THE EDUCATIONAL REFORM IN PERU AND THE CHALLENGE TO TRANSFORM
TEACHER EDUCATION PROGRAMS. A STUDY OF FUTURE TEACHERS’
OPPORTUNITIES TO LEARN, KNOWLEDGE FOR TEACHING, AND BELIEFS RELATED
TO MATHEMATICS TEACHING
By
Giovanna Moreano Villena

A DISSERTATION

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Curriculum, Teaching, and Education Policy – Doctor of Philosophy
2013

ABSTRACT
THE EDUCATIONAL REFORM IN PERU AND THE CHALLENGE TO TRANSFORM
TEACHER EDUCATION PROGRAMS. A STUDY OF FUTURE TEACHERS’
OPPORTUNITIES TO LEARN, KNOWLEDGE FOR TEACHING, AND BELIEFS RELATED
TO MATHEMATICS TEACHING
By
Giovanna Moreano Villena
Peruvian students at both the elementary and the secondary level have shown poor results
in evaluations of mathematics achievement in recent years. In addition, research has found that
teachers are not implementing the curriculum as expected; traditional practices of teaching are
still present in classrooms. These problems have driven a school curriculum reform which
recommends a student-centered instructional approach for mathematics teaching. Within this
frame, mathematics teaching requires more attention to students’ thinking, through problem
solving, reasoning, and communication of their thinking. The principles and processes adopted
by the curriculum call for change, and teacher education programs have to answer to these
demands by preparing teachers with professional competencies that allow them to enact these
ideals in their classrooms. The reform poses the need for future teachers to adopt a different view
of mathematics teaching and requires that future teachers have the pedagogical knowledge
necessary to develop the processes and content mandated in the curriculum.
This study addressed the research question: How do teacher education programs in Peru
respond to the challenges posed by the current educational reform for mathematics teaching, and
to what extent are such responses fulfilling the reform’s demands? The study examined the
opportunities to learn to teach mathematics provided by five teacher education programs in Lima
to future elementary teachers. The mathematical pedagogical content knowledge and beliefs

related to mathematics teaching of future teachers were also examined and considered as
outcomes of the opportunities to learn. Crossectional data of two cohorts (first-year students and
fifth-year students) in each institution were collected in order to examine any effect of the cohort
on the outcome variables. Data source were mainly institutional documents and a questionnaire
for future teachers. Institutional documents (syllabi of mathematics-related courses) were used to
analyze the intended curriculum, and the future teacher questionnaire collected information on
opportunities to learn, beliefs, and knowledge. The instruments and study framework were
adapted from the TEDS-M comparative study.
Regarding opportunities to learn, the findings show differences in the curriculum to
which future teachers were exposed in order to learn to teach mathematics; while some
institutions mostly covered the mathematics topics needed to teach the school curriculum, other
institutions were not providing the content knowledge required to teach mathematics. The
findings also showed that future teachers were not exposed to content on mathematics education
pedagogy that would allow them to have a more critical view of mathematics teaching.
Regarding the outcome variables, knowledge and beliefs, there were no significant
differences by cohort but there were by institution. Student teachers from both cohorts showed
similar patterns of beliefs about mathematics’ nature, learning, and achievement, which mostly
were compatible with the reform ideals. Findings related to mathematical pedagogical content
knowledge showed low performance of student teachers from both cohorts; this was true even
for cases of programs that consistently seemed to provide future teachers with opportunities to
learn to teach mathematics. These results are critical for the purposes of the reform since they
suggest that future teachers are graduating without the knowledge needed to implement the
curriculum, and consequently to enhance students’ learning.

ACKNOWLEDGEMENTS

I would like to acknowledge the support of my dissertation committee. Sincere
appreciation goes to my advisor and chairperson, Dr. Maria Teresa Tatto, for her support and
encouragement through all the milestones that had to be reached through the program. Her
insights helped me to open my mind to understand the teacher leaning process in a bigger context
and also to develop pertinent and useful research for the Peruvian context. Much appreciation
goes to Dr. Mary Kennedy whose work helped me to think critically about the complexity of
teacher learning; likewise, her valuable and pertinent feedback as well as her faith in my work
motivated me to keep moving in developing this study. I also owe thanks to Dr. Ralph Putnam
for his guidance, advice, and interest in my scholarly work; the many meetings we had through
the program helped me to better understand the processes involved in mathematics teaching.
Finally, I would like to thank to Dr. Peter Youngs for his helpful comments, suggestions, and
kind support throughout the dissertation process.
I would like to recognize additional faculty and colleagues who have been supportive
with my work in this research. Special thanks go to Dr. Richard Houang for his willingness to
assist me with the design of the study and the statistical analysis despite not being part of the
dissertation committee. In the same way, my appreciation to Cheng-Hsien Li for his prompt
answer and availability to meet with me at any time I needed help with the statistical analysis. I
extend my appreciation to Dr. Douglas Campbell for helping me in the process of writing not
only in my dissertation but in all my academic work in MSU. My sincerest gratitude also goes to
the persons who helped me in the process of data collection in Lima: Liliana Miranda, Sonia
Leon, and all the program coordinators of the institutions that participated in my study.
iv

Much appreciation also goes to my international graduate fellows in the program who
made more bearable the overwhelming process of dissertation work and helped me to keep my
spirit up with the encouragement, companionship and laughter they provided me during the last
five years. Finally, I would like to thank my family in Peru, my parents and siblings for their
permanent support and faith in me; in particular to my mother for her unwavering moral support
and constant prayers. Having her always reminding me the God’s promises gave me the strength
to keep moving forward in this challenging program. Definitively, the completion of this
dissertation would not have been possible without him.

v

TABLE OF CONTENTS

LIST OF TABLES…………………………………………………………………………..

ix

LIST OF FIGURES …………………………………………………………………...……

xiii

CHAPTER 1: INTRODUCTION…………………………………………………………...

1

CHAPTER 2: THE EDUCATIONAL REFORM IN PERU AND THE TEACHER
EDUCATION PROGRAMS……..........................................................................................
The Peruvian Educational System………………………………………………………
The Problem of Academic Achievement in Peru……………………………………….
The Current Educational Reform in Peru……………………………………………….
The School Mathematics Curriculum………………………………………………..
The Implementation of the Reformed Curriculum and its Challenges for
Teachers and Future Teachers…………………………………………………...
Teacher Education Policy in Peru……………………………………………………
The Context of Teacher Education Policy…….. ……………………………….
The Policy……………………………………………………………………….

6
6
8
10
11
15
19
20
22

CHAPTER 3: TEACHER LEARNING IN MATHEMATICS AND TEACHER
EDUCATION PROGRAMS: A LITERATURE REVIEW……………………………...…
The Nature of Teacher Learning………………………………………………………..
Future Teachers’ Knowledge…………………………………………………………...
Future Teachers’ Beliefs ……………………………………………………………….
Opportunities to Learn to Teach Mathematics in Teacher Education Programs………
Curriculum………………………………………………………………………….
Classroom Participation…………………………………………………………….
Reflection…………………………………………………………………………...

28

CHAPTER 4: METHOD……………………………………………………………………
Sample and Settings……………………………………………………………………
Data Sources…………………………………………………………………………...
Program Documents……………………………………………………………….
Future Teacher Questionnaire……………………………………………………..
Questionnaire Variables………………………………………………………………..
Program Variables………………………………………………………………….
Course Topics………………………………………………………………...
Activities for Learning to Teach Mathematics………………………………..
Student Variables…………………………………………………………………..
Mathematics Pedagogical Content Knowledge Index………………………..
Future Teachers’ Beliefs……………………………………………………...
Beliefs about the nature of mathematics…………………………………
Beliefs about learning mathematics……………………………………...

48
50
53
53
54
57
57
58
59
61
62
67
68
69

vi

28
30
36
41
42
45
46

Beliefs about Mathematics Achievement………………………………..
Data Analysis…………………………………………………………………………..

70
71

CHAPTER 5: RESULTS……………………………………………………………………
Research Question 1: Characteristics of the Teacher Education Programs…………...
The Curriculum of Elementary Education in the Teacher Education Programs…...
The Teacher Education Programs’ Philosophy…………………………………….
The Student Population in the Teacher Education Programs………………………
Research Question 2: Opportunities to Learn in Teacher Education Programs……….
The Intended Curriculum in the Teacher Education Programs……………………
School Mathematics Topics…………………………………………………..
Mathematics Education Pedagogy Topics……………………………………
The Implemented Curriculum in the Teacher Education Programs……………….
Course Topics in Teacher Education Programs………………………………
School Mathematics Topics……………………………………………..
Mathematics Education Pedagogy Topics………………………………
Activities for Learning to Teach Mathematics……………………………….
The Effect of Institution on the Activities for Learning to Teach
Mathematics Scales…………………………………………………….
Post-Hoc Tests for Differences Among Institutions……………………
Reflection in Teacher Education Programs…………………………………...
The Relationship between Opportunity to Learn to Teach Mathematics
Scale…………………………………………………………………………..
Research Question 3. Future Teachers’ Mathematical Pedagogical Content
Knowledge and Teacher Education Programs………………………………………...
Future Teachers Performance in the MPCK Items……………………………….
Future Teachers Performance in the MPCK Index……………………………….
The Effect of Cohort and Institution in on the MPCK Index…………………….
Post Hoc Tests for Differences between Institutions…………………………….
Research Question 4. Future Teachers’ Beliefs and Teacher Education Programs…...
What Future Teachers Believe about Mathematics Teaching…………………….
The Relationship between Future Teachers’ Beliefs and Mathematics
Pedagogical Content Knowledge………………………………………………....
The Effect of Institution and Cohort in Future Teachers’ Beliefs………………..
Post-Hoc Tests for Differences between Institutions…………………………….
Research Question 5. The Opportunities to Learn Associated with Mathematics
Pedagogical Content Knowledge and Beliefs about Mathematics Teaching………….
Course Topics, Mathematics Pedagogical Content Knowledge, and Beliefs for
Mathematics Teaching ……………………………………………......................
Activities for Learning to Teach, Mathematical Pedagogical Content
Knowledge, and Beliefs about Mathematics Teaching…………………….…….

76
76
76
79
83
88
88
89
94
99
99
99
101
103

CHAPTER 6: REFORMING TEACHER EDUCATION IN PERU: REFLECTION AND
A VIEW TO THE FUTURE………………………………………………………………..
vii

105
107
109
111
114
114
117
118
119
122
122
125
128
130
134
134
136

139

Knowledge and Beliefs in the National Context of the Reform……………………...
Knowledge and Beliefs in the Teacher Education Programs…………………………
Knowledge and Beliefs in the Process of Learning to Teach………………………...
Directions for Future Research……………………...………………………………..
Limitations……………………………………………………………………………

141
143
146
153
155

APPENDICES………………………………………………………………………………
Appendix A. Future Teacher Questionnaire………………………..………………..
Appendix B. Coding for MPCK items…………………...……….…………………..
Appendix C. Future Teachers’ Performance in MPCK Items by Cohort and
Institution………..........................................................................................................
REFERENCES………………………………………………………………………………

156
157
183
189

viii

194

LIST OF TABLES

Table 2.1: Structure of Peruvian Regular Basic Education.…………………………..….

6

Table 2.2. Percentage of Students by Achievement Level in the Census Evaluation
2007-2011…………………………………………………………………..…………….

9

Table 2.3. Processes Considered for Mathematics Teaching in the National School
Curriculum ………………………………………………………………………….……

13

Table 2.4. Structure of the Mathematics Elementary Curriculum ………..……………...

14

Table 2.5. Dimensions, Factors, Criteria, and Standards for the Accreditation of
Schools of Education Set by the CONEAU…………………………….............................

25

Table 4.1. Background of the Participating Institutions………………………………….

52

Table 4.2. Courses of Mathematics and Mathematics Pedagogy in Teacher
Education Programs………………………………………………………………………

54

Table 4.3. Structure of the TEDS-M Future Teacher Questionnaire Used in Peru….……

55

Table 4.4. Participating Institutions and Sampling Coverage……………………….……

57

Table 4.5. List of Program Variables…………………………………………….……….

58

Table 4.6. Items on Course Topics in the Questionnaire………………………..………..

58

Table 4.7. Scale and Items of Opportunity to Learn Included in the Questionnaire….…..

60

Table 4.8. Reliability of Opportunity to Learn Scales……………………………………

61

Table 4.9. List of Future Teacher Variables……………………………………………....

62

Table 4.10. Items of MPCK Evaluated in the Questionnaire……………………………..

66

Table 4.11 Interrater Reliability for MPCK Items…………………………………….…..

67

Table 4.12. Beliefs about the Nature of Mathematics: Scales and Items……………….....

68

Table 4.13. Beliefs about Learning Mathematics: Scales and Items……………………....

69

ix

Table 4.14. Beliefs about Academic Achievement as Fixed Ability: Items………………

70

Table 4.15. Reliability of Beliefs Scales…………………………………………….……

71

Table 4.16. Research Question, Statistical Analysis, and Data……………………….…..

72

Table 5.1. The Curriculum for the Elementary Teacher Education Programs………….....

77

Table 5.2. The Teacher Education Programs’ Philosophy and the Reform………….........

79

Table 5.3. Future Teachers’ Demographics and Socio-Economic Status by Institution.....

84

Table 5.4 Percentage of Endorsement to Reasons for Becoming a Teacher and
Expectation for Teaching as Career………………………………...………………….….

86

Table 5.5.Mathematics and Mathematics Pedagogy Courses Used in the Analysis….…...

88

Table 5.6. School Mathematics Content in the Syllabi of Teacher Education Programs....

91

Table 5.7. Advanced Mathematics Content in the Syllabi of Teacher Education
Programs………………………………………………………………………………….

93

Table 5.8. Mathematics Education Pedagogy Topics in the Syllabi of Teacher
Education Programs…………………………………………………………………….....

95

Table 5.9. Instruction of Mathematics Content in the Syllabi of Teacher Education
Programs…………………………………………………………………………………..

97

Table 5.10. Percentage of 5th Year Students who Studied School Mathematics
Topics….............................................................................................................................

100

Table 5.11. Percentage of 5th Year Students Who Studied Mathematics Education
Pedagogy Topics…………………………………………………………………………… 102
Table 5.12. Means and Standard Deviation of Activities for Learning to Teach
Mathematics Scales……………………………………………………………..….....…... 104
Table 5.13. Mutivariate Analysis of Variance for the Learning to Teach Mathematics
Scales……………………………………………………………………………………... 106
Table 5.14. Univariate Analysis of Variances for the Learning to Teach Mathematics
Scales……………………………………………………………………………………... 106
Table 5.15. Multiple Comparisons for Activities for Learning to Teach Mathematics
Scales……………………………………………………………………………………..
x

108

Table 5.16. Means and Standard Deviations for Reflection Scale………………………..

109

Table 5.17. One-Way Analysis of Variance for Reflection Scale…………………...…… 109
Table 5.18. Multiple Comparisons of Reflection Scales………………………..………... 110
Table 5.19. Intercorrelations between Opportunity to Learn to Teach Mathematics
Scales…………………………………………………………………………………..…. 111
Table 5.20. Future Teachers’ Performance in MPCK items ……………...…………..….. 115
Table 5.21. Means and Standard Deviation of the MPCK index by Cohort and
Institution……………………………………………………………………………..…... 117
Table 5.22. Univariate Analysis of Variance for MCPK Index…………………………..

119

Table 5.23. Multiple Comparisons for MPCK Index………………………………........... 119
Table 5.24. Means and Standard Deviation of Beliefs Scales…………………………….. 123
Table 5.25. Intercorrelation for Beliefs Scales and MPCK Index by Cohort…………….

126

Table 5.26. Mutivariate Analysis of Variance for Beliefs Scales……………….………..

128

Table 5.27. Univariate Analysis of Variances for Beliefs Scales……………….………..

129

Table 5.28. Multiple Comparisons for Beliefs Scales…………………………………..... 131
Table 5.29. Interracorrelation Between Course Topics, Knowledge, and Knowledge
for Exit Student Teachers………………………………………………………………… 135
Table 5.30. Intercorrelation Between Learning to Teach Scales, MPCK Index,
and Beliefs Scales for Exit Student Teachers…………………………………….……..... 136
Table B1. Item “Jose”………………………………………………………………… 183
Table B2. Item “Difficulty”………………………………………………………………. 184
Table B3. Item “Paper clip”………………………………………………………………. 185
Table B4. Item “Jaime”…………………………………………………………………… 187
Table B5. Item “Diagram”………………………………………………………………… 188
Table C1. Future Teachers’ Performance for Item “Jose” ( MFC505) by Cohort
and Institution……………………………………………………………………............... 189
xi

Table C2. Future Teachers’ Performance for Item “Difficulty” (MFC502B) by Cohort
and Institution……………………………………………………………………………... 190
Table C3. Future Teachers’ Performance for Item “Paper clip” (MFC513) by Cohort
and Institution.…………………………………………………………………………….. 191
Table C4. Future Teachers’ Performance for Item “Jeremy” (MFC208A) by Cohort
and Institution.…………………………………………………………………………….. 192
Table C5. Future Teachers’ Performance for Item “Diagram” ( MFC208B) by Cohort
and Institution…………….……………………………………………………………….. 193

xii

LIST OF FIGURES

Figure 4.1. Item “Jose” in the Questionnaire…………………………………………..

63

Figure 4.2. Items “Pencil” (a) and “Difficulty” (b) in the Questionnaire……………… 64
Figure 4.3. Item “Paper clip” in the Questionnaire…………………………………….

65

Figure 4.4. Items “Jaime” (a) and “Diagram” (b) in the Questionnaire ……………….

65

xiii

CHAPTER 1
INTRODUCTION
Students’ low mathematics academic achievement in national evaluations uncovers
problems of the poor quality of educational services offered in Peruvian schools (Benavides &
Mena, 2010; Cueto, 2007). Among many other implications, these results suggest that students
are not being exposed to learning experiences that allow them to enhance the skills needed to
function successfully in society ( Ministerio de Educación, 2008). Poor achievement also
suggests that Peruvian teachers are not endowed with the knowledge needed to assist students’
mathematics learning through schooling. Deficiencies in teachers’ knowledge related to
mathematics and reading have been demonstrated in teacher evaluations carried out by the
Ministry of Education (MED) (Cuenca, 2012). Likewise, research has shown that teachers still
endorse views of teaching that do not favor students’ inquiry and conceptual understanding in
mathematics ( Ministerio de Educación, 2008).
In response, the MED has developed a reform that requires changes in the national school
curriculum and policies related to the preparation of in-service and pre-service teachers in order
to improve students’ learning. The reformed school curriculum is based on a constructivist view
of learning and, concerning mathematics, it establishes three processes that teachers have to
develop in working with their students: problem solving, mathematical communication, and
reasoning and demonstration (Ministerio de Educación, 2009a). These three processes suggest
that teachers need to think about mathematics teaching in a different way, specifically different
from the teaching paradigm that emphasizes learning mathematics by memorizing procedures,
following rules, drilling, and getting the right answer. Instead, the curriculum requires that in
learning mathematics, students explore phenomena, question facts, communicate arguments,
1

develop strategies, and solve problems. In addition, the elementary school mathematics
curriculum requires that teachers demonstrate adequate mathematics knowledge for teaching
number, relations, and operations; geometry and measurement; and statistics so that students can
gain conceptual understandings of these topics.
The reform also demands well qualified teachers who can carry out their tasks
professionally ( Ministerio de Educación, 2007). In this line, policies that include a teacher
career ladder and accreditation of teacher education programs are being implemented to assure
quality teaching in Peruvian classrooms. Given this context, teacher education programs are
called to play their part and respond to educational problems by providing future teachers with
the learning experiences that allow them to develop skills and dispositions needed to implement
the curriculum and achieve the reform goals.
The literature recognizes the existence of two key elements in the process of learning to
teach: knowledge and beliefs. Shulman (1986) identified three key dimensions involved in
teachers’ knowledge: content knowledge, pedagogical content knowledge, and curriculum
knowledge, all of which need to be mastered to assure professional competence in teaching.
Researchers have provided evidence on how teachers’ competence in these knowledge
dimensions are related to the way teachers teach mathematics and correspondingly how such
knowledge affects students’ learning (Hill et al., 2008). In the mathematics field, mathematics
pedagogical content knowledge (MPCK) is especially important for teaching, because it allows
teachers to focus on students’ thinking, which in turn will allow them to build mathematics
conceptual understanding in students (An, 2004).
Regarding beliefs, an extended body of research and theory also confirms their
importance for teaching and for the success of any reform (Philipp, 2007). Changes in instruction
2

will rarely take place unless teachers examine their views of the nature, learning, and
achievement of mathematics, and become aware of how they can hinder the implementation of
practices that align with the reform curriculum (Cohen, 1990; Hill et al., 2008).
To have these two domains fulfilled, teacher education programs must arrange a set of
conditions that allow future teachers to gain the knowledge required and develop the views of
teaching necessary to embrace the view of mathematics as posed in the curriculum (Schmidt,
Blömeke, & Tatto, 2011). Coursework, activities, teaching methods, reflection, and practical
experience, among others resources become opportunities to learn to teach mathematics provided
to future teachers during their stay in the program.
Given this, the current study addressed the following overall research question: How do teacher
education programs respond to the challenges posed by the current educational reform for
mathematics teaching, and to what extent are such responses fulfilling the reform’s demands?
This overall question is addressed through five more specific research questions:
1. What are the characteristics of the teacher preparation programs?
2. What opportunities to learn do future teachers have during their preparation?
3. How do future teachers perform on a measure of MPCK?
4. What are beliefs about mathematics’ nature, learning and mathematics achievement stated by
future teachers?
5. What opportunities to learn are associated with future teachers’ beliefs and MPCK?
To develop this study, I used as a basis the Teacher Development Study of Mathematics
(TEDS-M), a comparative study about the preparation of elementary and secondary mathematics
teachers carried out in seventeen countries (Tatto et al., 2012). I adapted its design and
instruments in order to collect data in Peru in a sample of five teacher education programs in
3

Lima during the months of June-October, 2012. As in TEDS-M, the sample was formed by
future elementary teachers enrolled in the last year of their teacher preparation program.
Additionally, I surveyed future elementary teachers enrolled in the first year of their program so
that comparisons of knowledge and beliefs might be done between the two cohorts, with the
purpose of examining possible changes in these two outcomes. Statistical procedures, including
analysis of variance, were performed to examine the variability reported by institutions and by
cohort for each of the following variables: opportunities to learn, knowledge, and beliefs.
Comparisons by cohort allowed hypothesizing about the effectiveness of the program to develop
the knowledge and beliefs needed for mathematics teaching. Correlation analysis by cohort was
also performed to identify patterns in the way these variables were associated to each other.
Also document analysis, based on program documents (program plans, institutional future
teachers profiles, course syllabi related to mathematics), was done to examine the alignment of
the program with the purposes of the reform, and also to examine the content future teachers
were intended to be taught so that we can have an approximation on how coursework is
associated with future teachers’ knowledge and beliefs.
This study provides useful feedback on the implementation of educational policies related
to the preparation of future teachers in Peru. Research on the topics explored in this study has not
been developed before in the Peruvian context; however, these topics are pertinent due to the
reforms implemented in recent years to improve the quality of teacher preparation programs,
such as institutional accreditation and evaluation. Policy makers and practitioners need to
recognize in its real dimensions the importance of knowledge and beliefs for the learning to
teach process, so they can make sure these two elements are addressed properly in any effort
aimed to improve teacher preparation. Likewise, the results also call for teacher education
4

programs to design an appropriate curriculum to prepare future teachers to successfully
implement the school mathematics curriculum mandated by the reform.
Teaching is a complex job, and learning to teach cannot be done completely during the
five years of a teacher preparation program; nevertheless future teachers must be provided with
the base knowledge and dispositions that will allow them to learn in the field. As argued by Hill
(2010), “Teacher education programs must be focused where they will be most useful, and
knowing which topics and tasks teachers find to be challenging provides one source of guidance”
(p. 514). Likewise, the results can help to identify what learning experiences are associated with
reform beliefs about mathematics and with the required knowledge for reformed mathematics
teaching. In this way, teacher education programs can review the design of coursework and
activities based on an informed decision-making process (Swars, Smith, Smith, & Hart, 2009).
This study is organized into six chapters. In the first Chapter I present an overview of the
study. In Chapter 2 I describe important aspects of Peruvian education and frame the research
problem. In the third chapter I review the literature on the three main variables for this study:
opportunity to learn, knowledge, and beliefs. In Chapter 4 I explain the methodology, procedures
of data collection, and statistical analysis. In Chapter 5 I present the results for the five research
questions, and finally in Chapter 6 I present the discussion of my results and their implications
for policy and practice. Appendices with instruments, coding books, and other materials are
provided at the end.

