hours were spent in
association with Math Plus. It should be noted here that Mrs. Smith really acted as a
school teacher leader in name only. As far as the researcher could tell, the most Mrs.
Smith did in the way of mathematics reform was attend some of the professional
development opportunities that were provided by Math Plus. She also attempted to
implement a few activities in her own classroom, but it was apparent that she was not
comfortable at all with implementing any of the reform initiatives that she was supposed
to be advocating. Mrs. Smith was not observed fulfilling the above listed responsibilities
of a school teacher leader.
66
When asked to give a description of mathematics in general, Mrs. Smith stated
that mathematics was ?looking at patterns, looking at numbers, looking at various steps
and processes whether it?s solving equations, logarithms, whatever.? She also commented
that mathematics was ?realizing that there is a pattern, there is logic to things.? When
asked to discuss what she thought was the best way for students to learn mathematics,
Mrs. Smith responded by saying that ?not all students learn the same way.? She further
elaborated by saying that ?there are some who need constant repetition; there are some
who need hands-on; and there are some who say show me what works, how it works, and
then leave me alone.? In addition, Mrs. Smith replied that ?while a lot of them these days
do need hands-on, I don?t think everybody does.? Upon being asked to discuss her
feelings about the most effective way to teach mathematics, Mrs. Smith immediately
responded that it depended on what was being taught. She also commented, ?I try to
relate to something that they know something about.? When Mrs. Smith was asked to
describe a typical mathematics lesson in her classroom. She answered,
Again, it depends on what level we?re talking about. Some days, it will be lecture.
Some days it will be getting together to work on some problems to practice some
topic we?ve done. Occasionally, it will be a lab or some kind of hands-on.
Probably, there?s more lecture with the calculus and pre-calculus than there is
anything else, but they do, when they?re practicing a new topic or whatever, they
will sit together and do some group work and some teaching each other kinds of
things with those.
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Since Mrs. Smith specifically addressed her upper level classes, the researcher probed
further by asking her to describe a typical mathematics classroom in her remediation
class. She responded,
A lot of that is practice and drill. Positive slope looks this way. Negative slope
looks this way. How can you remember that? Those kinds of things. It is a real
skill because that?s what they have to do to pass the graduation exam. So it is very
intense, what I would call, drill on a particular skill, and we pull in any kind of
thing that we can to help them remember things for the graduation exam.
According to her responses on the beliefs survey, Mrs. Smith believed
mathematics is an important subject that should be available to all students because it is
something that will continue to be used even once the students are out of school. Mrs.
Smith agreed that students need good mathematical problem-solving skills in order to be
successful in the future. She also agreed that in order to formulate these problem-solving
skills students must be able to follow directions. Additionally, Mrs. Smith agreed that
students should not only be able to obtain correct answers, but they should also
understand the mathematical concepts involved in getting to that right answer. She also
felt that students should understand important mathematical concepts before they ever
attempt to memorize definitions and facts. Finally, Mrs. Smith responded that students
should be allowed to figure things out on their own rather than depending on
demonstrations/explanations given by their teacher. Refer to Appendix D Table 21 to
view all of Mrs. Smith?s responses to the beliefs survey.
When asked to discuss her anticipated role as the cooperating teacher, Mrs. Smith
responded,
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Well, I see my role as helping them not make mistakes, especially in front of the
students. Helping them prepare and to be prepared when they stand up in front of
the students. Helping them to have a good plan and to have a good idea, a good
understanding of our student population.
She further elaborated by saying
I don?t think it is my job to tell them how to teach because what works for me
might not work for them. Because to me, your teaching is also a function of your
personality. Ultimately, to guide them as best I can, but never to tell them this is
the way it is to be done.
Upon being asked to discuss the anticipated role of the university supervisor, Mrs.
Smith replied that she felt that the university supervisor should be more up to date on the
research and the theory on what should work in the classroom. She also described the
university supervisor as a supervising teacher; one who could give guidance and help.
When questioned about her expectations of the pre-service teacher, Mrs. Smith
immediately stated, ?I expect the intern, to first of all, obey the school rules because that
gets us into trouble quickly.? Then, she added, ?I expect them to be on time, and prepared
with the lesson. I don?t expect them to be perfect because sometimes you don?t realize
that something won?t work until you try it.? Additionally, she commented, ?So, be well
prepared for your students. And be open to suggestions. I wouldn?t necessarily say
criticism because I don?t think that?s my role, but be open to suggestions for how things
might improve from me or other teachers.?
69
Finally, Mrs. Smith was asked to talk about if she thought there would be
problems if she and her pre-service teacher had different beliefs about teaching and
learning mathematics. Her reply was,
No, because I don?t claim to have all of the answers. And I will admit that I guess
age and fatigue can be a factor in some of the things that I do or don?t do. And I
will have all of the admiration in the world to somebody that comes in and is
enthusiastic and energetic who wants to try other things because even with my
students I don?t say ?This is the way to work the problem, and it must be worked
this way.? If you discovered something else that works, and it is mathematically
correct, I don?t want anything that is not; then go for it. And I feel that way. I
don?t claim to have all the answers. What works for me may not work for
someone else.
Spring 2006 and Fall 2006 Classroom Observations
During the Spring 2006 semester, Mrs. Smith taught two blocks of pre-calculus
and one block of remedial math. The Fall 2006 semester, she taught one block of pre-
calculus, one block of remedial mathematics, and one block of calculus.
Throughout the Spring 2006 semester, Mrs. Smith was observed on three separate
occasions. These three observations were spread out over a period of a month. The first
observation was of a Pre-Calculus class. The second observation was of a remediation
math class. The third observation was of a Pre-Calculus class.
Based on the classroom observations, a typical Pre-Calculus class looked
something like the following:
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At the beginning of the class, the students were expected to work three review
problems involving simplifying trigonometric expressions such as (tan x / sec x).
As the students worked, Mrs. Smith walked around and gave the students ?hints?
about how to handle the problems. As the students started finishing up, she asked
individual students to work the problems on the board. Once this had been
completed, Mrs. Smith told the class that they would have a test in two days
covering fundamental trigonometric functions, solving equations over the set of
complex numbers, and rectangular and coordinate form. The teacher provided the
students with several examples of problems that the students could expect to see
on the test. Some of the problems, Mrs. Smith worked on the board and other
problems were worked by individual students. Throughout this whole process,
Mrs. Smith was constantly asking questions such as, ?Are you really going to
leave your answer that way?? and ?What?s the next step??. Apparently, the
students were not completely simplifying their answers, and this was a big deal
with the teacher. Also, it appeared that her emphasis was on the students
completing all of the steps to solve a problem. She never really asked them to
explain to her why things worked the way they did.
A typical remediation class resembled the following:
The lesson began by reviewing how to simplify ratios. For the class, the teacher
took the students through problems step-by-step. She did not skip steps when
working through the problems. She also said things like, ?First, we do?Then we
do??. Also, she approached each problem solving process like the students had
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never seen these problems before. Again, everything was very procedural in
nature.
Once the teacher was satisfied that the students could simplify simple ratios, she
moved on to solving proportions. She worked through some basic proportion problems
and then moved on to some more complicated problems where the distributive property
was required. Then, she gave the students some proportional word problems that required
the students to find things such as the number of gallons of punch needed for a party with
a certain amount of people attending. This appeared to be Mrs. Smith?s attempt to make
these problems apply to the students? lives. These problems also gave Mrs. Smith the
opportunity to discuss reasonable answers with the students. In this discussion, they
talked about the possibility of getting answers that involved half of a person, etc. After
working through a few of these problems, the students went to lunch.