5

CHAPTER 2
THE EDUCATIONAL REFORM IN PERU AND THE TEACHER EDUCATION
PROGRAMS
In this chapter I introduce some aspects of the Peruvian educational system, the problem
of academic achievement in schools, and the policy mandates related to the preparation of future
teachers that aim to address such problem. This context frames the rationale, problem, and
relevance of the study.
The Peruvian Educational System
The General Law of Education approved in 2003 establishes that the Peruvian
educational system encompasses two stages: basic education and superior education. Basic
education is mandatory and free of cost when it is provided by the government. The law also
establishes that basic education is organized as regular basic education, special basic education,
and alternative basic education. Regular basic education, which is targeted to students who go
through the educational process according to their physical, emotional and cognitive
development, encompasses three levels, initial, primary, and secondary education. Table 2.1
shows the structure of the regular basic education system.

Table 2.1: Structure of Peruvian Regular Basic Education
Level
Initial
education
Elementary
education
Secondary
education

Cycle
I
II
III
IV
V
VI
VII

Grade
Children of 0-2 years-old
Children of 3-5 years-old
1st and 2nd grade
3rd and 4th grade
5th and 6th grade
1st and 2nd year
3rd, 4th, and 5th year

6

As shown in the table, initial education is the first level; it serves students under six years
old. Elementary or primary education is the second level of regular basic education; it lasts six
years. Secondary education is the third level; it lasts five years. Regular basic education is
organized in seven cycles, which means that learning achievement must be developed through
those cycles ( Ministerio de Educación, 2009a). Each cycle is thought of as a basic unit in time,
which takes into account the pedagogical and psychological conditions of students and organizes
education by age and grades so that pupils, in an articulate way, can achieve the learning
expectations set by the curriculum (Ministerio de Educación, 2009a).
Once Peruvian students have completed their secondary education, they can pursue
superior education, which can take place in universities or in non-university institutions (in this
latter category there are institutions which target their curriculum to specific areas, such as
technology, arts, pedagogy, among others). Superior institutions, public and private, set their
own requirements for students to be enrolled. These requirements mostly include an admission
test that prospective students have to take and pass to enter the institution and start their studies.
It is worth mentioning that the Peruvian education system does not include taking a national test
of academic skills at the conclusion of secondary education to certify the learning acquired
through basic education (i.e. the SAT in the US); for this reason each higher education institution
designs and administers its own test to determine which students are eligible to initiate their
superior studies and which are not. Usually, the tests to enter universities are more complex than
the ones to be admitted to non-university institutions. For this reason, across the country there
are private institutions called “academias,” which prepare graduates of secondary schools to take
the exam required by the universities. The preparation includes taking courses on reading

7

comprehension, verbal reasoning, humanities, mathematical reasoning, advanced mathematics
and science, and other areas which are not covered properly in the secondary school curriculum.
The existence of academias implies that basic education, provided by public or private
schools, is not good enough for students to pursue post-secondary education and represents
evidence of the deficiencies of Peruvian education. In the next section I explain more about the
shortcomings in students’ academic achievement.
The Problem of Academic Achievement in Peru
Peruvian students’ low levels of academic achievement have been consistent through
pencil and paper assessments developed with census and representative samples of students
(Benavides & Mena, 2010; Cueto, 2007). These evaluations are not high-stakes, but they have
been used with the purpose of identifying critical areas of students’ performance, especially in
the areas of mathematics and reading comprehension.
The last assessment of primary education using a representative sample of second and
sixth grade students was the National Evaluation of 2004 (Evaluacion Nacional 2004 or EN2004
in Spanish). In mathematics, the assessment examined the students’ skills related to solving
problems, mathematical communication, and the application of algorithms in the content of
number and quantity, algebra and functions, space and form, and statistics and probability. In
reading, they were assessed on information gathering, elaboration of inferences, and reflection
about the text. Results for the national sample of primary students indicated that only 9.6% of
second grade students and 7.9% of sixth grade students showed an optimal level of the
mathematical skills described. Poor results were also obtained in the evaluation of reading skills;
the EN2004 reported that only 15% and 12% of the students in second and sixth grade,
respectively, developed the basic abilities to read in a comprehensive manner; therefore, a big
8

portion of Peruvian students do not obtain the necessary tools to continue their literacy learning
process.

1

In recent years the Ministry of Education (MED) changed their strategy of measuring of
students’ academic achievement and started to implement nationwide yearly assessments of
second grade students. These evaluations, initiated in 2007, also assessed mathematics and
reading comprehension skills. In mathematics, the assessment covered students’ ability to reason
with numbers, relations, and operations to solve problems. In reading, students were assessed on
their ability to elaborate meaning from several kinds of texts. The results allowed placing
students at three levels, based on their achievement at the end of second grade: level 2 (students
who achieved learning expectations set for second grade), level 1 (students who did not achieve
learning expectations set for second grade but were in the process of achieving them), and below
level 1 (students who did not achieve the learning expectations and had difficulties in answering
even the easiest questions of the test). Table 2.2 shows the official results reported by the MED
across five years.

Table 2.2. Percentage of Students by Achievement Level in the Census Evaluation 2007-2011
Subject

Level of
2007
achievement
2
15.9
Reading
1
54.3
Below 1
29.8
2
7.2
Mathematics
1
36.3
Below 1
56.5
Source: Ministry of Education of Peru

2008

2009

2010

2011

16.9
53.1
30.0
9.4
35.9
54.7

23.1
53.6
23.3
13.5
37.3
49.2

28.7
47.6
23.7
13.8
32.9
53.3

29.8
47.1
23.2
13.2
35.8
51.0

1

The National Evaluation 2004 also evaluated students of the third and fifth year of secondary
education. Poor results were also found for that sample.
9

Again, the results reveal that most Peruvian students do not achieve the expected learning
for the grade (level 2), which means that they cannot develop mathematics and reading skills
needed for supporting more complex learning in the next grades. Mathematics is the subject with
the most critical results, since in all evaluations about 50% percent of students or more are at the
lowest level (below 1), which means that they could not answer the easiest questions on the test.
Despite these negative results, it is possible to observe an increase in the percentage of students
who reach the expected learning levels in mathematics and reading over the years. However, a
slowdown effect in the last two assessments is also is noticeable.
The alarming results reported by the academic achievement evaluations through the last
decade call for attention on the deficiencies in teacher training (Benavides & Mena, 2010) since
teachers are the ones who deliver instruction in the classrooms. The Ministry of Education
responded to these demands by posing a set of policy mandates to improve students’ academic
achievement and the quality of teaching in Peruvian classrooms.
The Current Educational Reform in Peru
The National Educational Project (Proyecto Educativo Nacional or PEN in Spanish),
signed in 2007, is a policy document that set the basis for solving the structural problems that
affect Peruvian basic education and that established guidelines until 2021. The PEN posed six
strategic objectives: (1) equitable educational opportunities and results for all; (2) students and
educational institutions that achieve quality learning; (3) well-qualified teachers who carry out
tasks professionally; (4) decentralized and democratic management that achieves results and is
fairly financed; (5) high quality education that is a favorable factor for national development and
competitiveness; and (6) a society that educates its citizens and commits them to their
communities (Ministerio de Educación, 2007).
10

The third objective is the particular relevance to this study. The PEN considers teachers
as a key factor for change and proposes two main actions: the quality improvement and
accreditation of teacher preparation programs, and the implementation of the teacher career
ladder that drives teacher professional development (Ministerio de Educación, 2007). The aim of
this study is the preparation of future elementary teachers to teach mathematics, so I will go
more in depth in the mandates related to the preparation of future teachers.
In the next section, I go over the context of teacher education in Peru, the policy
addressed to improve teacher preparation programs, and the status of its implementation. Due to
the fact that the efficacy of teacher education programs has to do with their graduates’ ability to
assist students in the learning process, first it is important to understand the school curriculum
since this document addresses what future teachers need to do to assure students’ learning. Thus,
I describe the principles that support the current curriculum, and I analyze the challenges that this
curriculum potentially creates for teachers and future teachers before dwelling on the teacher
education policy.
The School Mathematics Curriculum
The tasks that elementary in-service and pre-service teachers have to carry out are
described in the National Curriculum Design (Diseno Curricular Nacional or DCN in Spanish),
which was approved in 2009, as recommended by the PEN. The policy document mandated
“establishing a common, intercultural, inclusive, and cohesive curriculum that allows having
regional curriculums” (Ministerio de Educación, 2007, p.15). As a consequence, the
corresponding organizations revised the curriculum and came up with an articulated curriculum,
as the first attempt to integrate the content and competences through all levels of basic education
(pre-school, elementary, and secondary).
11

Pedagogically, the DCN kept the constructivist approach to learning that it had adopted in
the curriculum reform that took place in the nineties. Six psycho-pedagogical principles support
students’ learning: self-construction of learning, communication and classroom interaction,
meaningful learning, organization of learning, comprehensiveness of learning, and learning
evaluation. Based on these principles, the document encourages adopting student-centered
teaching practices, and it provides the foundation for teaching that promotes critical thinking,
creativity, freedom, and the active participation of students, while discouraging practices that
emphasize rote learning, dictation, and memorization (Ministerio de Educación, 2009a) .
Regarding mathematics teaching, the DCN supports the view of mathematics as problem
solving, because it assumes that the purpose of learning mathematics is to provide students with
tools to face daily life situations. Likewise, the DCN stated that “being mathematically
competent means having the skills for using knowledge with flexibility and to apply that
knowledge in several contexts” (Ministerio de Educación, 2009a, p. 186). To this purpose, the
DCN also demands that students develop a positive attitude toward solving problems.
Consequently, the goals of school mathematics go beyond the traditional goals of teaching
mathematics, which are mastering procedures, rules, and computational skills. Instead, the DCN
proposes three processes for developing mathematical skills: problem solving, mathematical
communication, and reasoning and demonstration. Table 2.3 presents the definition of each
process.

12

Table 2.3. Processes Considered for Mathematics Teaching in the National Curriculum Design.
Process
Problem Solving

Definition
“It requires that students manipulate mathematical objects, activate
their own mental capacity, put in practice their creativity, reflect and
improve their thinking process, and apply and adapt mathematical
strategies in different contexts.” (p. 187)
Mathematical
“It implies to organize and consolidate mathematical thinking to
Communication interpret, represent (diagrams, graphs, and symbols), and formulate
with coherence and clarity the relationship between concepts and
mathematical variables; communicate arguments and knowledge
acquired; and acknowledge connections between mathematical
concepts and apply mathematics to problematic situations.” (p. 187)
Reasoning and
“It includes developing ideas, exploring phenomena, justifying
Demonstration
results, developing and analyzing mathematical conjectures, and
formulating conclusions.” (p. 186)
2
Source: Ministry of Education (2009).

Introducing these processes reflects that students must interact with mathematics in a
more complex way; students should do, think, and talk mathematics. Thus, the DCN wants
students to develop fundamental mathematical ideas by pushing students’ mathematical thinking
in order to test their conjectures, explain their procedures, and communicate their answers. In
sum, the curriculum consistently recommends a student-centered instructional approach for
mathematics teaching. With this framework, mathematics teaching requires more attention to
students’ thinking, through problem solving, reasoning, and communication of their thinking.
Regarding content, the articulated mathematics curriculum for basic education
demonstrates consistency, since it has curriculum organizers which maintain coherence
throughout its three levels. Likewise, taking the idea of a spiral curriculum, the curriculum
2

Although not mentioned in the DCN, such processes and definitions are similar to those
considered by the National Council of Teachers of Mathematics (NCTM) in its publication
“Principles and Standards for School Mathematics” (2000), which provides the guidelines for
mathematics teaching in American schools.
13

organizers become more complex as students move on in their schooling, but at the same time
curriculum organizers allow students to strengthen their prior knowledge. Thus, in initial
education the curriculum organizers are (1) number and relations; and (2) geometry and
measurement; in elementary school they are (1) number, relations, and operations; (2) geometry
and measurement; and (3) statistics. Finally, in secondary education the organizers are (1)
number, relations, and functions; (2) geometry and measurement; and (3) statistics and
probability. Table 2.4 provides detailed descriptions of the curriculum organizers for the
elementary mathematics curriculum.

Table 2.4. Structure of the Mathematics Elementary Curriculum
Curriculum
Organizers
Number, relations,
and operations

Description

“Knowledge of numbers, number systems, and number sense.
This involves skills to decompose natural numbers, use forms
of representation, and understand the meaning of operations,
algorithms and estimations. Also it involves establishing
relations among numbers and operations to solve problems.”
(p. 188)
Geometry and
“It is expected that students examine and analyze the forms,
measurement
characteristics, and relations of two- and three-dimensional
geometric shapes; interpret the spatial relations using
coordinate systems and other representational systems, apply
transformations and symmetry; understand measurable
attributes of objects, units, systems, and the measurement
process; apply techniques, instruments, and formulas to
determine measures.” (p. 188)
Statistics
“Understand notions of statistics to collect, organize data, and
represent and interpret statistical tables and graphs. Statistics
allows students to treat mathematically uncertain situations
and evaluate the likelihood of particular results. Likewise,
students must be able to make relevant decisions before
random phenomena.” (p. 188)
Source: Ministry of Education (2009).

14

The learning expectancies or competences for mathematics emphasize problem solving
around these topics and its use for daily life. Performance expectations such as identify,
formulate, and interpret, mentioned in the document, provide more evidence that the
mathematics curriculum intends that students develop high order skills rather than merely skills
related to the mastering of procedures and rules.
Here it is important to highlight that the current curriculum is the product of the
curriculum reform of 1996. Since then the curriculum has shown its alignment with
constructivist ideals for the teaching-learning process. However, it was not until the new versions
of the DCN, specifically the version released in 2001, that the DCN adopted the approach of
problem solving for mathematics learning. As part of this reform, more skills were added to the
curriculum. The intention as explained before was going further than routines and basic skills,
and seeking to develop higher order skills related to problem solving, reasoning and
demonstration, and communication. Regarding content, statistics and other number related topics
such as proportionality were included in the curriculum structure.
The sophistication of the DCN agrees with the international trend to transform the way in
which mathematics is taught and learned (National Council of Teachers of Mathematics, 2002;
Roesken, 2011; Sowder, 2007; UNESCO, 2012). Likewise, as identified in international
contexts, the implementation of curriculum calls for the need for educational change in which
teachers play an important role (Kennedy, 2005; Smith, 1996). They are the ones who execute
the reforms in the classroom, and therefore students’ learning depends on their competence, at
least from a top-down reform point of view.
The implementation of the reformed curriculum and its challenges for teachers and
future teachers. The implementation of a new mathematics curriculum requires that teachers
15

have thorough knowledge of mathematics content and its pedagogy to teach the reformed
curriculum and that teachers endorse the principles and views that support the curriculum;
otherwise, teachers’ practices will hardly align with the reform expectations (Kennedy, 2005;
3

Philipp, 2007; Roesken, 2011; Sowder, 2007). Research findings in the Peruvian context
provide evidence about limitations in mathematics teachers’ knowledge and beliefs.
The knowledge for teaching of most Peruvian teachers has been deemed as deficient in
several evaluations (Cuenca, 2012). The National Evaluation 2004 included a test for the
teachers of the students who were being assessed. The teachers of these students were asked to
answer questions about mathematics and reading voluntarily. The response rate was high, and
94% of the students’ teachers agreed to participate (Cueto, 2007). The results reported by the
MED (2005) uncovered deficiencies in teachers’ reading and mathematics skills. Regarding
mathematics, the focus of this study, teachers showed difficulty in solving mathematical
problems that require the implementation of alternative strategies rather than calculations or
following procedures. Teachers mostly were able to solve problems of an algorithmic nature, and
to solve equations and operations with fractions of low cognitive demand. About reading, the
results reported that most teachers were able to respond to questions that demand literal
understanding, meaning questions that demand minimal inferential skills, since ideas can easily
be extracted from the text (Ministerio de Educación, 2005). However, the same sample of
teachers showed problems when they were asked to interpret the relationship among two or more
ideas, to assess the coherence of texts, and to make inferences from the texts.

3

The importance of knowledge and beliefs for teacher learning is broadly discussed in the next
chapter.
16

Other evaluations were conducted with the purpose of starting policies of professional
development. In 2006 the MED carried out a census assessment of teachers at several levels who
had tenure and a contract to teach in public schools. The coverage was 66%, which represented
174,491 teachers. The evaluation included a paper-pencil test and covered the areas of reading
comprehension, mathematics, and curriculum knowledge. The content selected for this
assessment considered difficulty levels compared to the ones that could be solved by students at
4

the end of secondary school. The test set four levels of performance: level 0, level 1, level 2,
and level 3; level 0 represented poor skills and level 3 represented higher skills. Results showed,
again, that a large percentage of teachers lacked the reading and mathematics skills needed to
teach. In reading comprehension, 48.5% of teachers taking the test were placed in the two lowest
levels, meaning that they had insufficient skills. In mathematics, this percentage was higher;
85.7% of the teachers were placed in the two lowest levels (Cuenca, 2012). Clearly there is a
need to improve teachers’ knowledge to improve, in turn, students’ academic achievement.
In relation to teachers’ beliefs about the nature of mathematics and its learning, a
qualitative study conducted by Moreano (2011, April) with a small sample of primary teachers in
Lima found that beliefs supported by teachers do not completely reflect the spirit of the reform,
but do show some trace of them. For instance, while teachers supported teaching mathematics
with problems and recognized that mathematical problems can be solved using several strategies,
they also strongly believed that mathematics learning requires constant practice, and they
emphasized mastering the basic operations as the goal to be achieved during the academic year.

4

Here it is important to mention that the teacher union questioned this evaluation because it did
not include performance evaluation. Likewise, the tests’ technical qualities were observed by
practitioners, researchers, and teacher union.
17

For mathematical communication, all teachers supported the fact that students should share
explanations with their peers, but teachers did not consider other purposes of mathematical
communication, such as organizing and consolidating mathematical thinking through
communication and expressing mathematical ideas clearly. Finally, teachers’ views of reasoning
and demonstration showed little alignment with the reform. Participating teachers wanted
students to engage in reasoning during class rather than automatically following procedures for
problem solving. To do this, teachers posed several strategies to avoid students only following
procedures, but teachers were not aware that students gain deep understanding of mathematics
through analysis of conjectures, justification of results, and formulation of conclusions.
A qualitative study conducted by the Ministry of Education (2008), with a sample of
eight elementary teachers in public schools in Lima, also explored the appropriation of the new
pedagogical paradigm claimed by the national school curriculum. The study found a gap between
policy and teachers’ beliefs related to mathematics teaching and learning. Findings suggested
that teachers supported the conception of mathematics as a set of rules and procedures, the use of
key words to identify the procedure for problem solving, and the relevance given to the four
basic operations.
The implementation of the school mathematics curriculum can be hindered by the
deficiencies in teachers’ knowledge and their traditional beliefs about mathematics teaching.
This could be observed in the teachers’ performance reported in the mentioned studies. Teachers’
practices did not include activities that allowed students to do, think, and talk mathematics in
thoughtful ways. Instead, teachers’ practices emphasized learning procedures and definitions,
and they overlooked providing students with more opportunities to ask questions, make
arguments, or explain answers. Likewise, the activities implemented by teachers, in both studies,
18

did create potential situations for promoting conceptual understanding of mathematical content,
but teachers did not take full advantage of such situations to encourage deeper mathematical
thinking. In other situations, teachers took the role of the provider of knowledge by lecturing
about the reasoning and by showing the demonstration involved in a problem situation instead of
letting their students think by themselves. With these deficiencies, the demands of the curriculum
can paradoxically create more difficulties and challenges for teachers’ work, to the detriment of
students’ learning.
The implementation of the curriculum also poses a challenge for teacher education
programs. These institutions must supply future teachers with what is required to implement the
curriculum mandates; they are compelled to facilitate changes required by the reform during the
time future teachers are in the program (Swars et al., 2009). In the Peruvian case, teacher
education programs need to provide future teachers with the knowledge, skills, and attitudes
necessary to implement teaching that values meaningful understanding of mathematical concepts
by means of problem solving so that they can develop more thoughtful and rigorous practices
that meet the standards set by the reformers. In the next section I describe and analyze the
context of teacher education programs and the related policy mandates.
Teacher Education Policy in Peru
For many years, teacher education has been under public scrutiny due to the deficient
quality of teaching observed in Peruvian classrooms. Problems that have characterized the
preparation of teachers include graduates being unable to support students’ learning in their
communities, study plans that do not match the school’s curriculum, inadequate infrastructure to
support future teachers’ learning, etc (Ministerio de Educación, 2006). This situation has driven
the implementation of educational policies with the hope of improving the quality of teacher
19

preparation. In this section, I describe the context in which the policies are framed and then
describe the policy mandates included in the PEN.
The context of teacher education policy. In Peru, there are mainly two routes for
teacher preparation: Universities and Superior Pedagogical Institutes (Instituto Superior
Pedagogico or ISP in Spanish); both can belong either to the private or the public sector.
Universities are autonomous, which means that they are managerially and academically
independent as prescribed by the University Law (Ley Universitaria). Thus, universities that
have a school of education that prepares persons to teach at the pre-school, primary, and
secondary levels can decide on the curriculum of teacher preparation that matches their own
philosophy and characteristics. In universities, the study plan lasts five years, and its termination
leads to obtaining the bachelor’s degree; after this, graduates need to defend a dissertation or take
an exam in order to become licensed to teach. For their part, ISPs are institutions exclusively
devoted to teacher preparation. They are not autonomous; instead, the MED designs a national
curriculum that such institutions must implement; however ISPs can also make some adaptations
according to their needs and vision. The completion of the study plan in ISPs also takes five
years, but their graduates only receive a graduate certificate; in order to earn a pedagogical
certificate, they have to present a thesis.
Until the 1990’s, the state had control over the institutions of teacher preparation, but the
severe economic crisis which the country was going through made it difficult for the government
to satisfy the demands of the educational sector by supplying schools with well prepared
teachers. In order to face this problem, in 1996 the government of the President Alberto Fujimori
established the D.L. 882 Law of Promotion of Investment in Education to allow the participation
of the private sector in the delivery of educational services and, in this way, to broaden the offer
20

5

and coverage of tertiary education for teacher preparation. Under this law, many schools of
education (in brand new universities) and ISPs, public and private, were established in several
regions across the country. However, contrary to expectations, the teacher education programs
grew excessively and prepared more teachers than were required to satisfy the needs of the
market. The problem was exacerbated due to the lack of regulations that assure both the
recruitment of the best teacher candidates and the preparation of well qualified teachers. So,
while this law contributed to increase the teaching force it also brought unintended consequences
that worsened the situation of teacher preparation programs.
The MED, in the frame of the PEN, took actions to control the number of future teachers
who graduate every year and to assure that the most qualified teacher candidates enter ISPs
which are under its jurisdiction. Thus in 2007 the MED, under the administration of the President
Alan Garcia, assumed responsibility for the selection of future teacher candidates who planned to
enroll in ISPs. Since then, the MED has elaborated a national test for these candidates and fixed
a minimum grade average to pass the test. The “fourteen grade,” on a 10 to 20 grade scale, was
set as the cut point to decide whether a prospective future teacher could move to the second
phase of the selection done in each region of the country. The impact of this law was observed
since the very beginning. In the first evaluation, 17,950 prospective teachers enrolled nationwide
to take the test, and only 424 passed it (Ugarte, 2011). In the following years the number of
applicants who studied in ISPs has decreased, as well as the number of students who passed the
evaluations. Consequently the number of ISPs decreased in several regions of the country; many

5

The law affected all levels of education basic education, technical training, and tertiary
education.
21

of them closed due to the lack of teacher candidates, or they have changed to provide technical
training (Benavides & Mena, 2010).
The policy. The need of improving students’ academic achievement and the importance
of increasing the value of the teaching profession led the MED to establish policies to improve
teacher education programs. Thus, the PEN posed as one of its objectives “to improve and
restructure the systems of initial and continuing training of teachers” (Ministerio de Educación,
2007, p. 84). To accomplish this goal, the PEN set accreditation as a nationwide policy to assure
the quality of the services offered by institutions of teacher preparation (university and nonuniversity).
The MED decided to adopt the model of accreditation to assure the quality of teacher
preparation because, as suggested by specialists consulted by the MED, the accreditation would
“(1) Assure the preparation of competent professionals of education so that all students (children,
youths, and adults) achieve essential learning at several levels and modalities; (2) Assure the
preparation of professionals of education able to contribute to the development of schools … (3)
Build a streamlined, pertinent, efficient, effective national system of teacher preparation and
professional development” (Ministerio de Educación, 2003, p.133). Likewise, the
implementation of the accreditation policy in Peru also matches the accountability movement
taking place in recent years in many countries which introduced private sector management
principles into public institutions (Kjaer, 2004). In the education sector, international
organizations such as the World Bank, which has provided funding to support Peruvian
educational reforms, have strongly recommended setting up a culture of institutional evaluation
and developing accountability systems. Specifically, the Bank recommended, “improvement in
quality control systems and quality management systems geared at generating confidence in the
22

taxpaying public and in the economic authorities that the spending can be well managed” (The
World Bank, 2007, p. xiv).
According to the PEN, schools of education as well as pedagogical superior institutes
(ISP) must be evaluated periodically, and must be accountable for the evaluation results
(Ministerio de Educación, 2007). To implement this policy, the PEN mandates activating the
National System of Evaluation, Accreditation and Certification of Educational Quality (Sistema
Nacional de Evaluacion, Acreditacion y Certificacion de la Calidad Universitaria or SINEACE
in Spanish), whose creation was authorized in the General Law of Education N° 28044, signed in
2003 during the administration of President Alejandro Toledo. The law also authorized the
functioning of organizations in charge of defining the criteria and standards to assure educational
quality in institutions that provide tertiary education (university and non-university) for teacher
preparation. Thus two organizations have been created to carry out this task: the Council of
Evaluation, Accreditation, and Certification of the Quality of University Higher Education
(Consejo Nacional de Evaluation, Acreditacion y Certificacion de la Calidad de la Educacion
Superior Universitaria or CONEAU in Spanish) to evaluate universities and their programs,
including schools of education, and the Council of Evaluation, Accreditation, and Certification
of the Quality of Non University Higher Education (Consejo de Evaluación, Acreditación y
Certificación de la Calidad de la Educación Superior No Universitaria or CONEACES in
Spanish) to evaluate non-university programs such as ISPs.
The model of quality of accreditation for schools of education was released in March 13,
2009, by the CONEAU. It has three dimensions: program management, professional preparation,
and support for professional preparation/professional training (Ministerio de Educación, 2009b) .
The first one “aims to assess the efficacy of institutional and administrative management,
23

including mechanisms to examine the degree of the coherence and accomplishment of its
mission and objectives” (p. 3). The second “aims to assess the preparation of students in the
process of teaching and learning, research, university extension and outreach, as well as the
results that can be observed in their insertion in the world of work and in their performance” (p.
3). The last dimension refers to “the support for professional training; assure the managerial
capacity and participation of human and material resources as part of the teaching-learning
process” (p. 3). It is expected that both program management and the support dimension provide
support for the professional training of future teachers. The model of quality considers that each
dimension of the accreditation is, in turn, divided into factors, criteria, and standards to facilitate
the assessment. Table 2.5 details each of these elements.

24

Table 2.5. Dimensions, Factors, Criteria, and Standards for the Accreditation of Schools of
Education Set by the CONEAU.
Dimension

Factor

Program
Planning, organization,
management direction, and control

Professional
training/
professional
preparation

Teaching - learning

Research
University extension
and outreach
Teachers
Support for
professional
training/
professional
preparation

Facilities and
equipment
Welfare
Financial resources

Stakeholders
Source: Ministry of Education (2009)

Criteria
Strategic planning
Organization, direction, and control
Educational project - curriculum
Strategies of teaching - learning
Development of teaching-learning
activities
Learning evaluation and actions of
improvement
Current students and alumni
Production and evaluation of research
projects
Production and evaluation of university
extension and outreach projects
Teaching and tutoring work
Research work
University extension and outreach
projects work
Facilities and equipment for teachinglearning, research, University extension
and outreach, administration and
welfare
Implementation of welfare programs
Financing of the program
implementation
Relationship with stakeholders

N° of
Standards
5
9
13
2
4
2
11
9
10
10
5
3

2
6
3
3

The CONEACES has its own regulations for the accreditation of ISPs. The technical
document also considers three dimensions of the accreditation: institutional management,
academic processes, and support services for professional preparation. All these dimensions, as
well as their factors and standards, are similar to the quality model posed by the CONEAU in
meaning and purpose.
25

The assumption of this policy, or theory of change as it was called by Weiss (1997), is
that future teachers who are prepared in institutions that have systems of quality assurance will
be able to function professionally in their schools and satisfy the demands of Peruvian basic
education. Moreover, it is expected that the process of evaluation and accreditation will allow
reducing the number of teacher education institutions, which as explained before have
proliferated during the last decades.
Criticisms of the quality model are many. Practitioners mainly argue that the elaboration
of the model of quality did not involve the MED departments in charge of supervising the
functioning of ISPs (Cuenca, 2012) and that, instead, they were elaborated by professionals out
of the education sector. The absence of crucial factors related to teacher learning and the
emphasis on organizational factors have also been addressed as weaknesses. This might have to
do with the fact that, as stated in the law, the dimensions, factors, and indicators of the model
have been taken from the accreditation system of other professional programs. Still other voices
call for the integration of methodologies and standards for schools of education and ISPs, since
they have the same purpose (Benavides & Mena, 2010).
Currently, accreditation has a mandatory character, and institutions were given a time
period to be accredited after the publication of the model. However, as reported by the National
Council of Education (Consejo Nacional de Educación, or CNE in Spanish), the process of
accreditation nationwide is moving at a slow pace (Benavides & Mena, 2010; Consejo Nacional
de Educación, 2011). It seems that despite the fact that the norm and the accreditation model
have been delivered, other important aspects related to its implementation are not working
properly. For instance, after the CONEACES approved the model of accreditation, other
regulations were approved by the MED, which requested the ISPs take previous actions such as
26

internal evaluations before initiating the accreditation process (Cuenca, 2012). Other issues, such
as the uncertainty of ISPs due to the reduction of the number of students and the lack of
accrediting organizations, have contributed to the standstill of the process.
The CNE is currently working on more specific documents that establish the criteria for
good teaching and the professional profiles of graduates. Parallel, corresponding organizations
are working intensively in the development of learning standards, as mandated by the PEN
(Benavides & Mena, 2010; Consejo Nacional de Educación, 2011). It is expected that pitfalls in
the model of accreditation can be improved with the establishment of these learning and
professional standards. Thus, joint and coordinated work among several sectors involved in the
problem, including teacher education programs, could take place to improve the initial formation
of teachers in Peru.

27

CHAPTER 3
TEACHER LEARNING IN MATHEMATICS AND TEACHER EDUCATION PROGRAMS:
A LITERATURE REVIEW
The reformed mathematics curriculum requires that future teachers learn new and
sophisticated ways of teaching (Wilson & Berne, 1999). Research on teacher learning provides a
pertinent framework-based cognitive science which can help to understand how teachers learn to
teach and how teacher education programs can support future teachers’ learning, so graduates
can carry out mathematics education in their classrooms in a way that meets the reform’s
expectations. In this chapter I briefly introduce the nature of teacher learning which allows me to
argue about the importance of addressing future teachers’ beliefs and knowledge during teacher
preparation and providing pertinent learning opportunities to facilitate their acquisition. Then, I
revise the literature regarding each of these elements: knowledge, beliefs, and opportunities to
learn involved in mathematics education. This review is organized considering knowledge and
beliefs as outcomes of the process of learning to teach and the opportunities to learn as part of
such process that takes place in the teacher preparation programs.
The Nature of Teacher Learning
Teacher learning is mostly recognized in the literature as a process by which teachers
acquire knowledge, skills, and dispositions which allow them to enact instructional practices in a
professional way (Kennedy, 1991; Schmidt, Blömeke et al., 2011). In this process future teachers
are not mere recipients of techniques and strategies of teaching; rather, they are active learners
and participate with their prior knowledge about teaching. Individuals entering schools of
education come equipped with scripts, tacit knowledge, and school experiences that constitute
their initial ideas or beliefs about teaching (Polanyi, 1983; Stigler & Hiebert, 2009). This set of
28

ideas becomes the basis of their learning and a reference to consider any new way of teaching
they are exposed to. As Kennedy (1991) argued, applying cognitive theories of learning to
teachers, “teachers, like other learners, interpret new content through their existing understanding
and modify and reinterpret new ideas on the basis of what they already know or believe” (p. 2).
Thus, it is important that teacher education programs recognize this characteristic of teacher
leaning if the goal is to instill new ways of teaching among future teachers especially if such
prior knowledge is different from what the reform expects.
Teacher learning also takes place in context. Roesken (2011) asserted, “teacher learning
is rather unlearning than new learning, never occurring isolated but in an educational setting or
context” (p. 45). Situated cognition theory has highlighted two elements that count in this
educational setting: the need of learners to have learning experiences that allow them to perform
as practitioners in their field, and the importance of learners having opportunities to interact with
peers and other meaningful persons who can maximize their ways of learning and thinking
(Putnam & Borko, 2000). All these opportunities to learn can help future teachers to revise their
views of teaching and, in terms of Roesken, unlearning if these are not aligned with the reform
recommendations.
The teacher learning process also needs to consider the ultimate goal of teaching which is
connecting students with the academic content mandated by the school curriculum (Kennedy,
1991). Assuring students’ learning in those content requires that teachers are familiar with the
particularities of the subject areas included in the curriculum and demonstrate a deep
understanding of them since “different kinds of subject matter require different kinds of
pedagogies” (Kennedy, 1991, p. 2). In the case of mathematics education, teacher preparation

29

programs need to support the teacher’s task of teaching mathematics by providing them with the
necessary knowledge to teach it (Sowder, 2007).
It emerges that teacher education programs need to pay attention to future teachers’
beliefs and knowledge because both are the basis for effective mathematics teaching. It also
emerges that teacher education programs must provide learning opportunities to future teachers,
so they can acquire the knowledge needed for teaching mathematics and endorse the beliefs that
align with the reforms. In the following sections I analyze each of these factors and their
connection with learning mathematics teaching.
Future Teachers’ Knowledge
The current reform requires that future teachers demonstrate more sophisticated
knowledge that allows them to teach mathematics by emphasizing problem solving,
mathematical communication, and reasoning and demonstration and in that way assure that
students can build their learning on solid mathematical ideas. Developing and demonstrating this
knowledge becomes challenging and demanding for future teachers if we consider the demands
on teacher knowledge that dominated in prior mathematics curriculum: knowledge on contents
that teachers could manage and knowledge on pedagogical strategies that had practicing
procedure as the basis for students’ learning (Smith, 1996). The new context for mathematics
teaching is tremendously different; for this reason, improving future teachers’ knowledge is an
urgent task for teacher preparation programs if they are expected to contribute to improving
Peruvian students’ mathematics achievement.
Given this, it is important to define what kind of knowledge future teachers need in order
to be effective in their teaching. The literature is not conclusive about this and discussions to
define teacher knowledge are still going on (Roesken, 2011; Tatto, Schwille et al., 2008); in this
30

context, theorists and practitioners have identified several classifications of knowledge involved
in teaching (Borko & Putnam, 1996). The most salient is that proposed by Shulman (1986), who
identified three domains of knowledge that teachers need to master. Content knowledge is
defined as “the amount and organization of knowledge per se in the mind of the teacher” (p. 9).
Pedagogical content knowledge refers to ways of representing content knowledge or making it
accessible to novices; it means “the ways of representing and formulating the subject that make it
comprehensible to others” (p. 9). Finally, curricular knowledge refers to the teachers’ familiarity
with content across the curriculum and with the materials that are needed for instruction (texts,
software, etc). These dimensions are focused on cognitive knowledge, or in the words of
Grossman (1995), theoretical and abstract knowledge. Other proposals recognize the emotional
dimension of teacher knowledge which includes self-efficacy, identity, and emotions and that
also have undeniable effects on teaching (Grossman, 1995; Roesken, 2011). Due to the purposes
of this study, however, I will focus on the cognitive domain of teacher knowledge.
The classification proposed by Shulman became crucial for the study of teachers’
knowledge because it expanded the initial demands for teachers’ knowledge that traditionally
had been focused on mastering subject knowledge to include other domains of knowledge that
are key for teachers performance. The concept of pedagogical content knowledge revolutionized
the idea of pedagogy, which was traditionally related to classroom management, lesson planning,
and assessment, by making it content-specific. Thus, pedagogical content knowledge will allow
teachers, for instance, to plan a lesson for a specific subject matter in a way that content is
accessible to students. This classification did not diminish the importance of content knowledge,
which as Shulman (1986) argued is strongly connected to the other domains of knowledge. In the
same line, Borko and Putnam (1996), in their literature review on learning to teach, argued that
31

without content knowledge, teachers will hardly “learn powerful strategies and techniques for
representing the subject to students and for attending and responding to students’ thinking about
the subject in ways that help support their meaningful learning” (p. 700).
The research on mathematics education has taken Shulman’s ideas to define the
knowledge needed to mathematics teaching especially in two domains: mathematics content
knowledge (MCK) and mathematics pedagogical content knowledge (MPCK). MCK implies the
knowledge of mathematics content that allows individuals to solve mathematical problems. Ball,
Thames, and Phelps (2008) established differences between the common content knowledge that
any adult might have to solve mathematical problems and the specialized content knowledge that
a teacher should have to solve mathematical tasks. This specialized knowledge involve
conceptual understanding of mathematics content that enables teachers to do what is requested of
students such as posing hypotheses, reasoning mathematically, understanding and represent
mathematics contents among others (Baumert et al., 2010). The depth and breadth of teachers’
mathematics content knowledge also has been under discussion. Some argue that this knowledge
should limited by the contents included in the curriculum, others voices such as the National
Mathematics Advisory Panel (2008) considers that “teachers must know in detail and from more
advanced perspective the mathematical content they are responsible for teaching” (p. 37). While
this discussion remains open, what is not debatable is that teachers must know the mathematical
content with conceptual understanding to be able to teach them.
Here the concept of profound understanding of fundamental mathematics by Ma (1999)
becomes pertinent to teachers’ knowledge. She argued that, beyond being able to solve
mathematical problems and provide the rationale behind them, “a teacher with profound
understanding of fundamental mathematics is not only aware of the conceptual structure and
32

basic attitudes of mathematics, but is able to teach them” (p. xxiv). This concept drives
connections between content and pedagogy, which is the basis of MPCK.
Mathematics pedagogical content knowledge has to do with the knowledge of how
students learn mathematics and how to teach specific mathematical contents. Researchers have
identified an array of aspects that are involved in MPCK. An (2004) recognized three aspects:
planning for instruction, mastering instructional delivery, and knowing students’ thinking. For
the last aspect she highlights four specific features: building on students’ mathematics ideas,
addressing students’ misconceptions, engaging students in mathematics learning, and promoting
students’ thinking about mathematics. Alternatively, Krauss, Baumert, and Blum (2008)
identified three aspects that define MPCK: knowledge of mathematical tasks to be able to prompt
students learning, knowledge of students’ thinking to identify their misconceptions and
difficulties related to particular content, and knowledge of mathematical representation to
support students’ understanding on mathematics concepts. Both classifications build on the
importance of having teachers, and future teachers, understanding how students learn and think
about mathematics.
The Cognitively Guided Instruction (CGI) study conducted by Carpenter, Fennema,
Peterson, Chiang, and Loef (1989) is a representative work about the effects that focusing on
students’ thinking has on teachers’ knowledge. In an experimental study, these researchers
explored how teachers develop understanding of addition and subtraction word problems. They
wanted to see whether teachers in the experimental group, who received the CGI workshop,
adopted practices that were taught in the program, and if these teachers increased their
knowledge of students’ thinking. The findings showed that by attending to students thinking,
teachers in the experimental group were able to learn subject matter knowledge and pedagogical
33

content knowledge related to problems of addition and subtraction, compared to the control
group; they learned how students reason about mathematics and along the way, learned more
about the kinds of problems their kids were expected to do. Thus, the acquisition of this
knowledge helped to change teachers’ practices, and consequently had a positive impact on
students’ academic achievement; thus developing MPCK focusing on students’ thinking turns
critical for the accomplishment of the current reform recommendations.
It is worth noticing that MCK and MPCK together has been recognized in the literature
as mathematical knowledge for teaching (MKT), a concept coined by Ball et al. (2001) and that
refers the mathematical content knowledge that is uniquely relevant to teaching and that has
prompted the study of mathematics teaching and teacher learning.
Research has provided abundant evidence on how the domains of knowledge identified
by Shulman (1986) affect teachers’ practices. For instance, Borko, et al. (1992) presented the
case of Ms. Daniels, a college senior who had shown great performance in mathematics courses
and was attending professional coursework to learn how to use meaningful situations for
teaching. During a lesson about the division of fractions, Ms. Daniels tried to implement one of
those activities, but found herself making a mistake in the operation, and she did not know how
to restart the solution of the problem. As a consequence, she abandoned her initial intention and
recommended students follow the rule for division of fractions. This case shows that although
teachers can master the mathematical content, limitations in pedagogical content knowledge can
undermine meaningful mathematics learning by leading teachers to overemphasize following
rules and procedures, which is precisely the type of teaching the reform was intended to avoid to
avoid (Prawat, Remillard, Putnam, & Heaton, 1992).