At the conclusion of each of the above mentioned observations, the researcher
completed the Reformed Teaching Observation Protocol (RTOP) for each respective
lesson. The RTOP is an observational instrument that was designed to provide a
standardized means of determining to what degree mathematics instruction reform had
taken place (AzTEC, 2002). The researcher averaged the scores for the five questions in
each section to determine an average score for each section of the RTOP. The researcher
then determined the median score of the three observations for each section of the RTOP.
When looking at the scores in Table 6, a score of 0 represents ?Never Occurred? and 5
represents ?Very Descriptive? (AzTEC, 2002). Upon reviewing the RTOP results in
Table 6, it can be concluded that Mrs. Smith does not exhibit qualities indicative of a
teacher who is enthralled in reform mathematics practices. As indicated by the
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descriptions above, most of the lessons were very procedural in nature. The focus was on
getting the right answer and not necessarily the understanding behind getting the right
answer. The only thing that mattered was that the students could work through the
problems to obtain the desired outcome. Additionally, it appeared that there was no real
application for the problems. The problems were just that, problems that were to be
worked while in math class. No relevance was given to the material.
Table 6
RTOP Averages and Median for Mrs. Smith?s Spring 2006 Classroom Observations
Lesson Design &
Implementation
Propositional
Knowledge
Procedural
Knowledge
Communicative
Interactions
Student/
Teacher
Relationships
Observation 1 1.2 2 0.2 0.6 0.6
Observation 2 0.4 2 0 0 0
Observation 3 0.4 2 0.8 0.4 0.4
Median 0.4 2 0.2 0.4 0.4
At the beginning of the Fall 2006 semester, Mrs. Smith was observed one more
time just to see if any major changes in teaching style had occurred since Spring 2006.
The observation was of a Pre-Calculus class. Similar to the Spring 2006 observations,
this particular Pre-Calculus class occurred as follows:
The lesson focused on quadratic functions. Mrs. Smith wanted them to be able to
find all of the following things for parabolas: direction of opening; equation of
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axis of symmetry; coordinates of vertex; x-intercepts; y-intercept; and a sketch.
The teacher worked through one example with the class. During this time, they
discussed various ways to accomplish the above tasks. Then, she gave them one
to work on their own. As the students worked, Mrs. Smith walked around and
looked to see how the students were working the problem. She commented that
about half of the class chose to find the intercepts first and half of the class chose
to use the transformation method. After the students worked the problems, Mrs.
Smith asked for volunteers to work the problem. She specifically asked for a
person who found the intercepts first. No one really volunteered. Rather than
calling on someone, she worked the problem herself. Once the work had been
done, Mrs. Smith asked the people who worked the problem with transformations
if they got the same thing. They all said yes except one boy. She began working
the problem, but they realized that his problem came from a mistake in
completing the square. Mrs. Smith then asked them if they wanted a problem with
a negative or one with a fraction. They chose the one with the negative.
At this point, Mrs. Smith gave the class another problem to work. She
asked the students to see how much they could fill in before she started working.
As the students worked, she again walked around and answered questions. Mrs.
Smith only let the students work for a few minutes before she began working it on
the board. As she was going through the problem, she periodically asked
questions such as, ?What is the y-intercept??. A majority of the time, the students
didn?t respond, but she kept on working.
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After completing this problem, she asked the students to find a partner.
She explained to the students that they were going to have a contest. Then, she
gave the students a piece of paper. The page that they were given was blank on
one side and had numbers on the other side. The numbers were all out of order,
had no pattern, were different fonts and sizes, etc. For this exercise, Person #1 had
to find numbers one through thirty in order. Then, Person #2 had to find numbers
forty through sixty in order. Once this had taken place, each person had to tally
how many numbers had been found. Mrs. Smith took up the pages they had
worked on and told them that they may do more with this activity later on in the
week. Then, the class went back to working on parabolas.
At this time, Mrs. Smith assigned the class one more problem to work. She
asked if anyone had a question. No one did, so she also assigned them problems
to work on for homework. Then, she passed back tests that the students had taken
the previous week.
At the completion of the observation, the researcher again completed the RTOP
for that observation. In the same fashion as the Spring 2006 observations, the researcher
averaged the scores for the five questions in each section to determine an average score
for each section of the RTOP. The scores can be found in Table 7. As with the Spring
2006 observations, upon reviewing the RTOP results in Table 7, it can be concluded that
Mrs. Smith still did not exhibit qualities indicative of a teacher who was enthralled in
reform mathematics practices. As with her other lessons, the one described above was
very teacher centered. Likewise, the emphasis was on being able to answer the questions
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correctly and not necessarily looking at why or how the problems were worked in the
manner they were worked.
Table 7
RTOP Averages for Mrs. Smith?s Fall 2006 Classroom Observation
Lesson Design &
Implementation
Propositional
Knowledge
Procedural
Knowledge
Communicative
Interactions
Student/
Teacher
Relationships
Observation 4 0.8 2 0.4 0.4 0.2
Lesson Design and Implementation
Mrs. Smith utilized a very traditional lesson design and implementation style. She
generally began class by going over homework problems from the night before. Then,
she moved on to the new content for the day. The lesson itself involved her working
various examples of the problems for the students. Then, once she felt like the students
had seen enough examples, she gave them a problem to work independently. While they
worked, she walked around and observed their progress and sometimes commented about
the mistakes they were making. Once she felt like they had had sufficient time to
complete the example problem, she worked the problem on the board. After she felt
confident that the students knew the new material, she assigned additional problems for
homework. Then, the students worked on those problems until the end of class.
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Communicative Interactions
Student-Led Discussions
As is evident from the scenarios above, there were hardly any student-led
discussions in Mrs. Smith?s classroom. Every now and then, she allowed a student to
work a problem for the class, but she always went back in and re-worked or added to it
rather than allowing the student to explain his/her work.
Teacher-Led Discussions
Due to her traditional teaching style as described above, most of the discussions
held in the classroom were teacher-led discussions. Most of the class period involved
Mrs. Smith showing the students how to work various problems. Periodically, she posed
a question to the class, but she rarely gave anyone a chance to answer before she
answered the question herself. Also, most of the questions referred to procedures for
answering the question and not about why the answer came out to be what it was. This
was apparent in the Fall 2006 Pre-Calculus class description provided above.
Procedural Knowledge
Knowing the steps to solving a particular problem was extremely important to
Mrs. Smith. Other than her working through every example step-by-step, most of her
discussion with the class was posed as ?First we do?.and then we do??. Then, if she
did ask the students a question, it was usually phrased as ?What do we do next??. She
never really asked them to elaborate on their answers or to explain how they arrived at
their conclusion if it was different than the rest of the class. Again, it all went back to the
fact that Mrs. Smith believed that there is one right answer to every problem.