34

In the same line, Baumert et al. (2010) examined the extent in which mathematics content
knowledge and mathematics pedagogical content knowledge contributed to explain differences
in the quality of teaching practices and students’ achievement. This was a one-year study with a
representative sample of 10th grade students in Germany and their mathematics teachers. The
design included the administration of MCK and MPCK tests for teachers, and mathematics
achievement test for their students. Documents such as test, examinations, tasks and homework
were also collected to assess teachers’ instruction. Multilevel structural equation models were
used to analyze the relationship between both domains of teacher knowledge and instructional
practices and students achievement. Findings showed that although MCK and MPCK were
highly correlated, MPCK had a greater impact on teacher practices and students’ academic
achievement. Strong MCK had direct effect on teachers’ ability to cover the school curriculum
but did not have direct effect on practices that aim to support students’ conceptual understanding
of mathematics. These findings allowed Baumert et al. (2010) to prove their hypothesis that
“PCK is inconceivable without a substantial level of CK but that CK alone is not a sufficient
basis for teachers to deliver cognitively activating instructions that, at the same time, provides
individual support for student’s learning” (p. 163).
Overall, research and theory show that future teachers require learning more than just a
set of techniques. The knowledge that they need to be effective in teaching is really complex.
Shulman’s (1986) categories have been eye opening in identifying knowledge domains, and
research has provided evidence on the importance of addressing both MCK and MPCK during
teacher preparation programs if the goal of improving students’ learning is to be accomplished.
In this way, the reform goals of helping students become mathematical problem solvers, learn to
communicate mathematically, and learn to reason mathematically might be reached.
35

Future Teachers’ Beliefs
The literature on teachers’ beliefs and their relationship with teaching is enormous.
Several authors have reviewed different definitions of teachers’ beliefs (Pajares, 1992; Borko &
Putnam, 1996; Philipp, 2007). Speer (2005) encompassed other authors’ definitions and claimed
that beliefs are “conceptions, personal ideologies, worldviews, and values that shape practice and
orient knowledge” (p. 365). This definition reveals two characteristics of beliefs that have made
them crucial for teaching: (1) beliefs determine how teachers interpret, decide, and act in
classrooms, and (2) beliefs determine how teachers use knowledge and information for teaching.
Both characteristics account for differences between mathematics teachers’ performance
(Roesken, 2011), and consequently in students’ learning (Fennema et al., 1996); thus, teacher
education programs must address them so that future teachers can have more alternative views of
teaching and adopt the ones that are compatible with the reform ideals that drive instructional
practices and that facilitate students’ conceptual understanding of mathematics.
The work of transforming future teachers’ beliefs about mathematics teaching should
start with understanding how future teachers have adopted the beliefs they have. As mentioned
earlier, future teachers enter schools of education with prior knowledge resulting from their own
experience. Lortie (1975) stated that teachers are socialized about their role in the classroom by
the “apprenticeship of observation.” He calculated that students spend about 13,000 hours in the
classroom during their schooling, and that experience provides them with ideas about teachers,
teaching and learning, and students. Regarding mathematics, Philipp (2007) claimed that “while
students are learning mathematics, they are also learning lessons about what mathematics is,
what value it has, how it is learned, who should learn it, and what engagement in mathematical
reasoning entails” (p. 275). All these ideas become a framework which can influence the process
36

of learning to teach that will take place during the program. It has been recognized by several
theorists that beliefs limit prospective teachers’ ability to learn and incorporate the messages
conveyed by teacher education programs, as well as the way they learn them (Cady, Meier, &
Lubinski, 2006; Richardson, 2003). So, one of the initial goals for teacher education should be to
identify the beliefs held by the prospective teachers and work during the program to change them
if necessary.
The literature offers several classifications that describe possible views of mathematics
that future teachers can support. Thompson (1992) compiled the classifications formulated by
Ernest (1988), Lerman (1983), and Skemp (1987). According to her review, Ernest (1988)
classified views of mathematics as a dynamic problem-driven discipline, a static but unified body
of knowledge, and a bag of accumulated rules, facts, and skills. Thompson also found that
Lerman (1983) supported that mathematics can be absolutist when it is considered as certain and
absolute, or mathematics can be fallibilist if uncertainty is accepted as a feature of the discipline.
Finally, according to Thompson’s (1992) review, Skemp (1987) proposed a categorization based
on the type of knowledge that mathematics emphasizes: mathematics can be instrumental if
mathematical knowledge emphasizes rules and facts without reasoning, or mathematics can be
relational if mathematical knowledge emphasizes understanding the conceptual structures
underlying mathematics. Although these typologies seem to classify beliefs in extreme poles, it
is very likely that teachers are not so radical in holding their beliefs, and that instead they show
more nuances in accepting some beliefs of one group and some of the other group. This is
because beliefs are organized in systems that allow that these discrepancies coexist (Thompson,
1992).

37

Several studies have reported the status of future teachers’ beliefs in teacher preparation
programs, in which it is possible to observe how these beliefs are organized. The study of
Mathematics Teaching in the 21st Century (MT21), by Schmidt, Blomeke, and Tatto (2011), was
implemented in six countries of different continents, and it measured beliefs on the nature of
mathematics considering four scales: creative (emphasis on problem solving), useful (emphasis
on the applicability of mathematics in daily life), formal (emphasis on the deductive character of
mathematics), and algorithmic (emphasis on the rules, procedures, and facts that characterize
mathematics). The study found that pre-service teachers mostly supported beliefs recognized by
the researchers as a dynamic view of mathematics (meaning creative and useful); on the other
hand, beliefs that recognized mathematics as the following of procedures were less supported.
Also, teachers who supported an algorithmic view of mathematics correlated negatively with
those who understood mathematics as a creative way of thinking. The Teacher Education and
Development Study in Mathematics (TEDS-M) (Tatto, Schwille et al., 2008; Tatto et al., 2012)
also explored the beliefs of pre-service teachers about the nature of mathematics (which was
conceived in two forms: mathematics as a set of rules and procedures, and mathematics as a
process of enquiry). The results related to the nature of mathematics showed that future teachers
mostly supported beliefs that conceive mathematics as a process of enquiry, compared to beliefs
that support mathematics as rules and procedures. However, the results also showed cases where
a large percentage of future teachers supported both views with the same intensity (i.e. in
Botswana, more that 70% of future teachers supported the idea of mathematics as procedures and
rules and the alternate idea at the same time). TEDS-M also explored beliefs about learning
(conceived in two forms: learning through following teacher direction and learning through
active involvement). The results showed that future teachers of all these countries were more
38

supportive of beliefs that involve inquiry than of those which connect mathematics learning with
following teachers’ directions. In this case differences in the percentage of endorsement between
both conceptions were more marked. Overall, results from both studies show that future teachers
recognize the importance of procedures and algorithms for mathematics learning, but at the same
time they recognize that mathematics can be a product of creativity and inquiry. Likewise, the
active role of students while learning is recognized where some rejection to the traditional view
of teachers is observed.
In the Peruvian context, the DCN established a view of mathematics where the meaning
making perspective is emphasized; the DCN clearly dismisses the emphasis on procedures and
facts in mathematics learning; in their description of the three processes mentioned in the DCN it
aligns with the conceptions of mathematics as problem solving, fallible, and relational
(Thompson, 1992). Having this in mind, teacher education programs should address the efforts
needed for future teachers to examine their beliefs and challenge their ideas about teaching, so
they can espouse more progressive views of mathematics to become compatible with reform
expectations (Ambrose, 2004; Richardson, 2003).
Research about changes of beliefs in prospective teachers has shown that change in their
beliefs about mathematics and their teaching is hard to produce but is possible (Ambrose, 2004;
Borko & Putnam, 1996; Swars et al., 2009). Results from these studies agree with some
conclusions about what is needed to facilitate change, which is summarized by Feiman-Nemser
and Remillard (1995) in the following suggestions: “First, teachers need an opportunity to
consider why new practices and their associated values and beliefs are better than more
conventional approaches. Second, they must see examples of these practices, preferably under
realistic conditions. Third, it helps if teachers can experience such practices firsthand as learners”
39

(p. 24). The literature also has supported that change is possible when prospective teachers are
involved in constructivist learning environments, are required to focus their attention on
students’ mathematical processes rather than on results (Fennema et al., 1996), and are engaged
in activities that include meaningful mathematical problems (Conner, Edenfield, Gleason, &
Ersoz, 2011). The exposure to these different activities, tasks, and interactions should result in
differences in beliefs as well as in ways of using knowledge to enact teaching (Felbrich, Müller,
& Blömeke, 2008).
Likewise, if a conceptual understanding rather than an algorithmic view of mathematics
is the goal for school mathematics, then future teachers themselves should develop this
conceptual understanding of mathematics so that they can have more opportunities to develop
alternate views of mathematics teaching (Ambrose, 2004). Finally, it is important to consider the
role of teacher educators in instilling the desirable beliefs. TEDS-M also assessed teacher
educators’ beliefs and found that the patterns of responses of teacher educators were similar to
the responses observed in future teachers. In the same way, a quasi-longitudinal study carried out
by Felbrich, et al. (2008) in Germany with two cohorts of future teachers (beginning the program
and ending the program) and their respective teacher educators found that beliefs of future
teachers at the end of the program were more compatible with those of the teacher educators than
were the beliefs supported by future teachers in the first year of the program.
Research also has identified that although these elements can make possible changes in
future teachers’ beliefs (from a traditional to a progressive view of mathematics), regression to
traditional beliefs can be observed during the first years in one’s teaching experience (Cady et
al., 2006; Swars et al. 2009). This regression might be due to the pressures to comply with the
curriculum and school demands. Thus beginning teachers might set aside their progressive
40

beliefs in order to adjust in the routines established by schools and parents, and to come back to
support more conservative view of mathematics (Felbrich et al., 2008; Potari & GeorgiadouKabouridis, 2009). These findings highlight the importance of teacher education programs
providing what is needed for future teachers to nurture the beliefs needed for implementing the
reformed curriculum and to make them stronger so that they can withstand the harshness that
characterize the first years of teaching.
Opportunities to Learn to Teach Mathematics in Teacher Education Programs
Feinam-Nemser and Remillard (1995) claimed that learning to teach happens through
cognitive processes (beliefs, attitudes, reflection, knowledge, dispositions) and opportunities to
learn. So, “Teacher preparation programs [should] intentionally provide specific structured
learning experiences or opportunities to equip those who desire to become teachers with the
knowledge and skills needed for classroom instruction and more generally to assume the
professional role of teacher” (Schmidt, Blomeke, & Tatto, 2011, p. 76). Such learning
experiences or opportunities to acquire mathematics knowledge for teaching and beliefs as
demanded by the current reforms should be embedded in “both the contexts of learning (e.g.,
programs, settings, interventions) and the social interactions within these contexts that promote
learning” (Feiman-Nemser & Remillard, 1995, p. 21). The resistant feature of beliefs requires
that teacher education programs include in their curriculum learning experiences and interactions
that favor the examination of pre-service teachers’ current beliefs. In the same way, acquiring a
sophisticated knowledge to teach mathematics needs proper tasks and activities that drive teacher
learning. In the next paragraphs, I synthesize the main opportunities for learning in teacher
education programs which can affect future teachers’ beliefs and knowledge: curriculum,
classroom interaction, and reflection.
41

Curriculum
The curriculum of teacher education programs provide the frame in which opportunities
to learn are arranged to support teacher learning. It mainly involves course work and student
teaching experiences.
Course work included in teacher education curriculum should help future teachers with
the demands that they will face in the classrooms in which they will teach once they are done
with their preparation. To this purpose course work should cover all dimensions of knowledge
required for teaching, which as explained earlier, are subject matter content knowledge,
pedagogical content knowledge, and curriculum knowledge (Shulman, 1986).
Curriculum mostly addresses two types of courses that allow future teachers to gain
knowledge in the cited dimensions: content courses and methods courses. Regarding content
courses, these mainly aim to provide future teachers with the mathematics content that they will
need to teach in elementary or secondary education. The content courses included in the program
plan of teacher education can have school mathematics and advanced level. As could be
expected, research has provided evidence of existing positive correlation between the number of
content courses taken and the mathematical content knowledge (Schmidt, Cogan, & Houang,
2011). However, the content of the mathematics courses could be more relevant than the number
of courses for developing the knowledge required for teaching (Youngs & Qian, 2013). Thus,
being exposed to advanced mathematics courses could represent opportunities to gain deep and
breadth knowledge of mathematics content. This was evidenced in the MT21 study (Schmidt,
Blömeke et al., 2011) in which studying courses on calculus and advanced mathematics had a
positive effect on the future teachers’ knowledge in the domains of number, algebra, geometry,
function, and data.
42

Content courses, however, seems to have limited effect on teachers knowledge for
teaching. Ball et al. (2001) argued that there might be a point at which teachers cannot use, for
teaching purposes, the knowledge gained by taking specialized courses in the area. Thus, a
threshold effect might exist about taking specialized courses in mathematics as suggested by
Wilson, Floden, and Ferrini-Mundy (2001). This limitation of content courses makes sense
considering the previous discussion on MCK and MPCK and call for the necessity to provide
opportunity to learn other dimensions of teachers’ knowledge during teacher preparation.
It is here where method courses become important. These kind of courses aim mainly to
endow future teachers with knowledge on how to teach specific topics, how students learn those
topics, how to plan and assess lessons, and how to use instructional materials all in relation to the
curriculum they are going to teach. For mathematics teaching, all this knowledge and skills
involved in method courses aim to “develop pre-service teachers’ understandings of
mathematics, mathematics pedagogy, and children‘s mathematical development, and to cultivate
a positive disposition toward teaching mathematics” (Wilkins & Brand, 2004, p.226). A
characteristic of method courses is that they usually imply practical work with students in real
classrooms (Conner et al., 2011; Wilkins & Brand, 2004). This practical work can prompt
mathematics pedagogical content knowledge among future teachers as reported in the MT21
study; stronger levels of MPCK were associated to opportunities for future teachers to be
involved in instructional interaction and practical experiences with students (Schmidt, Blömeke
et al., 2011). In the same line a study conducted by Youngs and Qian (2013) found that the more
content on method courses (such as: mathematics curriculum in school, methods of teaching
mathematics, methods for solving school mathematics problems, assessment, and cognitive
aspects of mathematics) the better future teachers performed in mathematics knowledge for
43

teaching scale. In these methods courses, opportunities for decomposition and approximation of
practice had positive effect on the mathematics knowledge for teaching demonstrated by future
teachers (Youngs & Qian, 2013).
Here it is important to address some concerns about the duality in curriculum that could
be created for content and method courses. Philipp et al. (2007) argued that “separation of
learning mathematics from learning about mathematics teaching oversimplifies the learning of
both critical components,” (p. 440). To reduce this risk, both content and method courses should
be treated in the curriculum in an articulated way. The goal of the reform, expressed in the
mathematics curriculum can be the articulating element for course work. Thus, if the current
curriculum reform is intended to reduce the predominance of the mathematics-teaching-bytelling paradigm, designers of teacher education programs should avoid implementing curricula
that make future teachers learn mathematics only through following procedures and learning
recipes that can work with their own students. Likewise, if future teachers are requested to work
with problem solving in mathematics teaching, then through their course work they should be
treated as problem solvers with a disposition toward mathematics as inquiry, rather than as
followers of procedures (Ambrose, 2004), and also they should learn how to implement activities
properly for solving problem.
Regarding student teacher experiences, the literature supports that courses are not enough
for future teachers to learn how to teach. It is important that teacher education programs provide
future teachers with experiences and placements in schools that allow them to put into practice
what they have learned in their courses. Such field experiences can be useful to try out the
effectiveness of strategies and the adequacy of lesson plans. The field experience can be useful
as well for future teachers to rethink their ideas and assumptions of what mathematics teaching is
44

about. In this line of thinking, Richardson (2003) argued for “engaging the pre-service teachers
in a structured practice situation such that the construct and propositional knowledge presented
in the academic classes may be observed in practice and examined in relation to their own
beliefs” (p. 13).
Likewise, research findings have reported that the student teaching experience allows
teachers to have a real sense of teaching and what is needed for teaching. Ambrose (2004)
developed a study with pre-service mathematics teachers who had the opportunity to deal with
real classroom situations, and she concluded that “when the prospective teachers had first-hand
experiences teaching mathematics, they realized that they needed to understand it well and began
to appreciate the value of their mathematics class” (Ambrose, 2004, p. 115).
Classroom Participation
As argued by Muijs (2004), “If classroom practices are a major factor in the development
of beliefs, it is plausible that significantly altering these environments can foster positive
mathematics-related beliefs” (p. 355). Thus if we want future teachers to engage their own
students in active learning and to treat them as constructers of their own learning, then teacher
education classrooms should look like places consistent with the reform agenda. The kind of
teaching supported by the reforms demand classroom environments that are supportive and
challenging; to this purpose, it is important to consider the social dimensions of learning, since as
theorists of situated learning argue, “Interactions with the people in one’s environment are major
determinants of both what is learned and how learning takes place” (Putnam & Borko, 2000, p.
5).
Applying these notions to teacher preparation programs, rather than to classrooms where
lectures by teacher educators dominate, it is desirable to have environments where students,
45

future teachers in our case, have more participation and interaction with their peers and can
examine and share their mathematical ideas through small and large group discussions. Being
involved in these environments might allow future teachers to witness that such classrooms are
possible, to see what it takes to create such classrooms, and consequently to embrace beliefs that
support a more active role of learners in argumentation, making conjectures, and discussion.
These experiences and activities are also important because they provide future teachers with a
sense of what they have to do with their own students. Contrarily, not much change in beliefs
will be observed if the environments where future teachers learn to teach do not correspond with
the ones required in classrooms by the reforms (Philipp, 2007).
Reflection
Future teachers also require opportunities to reflect (Crespo, 2000). It is hard to get
involved in reflective processes if pre-service teachers do not have any support that allows them
to have opportunities for intellectual exchange about the problems that can arise in classrooms
and about the possible solutions that can be implemented. Feiman-Nemser (2001) argued, “If
novices learn to talk about specific clarifications, share uncertainties, and request help, they will
be developing skills and dispositions that are critical in the ongoing improvement of teaching”
(p. 1030). The same is required for future teachers through all the activities in which they are
engaged in learning to teach. It is important that future teachers have the support of their teacher
educators, mentors, or supervisors to go through several stages of reflection (i.e., pose
hypotheses, look for evidence, propose and implement changes). Also, it is important that future
teachers are in a proper environment where they can sustain reflective dialogue with their peers.
Thus ongoing and meaningful reflection can help future teachers, because if they “appreciate the
importance of concepts in developing mathematical understanding, they might try to develop
46

their own conceptual understanding as well as think of ways to teach for conceptual
understanding” (Ambrose, 2004, p. 97).

47

CHAPTER 4

METHOD
This study examined how teacher education programs are responding to the challenges
posed by the current educational reform for mathematics teaching, through five research
questions:
1. What are the characteristics of the teacher preparation programs?
2. What opportunities to learn do future teachers have during their preparation?
3. How do future teachers perform on a measure of MPCK?
4. What are beliefs about mathematics nature, learning, and mathematics achievement
stated by future teachers?
5. What opportunities to learn are associated with future teachers’ beliefs and MPCK?
To answer the research questions, I used the framework employed by the Teacher
Education and Development Study in Mathematics (TEDS-M), conducted by Tatto, et al. (2008).
TEDS-M was a comparative study in seventeen countries of primary and secondary mathematics
teacher education; it examined how different countries prepare their teachers to teach
mathematics in primary and lower-secondary schools.

6

TEDS-M sought to examine three domains in the process of teacher preparation:
education policies, practices, and outcomes. The first component involved understanding the
national policy context through the analysis of the national curriculum, the teacher preparation
curriculum, and course syllabi. The second component, practices, involved exploring the
6

The participating countries in TEDS-M were: Botswana, Canada, Chile, Chinese Taipei,
Georgia, Germany, Malaysia, Norway, Oman, Philippines, Poland, Russian Federation,
Singapore, Spain, Switzerland (German speaking cantons), Thailand, and the United States.
48

intended and implemented curriculum of the programs and the opportunities to learn that the
institutions provide to future teachers. The last component, outcomes, entailed examining future
teachers’ knowledge and beliefs related to mathematics teaching. All these characteristics made
the TEDS-M framework pertinent to my research purposes, so I adopted and then adapted it to
my data collection process and subsequent analysis.
This was an exploratory mixed-method study, since I used both qualitative and
quantitative methods. The qualitative component involved the analysis of institutional
documents, and the quantitative component mainly used a cross-sectional survey design, since I
administered a survey to a sample of people at one point in time (Creswell, 2011). Likewise,
although I followed the TEDS-M framework, the research design of this study had some unique
characteristics. While TEDS-M focused on student teachers in the final year of teacher
preparation, and referred to them as “future teachers,” the design of this study included the
participation of two cohorts of student teachers: student teachers beginning the teacher education
program (first-year students) and students at the final stage of the program (fifth-year students).
In this document I also use the terms “entry cohort” and “exit cohort” to refer to these groups,
and both groups were considered “future teachers.” Having two groups in each institution
allowed me 1) to identify the “entrance” knowledge and beliefs of future teachers so that I could
identify how much work teacher preparation programs must do if future teachers are to
implement the National Curriculum Design (DCN); 2) to identify what the salient knowledge
and beliefs of future teachers are; and 3) to identify the effect of the cohort and the program in
these outcome variables (knowledge and beliefs).
TEDS-M also focused on two groups: primary level and lower secondary level. My study
focused on primary level future teacher programs (which prepare future teachers to teach from
49

first to sixth grade) because in Peru primary teachers are mostly generalists, and, therefore,
trained to teach all subject areas of the school curriculum. This entails a further challenge for
teacher education programs which, despite of the lack of specialization, have to provide the
foundations required to teach mathematics as demanded by the school curriculum.
Sample and Settings
I used a purposeful sampling design, specifically critical case sampling (Patton, 1990). I
decided to seek the most prestigious institutions that have teacher education programs in Lima,
public and private, which could likely represent the best case for how well teacher education
programs are responding to the reform mandates described in Chapter 2. Lima, as the capital
city, has the best educational opportunities for many careers, including the teaching profession.
Thus, in theory, teacher education programs in this city can be said to be the best in the country,
because they have more resources and qualified teacher educators, which are crucial for future
teachers to receive preparation according to school curriculum expectations. In addition, working
with these institutions would help me to get a sense for how much variability there is among
institutions that prepare future teachers.
I planned to identify ten potential institutions in order to get the participation of at least
five teacher preparation programs. Initially, I tried to collect data in the Pedagogical Superior
Institutes (ISP) located in Lima, due to the dependency of these institutions on the Ministry of
Education (MED). This dependency should, in theory, guarantee the implementation of curricula
that encourage the acquisition of knowledge and beliefs compatible with the ministry’s school
curriculum reform goals. However, during my initial fieldwork and communications with
representatives of the MED to select possible participating institutions, I was notified that only a
few ISPs had students graduating that year. The application of the law in 2008 regarding the
50

implementation of a national exam to enroll in ISPs, as explained in the previous chapters,
reduced the student population in these institutions, and consequently few or no students who
were enrolled were ready to graduate at the end of 2012. Due to this event I had to expand the
sample to include schools of education in universities.
Once I had selected 10 potential institutions, it took four months to collect all the data:
June, July, September, and October of 2012. I first sent letters of presentation to program
coordinators, and I met with them to request authorization to conduct my research. Procedures
approved by the Institutional Review Board (IRB) of Michigan State University (MSU) were
followed in the institutions that agreed to participate. It is important to note that the study
initially planned to focus only on fifth year students. Data for this cohort was collected in June
and July. The first-year cohort in the study was added later. IRB revision to authorize the new
process of data collection was requested and approved. The second phase of data collection was
carried out in September and October by a helper who was trained in the procedures followed in
the first phase of data collection.
Finally, six institutions in Lima agreed to participate in the study; one did not provide
7

sufficient information leaving the sample to five: institutions A, B, C, D, and E. Table 4.1
summarizes the main aspects of the background of these institutions.

7

This institution did not have first year students of primary education because the program was
closed and no more students for this specialty were admitted. Besides, this institution did not
provide the information requested, such as syllabi and study plans that would be useful to
characterize the institution. In the study I use fake names in response to commitment on
confidentiality provided to the institutions.
51

Table 4.1. Background of the Participating Institutions

A
Private
University
Urban

Sector
Level
Location
Religious
Yes
Background
Programs
Liberal
offered
Arts

Institutions
B
C
D
E
Public
Public
Public
Public
University University University Institute
Urban
Urban
Urban
Urban
No

No

No

Yes

Liberal
Arts

Liberal
Arts

Only
education

Only
education

Notice that each institution has some unique features and some shared features. For
instance, Institutions A is a private institution located in a middle class neighborhood in Lima;
thus its students might come from more advantaged environments, and its program is likely to
have more resources. By contrast, public institutions in Peru, are used to lack of resources and
facilities, and they recruit students of low socioeconomic status.
Institution E is the only institute in the group. This institute, unlike other institutes, has a
special agreement with the Peruvian government which entitles institution E to establish its own
curriculum. Commonly, Pedagogical Superior Institutes must use a curriculum designed by the
Ministry of Education.
Institutions A and E have a Catholic background; the first two are managed by secular
administrators and the latter is managed by a religious congregation, according to an agreement
with the Peruvian government. No particular religion is required for prospective students to
enroll in them. The remaining institutions are secular.
Two of the five institutions, D and E, specialize in Education; the former prepares future
teachers for basic education, and the latter prepares future teachers for basic education,

52

8

alternative education, and technical non-university education. The remaining universities offer
other programs in addition to education.
Data Sources
Data sources were formed by program documents and questionnaires completed by future
teachers.
Program Documents
I collected each institution’s program plan, the syllabi of all their courses on mathematics
(devoted to strengthening mathematical content knowledge) and all their courses on mathematics
education pedagogy (devoted to providing guidelines on how to teach mathematics in primary
education) included in the program plan, and professional profiles of their graduates. The
program plan and syllabi were provided by the program coordinator of each institution. This
document basically included the list of courses organized by semester and the equivalence of
courses in credits and hours. Finally, professional profiles of future teachers were downloaded
from the institutional websites of each program. Table 4.2 shows the list of the 20 syllabi
collected across the institutions.

8

During the last years, Institution E has opened some programs not related to teacher education,
but officially it still identifies itself as a university that prepares teachers.
53

Table 4.2. Courses of Mathematics and Mathematics Pedagogy in Teacher Education Programs

A
B
 Logical
 Mathematics I
mathematical  Mathematics II
reasoning
 Pedagogy for
 Mathematics I
mathematics
teaching in
 Pedagogy of
mathematics I
primary
education
 Pedagogy of
mathematics II

Institutions
C
 Development of
logical mathematical
thinking
 Pedagogy of
mathematics I
 Pedagogy of
mathematics II
 Pedagogy of
mathematics III

D
 Mathematics I
 Mathematics
II
 Pedagogy of
mathematics I
 Pedagogy of
mathematics
II

E
 Mathematics I
 Mathematics
II
 Mathematics
III
 Mathematics
IV
 Mathematics
V

The syllabi provided by the program coordinators do not belong necessarily to the classes
where students in the sample were enrolled.
Future Teacher Questionnaire
I used the TEDS-M future teacher questionnaire, which originally had four important
9

sections: student background, opportunities to learn, knowledge, and beliefs. Some changes to
the instrument were made so that it would fit the purposes of my study. Due to the study
considering two cohorts (first year and fifth year students), I prepared two versions of the future
teacher questionnaire, one for each cohort of students. Table 4.3 summarizes the structure of the
questionnaire used in this study, as well as the respondents for each section (see the complete
instrument in Appendix A).

9

The items of general background, beliefs, and opportunities to learn are published in IEA
(2012a). The items of knowledge are published in IEA (2012b). Both publications are available
in http://www.iea.nl/teds-m.html
54

Table 4.3. Structure of the TEDS-M Future Teacher Questionnaire Used in Peru
Section
General
Background
Beliefs
Knowledge

Opportunities
to Learn

Content
General information of the future teacher:
age, gender, SES, parental education,
motivation to become a teachers
Beliefs about the nature of mathematics,
about mathematics learning, and about
mathematics achievement
Six items of Mathematics Pedagogical
Content Knowledge
University- or tertiary- level mathematics,
school-level mathematics, mathematics
education pedagogy, general pedagogy,
learning through school-based experience,
teaching diverse students, and coherence of
teacher education program

Respondents
1st year students
5th year students
1st year students
5th year students
1st year students
5th year students

5th year students

Some clarifications regarding the instrument are important. First, the knowledge section
of the original TEDS-M future teacher questionnaire was devoted to assess mathematical content
knowledge and mathematical pedagogical content knowledge. TEDS-M used a complex
evaluation model to come up with valid and reliable measurements of these constructs (i.e,
several forms of the test were applied to cover as much as possible the spectrum of the variable).
The small sample size involved in this study could not allow applying the same procedure;
besides, the items were not public for the researchers. These limitations made it impossible to
use the same instrument to measure knowledge; however, I used a set of MPCK items for
elementary future teachers that were released in the TEDS-M publications (IEA, 2012b), and I
included them in the survey to have a rough proxy of this variable. I picked MPCK items
because this was the focus of the study and a critical area of teacher preparation in Peru. While
research has provided evidence about the limitations of Peruvian teachers’ content knowledge,

55

the area of pedagogical content knowledge used to be more neglected in the study plans of
teacher preparation programs.
Second, the questionnaire for the fifth year cohort included all the sections included in
the original questionnaire: background, beliefs, and opportunities to learn, and the selected items
of MPCK. The questionnaire for the first year included the sections on background, beliefs, and
the selected items of MPCK; the section about opportunities to learn was not included for this
group of respondents because the design of these questions asked future teachers to provide a
comprehensive evaluation of the learning opportunities they had during the program when they
were about to graduate. This section would not be meaningful for the first year students who had
not had the chance to take all the mathematics courses listed in the study plan of the institutions
in which they were enrolled.
Third, the survey, originally in English, had been translated into Spanish to be used in the
TEDS-M study. I used the Chilean version of the questionnaire, also in Spanish, to develop the
Peruvian version for my study, and I introduced some minor adaptations to items that asked
questions related to the Peruvian context.
In the selected institutions, all future teachers who were enrolled in their first and last
year of the elementary education program were invited to participate in the study. The
Questionnaire, which was anonymous, was administered in groups and took about an hour to be
completed. Student teachers who completed the survey received an incentive (a phone card) as a
token of appreciation for their time and interest in participating in the study. Data was collected
when first year future teachers were beginning the second semester of the program, and when
fifth year future teachers were ending the ninth semester of the program. The table 4.4 shows the
composition of the sample in each institution.
56

Table 4.4. Participating Institutions and Sampling Coverage
5th year students
1st year students
Institution
Surveys
Percentage
Surveys
Percentage
Enrolled
Enrolled
Completed
covered
Completed
covered
A
6
5
83.3
12
10
83.3
B
24
21
87.5
45
41
91.1
C
34
29
85.3
20
13
65.0
D
102
75
73.5
80
56
70.0
E
37
35
94.6
30
23
76.7
Note. The intended sample was the total of students enrolled in the semester when data was
collected.

Questionnaire Variables
The TEDS-M study developed scales to measure variables that were used in this study:
opportunities to learn, knowledge, and beliefs. In this section I describe the variables and the
psychometric properties of the scales developed for each variable. I also explain adaptations
made to the measures, if any. This description is organized by the nature of the variables:
program variables and future teacher variables.
Program Variables
In this study the variables related to the program were represented by the opportunity to
learn scales included in the future teacher questionnaire. These scales allowed describing how
institutions were doing in terms of exposing future teachers to content and processes during their
preparation. TEDS-M elaborated many scales of opportunity to learn, but I selected the ones that,
according to the literature, were more related to the outcomes expected from teacher education
programs: knowledge and beliefs. Table 4.5 below presents the scales of opportunity to learn
analyzed in this study which are further described later.

57

Table 4.5. List of Program Variables
Variable

Opportunities
to learn

Scale
Course topics
Solving problem
Class reading
Class participation
Instructional practice
Instructional planning
Assessment practice
Reflection on practice

Course topics. The questionnaire for future teachers explored the academic content of
teacher education programs in the courses on school level mathematics and mathematics
education pedagogy, as showed in Table 4.6.
Table 4.6. Items on Course Topics in the Questionnaire
Areas
School level mathematics

Mathematics education
pedagogy

Course topics
Numbers, measurement, geometry, functions, data
representation, calculus, validation, structuring, and
validation.
Foundation of mathematics, context of mathematics
education, development of mathematics, mathematics
instruction, developing teaching plans, mathematics
teaching, mathematics standards and curriculum,
affective issues in mathematics.

For each item, respondents had to answer whether they studied or did not study the topic
mentioned as part of their program. The questionnaire also explored whether future teachers had
ever studied tertiary-level mathematics, such as geometry, discrete structures and logic,
continuity and functions, and probability and statistics. This set of topics was not included in the
analysis because, as will be explained later, the analysis of the study plan revealed that the study
plan of the institutions was mainly focused on school mathematics courses, and no courses about
tertiary level mathematics were part of such study plans.
58

Activities for learning to teach mathematics. The teacher questionnaire also included
questions about learning activities to capture in what kind of learning strategies future teachers
were engaged in their classrooms and what strategies they learned to teach mathematics. All
items were measured with a four-point categorical scale of frequency: never, rarely,
occasionally, and often. Table 4.7 shows the list of the opportunity to learn scales.

59

Table 4.7. Scale and Items of Opportunity to Learn Included in the Questionnaire
Scale
Class
participation

Class reading

Solving
problem

Instructional
planning

Instructional
practice

Assessment
practice

Reflection on
practice

Items
Ask questions during class time
Participate in a whole class discussion
Make presentations to the rest of the class
Teach a class session using methods of my own choice
Teach a class session using methods demonstrated by the instructor
Read about research on mathematics
Read about research on mathematics education
Read about research on teaching and learning
Write mathematical proofs
Solve problems in applied mathematics
Solve a given mathematics problem using multiple strategies
Use computers or calculators to solve mathematics problems
Accommodate a wide range of abilities in each lesson
Create learning experiences that make the central concepts of subject matter
meaningful to pupils.
Create projects that motivate all pupils to participate
Deal with learning difficulties so that specific pupil outcomes are accomplished
Develop games or puzzles that provide instructional activities at a high interest
level
Develop instructional materials that build on pupils’ experiences, interests and
abilities
Explore how to use manipulative (concrete) materials or physical models to
solve mathematics problems
Locate suitable curriculum materials and teaching resources
Use pupils’ misconceptions to plan instruction
Explore how to apply mathematics to real-world problems
Explore mathematics as the source for real-world problems
Learn how to explore multiple solution strategies with pupils
Learn how to show why a mathematics procedure works
Make distinctions between procedural and conceptual knowledge when teaching
mathematics concepts and operations to pupils
Integrate mathematical ideas from across areas of mathematics
Analyze and use national or state standards or frameworks for school
mathematics
Analyze pupil assessment data to learn how to assess more effectively
Assess higher−level goals (e.g., problem-solving, critical thinking)
Assess low−level objectives (factual knowledge, routine procedures)
Build on pupils’ existing mathematics knowledge and thinking skills
Use teaching standards and codes of conduct to reflect on your teaching
Develop strategies to reflect upon the effectiveness of your teaching
Develop strategies to reflect upon your professional knowledge
Develop strategies to identify your learning needs
60

The opportunity to learn scales developed by TEDS-M proved to have high indices of
reliability, as measured by the sophisticated procedures and software (Tatto et al., 2012) Scales
also proved to be reliable for the Peruvian sample, calculated with Cronbach’s alpha, as showed
in the Table 4.8.
Table 4.8. Reliability of Opportunity to Learn Scales
Scale

Number
of Items

TEDS-M
reliability

Valid
N

Cronbach’s
Alpha

Class participation
5
.85
140
.82
Class reading
4
.83
138
.80
Solving problem
4
.78
140
.78
Instructional planning
9
.90
142
.86
Instructional practice
6
.89
141
.82
Assessment practices
5
.87
142
.76
Reflection
4
.93
141
.86
Note. Reliability analysis for OTL scales was calculated with the sample of exit students.

In this study, I organized these opportunities to learn variables in two categories: intended
curriculum and implemented curriculum to enrich analysis and interpretation. Remillard (2005)
defined intended curriculum as “teachers’ aims” (p. 213) and implemented (or enacted) as “what
actually takes place in the classroom” (p. 213). In this case, the intended curriculum was
represented in the syllabi, and the implemented curriculum was reported by future teachers in the
questionnaire.
Student Variables
The student variables were related to demographics and the outcomes expected from
teacher preparation programs: knowledge and beliefs; however, some questions on demographics
were also examined to have a sense of the students who were entering the teacher education
program. Table 4.9 show all the variables related to future teachers.
61

Table 4.9. List of Future Teacher Variables
Variable

Scales
Gender, age, books, parental education, motivation
Demographics
for entering the teacher career
Mathematical Pedagogical Content Knowledge
Knowledge
Index
Mathematics as a set of rules and procedures
Mathematics as a process of inquiry
Learning mathematics through following teacher
Beliefs
directions
Learning mathematics through active involvement
Mathematics as fixed ability
Descriptions below focus on the two outcomes: knowledge and beliefs, because they
were the focus of the study.
Mathematics pedagogical content knowledge index. The TEDS-M study developed a
framework to measure knowledge which included two areas: mathematics content knowledge
(MCK) and mathematics pedagogical content knowledge (MPCK). MCK was organized into two
domains to assure coverage of the construct: content and cognitive processes. The subdomains of
content were number and operations, geometry and measurement, algebra and functions, and
data and chance; the subdomains of cognitive processes were knowing, applying, and reasoning.
On the other hand, the subdomains for MPCK were mathematics curricular knowledge,
knowledge of planning for mathematics teaching and learning, and enacting mathematics for
teaching and learning.
As explained before, I selected six items released by TEDS-M that provided information
on the MPCK of future teachers and that covered the main domains of the Peruvian national

62

curriculum: number, geometry and measurement, and data. One of the six items was a PCK item
it had to be included in the questionnaire because it was related to a MPCK item.
The first item was about number and curricular knowledge and knowledge of planning,
and it required respondents to identify the two most difficult number-story problems from a set
of four verbal problems which could be solved using a single arithmetic operation with whole
numbers. The figure below shows the actual item, which was labeled as “Jose” for reference in
the rest of the document. The italics are to differentiate the problem from the question that future
teachers had to answer.
A <Grade 1> teacher asks her students to solve the following four story problems, in any
way they like, including using materials if they wish.
(a) Problem 1: [Jose] has 3 packets of stickers. There are 6 stickers in each pack.
How many stickers do [Jose] have altogether?
(b) Problem 2: [Jorgen] had 5 fish in his tank. He was given 7 more for his birthday.
How many fish did he have then?
(c) Problem 3: [John] had some toy cars. He lost 7 toy cars. Now he has 4 cars left.
How many toy cars did [John] have before he lost any?
(d) Problem 4: [Marcy] had 13 balloons. 5 balloons popped. How many balloons did
she have left?
The teacher notices that two of the problems are more difficult for her children than
the other two. Identify the TWO problems which are likely to be more DIFFICULT to solve
for <Grade 1> children.
Figure 4.1. Item “Jose” in the Questionnaire
The second and third items were related each other. The second item, labeled as “Pencil”,
asked students to solve a problem about data that used an unlabeled bar graph. The third item,
“Difficulty”, requested students to state the main difficulty that elementary students might have
with this type of problem. Figure 2 shows the two items included in the questionnaire.

63

The following problem was given to children in <primary> school:
The graph shows the number of pens, pencils, rulers, and erasers sold by a store in a week.

The names of the items are missing from the graph. Pens were the item most often sold.
Fewer erasers that any other item were sold. More pencils than rulers were sold.
(a) How many pencils were sold?
(b) Some <primary> students would experience difficulty with a problem of this type.
What is the main difficulty you would expect? Explain clearly with reference to the
problem.
Figure 4.2. Items “Pencil” (a) and “Difficulty” (b) in the Questionnaire

As can be seen, the item “Pencil” had to do more with MCK than with MPCK. It was
included because it provided context to the respondents to answer the next question but it was
removed from the analysis because MCK was not the intended focus of the study.
The fourth item, labeled in the study as “Paper clip,” explored notions of geometry and
measurement as well as knowledge of curriculum and planning. The item presented the case of a
teacher who was working with students on a first lesson on length measurement. The figure
below shows the item.
64

When teaching children about length measurement for the first time, Mrs. [Ho] prefers to
begin by having the children measure the width of their book using paper clips, then again
using pencils.
Give TWO reasons she could have for preferring to do this rather than simply
teaching the children how to use a ruler?
Figure 4.3. Item “Paper clip” in the Questionnaire

The fifth and sixth items, labeled as “Jaime” and “Diagram,” were related to each other
and covered the content domain of number and the subdomain of enacting. The item “Jaime”
introduced the case of a student who got confused with the results of arithmetic operations with
decimal numbers. As shown in the figure below, respondents had to identify why Jeremy was
puzzled by these results, and to propose a visual representation to explain the arithmetic
operation.

[Jeremy] notices that when he enters 0.2 × 6 into a calculator his answer is smaller than
6, and when he enters 6 ÷ 0.2 he gets a number greater than 6. He is puzzled by this,
and asks his teacher for a new calculator!
(a) What is [Jeremy’s] most likely misconception?
(b) Draw a visual representation that the teacher could use to model 0.2 × 6 to help
[Jeremy] understand WHY the answer is what it is?
Figure 4.4. Items “Jaime” (a) and “Diagram” (b) in the Questionnaire

As can be seen, all these items covered different domains of the MPCK. Likewise, the
items asked respondents to construct their answers, which could receive no credit =0, partial
credit= 1, or full score =2. Table 4.10 summarizes the main characteristics of the items.
65

Table 4.10. Items of MPCK Evaluated in the Questionnaire
Item
Label/Code
Jose
(MFC505)

Item
Content
Subdomain
Content
Domain
Identify two most
Number
Curriculum
difficult number
story problems
Pencil
Interpret
Number
Reasoning
(MFC502A) information on a
bar graph
Difficulty
Difficulty with a
Data
Curriculum/Plan
(MFC502B) data representation
problem
Paper clip
Two reasons for
Geometry Curriculum/Plan
(MFC513)
measuring with
paper clips
Jaime
Students’
Number
Enacting
(MFC208A) misconception in
using a calculator to
multiply 0.2 x 6
Diagram
Propose a visual
Number
Enacting
(MFC208B) representation to
model 0.2 × 6
Note. Item codes corresponded to the ones used in TEDS-M.

Item
Score
format
Constructed 2, 1, 0
response
Multiple
choice

1, 0

Constructed 2, 1, 0
response
Constructed 2, 1, 0
response
Constructed 2, 1, 0
response

Constructed 2, 1, 0
response

Due to all the knowledge questions included in the survey that demanded a constructed
response, answers had to be coded using the TEDS-M scoring guide for the released items (see
Appendix B). All responses to MPCK were coded by two reviewers; one of them was the author
of the study. An interrater reliability analysis using the Kappa statistics was performed to
determine the consistency among raters. Indices were calculated with SPSS. Results in Table
4.11 revealed almost perfect agreement between the coders. Differences were discussed to arrive
at 100% agreement.

66

Table 4.11 Interrater Reliability for MPCK Items
Label
Jose (MFC505)
Difficulty (MFC502B)
Paper clip (MFC513)
Jaime (MFC208A)
Diagram (MFC208B)

Valid
N
294
307
303
306
289

Percent
91.9
95.9
94.7
95.6
90.3

Kappa
statistic
1
.917
.910
.923
1

The five items selected were aggregated to make a scale that worked as a continuous
variable, which was named the MPCK index. The items with no answer (missing values) were
coded as “0” to preserve the cases in the sample at the time to calculate the index. Due to all
questions being partial credit items, the maximum score for the score was ten and the minimum
zero. I calculated the reliability of the MCPK index for the whole sample with valid observations
in the five items (N=308), and the Cronbach’s Alpha resulting was low (.40). This is because
correlations between items that formed the scale were small. The low reliability of these items
might have to do with the fact that the index covered, as described in Table 4.10, different
aspects (number, geometry, measurement) and different subdomains (curriculum, planning
enacting). This means that the items were not measuring only one construct. Likewise, the
performance of future teachers on the items also could have been influenced in this result. As
will be further described in the findings section, some items were too easy, and others were too
hard (see Appendix C for distribution of percent correct responses by item). The MPCK index
had limitations, but it still provided useful information about the knowledge future teachers could
demonstrate.

67

Future teachers’ beliefs. The future teacher questionnaire examined future teachers
beliefs related to the nature of the mathematics, the learning of mathematics, and academic
achievement, and it used a 6-point Likert scale (strongly disagree, disagree, slightly disagree,
slightly agree, agree, and strongly agree) to measure the extent to which future teachers endorsed
these beliefs.
Beliefs about the nature of mathematics. This scale was formed by 12 items that in turn
included two scales of beliefs: mathematics as a set of rules and procedures, and mathematics as
a process of inquiry, with six items for each scale. Respondents who obtained high scores on the
first scale emphasized the logical character of mathematics, while those who strongly supported
the second scale saw mathematics as “a means of answering questions and solving problems”
(Tatto et al., 2012, p. 155). In the rest of the study these scales are referred as “rules” and
“inquiry.” Table 4.12 shows the items included in these scales.

Table 4.12. Beliefs about the Nature of Mathematics: Scales and Items
Scale

Mathematics as
a set of rules
and procedures

Items
Mathematics is a collection of rules and procedures that prescribe how to
solve a problem.
Mathematics involves the remembering and application of definitions,
formulas, mathematical facts and procedures.
When solving mathematical tasks you need to know the correct procedure
else you would be lost.
Fundamental to mathematics is its logical rigor and preciseness.
To do mathematics requires much practice, correct application of routines,
and problem solving strategies.
Mathematics means learning, remembering and applying.

68

Table 4.12 (cont’d)

Mathematics as
a process of
inquiry

Mathematics involves creativity and new ideas.
In mathematics many things can be discovered and tried out by oneself.
If you engage in mathematical tasks, you can discover new things (e.g.,
connections, rules, concepts).
Mathematical problems can be solved correctly in many ways.
Many aspects of mathematics have practical relevance.
Mathematics helps solve everyday problems and tasks.

Beliefs about learning mathematics. This scale was formed by 14 items, which
measured two kinds of beliefs: learning mathematics through following teacher directions (8
items), and learning mathematics through active involvement (6 items). Respondents who scored
high on the first scale “tend to see mathematics learning as being heavily teacher-centered,” and
respondents who scored high on the second scale “tend to see mathematics as being active
learning” (Tatto et al., 2012, p. 156). In the rest of the study these scales are referred as “teacher
direction” and “active involvement.”