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Propositional Knowledge
From the lessons that were observed, propositional knowledge was not a big
priority in Mrs. Smith?s class. It was as if she just assumed that because the students
could follow the steps to get to the answer she was looking for, that they truly understood
what they had done to get that answer. The few times she asked about why an answer had
to be the way it was, the students froze up and looked everywhere but at Mrs. Smith. It
appeared that they did not know how to respond. Additionally, when Mrs. Smith did ask
these types of questions, she rarely gave the students any think time. She just asked the
question and then proceeded to answer it herself. For the researcher, this did nothing but
prove that Mrs. Smith had the propositional knowledge, but couldn?t really get her
students to relay the same information.
Student/Teacher Relationships
It appeared that Mrs. Smith was all business when it came to her classroom. She
was more concerned about getting her lesson taught than she was about trying to get to
know her students on a more personal level. This was particularly true with the
remediation classes. It was presumed by the researcher that the reason for this was
because Mrs. Smith felt solely responsible for these students being able to pass the
graduation exam. Every once in a while, Mrs. Smith attempted to phrase a problem so
that it was appealing to the students; however, they just snickered because the example
she gave was dated. To them, she appeared to be out of touch. Another thing that the
researcher noticed was that when Mrs. Smith helped an individual at his/her desk, she
spoke loud enough so the rest of the class could hear her whether they wanted to hear the
conversation or not. There were times when this was appropriate, but from what the
78
researcher could tell, there were many times it was not because the student was seeking
one-on-one attention. Respect for Mrs. Smith never really seemed to be an issue;
however, the same could not be said for Mrs. Smith?s respect for her students.
Mrs. Franklin
As mentioned earlier, Mrs. Franklin was the pre-service teacher placed with Mrs.
Smith. Mrs. Franklin is a White female in her early to mid twenties working on her
degree in Secondary Mathematics Education. Like Mrs. Smith, Mrs. Franklin is tall in
stature; even so, her presence in the room was somewhat apologetic. She came across as
being very self-conscience, guarded, anxious, and cautious. Mrs. Franklin is also a devout
Muslim. Her daily attire consisted of the traditional Muslim attire of hijab which means
that she wore clothing that covered all of her features except her face and hands. On her
head, she wore a headscarf that covered her hair, ears, neck, and upper chest. Her
clothing covered her all the way from her throat to her wrist and ankles so that her figure
was obscured. Also, she wore socks and shoes so that her feet were not exposed. The
issue of dress was an area of concern for Mrs. Franklin. In the Spring 2006 interview, she
revealed to the researcher that she was not only anxious about her internship experience
just because it was her internship experience, but she was also concerned how her
students would accept her because of her appearance. Obviously, she looked different
from her students. This concern definitely came out in her demeanor in the classroom. As
described above, she was very guarded and cautious when it came to things such as
classroom management, taking charge of the room, etc.
When asked to give a description of mathematics in general, Mrs. Franklin
responded, ?It?s about problem-solving and about how you can solve problems and
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something we do everyday and something really important in our life even in our daily
lives.? When asked to describe the best way students learn mathematics, Mrs. Franklin
commented,
Based on my experience, I think hands-on activities and visual things help us to
learn?really learn math when they see it. Also, connecting the things they are
learning to something they are interested in and something they do everyday, it
really helps them understand. Unless they know what they?re going to use it for,
they don?t really care about it. So, when you show them that they?re going to use
it more than one time in their classroom, they?ll be interested in learning it.
Upon being asked to discuss her feelings about the most effective way to teach
mathematics, Mrs. Franklin stated,
I think that students should take an active role in their learning, and the teacher
should not be a person that comes in and lectures the whole time and gives
homework. I think that students should do activities where they can discover the
concept that we are about to teach.
Based on her responses from the beliefs survey, Mrs. Franklin strongly agreed
that students should be allowed to figure out how to solve mathematics problems for
themselves. She agreed this could be done by the students applying their own personal
experiences to solving the problem at hand. Mrs. Franklin also strongly agreed that
teachers? demonstrations/explanations were the best way for students to learn.
Additionally, she strongly agreed that teachers should demonstrate and model
mathematical procedures prior to expecting their students to use them. She agreed this
could be done by the students applying their own personal experiences to solving the
80
problem at hand. In addition, Mrs. Franklin strongly agreed that teachers should allow
students to communicate their mathematical processes in ways that are relevant to them.
In the area of problem-solving, Mrs. Franklin strongly agreed that students should be
provided with informal experiences to explore mathematical concepts prior to them being
expected to master that concept. She also strongly agreed that students should have to
work with mathematical concepts in various contexts; therefore, the teacher should
provide varied and multiple experiences for students to work through problems.
Additionally, Mrs. Franklin strongly agreed that students must be able to follow
directions in order to sharpen their problem-solving skills. Concerning statements
pertaining to various mathematical procedures and understanding, Mrs. Franklin
disagreed that time should be spent practicing mathematical procedures before students
spend much time solving mathematics problems. She also strongly disagreed that
students will not understand a mathematical concept until they have memorized the
definitions and procedures associated with that concept. Refer to Appendix D Table 22 to
view all of Mrs. Franklin?s responses to the beliefs survey.
When Mrs. Franklin was asked to define the role of her cooperating teacher, she
stated,
I think she will not be my boss but more like my advisor. I know I?ll be the one
teaching, but I?m expecting her to always give me her feedback. I?m expecting
her to help me with all of my lessons in the sense of approving the lessons and
knowing if these lessons are going to work or not. I?m also hoping that she will
support me in the way that I want to teach even if hers is different.
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When Mrs. Franklin was asked to elaborate on the role of the university
supervisor, she replied,
I would expect them to tell me if something is majorly wrong with my lesson
plans before so that I can change my plans ahead of time and make sure
everything is right. I also expect them, when they come observe me, especially for
the first time, I know I?m not perfect, and this is my first ever experience at
teaching, I?m hoping that they?re not going to, I know I?m going to be doing stuff
wrong, so I hope they don?t just tell me that in a way that will make me hate what
I?m doing, but that they will encourage me to change to the right thing and do the
right thing?that will make me want to do it right. I think they are the ones, after
my cooperating teacher, I will go to if I have problems. I hope they will be a help
to me.
When asked to discuss her anticipated role as pre-service teacher, Mrs. Franklin
answered,
I believe that interns are teachers. When I had interns in school, I treated them just
like the regular teacher. I know that I don?t have the full power of the teacher in
the classroom. I don?t know if I can write up students or tell someone to go out in
the hall, I don?t know all of that yet, but I hope that I will be just like a teacher. I
hope that I will have the respect of a teacher that way I will give just like a
teacher. Feel like a teacher. Act like a teacher.
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Finally, Mrs. Franklin was asked to talk about if she thought there would be problems if
she and her cooperating teacher had different beliefs about teaching and learning
mathematics. Her reply was,
I don?t think it?s going to be a problem until she doesn?t allow me to teach the
way that I know how to teach. The way I know how to teach is the way I believe,
and I think that this is the best way I can teach. Maybe she can teach in a better
way, but it?s not the way I can teach. So, I don?t think if we have different beliefs
that it?s going to be a problem, but it?s going to be a problem if she expects me to
teach her way. I don?t know, and I have never used it. So, I think that would be
the problem. But, if she just has different beliefs and lets me do whatever I want
as long as it is right, and she approves of it, I think we?ll be fine.
Fall 2006 Classroom Observations
During the Fall 2006 semester, Mrs. Franklin taught one block of pre-calculus,
one block of remedial mathematics, and one block of calculus. Throughout the course of
the Fall 2006 semester, Mrs. Franklin was observed on three separate occasions. These
three observations were spread out over a period of three months. All three observations
were of Pre-Calculus classes. The first two observations resembled the following
scenario:
At the beginning of class, Mrs. Franklin had the groups go over their work from
the previous class meeting. It was apparent to the researcher that the students had
a worksheet that they were supposed to have worked on over the weekend. The
purpose of today was to get them to go over their work and help each other with
their problems they couldn?t work. As the students worked in groups, Mrs.
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Franklin walked around to make sure the students were working. For the most
part, the students remained on task; however, there were a few times Mrs.
Franklin had to say something to them.
After the students had time to work through their homework problems,
Mrs. Franklin had one group member from each group work one of the problems.
During these presentations, Mrs. Franklin placed herself at the back of the room.
Also, during the presentations, Mrs. Franklin asked questions such as: ?How do
you know that??; ?Why does that work??; ?How do you get to the next step??;
and ?Can you explain that??. As they were working through these problems,
problem #2 seemed to be an issue for the students. Even though the group was
unsure of their work, they agreed to work through the problem as far as they could
get. It took about four people, but they finally got it. Instead of Mrs. Franklin
doing the work, she got the students to work through the problem by helping each
other. The researcher noted that during the presentations, Mrs. Franklin had to call
the groups down several times. This activity didn?t hold their attention at all. The
researcher speculated that it was because many of them were still confused about
it. The homework discussion took an hour. They were going over six induction
proofs, two of which didn?t hold for the first step. So, they really only had to work
four problems. At the end of the homework discussion, Mrs. Franklin eventually
ended up working the final problem because no one in the class could do it. Then,
Mrs. Franklin gave each group a copy of an activity that they were to complete as
a group. There were four different activities, so each group had a different
problem. While the groups worked on their problem, Mrs. Franklin walked
around and monitored group progress and answered questions. Presentations of
these problems were planned, but due to time constraints, the presentations had to
be done in the following class meeting.
The final observation was different from the first two. It was interesting to the
researcher to see just how close to Mrs. Smith?s style this particular lesson was. It
occurred as follows:
Mrs. Franklin began class by asking the students to graph siny x= . She
specifically wanted them to find the amplitude, the period, the x-axis interval, the
maximum point and/or the minimum point. As the students worked on the given
problem, Mrs. Franklin walked around and encouraged the students to work the
problem. She also encouraged them to do it without using their notes; however,
she didn?t tell them they couldn?t use their notes.
After the students had time to complete the above problem, Mrs. Franklin
asked the class questions like: ?What is the amplitude??; ?Where do you find it??;
?What is the definition of amplitude??; ?What is the period??; etc. Sometimes
Mrs. Franklin questioned specific students. Sometimes she didn?t. Either way, she
didn?t really ask for explanations. This questioning continued until they had
completed the graph.
Then, she asked them to work on 2sin2yx= . As with the other problem,
Mrs. Franklin asked the students the same questions until they had completed the
graph. Then, she asked them to work on 2sin4yx=? . As with the previous
problems, Mrs. Franklin asked the students the same questions until they had
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completed the graph. She asked them what the negative did to the points on the
graph. The students said that it ?flipped? the graph.
Next, Mrs. Franklin drew a sin graph with an amplitude of 1 and
divisions
2
?
, ? ,
3
2
?
, and 2? . The students automatically said that it was the
graph of siny x= . Then, she drew a sin graph with an amplitude of 2 and
divisions ,,,
632
? ??
and
2
3
?
. The students said it was 2sin3y x= . Next, Mrs.
Franklin asked the following questions: ?What is the maximum point??; ?What is
the minimum point??; ?What is the amplitude??; and ?What is the period??. She
asked her questions pretty quickly. There wasn?t much wait time at all. This
seemed to frustrate some students. The researcher kept hearing ?Could you please
slow down??. The researcher wasn?t sure if Mrs. Franklin heard them or not
because she never really slowed down. Finally, Mrs. Franklin brought in some
discussion about shifts in the graphs. She related this back to what the student
already knew about shifts in parabolas. Then, she walked them
through ()sinyx?=+.
It should be noted that during these exercises, some students seemed
engaged; mainly the ones at the front of the room. Still, however, several students
were losing interest very quickly; mainly the students at the back.
Next, Mrs. Franklin asked the students to work through
2sin
4
yx
?
?
=?
?
??
?
?
on their own. As the students worked, Mrs. Franklin walked
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around and helped the students. Once the students had time to complete the graph,
they went over it as a class.
At this point, Mrs. Franklin gave the students a short break. During the
break, the students played a game. When they were called on, the students had to
come up with a 4-letter word that begins and ends with the same letter. They
couldn?t repeat what someone else had already said. If they did, they were out.
They kept going around the room until only one person was left.
After the game had been completed, Mrs. Franklin brought the class back
together and got them to work ( )3sin 2yx?= ? . This discussion was just like the
rest. Finally, Mrs. Franklin asked the students to work a few more problems with
a partner. Mrs. Franklin also made an attempt to summarize the day?s lesson.
Even though she summarized somewhat, several students were not paying
attention. As the students worked, Mrs. Franklin walked around and tried to keep
the students on task until the end of the period. Before the bell, Mrs. Franklin also
told the students about upcoming quizzes and tests.
At the conclusion of each of the above mentioned observations, the researcher
completed the RTOP for each respective lesson. When looking at the scores in Table 8, a
score of 0 represents ?Never Occurred? and 5 represents ?Very Descriptive? (AzTEC,
2002). It should be noted that for Mrs. Franklin, the first two observations had scores that
were somewhat higher than the last observation. In the first two observations, Mrs.
Franklin attempted to incorporate many of the techniques she had learned throughout her
methods courses. Even though the lessons may not have been implemented in the manner
Mrs. Franklin envisioned, the techniques were still present. Throughout this process, Mrs.
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87
Franklin?s university supervisor attempted to help many times. In addition to the three
required observations, Mrs. Franklin?s supervisor visited on several different occasions.
Even with the additional help, Mrs. Franklin gave in and ultimately implemented lessons
that were more in line with Mrs. Smith?s teaching style. It is my belief that even though
Mrs. Franklin received additional assistance from her university supervisor, Mrs.
Franklin eventually gave in and succumbed to a more traditional style of teaching that
was very similar to Mrs. Smith?s because Mrs. Smith did not encourage her to teach in a
reform manner. Overall, the scores indicate that Mrs. Franklin cannot be depicted as a
reform minded teacher.
Table 8
RTOP Averages and Median for Mrs. Franklin?s Fall 2006 Classroom Observations
Lesson Design
&
Implementation
Propositional
Knowledge
Procedural
Knowledge
Communicative
Interactions
Student/
Teacher
Relationships
Observation 1 1.6 2.2 1.2 1.6 1.6
Observation 2 1.8 2.4 1.8 2 2.6
Observation 3 0.8 1.6 0.8 1 0.6
Median 1.6 2.2 1.2 1.6 1.6
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Lesson Design and Implementation
Mrs. Franklin attempted to utilize many of the reform mathematics techniques
throughout her lesson design and implementation. The students were usually seated in
groups of three, four, or five as opposed to the rows that Mrs. Smith always had them
sitting in. This configuration, however, didn?t always guarantee that the students were
engaged in group discussions. There were times, especially in the last observation, when
it appeared that they were only seated that way for looks. Many times, Mrs. Franklin had
trouble with the positioning of the groups. Because she did so much work at the front of
the room, the groups should have been arranged so that all students could see; however
this was not the case. There were several times when students had their backs to the
board. During the times when Mrs. Franklin was doing most of the talking, these
configurations ultimately led to disengagement for those students.