Table 4.13. Beliefs about Learning Mathematics: Scales and Items
Scale

Learning
mathematics
through teacher
direction

Items
The best way to do well in mathematics is to memorize all the formulas.
Pupils need to be taught exact procedures for solving mathematical problems.
It doesn’t really matter if you understand a mathematical problem, if you can
get the right answer.
To be good in mathematics you must be able to solve problems quickly.
Pupils learn mathematics best by attending to the teacher’s explanations.
When pupils are working on mathematical problems, more emphasis should
be put on getting the correct answer than on the process followed.
Non-standard procedures should be discouraged because they can interfere
with learning the correct procedure.
Hands-on mathematics experiences aren’t worth the time and expense.

69

Table 4.13 (cont’d)

Learning
mathematics
through active
involvement

In addition to getting a right answer in mathematics, it is important to
understand why the answer is correct.
Teachers should allow pupils to figure out their own ways to solve
mathematical problems.
Time used to investigate why a solution to a mathematical problem works is
time well spent.
Pupils can figure out a way to solve mathematical problems without a
teacher’s help.
Teachers should encourage pupils to find their own solutions to mathematical
problems even if they are inefficient.
It is helpful for pupils to discuss different ways to solve particular problems.

Beliefs about mathematics achievement. The questionnaire included a scale of eight
items to examine whether future teachers “tended to see mathematics achievement as heavily
dependent on the ability of the student” (Tatto et al., 2012 p. 156). In the rest of the document
this scale is referred as “fixed ability.” The items are presented in the following table.

Table 4.14. Beliefs about Academic Achievement as Fixed Ability: Items
Scale

Items
Since older pupils can reason abstractly, the use of hands-on
models and other visual aids becomes less necessary.
To be good at mathematics you need to have a kind of
“mathematical mind.”
Mathematics is a subject in which natural ability matters a lot
more than effort.
Mathematics Only the more able pupils can participate in multi-step
as fixed
problem solving activities.
ability
In general, boys tend to be naturally better at mathematics
than girls.
Mathematical ability is something that remains relatively
fixed throughout a person’s life.
Some people are good at mathematics and some aren’t.
Some ethnic groups are better at mathematics than others.
70

Belief scales proved to be reliable measures for the TEDS-M study. Results with the
Peruvian sample were also positive; the Cronbach’s alpha for all scales was found to be
acceptable, as shown in the table below.

Table 4.15. Reliability of Beliefs Scales
Number
TEDS-M
Valid
of Items
reliability
N
Rules
6
.91
302
Inquiry
6
.94
295
Teacher directions
8
.86
291
Active involvement
6
.92
299
Fixed ability
7
.88
295
Note. Reliability analysis for beliefs was done with the whole sample.
Scale

Cronbach’s
Alpha
.77
.79
.79
.69
.76

Data Analysis
In this study I considered that future teachers’ beliefs and knowledge were outcomes of a
process developed by the teacher education program through opportunities to learn to which
future teachers were exposed. Likewise, given the context of the reform, I examined the extent to
which the processes (opportunities to learn) in teacher education prepared future teachers to
implement the mathematics school curriculum, and the extent to which the outcomes (knowledge
and beliefs) were aligned with the reform demands (i.e., future teachers’ beliefs go in the same
direction as the principles supported in the school curriculum).
Document analysis and statistical analysis with SPSS were carried out to answer all
research questions. Table 4.16 shows a summary of the analysis done by research question, and
later I provide further explanations on each one.

71

Table 4.16. Research Question, Statistical Analysis, and Data
Research
Question
1. What are the characteristics
of the teacher preparation
programs?
2. What opportunities to learn
do future teachers have during
their preparation?

3. How do future teachers
perform on a measure of
MPCK?
4. What are beliefs about the
mathematics nature, learning
and mathematics achievement
stated by future teachers?
5. What opportunities to learn
are associated with future
teachers’ beliefs and MPCK?

Analysis

Cohort

Document analysis and descriptive of
students’ demographics

1 and 5 year

Intended curriculum (analysis of
syllabi and program plan)
Implemented curriculum
(analysis of coursework, activities for
learning to teach, reflection in the
questionnaire)
Descriptive by institution of
opportunity to learn scale.
MANOVA, ANOVA, and multiple
comparisons
Descriptive of MPCKA index by
cohort/institution
Univariate ANOVA and multiple
comparisons
Descriptive of beliefs scales by
cohort/institution
Correlations between beliefs scales
and MPCK by cohort.
MANOVA and multiple comparisons
Correlations between beliefs and
knowledge with opportunities to learn
(course topics and opportunity to
learn scales)

5 year

st

th

th

st

th

st

th

1 and 5 year

1 and 5
year

th

5

year

To answer the first research question, I described the institutions participating in the
study in terms of the structure of their programs, their philosophy for teaching, and their
philosophy for mathematics teaching specifically. Thus the program plan, syllabi, and
professional profile of future teachers were used to characterize the program curriculum for
elementary future teachers and to identify some indicators of alignment between the institutional

72

philosophy and the reform purposes. I also analyzed the student population enrolled in each
institution by using questions on the future teachers’ background included in the questionnaire.
To answer the second research question I analyzed the intended and implemented
curriculum for learning to teach mathematics, in order to identify the learning opportunities
offered to future teachers in their institutions. Notice that in this study I organized the
opportunities to learn variables in two categories: the intended curriculum and the implemented
curriculum, to enrich analysis and interpretation. Remillard (2005) defined the intended
curriculum as “teachers’ aims” (p. 213), and the implemented (or enacted) curriculum as “what
actually takes place in the classroom” (p. 213). In this case, the teacher-intended curriculum was
represented in the syllabi, and the implemented curriculum was reported by future teachers in the
questionnaire. To analyze the intended curriculum, I used the syllabi I collected and I coded
them using the TEDS-M Framework for Coding Teacher Education Course, developed in 2008
by Teresa Tatto, Kiril Bankov, and Sharon Senk for the TEDS-M international study.

10

To

analyze the implemented curriculum, I used the section of opportunities to learn of the
questionnaire: course topics (school mathematics level and mathematics education pedagogy),
activities for learning to teach mathematics scales, and reflection.
To answer the third research question, I analyzed future teachers’ performance in the
MPCK index, to see the extent by which future teachers could addresses the processes (problem
solving, reasoning and demonstration, and mathematical communication) stated in the school
curriculum. To answer the fourth research question, I described what beliefs were reported by

10

The document is unpublished work and therefore is not included in the appendix.
73

future teachers regarding the nature of mathematics, learning, and achievement, and how these
beliefs were compatible or not with the reform goals for mathematics teaching.
The analysis of the implemented curriculum, the MPCK index, nd beliefs followed the
same procedure: (1) Performing basic descriptive statistics by institution (mean and standard
deviation) with the raw data for all the variables (opportunities to learn, knowledge, and beliefs).
Correlations between the variables also were performed to identify patterns when possible. (2)
Performing analysis of variance to determine the existence of differences in the variables by
groups. Here, the variables of interest became the dependent variables (opportunities to learn,
knowledge, and beliefs), and the variables that defined groups became independent variables
(cohort and/or institution). (3) Conducting post-hoc analysis to determine where the differences
were, specifically which groups were different. The last two steps implied the reduction of
sample size, since by default the analysis only included the observations that had information for
all the dependent variables.
While multiple analyses of variance, one-way or two-way, might have been applied for
the analysis of opportunities to learn, knowledge, and beliefs, other more advanced techniques
were performed in order to avoid type 1 error inflation. Thus, when the analysis involved more
than one dependent variable (i.e., beliefs had five scales and opportunities to learn had six scales)
and two factors or independent variables (cohort and institution), then a multivariate analysis of
variance (MANOVA) was performed. When the analysis involved only one scale (MPCK index)
and two factors (cohort and institution) as independent variables, then I performed a univariate
ANOVA.
Regarding post-hoc analysis, multiple comparisons were done using Fisher’s least
significance difference (LSD). There are several discussions about what is the most powerful test
74

to identify significant differences. Although LSD is considered one of the weakest tests, it was
chosen over the others because this allowed me to identify patterns that at some point were not
clear with other, more conservative tests, such as Bonferroni or Tukey.
For the fifth research question, I examined the association between the opportunities to
learn and the future teachers’ beliefs and knowledge by using Pearson correlations, so I could
identify the variables of the learning to teach process that could have some influence on the
outcome variables. I had initially planned to perform multiple regressions, controlling for the
effect of the institution on the dependent variables; however, the data had limitations on
performing these analyses, such as sample size in some institutions that made it hard to
accomplish it. Besides, contrary to expectations, there were not many significant correlations
found between the processes and outcomes variables, meaning between dependent and
independent variables, which is required to perform regression analysis.

75

CHAPTER 5
RESULTS
The results reported in this chapter are organized by research question, with a section for
each of the five research questions.
Research Question 1: Characteristics of the Teacher Education Programs
In this section, I describe the characteristics of each institution in three main areas: the
curriculum of the institution to prepare elementary mathematics teachers, the philosophy of the
program, and the student population enrolled in each institution. This description is based on
information collected from institutional documents (study plans, course syllabi, and websites)
and the future teacher questionnaire.
The Curriculum of Elementary Education in the Teacher Education Programs
Table 5.1 includes the distinctive characteristics and contexts of the elementary teacher
education program in each institution.

76

Table 5.1. The Curriculum for the Elementary Teacher Education Programs

A
Program
duration
N° of program
Credits
Credit hour
N° of program
courses

Institutions
C

B

D

E

5 years

5 years

5 years

5 years

5 years

192

220

222

210

225

50 min

50 min

50 min

50 min

60 min

54

80

71

74

71

th

Practicum

The 5
year

Degree at
completion

Bachelor

rd

rd

rd

st

From the 3
year on

From the 3
year on

From the 3
year on

From the 1
year on

Bachelor

Bachelor

Bachelor

Bachelor

Specialization

Generalists

Generalists

Generalists

TE program
Accreditation

In process

In process

In process

Generalists/
Alternate
basic
education
In process

Generalists
Accredited

All programs take five years to complete; during this period, the programs provide
specific numbers of credits and courses. The study plan of institution E includes more credits,
225; the other three public institutions (B, C, and D) also have over 210 credits. Institution A has
the least number of credits in its study plan, 192. In all these institutions, each credit is equivalent
to 50 minutes, except in institution E, where one credit is equivalent to 60 minutes. The number
of credits and the equivalence of credit hours might imply that future teachers in institution E
spend more time in classrooms as they prepare to become teachers, and that future teachers in
institution A might have less time for instruction. Regarding the number of courses, they are not
necessarily directly proportional to the number of credits; however, institution A also has the
least number of courses in its program, 54 courses.
77

Likewise, the institutions vary in the time they devote to the practicum, which involves
extended school experiences. Most of the institutions include extended school experience from
the third through the fifth year (Institutions B, C, and D). Institution A only requires a one-year
practicum, which takes place during the last year of the program. It is worth noting that the
institutions with more semesters devoted to the practicum (B, C, D, and E) are the ones with
more credits and courses in their program.
No specialization in a particular discipline (i.e., mathematics, literacy, science, or others)
was identified in the study plan of the participating institutions; all graduates become elementary
generalist teachers. However, the program of institution D devotes 25 credits (11 courses) of its
program to provide future teachers with a secondary area of specialization, in primary education
for adults (called alternative basic education). Future teachers in this institution must take classes
for this specialization in alternative basic education from the third year through the fifth year.
To graduate as future teachers in all institutions, students have to receive a passing grade
in all the courses required by the program (including the practicum). The completion of this
requirement allows the graduates from the university institutions (A, B, C, D) to earn a
bachelor’s degree.

11

Graduates from institution E also earn a bachelor’s degree, despite not being

at the university level.
Finally, in relation to accreditation, which is an important aspect to highlight due to the
policy mandates for teacher preparation, only institution E has been recently accredited for its
educational quality by the CONEACES, the entity in charge of accrediting non-university
institutions. The remaining participating institutions (A, B, C, and D) have not been accredited

11

Institution A also requires students to demonstrate initial knowledge in a foreign language.
78

yet by the corresponding entity CONEAU, despite the fact that the accreditation procedures were
released in 2008. These institutions reported to be working on the process of internal evaluation
required for institutional accreditation. However, some institutions, such as institutions A and D,
have accreditations provided by international institutions.
The Teacher Education Programs’ Philosophy
Teacher education programs set their philosophy about teaching through the kind of
curriculum they expose their students to, as well as the professional profile the program intends
to develop in their graduates. In this section I examine information in this regard that is available
on institutional websites, to see how aligned they are, at the intended level, with the reforms in
effect. Also, I examine the alignment of curriculum for teaching mathematics with the reform, as
expressed in the syllabi collected. Findings in this section are based only on available documents,
which can result in limited characterization of these institutions in these respects; despite this
limitation, it is still possible to identify some patterns across the institutions.

12

The results are

showed in Table 5.2.
Table 5.2. The Teacher Education Programs’ Philosophy and the Reform

Program
Curriculum
Future teachers
profile
Curriculum for
mathematics
teaching

A
Reform
oriented
Reform
oriented
Reform
oriented

No
Info
Reform
oriented

Institutions
C
No
Info
No
Info

No
Info
Reform
oriented

E
Reform
oriented
Reform
oriented

No
Info

Reform
oriented

No
Info

Reform
oriented

B

12

D

The documents are not cited directly as part of the results because that would reveal
institutional identities and confidentiality was part of the commitment with the participating
institutions.
79

Regarding the program curriculum, it could be expected that institutional documents
show that teacher preparation programs have adopted the constructivist principles for teaching
and learning as the Ministry of Education have done it. Evidence supports that this is the case for
institutions A and E. Institution A recognizes that the preparation of teachers is guided by social
constructivism and reflection, and that future teachers are active participants in their learning.
Institution E also supports a pedagogical approach that is related to constructivism, with future
teachers as learners involved in cooperative environments, and that is intended to be autonomous
and reflective. No detailed information about the curriculum orientation was found for the
remaining institutions.
Regarding the future teacher profile, the expectation is that teacher education programs
show that they are preparing teachers that meet the goals posed in the reform documents. The
reform expects that teachers are able to produce effective learning in students, and that they are
committed to their own professional development and the community in which they operate
(Ministerio de Educación, 2007). The documents collected show that the graduate profiles of the
participating institutions recognize these dimensions of the teaching profession, although they
address them in different ways. Thus, according to their institutional profiles, institution A and
institution B organize the teacher’s profile to include dimensions that cover the work of the
teacher in the classroom and also in his or her community; through these dimensions, future
teachers are expected to develop skills for educational research and practice, teaching processes,
school management and leadership, and personal and professional development. For its part,
institution D poses generic teaching skills as part of its graduate profile and organizes them in
three components: social, professional, and personal. Finally, institution E supports future
teachers’ profile in competences which include several dimensions of the teaching profession,
80

such as personal and professional development, awareness of the context of practice, educational
research, use of TICs, design of educational innovations, school management, learning
management, and educational evaluation.
It is important to delve into the dimensions of the professional profile that have to do with
the two outcomes of the teacher preparation programs: knowledge and beliefs, which, as the
literature reviewed supports, are crucial to assure student learning. Most of the institutions
address the disciplinary knowledge needed for teaching (content knowledge) by means of
general statements such as “master content for teaching to assure professional efficiency” and
“manage discipline content updated according to their level and specialty” but do not address
pedagogical content knowledge as part of their professional profile. Likewise, particular
dispositions for teaching (in the form of views, conceptions, and beliefs related to education,
teaching, and learning) are not included in the profile either.
These general statements on knowledge and omission of beliefs in the institutional
documents drive questions on to what extent the programs are aware of the importance of these
two outcomes for effective teacher preparation; however, more data other than the available
documents for this study would be needed to answer this question.
Finally, regarding the curriculum for mathematics teaching, Table 5.2 shows that
documents collected in institutions A, C, and E demonstrate that their teacher education
programs, at least in theory, were aligned with the problem solving approach for mathematics
teaching recommended by the national school curriculum as showed. These institutions address
in their syllabi different aspects that are related to the new way of understanding mathematics,
and that are included in their process of learning to teach.

81

Thus, institution A had syllabi addressing that the purpose of mathematics education is
fostering future teachers’ skills for analysis of alternatives, justification of statements, searching
for solutions, and the creation of new problems. Likewise, documents strongly recognized that
future teachers need to learn to teach mathematics through posing and solving problems. This
language related to problem solving is observed in its syllabi of mathematics pedagogy and
indicates that this institution systematically attempts to emphasize problem solving as a way of
learning mathematics and learning how to teach it. Institution C’s syllabi highlighted the
importance of future teachers attending to the development of students’ thinking and cognitive
structures to understand the development of mathematical skills through reasoning and
demonstration, mathematical communication, and problem solving at the primary school age.
The syllabi of this institution also highlighted the fact that future teachers have experiences that
allow them to observe, analyze, formulate hypotheses, and test different procedures as a way as
they are learning to teach mathematics. For its part, institution E’s syllabi emphasized the three
processes of the school mathematics curriculum: mathematical communication, solving problem,
and reasoning and demonstration as processes that they want their future teachers to develop
through their mathematics courses.
The syllabi of institution B pose some objectives which mention the importance of
students having the opportunity to solve problems, specifically in the courses on mathematics;
however, there is no mention of the other components of problem solving, such as
communication and reasoning, that would indicate more clearly that the program has taken
problem solving as a way to make future teachers learn to teach mathematics. Finally, institution
D does not describe their objectives or goals in terms on problem solving.

82

The fact that institutions A, C, and E clearly mention aspects of the reformed
mathematics curriculum would suggest that their programs are aware of the reform demands for
mathematics teaching, and accordingly they are preparing their graduates with what is required
to teach a mathematics curriculum oriented to solving problem. The status of the outcome
variables (knowledge and beliefs) in these institutions to be discussed later will help us to
examine the extent in which these intentions are being materialized in concrete results.
The Student Population in the Teacher Education Programs
The survey collected information about several characteristics of future teachers. In this
section I describe the future teachers’ characteristics in terms of demographics, socio-economic
status, and motivations for becoming an educator. Table 5.3 shows the demographic
characteristics of future teachers by age, gender, language, and some indicators of the future
teachers’ socioeconomic status.

83

Table 5.3. Future Teachers’ Demographics and Socio-Economic Status by Institution
Institutions

Total

A
1st year students’ age
5th year students’ age

B

C

D

E

18
(1.9)
23.3
(2.9)
93.3

19.3
(2.5)
23.6
(2.4)
91.9

19.3
(1.8)
23.1
(2.4)
95.2

19.2
(2.6)
23.0
(2.6)
92.4

19.3
(2.0)
22.7
(1.6)
96.6

19.25
(2.30)
23.16
(2.40)
93.5

Female
Always speak Spanish at
100.0
93.5
95.2
94.7
96.6
95.1
home
Books at home
8.8
None or few
6.7
8.1
9.5
12.2
1.7
69.1
11-100
66.7
67.7
66.6
71.8
67.3
16.2
101-200 books
6.7
17.7
23.8
9.9
25.9
20.0
6.5
0.0
6.1
5.2
5.8
More than 200
Parental education
50.3
Basic
0.0
59.0
51.2
55.5
42.6
27.2
Non tertiary
26.7
21.3
24.4
28.5
22.4
Tertiary
73.3
19.7
24.4
16.0
24.1
22.3
Note. Means and standard deviation (in parenthesis) for students’ age and percentages for
the remaining characteristics.

The results show that future teachers in the institutions participating in the study shared
similar characteristics in many of the categories listed in the table. Thus, the average age of firstyear students was 19.25 (SD=2.30), with institution A having the younger students enrolled in
this cohort (M=18, SD=1.9). The average for the fifth-year cohort was 23.16 (SD=2.40). These
results were expected, considering that Peruvian students mostly graduate from high school at 17
or 18 and spend about a year to prepare in academias to take the exam required for entering the
university. The average exit age also fits with the duration of the program, five years.
Regarding gender, education has been a career that attracts mostly women in many
countries, and institutions in Peru are no exception; Table 5.3 shows that 93.5% of the whole
sample was formed by female future teachers. The pattern was also observed by institution; in all
84

cases the vast majority of future teachers were female. Regarding how often the participant speak
Spanish at home, 95.1% of future teachers reported always spoke Spanish; variation was
observed by institutions, but in all cases the predominance of the Spanish language held.
Concerning socio-economical status, two indicators were used as proxies of such status:
books available at home and parental education level since the more economical resources of the
household, the more educational resources are likely to be at home; in the same way, higher
levels of parental education are related to higher socio-economic status. As expected, students
from the private institution (A) seemed to have more economical resources than their peers in the
public institutions (B, C, D, and E). Results show that the majority (69.1%) of students in all
institutions owned between 11-100 books at home; however, this highlights the case of the
private institution A, which presented the greater percentage of students having more than 200
books (20%). Regarding parental education, the survey examined this variable, considering eight
categories for both mother and father. These categories were collapsed into three levels: basic
(elementary and high school), non-tertiary (technical), and tertiary (undergraduate and graduate).
The percentage reported in the table represents the higher level of parental education at home
(either from the father or from the mother). Results show that 50.3% of future teachers came
from families where basic education was the highest level of parental education;
correspondingly, the percentages for non-tertiary and tertiary education were less. It is
noticeable, however, that this pattern corresponded only to student teachers from institutions B,
C, D, and E, which means public institutions. The majority of student teachers from institution A
(73.3%) were from families in which tertiary education was the highest parental educational
level. The remaining percentage, 26.7%, was formed by future teachers whose parents have nontertiary education.
85

Finally, Table 5.4 shows the motivations that future teachers had to pursue a teacher
career. Regarding reasons to become a teacher, future teachers were asked to assess the strength
of their possible reasons on a four-point scale: not a reason, a minor reason, a significant reason,
or a major reason. Only percentages for major or significant reason are provided in the table.