In addition to different seating arrangements, Mrs. Franklin also allowed her
students to work through various examples on the board. During these discussions, she
also asked higher order thinking questions such as ?How did you know that??, etc. The
only problem with these types of discussions was that more times than not, the lesson
moved much slower than it should have. Mrs. Franklin had a difficult time with pacing.
She just wasn?t getting the material covered fast enough to suite Mrs. Smith. It is
believed by the researcher that this is the reason that Mrs. Franklin?s style changed so
drastically during the last observation.
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Communicative Interactions
Student-Led Discussions
There were many times that Mrs. Franklin attempted to facilitate student-led
discussions. Most of these were initiated by allowing students to work problems on the
board. After working the problem, Mrs. Franklin then expected the student to explain the
work he/she had done. At this time, either other students or Mrs. Franklin asked questions
for further clarification. At this point, Mrs. Franklin had to be conscience of her role as
facilitator because there were many times she wanted to take over the discussion. When
the students were working in groups, there was not much discussion unless Mrs. Franklin
went to each group and asked the students in that group specific questions. Otherwise, the
students really preferred to work independently which was the way they were accustomed
to conducting class.
Teacher-Led Discussions
As stated previously, Mrs. Franklin really tried to get her students to carry the
conversation when problems were being discussed; however, there were times when she
struggled with getting the discussions to ?move along? in a timely fashion. This caused
frustration for both her and Mrs. Smith. As a result, by the end of the internship, Mrs.
Franklin had reverted to teaching the way Mrs. Smith taught which was very traditional.
At the beginning of the internship experience, Mrs. Franklin tried to pull the students into
the lesson and to get them to do most of the work. By the end, however, Mrs. Franklin
was working all of the example problems without asking for any discussion from her
students. It was strictly ?First we do?.then we do??. So, Mrs. Franklin went from
lessons that were somewhat student oriented to lessons that were strictly teacher oriented.
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Procedural Knowledge
There was no doubt that Mrs. Franklin had no problem comprehending the
material that she was teaching. Her problem was making her students understand. Mrs.
Smith stated many times in many conversations that Mrs. Franklin just couldn?t get down
on the students? level. This was very frustrating for Mrs. Smith to watch and very
frustrating for Mrs. Franklin to endure. It was obvious that she didn?t understand why the
students weren?t picking up on the information she was giving them. Part of the problem
is because she was giving them the information and not allowing them to figure things
out on their own. This was evident more toward the end of the internship experience than
it was at the beginning. At the beginning, Mrs. Franklin attempted to let the students
explore some topics, but when things didn?t move as fast as Mrs. Smith liked, Mrs.
Franklin began teaching the concepts very procedurally. Her style became very similar to
what Mrs. Smith had been doing.
Propositional Knowledge
As indicated previously, Mrs. Franklin made a valiant attempt to unleash her
students? propositional knowledge. Initially, she asked students to work problems on the
board and then explain their work. Then, she allowed students to ask questions for further
clarification. It was clear that her desire was for the student to provide the clarification,
but many times, she had to intervene because the student didn?t know how to precede
with his/her explanation. It is presumed that part of the reason for this is because Mrs.
Franklin appeared to be unsure about what questions to ask to get the student to analyze
the work that had been presented. As long as Mrs. Franklin was asking questions such as
?How did you arrive at your answer??, ?What does your answer mean??, etc, things were
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fine. Her breakdown seemed to occur during student presentations. One of the biggest
problems that was observed by the researcher was that Mrs. Smith didn?t know how to
help Mrs. Franklin with her questioning. Because of this, Mrs. Franklin?s university
supervisor stepped in and tried several times to help Mrs. Franklin revise her lessons so
that she could successfully implement the techniques she had learned and practiced in her
methods courses. Despite the additional help from her university supervisor, Mrs.
Franklin ended up teaching very procedural based lessons instead of pursuing exploration
techniques.
Student/Teacher Relationships
Mrs. Franklin genuinely seemed to care about her students; however, her self
confidence issues interfered with her getting too close to them. This was evident in her
classroom management skills. She wanted to be assertive with her students, but it seemed
to go against her nature. It appeared as though she didn?t like taking a stern tone of voice
with them or confronting them when they were misbehaving. It didn?t take her students
long to pick up on this. Her classroom management was something she had to work on all
semester. Other than that, it appeared as though the students liked having Mrs. Franklin
as a teacher. Of course, this could be because she was the ?nice? teacher compared to
what they were used to with Mrs. Smith.
Similarities and Differences Between Mrs. Smith and Mrs. Franklin
Upon analyzing all of the information pertaining to Mrs. Smith and Mrs. Franklin,
it appears that they were very similar in many ways. The main difference that was seen
by the researcher was that while Mrs. Franklin was more open to trying techniques she
had learned in her methods courses, Mrs. Smith was more hesitant. Mrs. Smith was more
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like the content-focused teacher described by Thompson (1992). She saw her role as
demonstrating, explaining, and defining mathematics for her students. She definitely
maintained an expository style. Mrs. Smith?s idea of attempting reform based instruction
was having her students play a game where they had to recall basic trigonometric
information. Mrs. Franklin began the internship experience more in line with the Platonist
view described by Thompson (1992): she attempted to emphasize students?
understanding of the logical relationships among various mathematical topics and the
logic underlying the mathematical procedures. At the conclusion of the internship
experience, however, she more like the content-focused teacher described by Thompson
(1992); hence, more like Mrs. Smith. Even though Mrs. Franklin attempted many new
techniques, when things didn?t go according to plan, she eventually ended up teaching the
class the same way Mrs. Smith always had done. It should be noted here that even though
Mrs. Smith and Mrs. Franklin were similar in many ways, their similarities didn?t
promote reform based teaching.
Outcome of the Internship Experience
At the conclusion of the internship experience, Mrs. Smith stated that as a
cooperating teacher she felt like she had many different roles such as: instructor in the art
of teaching; mother, after one of her observations; cheerleader, "Yes, you can do this";
coach "Try this play and see if it works"; timekeeper, "We can't take that many days on
that topic". In redefining her description of the role of the pre-service teacher, Mrs. Smith
commented,
At times she was the TEACHER, and every decision was hers. Other times I
would teach and then she would act as tutor and help and answer questions. She
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was busy most all the time. I especially liked and appreciated it when she acted as
aide and graded papers.
One final comment Mrs. Smith made was about whether or not she and her pre-service
teacher had differing beliefs about teaching and learning mathematics. She confidently
stated that she felt there was not a difference of opinion; however, she also made the
following comment:
I think the problem we did have was one of experience. In high school and
college, her experience was with upper level students and over-achievers like
herself. I had to constantly remind her that "once" was not enough for most of my
students. They needed review and restating before they were locked in on a
concept.
When asked to discuss the role of her cooperating teacher, Mrs. Franklin
commented that she felt Mrs. Smith had a good effect on her. She explained that Mrs.
Smith was helpful in developing her lessons by discussing things that were good and
things that were bad. Additionally, Mrs. Franklin stated that Mrs. Franklin helped her
connect different mathematical topics to previous topics in an effort to maximize learning
and to insure that all students were learning. Mrs. Franklin also elaborated on her role as
pre-service teacher by stating, ?I was first an observer then slowly I became a full time
teacher. I planned and executed many lessons using different activities and different
tools.? When the researcher asked who influenced her most throughout her experience,
Mrs. Franklin responded, ?My cooperating teacher. She helped me so much in all phases
of my development as a professional. When I felt that everything was going wrong she
was always there to support me. I felt that she was always there for me.? When asked to
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discuss whether or not she and her cooperating teacher had different beliefs about
teaching and learning mathematics, Mrs. Franklin responded, ?I think we had different
beliefs. My teacher is very traditional and does not like to use new ways because her
ways proved great success throughout all her years of teaching. I like to use and explore
new tools and activities in my teaching.? She further commented that this difference in
beliefs really did not cause a problem because they had good communication between
them.
Case 2: Mrs. Johnson and Ms. Walters
Mrs. Johnson is the White female cooperating teacher discussed throughout Case
Two. She has been teaching mathematics for almost seven years. Ms. Walters is the
White female pre-service teacher discussed throughout Case Two. She is working on her
degree in Secondary Mathematics Education.
The internship experience discussed throughout Case Two takes place at
Riverdale High School. Refer to Case One for a detailed description of Riverdale High
School. Table 9, which contains a demographic summary of Riverdale High School, has
been repeated here for the reader?s convenience.
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Table 9
Demographic Summary of Riverdale High School
System
Type
Grades
Serviced
Total
Population
Student/
Teacher
Ratio
Students Eligible
for Free or
Reduced-Price
Lunch
Racial
Background
City 9-12 1312 16 49% African American ? 59%
White ? 38%
Asian/Pacific Islander ? 2%
Hispanic ? 1%
American Indian/
Alaskan Native ? <1%
Mrs. Johnson
Mrs. Johnson is a White female in her mid thirties. She has two children; one in
elementary school and one in pre-school. She is somewhat of an average height, but her
personality is huge. Her bubbly, cheerleader-like personality makes her classroom an
inviting and fun place to be. Mrs. Johnson has been teaching high school for 6 ? years.
Six of those years have been spent teaching at Riverdale High School. She has a
bachelors degree in mathematics education.
Mrs. Johnson reported that over the past year, she has spent approximately twenty
to forty hours involved in some type of professional development. Of those hours, she
stated that six to ten of those hours were spent in professional development settings that
specifically focused on mathematics. Additionally, she reported that none of her
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professional development hours were spent in association with Math Plus. Even so, Mrs.
Johnson commented that she had been involved in implementing various aspects of Math
Plus at Riverdale High School.
When asked to discuss her views about mathematics, Mrs. Johnson stated, ?First
of all, the world is getting so technology based now that the kids have got to have it
(mathematics). They?ve got to know the algebra. They?ve got to know the geometry.
They?ve got to know those skills.? She further elaborated by saying, ?We show a lot of
skill and drill, and then we take it to some of the technology ways it?s used and all.?
When asked to talk about her idea of the best way for students to learn mathematics, Mrs.
Johnson replied,
Oh, it?s got to be a variety of ways. There is no set way for them to learn. There
are days that we do lecture. There are days we?re counting M&M?s. There?s no
one way for any student to learn math. A lot of it, they?ve got to see the skill so
lecture must be a part of mathematics, but I do believe that you have to
incorporate the real world.
When asked to discuss the most effective way to teach mathematics, Mrs. Johnson
responded by saying, ?Again, I think it is a combination of?you?ve got to have the skill
and drill, lecture, and then you?ve got to have the hands-on.? Mrs. Johnson agreed that
students should not only be able to obtain correct answers, but they should also
understand the mathematical concepts involved in getting to that right answer.
Additionally, she felt that students should understand important mathematical concepts
before they ever attempt to memorize definitions and facts.
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Upon being asked to describe a typical mathematics lesson in her classroom, Mrs.
Johnson?s response was
Typically in my classroom, we focus a majority of the time on a lot of skill and
drill. We have special occasions where we pull out our labs and break away from
the lecture. Normally, we start our class sometimes with a pop quiz to focus a lot
on those graduation exam objectives because they?re required by the state. We do
a lecture. We assign work. And, then a lot of times, I allow my kids to do the
group work. Now, you haven?t been able to see that because I?ve been left with
the two longest lessons in the book to teach. But, normally, we do a lot of group
interaction because I like?I?m not one of those teachers that makes them just
stay quiet the whole time and not breathe a sound because I like hearing the
interaction between them because a lot of times that lets me know what I?ve done
right and wrong with in the lesson. And by letting them have the group work, and
really listening to them talk about it, is where I get my feedback from more than I
do assessments I would say. I like groups.
Mrs. Johnson further described the learning environment in her classroom as ?part
traditional, part hands-on.? Mrs. Johnson responded that students should be allowed to
figure things out on their own rather than depending on demonstrations/explanations
given by their teacher.
As indicated by her responses on the beliefs survey, Mrs. Johnson strongly agreed
that students need good mathematical problem-solving skills in order to be successful in
the future. She also strongly agreed that in order to formulate these problem-solving skills
students must be able to follow directions. According to her responses, Mrs. Johnson felt
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like mathematics is an important subject that should be available to all students because it
is something that will continue to be used even once the students are out of school. Refer
to Appendix D Table 23 to view all of Mrs. Johnson?s responses to the beliefs survey.
In defining what she anticipated as being her role as cooperating teacher, Mrs.
Johnson replied,
I almost think of being a counselor or just being there to give them the motivation
to say ?It?s OK to try this, to try that?. And, then if they have any questions, just
to ask. I?m not the one to bite somebody?s head off, you know. And if you see
something wrong, let them know about it. That?s the one thing I am. I?m blunt.
You know, if something, and I?m not saying it?s in their teaching methods
because to each his own, I?m fine with that, but if there is a mistake, if they
should have dealt with something discipline-wise, or this or that, letting them
know to not be afraid to beat around the bush because here?s their learning
experience before they get thrown into the real world.
When asked about the role of the university supervisor, Mrs. Johnson stated that she
really wanted the university supervisor to come in and tell her and her pre-service teacher
exactly what was expected in terms of what the university wanted. Her comment was, ?I
want to know what they want to see in the classroom.?
When asked to elaborate on her expectations of a pre-service teacher, Mrs.
Johnson replied,
Willingness to try anything. I think the biggest thing they need to do is to be
willing to, you know, they might have worked all night on a lesson and be willing
to throw it out if need be because sometimes you have planned this whole lesson
and you?re up there teaching and it?s not working worth a flip. You?ve got to be
willing to just throw it out. Even though you do all this planning, you have got to
be willing to go on the fly if need to. So, just that willingness to be able to not
stick to a regiment. You?ve got to be kind of free out there.
Finally, when asked if she thought it would be a problem if she and her pre-
service teacher had different beliefs about teaching and learning mathematics, Mrs.
Johnson?s response was
No. Each person can do something different and still be able to convey that
message. We?re individual teachers just like the students are individual learners.
Different types for different types.