Table 5.4. Percentage of Endorsement to Reasons for Becoming a Teacher and Expectation
for Teaching as Career
Institutions

Total

A
Major or significant reason to
become a teacher
Talent for teaching
Working with young people
Influence on next generations
Teaching as challenging job
Attracted by teacher salaries
Long term security
Availability of teaching positions
Future in teaching
Expected to be a lifetime career
Could possibly be a lifetime career

B

C

D

E

86.7
80.0
80.0
86.7
0.0
20.0
13.4

90.0
81.3
89.8
78.7
5.0
27.9
11.9

78.1
76.9
97.6
90.4
5.3
12.8
15.4

92.1
78.0
90.4
81.6
9.9
38.8
21.1

87.7
70.7
92.9
84.5
1.7
12.0
6.9

88.7
77.2
91.2
83.1
6.1
26.9
15.3

60.0
33.3

59.7
30.6

50.0
47.6

57.3
26.0

72.4
25.9

59.7
30.2

The results show that future teachers mostly adhered to intrinsic reasons for becoming a
teacher, such as having the talent needed for teaching (88.7%), the opportunity to work with
young people (77.2%), the ability to influence future generations (91.2%), and the challenging
character of teaching (83.1%). On the contrary, future teachers adhered in less degree to extrinsic
reasons for teaching, such as availability of teaching positions (15.3%), teacher salaries (6.1%),
and security associated with the teaching career (26.9%). Regarding their willingness to take
teaching as a lifetime career, the results show that the majority of future teachers in all
institutions had the intention of staying in the career (60%), and an important percentage of
86

future teachers considered that as a very likely event (30.3%). These results imply that these
future teachers were committed to teaching over the years.
All in all, the findings show that the programs of teacher education share similarities in
their program plan, such as lasting 5 years, preparing generalist teachers, and providing bachelor
degrees to their graduates. Differences were observed in the number of credits and courses as
well as the years for the practicum; implications of the course work for the development of
mathematical pedagogical content knowledge and beliefs related to mathematics teaching will be
analyzed later. The philosophy of the teacher education programs expressed in the institutional
documents shows that institutions are not fully aware of the reform demands for mathematics
teaching. Although the data source was limited, it was clear that institutions have grasped the big
ideas about the way learning is thought over in the curriculum but have failed in incorporating
the specific ideas related to mathematics teaching; thus, only some of them addressed the
principles of the current school mathematics curriculum. It is also noticeable identifying that
pedagogical content knowledge was not part of the professional profiles on the institutions and
that, instead, attention to other aspects not related to student learning were part of such profiles.
Finally, in relation to student population, the institutions seemed to recruit students
having the same profile in terms of gender, age, language, and motivations to pursue a teaching
career. Differences in student recruitment were observed, however, in socio-economic status
indicators. The higher levels of parental education among future teachers in institution A, a
private school, suggest that this program attracts student with more resources and better basic
education, which might imply a stronger knowledge base than their counterparts on which to
build more knowledge for mathematics teaching. Thus results in the outcome variables could be
related to some kind of recruitment effect; I will further discuss this in the last chapter.
87

Research Question 2: Opportunities to Learn in Teacher Education Programs
In this section I examine the opportunities to learn to teach elementary mathematics
provided by the institutions to future teachers. Specifically, I examine the intended curriculum
based on the syllabi, and the implemented curriculum based on students’ reports. As I describe
each one of these components of opportunities to learn, I seek to identify patterns and variations
by institutions on such learning experiences.
The Intended Curriculum in the Teacher Education Programs
Overall, institutions planned to provide future teachers with mathematics and
mathematics pedagogy content as observed in the 20 syllabi collected showed in the Table 5.5.
Table 5.5. Mathematics and Mathematics Pedagogy Courses Used in the
Analysis

Institution A

Mathematics
courses
Logical mathematical
reasoning
Mathematics I

Institution B

Mathematics I
Mathematics II

Institution C

Development of
logical mathematical
thinking

Institution D

Mathematics I
Mathematics II

Institution E

Mathematics I
Mathematics II
Mathematics III
Mathematics IV
Mathematics V

Mathematics
pedagogy courses
Pedagogy of
mathematics I
Pedagogy of
mathematics II
Pedagogy for
mathematics teaching
in primary education
Pedagogy of
mathematics I
Pedagogy of
mathematics II
Pedagogy of
mathematics III
Pedagogy of
mathematics I
Pedagogy of
mathematics II

Total
4

3

4

4

5

88

Some differences; however, were observed in the way institutions organized the
mathematics curriculum to attend both areas of teacher knowledge. Institutions A and E had the
same organization of courses for mathematics education: two courses on mathematics content
and two courses on mathematics pedagogy. Institution B also has two courses on mathematics
content, but it has only one course on mathematics pedagogy; besides, it is the institution with
the lowest number of mathematics courses. Institution C has only one course on mathematics
content and three courses on mathematics pedagogy. Finally, although institution E seems to
teach only mathematics courses, it has five courses named “mathematics”, a quick look at its
syllabi reveals that both mathematics content and pedagogy are combined in each course.
These differences in the arrangement of courses could suggest the kind of content that
institutions plan to expose their future teachers but more analysis is necessary to identify the kind
of topics that are being included in each syllabus. This section presents the results for the syllabi
analysis for two kinds of topics: school mathematics topics and mathematics education pedagogy
topics. Here it is important to note that the goals and content mentioned in the syllabi represent
the intended curriculum, which reflects the intentions of each institution for learning
mathematics and mathematics pedagogy. No claim about the implementation of the syllabi in the
classroom (i.e. the curriculum in use) can be made from the analysis of documents, since such
information was not collected.
School mathematics topics. To analyze the school mathematics topics included in the
syllabi, I used as a reference the TEDS-M Framework for Coding Teacher Education Course by
Maria Teresa Tatto, Kiril Bankov, and Sharon Senk for the TEDS-M study. This framework
provides a list of codes to analyze the content included in the subjects of mathematics,
mathematics pedagogy, and general pedagogy included in the curriculum of teacher education
89

programs. In this study I only used the list of codes corresponding to the section on mathematics
content to analyze the syllabi.
The codes for mathematics topics in the TEDS-M framework are based on the Trends in
International Mathematics and Science Study (TIMSS) standards (Tatto, Bankov, & Senk, 2008),
which includes seven categories (I) numbers; (II) functions, relations, and equations; (III)
geometry; (IV) measurement; (V) data representation, probability and statistics; (VI) elementary
analysis; and (VII) validation and structure. These categories are compatible with the content
organizers included in the Peruvian national curriculum design which makes the framework
pertinent for this study.
For coding each syllabus, I verified whether the mathematical content listed in the TEDSM framework was present or not present in the syllabus. Table 5.6 presents the results of coding
22 topics included in the first five categories mentioned before. The check symbol shows
whether a specific topic was included in the syllabi of the institution; the numbers in the row
“total” under each category represent the total amount of topics (or number of checks) in each
category.

13

13

As mentioned before, the TEDS-M framework also includes elementary analysis, validation,
and structure as themes for coding, but they are not included in Table 5.6. The reason is that no
content related to elementary analysis was found and validation and structure include logical
connectives and truth tables which are not part of the curriculum for elementary mathematics.
However, due to topics related to logic being identified in all the syllabi it was coded as
advanced mathematics as will be showed in Table 5.7.
90

Table 5.6. School Mathematics Content in the Syllabi of Teacher Education Programs

Topics

A

I. Numbers
1. Whole numbers
2. Fractions and decimals
3. Integer, rational, and real numbers
4. Other numbers, number concepts, and number
theory
5. Estimation and number sense concepts
6. Ratio and proportionality
Total
II. Functions, relations and equations (algebra)
1. Patterns, relations and functions
2. Equations and formulas (Inequality)
3. Trigonometry and analytic geometry
Total
III. Geometry
1. 1-D 2-D coordinate geometry
2. Euclidean geometry
3. Transformational geometry
4. Congruence and similarity
5. Construction with straightedge and compass
6. 3-D geometry
7. Vector geometry
8. Simple topology
Total
IV. Measurement
1. Measurement units
2. Computations and properties of length, perimeter,
Area, and volume
3. Estimation and error
Total
V. Data representation probability and statistics
1. Data representation and analysis
2. Uncertainty and probability
Total
Total number of content included (out of 22)

91

B








Institutions
C
D






6


3



5










2

2





2
















E



3


5







2






2







5

0

3






1

2




3


2
18

3



0


0
6

1
13

4

2


0
8

1
14

The results show variations regarding the treatment of mathematical content across the
institutions. Thus, institutions A, E, and C, in that order, are the ones that, in total, plan to cover
the most topics from the framework (18, 14 and 13 respectively out of 22). By contrast,
institutions B and D include fewer topics on school mathematics in their courses (six and eight
respectively). Given that the categories proposed by the TEDS-M framework resemble the
national curriculum organizers (number, geometry and measurement, and statistics), it is possible
to argue that the syllabi of institutions A, C, and E, in particular the first one, align more with the
expectations posed by the curriculum in terms of mathematics content; this means that future
teachers from these institutions might be exposed to more school mathematics content, and
therefore could have more mathematical content knowledge to teach these topics. On the
contrary, syllabi of institutions B and D show little alignment with the school curriculum as they
are not planning to cover all topics needed to teach mathematics; this also implies that their
graduates could end the program without a strong mathematics content knowledge.
Table 5.6 also shows clearly more concentration on topics related to numbers and
functions across all institutions; it means that all programs mostly included those topics in their
syllabi. The topics of geometry receive less attention, especially in institution B whose syllabi
did not include any topic from this category. The topics of measurement also seem to receive less
attention from the institutions, especially from institutions B and D that include only one topic
and no topic respectively in their syllabi. Finally, regarding data representation and statistics,
only institution A covers fully the topics of the framework, while institutions C and E made it
partially. Data revealed also that institutions B and E did not include these themes in their
syllabi. These findings show that the curriculum for mathematics education emphasize mainly
topics related to numbers and functions and neglect other areas that are also part of the school
92

curriculum and that need to be mastered by future teachers. The absence of topics on geometry,
measurement, and data representation in their syllabi, especially for institution B and D, implies
that programs of education are not properly considering the national curriculum mandates for
mathematics teaching at the time they plan their curriculum.
The syllabus analysis allowed identifying that some institutions included in their syllabi
content that might be considered advanced level. Thus, although advance mathematics content
was not the focus of this study, I used the corresponding TEDS-M framework to code them to
see how recurrent this was in the institutions. The TEDS-M framework for advanced
mathematics considers twenty topics, such as axiomatic geometry, non-Euclidean geometry,
differential geometry, topology, abstract algebra, calculus, differential equation, functional
analysis, discrete mathematics, graph theory, and game theory, among others. Table 5.7 shows
the advanced content topics that could be identified in the syllabi.

Table 5.7. Advanced Mathematics Content in the Syllabi of Teacher Education Programs
Topics
1. Set theory
2. Mathematical logic (truth table, symbolic logic,
propositional logic, set theory, binary operations)
3. Analytic/coordinate geometry (e.g. equations of
lines, curves, conic sections, rigid transformation)
Total

A


Institutions
B
C
D




E








2




2





3

3

2

The table shows that institutions emphasize mainly three advanced content areas: set
theory, mathematical logic, and analytic/coordinate geometry. Among these, set theory and logic
are intended to be taught in all institutions. Likewise, content on analytic geometry were
included in the syllabi of institutions C and D. Overall these results show that advanced
93

mathematics, as could be expected, is not usual in the programs for elementary education in the
participating institutions.
Mathematics education pedagogy topics. The syllabi analysis allowed identifying two
kinds of topics in this area: topics that aim to provide future teachers with an overview on what is
involved in mathematics teaching and topics that aim to provide future teachers with knowledge
on how to teach specific mathematics content. Regarding the first kind of topic, I used item 4 on
the section of opportunities to learn included in the future teacher questionnaire as a rubric to
code the contents on this area (See Appendix A). Table 5.8 shows the mathematics education
pedagogy topics found in the syllabi.

94

Table 5.8. Mathematics Education Pedagogy Topics in the Syllabi of Teacher Education
Programs
Topics
A
1) Foundations of Mathematics (e.g., mathematics
and philosophy, mathematics epistemology,
history of mathematics)
2) Context of Mathematics Education (e.g., role of
mathematics in society, gender/ethnic aspects of
mathematics achievement)
3) Development of Mathematics Ability and
Thinking (e.g., theories of mathematics ability
and thinking; developing mathematical concepts;
reasoning, argumentation, and proving;
abstracting and generalizing; carrying out
procedures and algorithms; application;
modeling)
4) Mathematics Instruction (e.g., representation of
mathematics content and concepts, teaching
methods, analysis of mathematical problems and
solutions, problem posing strategies, teacherpupil interaction)
5) Developing Teaching Plans (e.g., selection and
sequencing the mathematics content, studying
and selecting textbooks and instructional
materials)
6) Mathematics Teaching: Observation, Analysis
and Reflection
7) Mathematics Standards and Curriculum
8) Affective Issues in Mathematics (e.g., beliefs,
attitudes, mathematics anxiety)
Total number of content (out of 8)

Institutions
B
C
D

E
































3

3


7

2

3

The table shows that Institution A planned to cover more topics on mathematics
education pedagogy, 7 out of 8. The syllabi belonging to the other institutions included a smaller
number of topics, mostly three topics of the eight proposed. These results indicate that most
institutions are not providing future teachers with the knowledge required to have a deep
understanding of what teaching mathematics involve.
95

Observing the kind of topics addressed in the syllabi, it highlights that institutions aimed
to offer their future teachers topics that would help them to perform a mathematics lesson
(represented in topic 4, 5, and 6) and that less emphasis is put on topics that would help them to
develop a critical view of mathematics as part of the school curriculum (topics 1, 2, and 3).
It also highlights that institutions are failing in planning content that allow future teachers
to understand the principles that support the current reform for mathematics teaching since topics
such as development of mathematics ability and thinking (topic 3) and topics on standards and
curriculum (topic 7) are not being included in their syllabi. Topic 3 can help future teachers to
recognizing the cognitive aspect of mathematics learning which has inspired the current
curriculum but it is only addressed by institutions A and C. Accordingly, these aspects were
explicitly highlighted in their syllabi of mathematics and mathematics pedagogy. Topic 7, which
can help future teachers to learn about the national curriculum and its mandates for mathematics
teaching, as well as problem solving in mathematics, is only addressed by institutions A, D, and
E. The absence of these topics in the mathematics pedagogy courses is critical since future
teachers need to become familiar with the curriculum characteristics so that they can know and
learn the content, principles, and approach that the MED has taken. In the same way, as
explained before, the national curriculum design has adopted problem solving as an approach to
learning mathematics, and it addresses how to deal with problem solving, which may help future
teachers to implement the curriculum as expected.
The second kind of topics found in the syllabi is related to the instruction of specific
mathematics content. Syllabi collected showed that institutions planned to teach future teachers
the pedagogy related to number, geometry, etc. To code these topics I used the mathematics
content listed in Table 5.6 as a reference to examine the extent to which institutions planned for
96

future teachers to learn how to teach the mathematics content they had learned. Table 5.9 shows
the results of this coding process.
Table 5.9. Instruction of Mathematics Content in the Syllabi of Teacher Education Programs
Topics
Institutions
A
B
C
D
E
I. Numbers
1. Whole numbers








2. Fractions and decimals


3. Integer, rational, and real numbers

4. Other numbers, number concepts, and number


Theory




5. Estimation and number sense concepts



6. Ratio and proportionality
Total
6
3
4
2
5
II. Functions, relations and equations (algebra)
1. Patterns, relations and functions



2. Equations and formulas
3. Trigonometry and analytic geometry
Total
0
1
0
0
2
III. Geometry


1. 1-D 2-D coordinate geometry





2. Euclidean geometry

3. Transformational geometry

4. Congruence and similarity

5. Construction with straightedge and compass

6. 3-D geometry

7. Vector geometry
8. Simple topology
Total
4
0
3
1
4
IV. Measurement



1. Measurement units
2. Computations and properties of length,




perimeter, area, and volume

3. Estimation and error
Total
3
1
2
0
2
V. Data representation probability and statistics


1. Data representation and analysis

2. Uncertainty and probability
Total
1
0
1
0
1
14
5
10
3
Total (out of 22)
14
97

Findings indicate that institutions A and C are the ones that plan to teach the pedagogy of
more mathematics content, especially institution A. In the same way, institutions B and D do not
include many of these topics. Likewise, institutions mostly included topics about the instructions
of numbers and geometry in their mathematics pedagogy courses. By contrast, syllabi did not
provide evidence that teacher education programs plan to expose future teachers to the
instruction of topics on function, measurement, and statistics.
These results maintain somehow the pattern found in the analysis of mathematics content.
However, a comparison between Table 5.6 and Table 5.9 indicate that there is no complete
alignment between the mathematics content and the pedagogy of such content with the exception
of institution E which is explained later. Setting aside this case, future teachers from other
institutions might be exposed to a certain amount of mathematics content, but they might not be
taught how to teach that content. For instance, Table 5.6 shows that all institutions include topics
on functions, relations, and equations in their syllabi; however, none of this content is included in
the pedagogy courses. Only institution C includes a topic from this category.
Regarding institution E, the complete alignment between school mathematics and
mathematics education pedagogy has to do with the way this institution organizes its syllabi.
This institution has five courses on mathematics, and all of its syllabi have a section on pedagogy
in which it is stated that students learn the pedagogy of the mathematical content listed in the
syllabus. Because this explanation was unclear, I actually called representatives of this institution
to ask about the pedagogical treatment of the mathematical content. Their response confirmed
what was written in the document; thus the checks in Table 5.9 are the same as in Table 5.6.

98

The Implemented Curriculum in the Teacher Education Programs
The future teacher survey included questions about course topics, activities oriented to
help future teachers in learning to teach mathematics, and reflective experiences that characterize
their preparation. In this section, each of these variables is examined in order to know what
opportunities to learn were offered to the cohort of fifth year students in each institution.
Course topics in teacher education programs. The survey asked future teachers to
answer whether they studied or did not study a list of topics related to school mathematics and
mathematics education pedagogy. The questionnaire also explored themes on tertiary level
mathematics, but they were not analyzed since, as stated before, the study plan of the
participating institutions did not include courses on advance mathematics apart from the ones
devoted to teach school level mathematics.
Here it is important to mention that the results of the intended curriculum based on the
syllabi might be different than the implemented curriculum based on students’ reports; thus it is
likely that both results do not match. Likewise, the syllabi provided by the institutions do not
necessarily belong to the classes where students in the sample were enrolled, and also student
teachers report might not be accurate since they have to remember what happened a semester ago
or more.
School mathematics topics. As explained in the instrument section, student teachers were
asked to report whether they studied or did not study “numbers,” “measurement,” “geometry,”
“functions, relations and equations,” “data representation, probability, and statistics,” “calculus,”
and “validation, structuring, and abstracting.” The first five topics correspond with the ones
recommended in the national school curriculum for elementary education, and the last two topics
are most suitable for future secondary teachers but were also included in the survey. The analysis
99

and interpretation were done for the first five topics. Table 5.10 shows the percentage of fifthyear student teachers who reported having studied these topics in each institution, and also for
the total sample.
Table 5.10. Percentage of 5th Year Students who Studied School Mathematics Topics
Institutions

Numbers
Measurement
Geometry
Functions, relations, and equations
Data representation, probability, statistics
Calculus
Validation, structuring, and abstracting

A
100.0
100.0
90.0
80.0
90.0
20.0
30.0

B
77.5
72.5
40.0
75.6
77.5
22.0
27.5

C
100.0
84.6
23.1
53.8
84.6
0.0
7.7

Total
D
100.0
74.5
43.6
53.8
84.6
0.0
7.7

E
100.0
100.0
87.0
100.0
100.0
47.8
43.5

93.7
80.9
51.1
80.4
88.0
27.3
25.4

The percentage corresponding to the total sample shows that “numbers” (whole numbers,
fractions, decimals, integers, rational and real numbers, number concepts, number theory,
estimation, ratio and proportion) is the most frequently covered topic in the participating
institutions. “Data representation, probability, and statistics” is the second group of topics most
covered in the institutions. “Measurement” (i.e. measurement units, computations and properties
of length, perimeter, area, and volume; estimation and error) and “functions, relations, and
equations” (algebra, trigonometry, and analytic trigonometry) are the third and fourth themes
most covered, respectively for the total sample. Finally, “geometry” (1-D and 2-D coordinate
geometry, Euclidean geometry, transformational geometry, congruence and similarity and
others) is the content for elementary mathematics that is less covered in the aggregated sample.
The variation in future teachers’ report by institution does not allow for easily identifying
patterns in Table 5.10, except that the largest percentages are often those from institution E,
100

whose students reported having studied topics for elementary mathematics and for advanced
mathematics at a higher percentage. Institution A is the second institution that covers these five
themes well. Regarding the remaining institutions, B, C, and D, the majority of students reported
having studied the topics of “numbers,” “ measurement,” “fucntions,” and “data representation,”
but they had a lower percentage in “geometry.” These findings indicate that not all future
teachers have the same opportunities to develop the content knowledge required to teach
elementary mathematics but that acquiring such knoweldge will depend on the institutions in
which they are enrolled.
From Table 5.10, it also highlights that students are mostly exposed to topics required for
teaching elementary school mathematics (the majority of students reported having studied them),
rather than topics of advanced mathematics (a smaller percentage of students reported having
studied them). These results are compatible with the ones founded in the syllabi, as future
elementary teachers are not exposed to advanced mathematics.
Mathematics education pedagogy topics. The topics for mathematics education
pedagogy examined in the questionnaire were “foundation of mathematics,” “ context of
mathematics education,” “ development of mathematics thinking,” “ mathematics instruction,”
“developing teaching plans,” “mathematics teaching (observation, analysis, and reflection),”
“mathematics standards and curriculum,” and “affective issues in mathematics.” Table 5.11
shows the percentage of students who stated they had studied these topics, by the institutions and
for the total sample.

101

Table 5.11. Percentage of 5th Year Students Who Studied Mathematics Education Pedagogy
Topics.

Foundations of math
Context of math education
Development of math ability and thinking
Math instruction
Developing teaching plans
Math teaching
Math standard and curriculum
Affective issues in math

A
60.0
50.0
80.0
100.0
100.0
100.0
70.0
60.0

Institutions
B
C
58.5
61.5
39.0
38.5
36.6
76.9
65.0
84.6
57.5
84.6
46.3
76.9
43.9
23.1
36.6
23.1

Total
D
60.7
40.0
49.1
80.0
77.8
78.6
67.9
41.1

E
60.9
60.9
52.2
73.9
87.0
65.2
56.5
50.0

60.1
43.7
50.7
76.6
75.7
68.5
55.2
40.8

The first thing to note is that the coverage percentages of mathematics education
pedagogy topics are not as large as the covererage percentages reported for the school
mathematics topics; Table 5.11 shows that the reported coverage for many topics in the
mathematics pedagogy area were below 60%. This means that, apparently, future teachers are
less exposed to topics that can help them have a comprehensive view of what is involved in
mathematics teaching, beyond the mathematical content itself. Given this pattern of response,
Table 5.11 shows that more fifth-year students have been exposed to topics related to
“mathematics instruction” (i.e. representation of mathematics topics, analysis of problems, etc),
“developing teaching plans” (i.e., sequencing the mathematics content, selection of educational
materials, etc), and “mathematics teaching” (observation, analysis and reflection); all these are
topics that can help future teachers to develop their practical knowledge for mathematics
teaching. By contrast, the topics least often experienced by students but just as crucial for the
purposes of the reform, are “foundation of mathematics,” “development of mathematics ability
and thinking,” and “mathematics standards and curriculum.” The first topic can allow future
teachers to examine the philosophy and epistemology of mathematics so have a broader view of
102

this subject area. The second and third topic can help future teachers to understand the basis for
mathematics teaching and gets them closer to the spirit of the reform. The low emphasis put for
the teacher education programs to topics that would help future teachers to have a complete
picture of the nature of the reformed curriculum, also identified in the syllabi analysis section,
leads to question about the extent in which institutions are really aware of the curriculum
demands and the need of addressing them as part of the future teacher preparation.
The second noticable pattern here is that institution A again has greater percentages than
do other institutions for many of the mathematics pedagogy topcis listed in Table 5.11 while
institution B has the smallest percentage coverage for most of the topics, a pattern that was
observed also in the syllabi analysis section. The percentages reported for the students of the
remaining institutions (C, D, and E) are hard to differentiate. Comparing means could be more
useful for purposes of comparing by institutions. To facilitate this kind of analysis, TEDS-M
proposed scales in which some topics are conceptually grouped and aggregated to form a scale
(i.e., in school level mathematics, the topics of numbers, geometry, and measurement are part of
a scale). This analytic approach helps to get institutional means for each scale and to perform
comparisons by using MANOVA. Multivariate analysis was performed, but the results were not
reliable and therefore are not reported here.
Activities for learning to teach mathematics. In examining opportunities to learn, the
TEDS-M survey included items about the kind of learning activities in which future teachers
engaged (class participation, class reading, and solving problems). Other TEDS-M items asked
future teacher whether they engaged in activities that provided opportunities to learn particular
pedagogical practices (i.e. instructional practice, instructional planning, and assessment practice).