Spring 2006 and Fall 2006 Classroom Observations
During the Spring 2006 semester, Mrs. Johnson was observed on three separate
occasions. These three observations were spread out over a period of a month. All three
observations were of Algebra I classes. A typical lesson looked like the following:
Mrs. Johnson began class by giving back a test. The test was on factoring. There
was some discussion about the test, but not very much. After the teacher took up
the test, she gave them a pop quiz on material that was covered two weeks ago
(multiplication of polynomials). The quiz had five multiple choice questions. It
didn?t take the students long at all to complete the quiz.
Once the students completed the quiz, the teacher began the discussion on
Chapter 10.1 (Graphing Quadratic Functions). Mrs. Johnson began the discussion
by reminding the students that quadratics have
2
x in them. She then asked the
class what kind of graph they would get from the equation 3 1y x=+. A student
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100
responded and said that the equation is that of a line. Mrs. Johnson then explained
that at this point they would begin to add degrees to the equation of a line. She
also stated that when you add degrees, you add curves to the graph. At this point,
Mrs. Johnson related doing arithmetic to passing a bill through Congress. She said
that it may or may not pass. Then, she reassured her students that working with
quadratics was not hard. She told them that working with quadratics was just long
and dangerous. At this point in the lesson, Mrs. Johnson began discussing the
various components of parabolic graphs such as axis of symmetry, symmetry,
minimum point, maximum point, etc.
Next, Mrs. Johnson began leading the class through the steps of generating
an accurate graph for a quadratic function. Here, she stated, ?Now we?re going to
look at all of the steps. Don?t try to skip steps. Don?t do your arithmetic in your
head! These must go in order.?. She then proceeded to demonstrate her order of
steps by working through a problem with the students. After she worked through
one problem with the class, she posted another problem for them to work. The
students were not overly excited. Mrs. Johnson just said, ?Practice makes
perfect.? With the second problem, Mrs. Johnson told the students, ?Now on this
problem, you tell me the steps.? From here, the class began working. Throughout
the whole process, the teacher asked very procedural questions. Once the class
had completed the second example, Mrs. Johnson assigned a problem for the
students to work individually. After the class discussed this last problem, Mrs.
Johnson assigned homework problems for the students to work on. As they
worked, she passed out progress reports.
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Throughout the class, all of the notes Mrs. Johnson used were typed on an
overhead transparency. All Mrs. Johnson had to do was uncover the various parts of the
transparency as she went through the notes. She did, however, work the problems on a
separate transparency or on the board.
At the conclusion of each of the above mentioned observations, the researcher
completed the Reformed Teaching Observation Protocol (RTOP) for each respective
lesson. When looking at the scores in Table 10, a score of 0 represents ?Never Occurred?
and 5 represents ?Very Descriptive? (AzTEC, 2002). Most of the emphasis in Mrs.
Johnson?s lessons was placed on getting the steps correct. She was extremely strict about
not allowing her students to skip steps when working the problems. Throughout the
lessons, there were hardly any discussions about why the problems worked the way they
did. Even though she allowed her students to demonstrate their work for the various
problems, it was apparent to the researcher that the lessons were very much teacher
centered. As indicated by the median scores listed in the table, Mrs. Johnson?s lessons do
not indicate that she is designing lessons that would be considered reform based.
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Table 10
RTOP Averages and Median for Mrs. Johnson?s Spring 2006 Classroom Observations
Lesson Design
and
Implementation
Propositional
Knowledge
Procedural
Knowledge
Communicative
Interactions
Student/Teacher
Relationships
Observation 1 0.6 2.4 0.4 1 1
Observation 2 1.2 2.2 0.6 1.2 1.6
Observation 3 0.6 2.2 0.4 1 1
Median 0.6 2.2 0.4 1 1
At the beginning of the Fall 2006 semester, Mrs. Johnson was observed one more
time just to see if any major changes in teaching style since Spring 2006 had occurred. As
with the Spring 2006 observations, this observation was of an Algebra I class. Unlike the
Spring 2006 observations, this lesson was somewhat different just because Mrs. Johnson
made an honest attempt to incorporate more student centered activities. One of the things
that she added was the Problem of the Week (POW). This was a problem/situation that
was given to the class at the beginning of the week. They had all week to work through
the problem individually or with a classmate. At the end of the week, they had to turn in
all work along with a written explanation of what had been done. After discussing the
problem of the week, Mrs. Johnson gave the students a handout that guided them through
the content for the day. They were supposed to work with a partner to work through the
information on the sheet. After sufficient time had passed, they discussed the work as a
class. At this point, the lesson began looking more like the three that had been observed
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in the Spring. Everything was very procedural. Again, the focus was on the steps. As
displayed in Table 11, even though the scores are higher, they still don?t indicate that
Mrs. Johnson plans lessons that are indicative of reform based teaching.
Table 11
RTOP Averages for Mrs. Johnson?s Fall 2006 Classroom Observation
Lesson Design
and
Implementation
Propositional
Knowledge
Procedural
Knowledge
Communicative
Interactions
Student/Teacher
Relationships
Observation 4 1.6 2.2 1.2 2 2.4
Lesson Design and Implementation
The lessons designed by Mrs. Johnson were very structured. In general, the class
began with a small quiz that was supposed to act as a review for graduation exam
objectives. Then, there was some discussion of the previous night?s homework problems.
After that, Mrs. Johnson spent the remainder of the class time discussing at least one if
not two new sections from the textbook. The word discussion is somewhat deceiving in
this situation because the discussion was generally one-sided. Mrs. Johnson did most of
the talking, and the students did all of the listening. While going over the new concepts,
Mrs. Johnson worked through an example problem and then gave the students a problem
to work individually. When Mrs. Johnson thought the students had enough time to work
through the problem, she worked through it with the class. At times, she asked various
students to tell her what they had done. All throughout the lesson, Mrs. Johnson was very
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concerned with the step-by-step processes. She was very adamant that the students follow
the proper steps for working the problems. This pattern of working an example problem,
giving an individual problem, and discussing the problem continued until class was
almost over. At that time, Mrs. Johnson assigned homework problems for the students to
work through until the bell rang.
In the Fall 2006 semester, after participating in a two week summer institute
hosted by Math Plus, Mrs. Johnson changed some of her lessons. It appeared that she
tried to incorporate some Math Plus approaches in her class by including a Problem of
the Week (POW) for the students to work on throughout the week. It also appeared that
she tried to incorporate more group work in her class; however, the tasks were very rote
in nature. As with her previous lessons, she continued the pattern of working an example
problem, giving the students an individual/group problem to work, and then discussing
the problem. The only difference was that the students had a worksheet to write down all
of the things they discussed. It was very guided.
Communicative Interactions
Student-Led Discussions
As is evident from the above scenario, the student-led discussions in Mrs.
Johnson?s classroom were few and far between. The few times she did allow students to
verbally participate in the lesson, she focused strictly on the steps to solving the problem.
There was very little discussion about how the student(s) arrived at the answer.
Teacher-Led Discussions
Even though Mrs. Johnson made her lessons appear student-centered, they were
mainly teacher-centered. Most of the period involved Mrs. Johnson working examples of
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the various problems from the new sections. Several times throughout her lessons, Mrs.
Johnson asked questions, but she didn?t really intend for the students to answer her. As a
matter of fact, most of the questions referred to procedures for answering the questions
and not about the mathematics involved in working through the problems. Even though
Mrs. Johnson made an attempt to incorporate more Math Plus techniques in her
classroom during the Fall 2006 semester, the lessons were still very much teacher-
centered.
Procedural Knowledge
Knowing the steps to solving the various problems appeared to be imperative to
Mrs. Johnson. This was very apparent in her questioning techniques. Most of her
discussion with the class was spent asking questions such as ?For step one we do?? and
?For step two we do??. When she did ask the students a question, it was generally
phrased like ?What do we do next??. She never really asked them to elaborate on their
answers or to explain how they arrived at their conclusion if it was different from the rest
of the class. When Mrs. Johnson allowed a student to ?work? through a problem in front
of the class, skipping steps was not allowed. All steps had to be shown or Mrs. Johnson
did not allow the student to continue working the problem.
Propositional Knowledge
From the lessons that were observed, propositional knowledge took a back seat to
procedural knowledge. As stated above, knowing the steps to solving a problem was
imperative in this class. It appeared to the researcher that Mrs. Johnson assumed that
because the students could follow the steps to get to the answer she was looking for, that
they also understood the methods they had used to arrive at that particular answer. At the
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beginning of the Fall 2006 semester, it seemed that Mrs. Johnson had made an effort to
incorporate more thought provoking, exploratory type activities into her lesson by adding
in the POW and more group activities. After further inspection, however, the group work
more or less turned into a guided activity where the students simply completed a
worksheet by filling in the blanks and the POW was more of the ?First we do??, ?Then
we do?? type of discussion.
Student/Teacher Relationships
Mrs. Johnson had a wonderful rapport with her students. It didn?t matter if they
enjoyed math class or not, the students loved coming to her room. As mentioned
previously, Mrs. Johnson had a very bubbly attitude that could be quite contagious. She
was very much like a cheerleader. All throughout her lessons, she reassured the students
that everything was going to be alright, especially if they followed her prescribed steps
for solving the problem. Even if a student did something that was incorrect, she always
seemed to find a way to correct the student without embarrassing him or her. She also
had incentives for students who performed well on assignments such as allowing them to
write their name on the ?A Board? when making an ?A? on a test. This was a big
motivation for her students.
Ms. Walters
As mentioned earlier, Ms. Walters was the pre-service teacher placed with Mrs.
Johnson. Ms. Walters is a White female in her early twenties working on her degree in
Secondary Mathematics Education. Like Mrs. Johnson, Ms. Walters has a very bubbly
and energetic personality. In addition to her cheerful personality, Ms. Walters could also
be described as somewhat easy going. She always took things in stride, so it didn?t appear
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that many things bothered her. She also appeared to have a positive attitude about most
things. Because of this, she fit in well with her students, and the students really enjoyed
being around her.
When asked to define mathematics, Ms. Walters responded, ?It is about learning
about numbers and how they operate. And not even just numbers, different functions that
can be related in the classroom and outside. I feel that mathematics is about everything in
life.? Ms. Walters stated that her experiences throughout school helped shape her beliefs
about mathematics. She further commented, ?I think my beliefs about it have come from
experiences throughout my school and through me just sitting down and doing problems
and trying to understand myself.?
Upon being asked to discuss her feelings about the most effective way to teach
mathematics, Ms. Walters commented,
The best way I feel is through communication, talking out the problems and stuff
like that. But, I also feel that there does need to be some structure for students to
go by. But for them to really get the concept and understand it, talking to each
other and talking about it, even if it?s at home with their parents, is where it really
gets embedded in their knowledge.
When asked to elaborate on the best way to teach mathematics, Ms. Walters stated,
This is hard because of all the Math Plus stuff I?m learning. I think, like I said
before, with communicating and having structure. I think it is a combination of
group work and even individual work out of the book because I do feel that the
students need that structure of practicing, but I also feel that they need group work
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to get other students? ideas and to learn what other people are thinking. So, I think
it is a combination of both. I don?t think that one way is necessarily better.
As indicated by her responses on the beliefs survey, Ms. Walters strongly agreed
that students should be allowed to figure out how to solve mathematics problems for
themselves rather than depending on teacher demonstrations/explanations. She agreed
this could be done by the students applying their own personal experiences to solving the
problem at hand. Additionally, Ms. Walters agreed that teachers should allow students to
communicate their mathematical processes in ways that are relevant to them.
Furthermore, Ms. Walters disagreed that students learn best by teacher demonstrations
and explanations. She also disagreed that teachers should demonstrate and model
mathematical procedures prior to expecting their students to use them. Refer to Appendix
D Table 24 to view all of Ms. Walters? responses to the beliefs survey.
When Ms. Walters was asked to define the role of her cooperating teacher. she
stated,
I feel that her role is going to be to show me different ideas and different ways of
teaching and to really show me what it?s like to be a teacher in a school everyday.
So, I feel that she is sort of my guide to show me what could be expected in my
own classroom.
When Ms. Walters was asked to elaborate on the role of the university supervisor,
she replied,
I guess I see her as a kind of, this may sound weird, but as a parent to make sure
that I?m on the right track. To make sure what I?m learning is what she hopes for
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me to learn. And to make sure that I feel comfortable in what I?m trying to do. So,
I guess kind of as a parent looking over me to make sure I?m on the right track.
When asked to discuss her anticipated role as pre-service teacher, Ms. Walters
answered,
Well, I see myself still as a student, you know learning, but hopefully once I get
in there, I?ll see myself as hopefully becoming a professional. Not a professional
teacher, but becoming someone who is getting into their career. While I?m still
learning, hopefully, I?ll see myself as a teacher which is scary. I just
hope?because I?m scared to death, but really excited about it?hopefully when I
get in there, I?ll be even more excited, and I?ll be a teacher.
Finally, Ms. Walters was asked to talk about if she thought there would be problems if
she and her cooperating teacher had different beliefs about teaching and learning
mathematics. Her reply was,
I don?t think it will be a problem as long as we are tolerant of each other?s
different opinions. And if I have a cooperating teacher that isn?t willing to listen
to my opinions and what I?m hoping to do, that might be a problem if you can?t be
somewhere where your opinion is respected or just listened to, you can?t grow
from that. As long as they listen to me and are open to it, they don?t necessarily
have to let me do every single thing I believe in, but as long as they let me explore
what I believe then that?s fine if they have different beliefs and vice versa. I have
to respect what they believe in.
Fall 2006 Classroom Observations
During the Fall 2006 semester, Ms. Walters was observed on three separate
occasions. These three observations were spread out over a period of three months. All
three observations were of Algebra I classes. A typical lesson looked like the following:
Ms. Walters began class by giving the students a homework check. She called out
certain problems and all they had to do was write down their final answer. They
had very little time to write, so they had to have the problems worked ahead of
time. As soon as the students finished, three minutes tops, they turned them in and
Ms. Walters went over the answers.
It seemed that the class worked on 1-step inequalities during the previous
class meeting because Ms. Walters began the discussion by asking the students
for their ?rules?/?definitions? for adding, subtracting, multiplying, and dividing
with inequalities. She then asked the students for some specific examples. A
student gave87 . Ms. Walters said this was true, but in order for the rule
to hold they had to subtract the same number from both sides, so another student
changed the original response to87
82?