103

The table below shows the means and standard deviations of the scales that measured these
variables in each institution.
Table 5.12. Means and Standard Deviation of Activities for Learning to Teach
Mathematics Scales.
Institutions
Total
A
B
C
D
E
Class participation
3.48
2.72
2.92
2.97
3.35
2.99
(0.58)
(0.73)
(0.41)
(0.55)
(0.53) (0.64)
Class reading
2.95
2.37
2.42
2.59
2.86
2.58
(1.07)
(0.87)
(0.81)
(0.71)
(0.69) (0.80)
Solving problem
2.98
2.53
2.65
2.46
3.30
2.67
(0.73)
(0.73)
(0.66)
(0.74
(0.41) (0.74)
Instructional practice
3.53
2.88
3.33
3.15
3.44
3.16
(0.63)
(0.65)
(0.44)
(0.49)
(0.48) (0.58)
Instructional planning
3.29
3.05
3.21
3.15
3.42
3.18
(0.71)
(0.54)
(0.32)
(0.57)
(0.52) (0.55)
Assessment practice
3.18
2.92
2.92
2.99
3.13
3.00
(0.52)
(0.60)
(0.53)
(0.59)
(0.55) (0.58)
Note: N=143. Standard deviation in parentheses. Scale measured with Likert scale:
1= never, 2 = rarely, 3 = occasionally, and 4 = often.

Regarding the activities in which future teachers were involved, Table 5.10 shows that
students from all institutions reported having occasional opportunities to participate in
classrooms (M=2.99. SD=0.64), which means that on average they had occasional chances to ask
questions, participate in whole class discussions, make presentations, and teach lessons to the
rest of the class. Regarding “class reading,” the average mean was lower (M=2.58, SD=0.80);
students might have had fewer opportunities to be exposed to activities that included reading
research about mathematics and mathematics pedagogy, as well as analyzing videos or cases on
exemplary teaching. Activities related to “solving problems,” such as writing mathematical
proofs and solving problems using several strategies, were also reported as something happening
rarely or only occasionally in their classrooms (M=2.67, SD=0.74).
104

Regarding pedagogical practices, the average means are slightly greater than the ones
described above, especially for “instructional practice” and “assessment practice,” but it can still
be concluded that future teachers are occasionally engaged in activities that help them to learn
how to enact these practices. Thus instructional practice, which addresses opportunities to learn
to teach mathematics in a comprehensive way (by means of problem solving, applying strategies,
differentiating between concepts and procedures, etc.) has an average mean of 3.16 (SD=0.58).
Instructional planning, which examines the opportunities to learn to develop materials, explore
the use of manipulatives, and design lessons that meet students needs and interests, has a
marginal mean of 3.18 (SD=0.55). Finally, assessment practice (which includes assessing
mathematics learning goals, examining the national mathematics curriculum, and analyzing
students’ academic outcomes to improve assessment) got an average mean of 3.00 (SD=0.58).
Table 5.10 also shows some variation in the institutional means in each one of the
learning- to-teach scales, and among them, institutions A and E have the greater means;
however, significant differences cannot be asserted unless more statistical analysis is done.
The effect of institution on the activities for learning to teach mathematics scales. I
performed one way multivariate analysis of variance (MANOVA) to examine the extent to
which variation of learning to teach scales implied differences among the institutions. The
assumption of homogeneity of the covariance matrices required for MANOVA analysis as
assessed by Box’s M test was met (p = .071). Table 5.13 presents the results of the multivariate
test.

105

Table 5.13. Mutivariate Analysis of Variance for the Activities for Learning to Teach
Mathematics Scales.
2

Source

Wilks' Λ

df

Error

F

p

η

Institution (I)

.648

24

430.306

2.377

.000

.103

Note. Multivariate F ratio was generated by Wilks Lambda statistic.
The results show that there was a statistically significant difference between the
institutions on the combined dependent variables, F (24, 430.306) = 2.377, p < .001; Wilks' Λ =
2

.648; partial η = .103. The univariate main effects of institution (dependent variable) on the six
scales of opportunities to learn (independent variables) were examined with univariate analysis
of variance (ANOVA), and the results are presented in Table 5.14.

Table 5.14. Univariate Analysis of Variance for the Activities for Learning to Teach
Mathematics Scales.
Source

Dependent
Variable
Class
participation
Class
Reading
Solving
problem
Instructional
practice
Instructional
planning
Assessment
practice

Institution

2

df

SS

MS

F

p

η

4

7.411

1.853

5.326

.001

.968

4

4.188

1.047

1.710

.152

.512

4

12.228

3.057

3.057

.000

.968

4

6.587

1.647

5.503

.000

.973

4

1.740

.435

1.488

.586

.451

4

.963

.241

.710

.152

.225

Note. N=133
Univariate ANOVAs showed that class participation scores ( F (4, 128) = 5.326, p <
2

2

.008; partial η = .968), solving problem scores (F(4, 128) = 3.057, p < .008; partial η = .968),
106

2

and instructional practice scores (F(4, 128) = 5.503, p < .008; partial η = .973.) were
significantly different statistically between the future teachers from different institutions, using a
Bonferroni adjusted α level of .008. This correction was set by dividing 0.05 by 6, because there
were six dependent variables.
Post-Hoc tests for differences among institutions. Table 5.15 shows the multiple
comparisons for scales where the null hypothesis of the univariate ANOVA means being equal
throughout the sample was rejected. These comparisons are based on least significant differences
(LSD). Means and standard deviations are reported again because these are calculated based on
the default list-wise deletion used to perform the univariate ANOVA. The cells in the matrix of
comparisons represent the value of the mean of the institution on the row minus the mean of the
institution in the column, i.e., 0.76 is the result of 3.48 (A) minus 2.72 (B). Mean differences are
reported only if they are significant at the .05 level; otherwise they are reported as “n.s.” Positive
values in the cells imply that the institutional mean on the row was greater than the institutional
mean, in the column, and negative values imply that the institutional mean in the column was
greater that the institutional mean in the row.

107

Table 5.15. Multiple Comparisons for Activities for Learning to Teach Mathematics Scales
Scales

Institution n
M
SD
A
B
C
D
A
10 3.48 0.58
0.76
0.55
0.51
C
38 2.72 0.73
n.s
n.s
Class
D
12 2.93 0.43
n.s
participation
E
52 2.97 0.53
F
21 3.31 0.52
A
10 2.97 0.73
n.s
n.s
0.48
C
38 2.50 0.71
n.s
n.s
Solving
D
12 2.73 0.64
n.s
problem
E
52 2.46 0.74
F
21 3.31 0.41
A
10 3.53 0.63
0.66
n.s
n.s
C
38 2.87 0.65
-0.46
-0.30
Instructional
D
12 3.33 0.44
n.s
practice
E
52 3.17 0.49
F
21 3.44 0.49
Note. Differences are significant at p < .05 level. Non significant differences: n.s.

E
n.s
-0.59
n.s
-0.34
n.s
-0.81
-0.58
-0.85
n.s
-0.57
n.s
n.s
-

Overall, most significant differences involved comparisons between institutions A and
the others, and between E and the others. This implies that in many cases their institutional
means were greater than the ones with which they were compared. Thus, students from
institution A reported having more opportunities to be involved in “class participation” than their
counterparts in institutions B, C and D; the same was true with students in institution E in
relation to students in institution B and D. In “solving problem,” the institutional mean for E
was significantly different from institutions B, C, and D, meaning that on average its students
were engaged more often in activities related to mathematically solving problems. Institution A
also had a greater mean than institution E on this scale. In “instructional practice,” it is noticeable
that institution B had the smaller means than all the other institutions, including A and E; this
implies that students in institution B were less exposed to activities that would help them to learn
108

to teach to use mathematics to solve daily problems, to explore strategies for solving problems,
and to distinguish between procedural and conceptual approaches among others.
Reflection in teacher education programs. The questionnaire examined whether fifthyear students developed strategies for self-reflection and critical analysis of their performance
and learning needs. Table 5.16 shows descriptive statistics for this variable.

Table 5.16. Means and Standard Deviations for Reflection Scale
Institutions
Total
A
B
C
D
E
Reflection
3.20
2.84
2.69
3.21
3.35
3.08
(0.82)
(0.70)
(0.73)
(0.49)
(0.64) (0.66)
Note. N=141. Standard deviation in parentheses. Scale measured with Likert scale:
1= never, 2 = rarely, 3 = occasionally, and 4 = often.

Results show that students from all institutions reported to have been involved in these
kinds of activities occasionally. Likewise, the means of the institutions E, D, and A seems to be
greater than the others, which would imply that students from the former institutions had more
opportunities for reflection on teaching.
A one-way ANOVA was used to test for differences among the five institutions. The
homogeneity of variance, as assessed by Levene’s Test, was met (p =.55), and the results are
shown in Table 5.17.
Table 5.17. One-Way Analysis of Variance for Reflection Scale
Source
Between Groups
Within Groups
Total

df
4
136
140

SS
6.836
53.306
60.142

MS
1.709
.392

109

F
4.360

p
.002

2

η
0.114

The results show that not all group means were equal (at least one institutional mean in
2

this scale was different), F (4, 136) = 4.36, p = .002, partial η = 0.114. Thus, some institutions
are exposing their students more than others to activities so that they become inquiring
pedagogues. Table 5.18 shows the multiple comparisons between institutions, to identify where
the differences were.

Table 5.18. Multiple Comparisons of Reflection Scales
Institution n
M
SD
A
B
C
D
A
10
3.20
.82
n.s
n.s
n.s
C
41
2.84
.70
n.s
-.36
Reflection
D
12
2.69
.73
-.52
E
56
3.21
.49
F
22
3.35
.64
Note. Differences are significant at p < .05 level. Non significant differences: n.s.

E
n.s
-.51
-.66
n.s
-

These results show significant differences for institutions D and E when compared to
institutions B and C. Thus, students in the latter institutions were more consistent in stating that
they were exposed occasionally to reflective activities. It is worth noting that the institutional
mean for A was one of the larger ones; however, it also reported greater variability in responses,
and this did not help to establish differences with the remaining institutions.

110

The relationship between opportunity to learn to teach mathematics scales. The
activities for learning to teach mathematics and reflection scales are part of the opportunity to
learn scales assessed by TEDS-M. Table 5.19 shows the intercorrelations between these scales.

Table 5.19. Intercorrelations between Opportunity to Learn to Teach
Mathematics Scales

1
2
3
4
5
6
7

Measures
Class
participation
Class
reading
Solving
problem
Instructional
practice
Instructional
planning
Assessment
practice
Reflection

1
1

2

3

4

5

6

7

.61** 1
.52** .59** 1
.49** .51** .42** 1
.41** .47** .33** .75** 1
.43** .39** .30** .63** .62** 1
.34** .54** .33** .63** .57** .56** 1

Overall, there are positive moderate intercorrelations between all the scales of
opportunity to learn assessed in the questionnaire which makes sense considering that the means
reported by future teachers in each scale are similar.
All in all, the analysis of the intended and implemented curriculum provided evidence
that institutions are not being consistent with the reform demands which require future teachers
to develop the solid content knowledge and pedagogical content knowledge to teach
mathematics. The course topics related to mathematics education and the learning experiences
show some limitations in the opportunities to learn future teachers receive in their programs.

111

The data showed that future teachers are not exposed to all the content required for
teaching elementary school mathematics, at least not in all institutions. While the national
curriculum considers three main topics: “number, relations and operations”, “geometry and
measurement”, and “statistics”, all institutions emphasize mostly “numbers, relations and
operations,” as found in the syllabi and questionnaires. The remaining school mathematics topics
showed greater variation in the syllabi than in the questionnaires. For instance, “geometry and
measurement”, and “statistics” were mostly absent in institution B and D’s syllabi while student
teachers in these institutions did report having studied them. This mismatch could be due to data
coming from different sources: syllabi and self-report and that in the case of the questionnaire
future teachers were asked to remember something that happened years ago. Despite this
possible explanation, it calls to attention the fact that documents do not show that institutions are
taking into account the curriculum mandates to organize the learning experiences future teachers
will be exposed.
Analysis of intended and implements curriculum also shows some flaws in the topics
about mathematics pedagogy; future teachers have few opportunities to learn about the
foundations for mathematics teaching and also few chances to study the curriculum as found in
the syllabi and as it was reported by student teachers. Studying these topics of mathematics
pedagogy has important implications for future teachers’ knowledge, and its absence in the
mathematic curriculum can hinder future teachers’ chances to learn more about current views of
mathematics present in the curriculum.
I also expected some alignment between the mathematics topics and mathematics
instruction of those topics, the alignment between the two kinds of topics on the syllabus would
indicate that, in theory, institutions recognize the importance of addressing mathematics content
112

knowledge and mathematical pedagogical content knowledge and that they are planning to
provide future teachers with that knowledge. This was not the case for institutions B and D, in
which to the problem of limited coverage of some mathematical topics, was added little
alignment between school mathematics topics and the instruction of such topics. Deficiencies in
both domains hinder the possibility of future teachers to facilitate the development of problem
solving, reasoning and demonstration, and mathematical communication recommended in the
reform mathematics curriculum. Institutions A, C, and E planned to cover more topics on
mathematics content and more topics on how to teach that content but still the alignment was not
complete. This could be because courses on mathematics pedagogy do not only include
instruction on mathematics topics but also others like planning, assessment, etc.
Regarding the activities for learning to teach mathematics examined through the
questionnaire, future teachers reported that they mostly had the opportunity of being exposed to
meaningful learning experiences (solving problem activities in lesser extent), and reflection.
Exposure to these learning experiences is crucial for the process of learning to teach
mathematics; however, the extent in which they are beneficial for the acquisition of
mathematical pedagogical content knowledge and beliefs related to mathematics teaching will be
examined in research question five.

113

Research Question 3: Future Teachers’ Mathematical Pedagogical Content Knowledge
and Teacher Education Programs
As explained in the methodology section, the survey used in this study included five
items on mathematical pedagogical content knowledge (MPCK), which were aggregated to get a
total score that is named the MPCK index. The variable was used to examine the knowledge for
teaching mathematics of the first-year students (entry students) and the fifth-year students (exit
students) and how students varied across institutions. Differences by institution and cohort were
also examined. The knowledge acquired by future teachers when they are exiting their program
would indicate the extent to which their teacher education program has accomplished its main
goal, and the knowledge demonstrated at the beginning of the program would indicate the
potential knowledge future teachers have for learning to teach mathematics. Expectations are that
the exit cohort gets greater scores than the entry cohort since items are about specialized
knowledge that future teachers about to start their career should master.
Future Teachers’ Performance in the MPCK Items
Table 5.20 shows the performance of Peruvian future teachers in the items included in the
questionnaire. In the table, the column “full” represents the percent of students who got the full
credit (2), the column “partial” represents the percentage of students who got the partial credit
(1), and the column “no score” represents the percentage of future teachers who did not answer
the item or who did answer incorrectly (0). Likewise, the table also shows the performance
demonstrated by the international sample and Chilean sample in those items. It was done with
the purpose of establishing a benchmark to qualify the future teachers’ performance.

14

The performance of future teachers by institutions is provided in Appendix C.
114

14

The

Chilean sample was used for this comparison because of the similar problems Chile and Peru
face to improve students’ academic achievement. International evaluations of academic
achievement such as PISA show both countries in the low positions in the ranking, although
Chile has always shown a better performance.
Table 5.20. Percentage of Future Teachers by Item and Score
Item

Full
(2)
MFC505
Jose

Sample

International
Chile
st
Peru 1 year
th

MFC502B
Difficulty

Peru 5 year
International
Chile
st
Peru 1 year

MFC513
Paper clip

Peru 5 year
International
Chile
st
Peru 1 year

th

th

MFC208A
Jeremy

Peru 5 year
International
Chile
st
Peru 1 year

MFC208B
Diagram

Peru 5 year
International
Chile
st
Peru 1 year

th

Score
Partial
(1)

No score
(0)

76.7
76.9
70.9

19.9
17.1
22.4

3.4
5.9
6.7

65.7

22.4

11.9

22.7
11.4
1.2

50.5
62.2
73.9

26.9
26.5
24.8

0.0
9.1
9.1

63.6
39
34.9

36.4
51.9
56.1

0.0

10.9

89.1

0.0
20
6.4
11.5

25.2
11.6
2.4
10.9

74.8
68.5
91.3
77.6

10.5

3.5

86.0

15.9
3.8

16.4
4.7

67.7
91.4

1.2

2.4

96.4

th

Peru 5 year
4.2
3.5
92.3
Source: TEDS-M almanacs for international and Chilean sample and Peruvian data.

115

The rates responses in all samples show a pattern that remains for all the groups. Where
international future teachers did well, Peruvian and Chilean future teachers also did well (item
“Jose”) or relatively well (item “Difficulty”). Also, where international sample show difficulties,
Chilean and Peruvian future teachers also had difficulties (items “Paper clip”, “Jeremy”, and
“Diagram”). Also, results show that future teachers performed less well than the international
sample and the Chilean sample; Peruvian students in the first and fifth year had more difficulties
that students in these sample to properly solve the items of the questionnaire.
Regarding the Peruvian sample, the rate responses for the items showed in Table 5.18 and
the items’ characteristics described in Table 4.10 provide the frame to interpret their performance
in terms of mathematics pedagogical content knowledge. Thus, it is possible to state that in
general future teachers from both cohorts were mostly able to identify when some problems or
activities could be hard for students, but they had trouble in providing thoughtful and precise
answers that showed their knowledge on why this should be the case. They could not provide
pedagogically valid reasons to plan and select the appropriate activities or methods for
representing mathematical ideas; they had trouble diagnosing students’ misconceptions; and they
showed limitations in representing mathematical concepts or procedures.
Specifically, most future teachers could identify the elements of single-step story
problems that created difficulty for first grade students, as measured by the “Jose” item. They
had trouble in identifying how language could make it hard for primary students to solve a
problem on data as in the “Difficulty” item. Limitations were also noticeable for finding
thoughtful reasons that support certain activities in measurement (the “two clips” item). Future
teachers could not provide a clear diagnosis on student confusions with the results of arithmetic
operations with decimal numbers, as observed in the “Jeremy” item; mostly, they attributed the
116

mistake to his lack of understanding decimal numbers. Finally, future teachers had trouble
representing mathematical concepts related to multiplication with decimal numbers (the
“Diagram” item); the majority could not come up with a diagram to explain this notion, and
instead they provided as an answer an equation or written calculation of 2 x 0.6.
Future Teachers Performance in the MPCK Index
Table 5.21 provides a summary of the main descriptive for the MPCK index which, as
already mentioned, is the aggregate of the five items included in the questionnaire; thus the
maximum score for this index is 10 and the minimum is 0.

Table 5.21 Means and Standard Deviation of the MPCK index by Cohort and Institution

1st year
5th year

A
3.00
(1.41)
4.10
(1.19)

Institutions
B
C
2.71
3.65
(1.14)
(1.61)
2.53
4.15
(1.38)
(1.57)
st

Total
D
2.56
(1.38)
2.28
(1.27)

E
3.11
(0.86)
3.13
(1.28)

Note. Standard deviation in parenthesis. 1 year students = 165 and 5
143

th

2.90
(1.36)
2.79
(1.47)
year students =

Results show that in general future teachers in both cohorts showed low performance in
the items that formed the MPCK index if the maximum score is considered as a benchmark. The
total column shows that in average the first-year cohort scored 2.90 (SD=1.36) and the fifth-year
scored 2.79 (SD=1.47). Although these results are low and indicate that items were hard for
student teachers, the findings reported in Table 5.18 show that the items were also hard for the
international and Chilean sample. For a better appreciation of the performance of future teachers
I computed the mean for both samples. To this purpose I first calculated the weighted average for
each item, based in the rate response and the score, and then I added the average of all items. I
117

followed this procedure for each sample. The resulting means were 4.26 for the international
sample and 3.37 for the Chilean sample. So, results confirmed that, in overage, Peruvian future
teachers from both cohorts performed less well than their counterparts in Chile and the
aggregated international sample.
A detailed analysis of the results for the Peruvian sample shows than in average the entry
cohort scored higher than the exit cohort. This difference favorable to the first year cohort is due
to fifth year students in institution B and D scored lower and that those institutions had the
bigger samples which could pull the grand mean down in the exit cohort. It is also noticeable
that future teachers from institutions A, C, and E got better results in the scale in both cohorts
with an increase from the entry to the exit cohort. Furthermore, students from institutions A and
C in the exit cohort got scores (4.10 and 4.15 respectively) that are similar to the one obtained for
the international sample (4.26) in the MPCK index.
In the next section, I further examine the performance of Peruvian future teachers in the
MPCK index by institution and cohort.
The Effect of Cohort and Institution on the MPCK Index
Univariate analysis of variance was performed to examine the effect of cohort and
institution on the MPCK index, and to establish whether there were significant differences in the
dependent variable (MPCK index) according to the different levels of the independent variables.
The equality of error variances was tested to examine whether the error variance of the
dependent variable (MPCK index) was equal across groups; analysis found that there was
homogeneity of variances, as assessed by Levene’s test (p= .072). Table 5.22 presents the results
of the main effects and the interaction effects of the MPCK index.

118

Table 5.22. Univariate Analysis of Variance for MCPK Index
Source
df
SS
MS
Institution (I)
4
76.692
19.173
Cohort (C)
1
2.305
2.305
Institution * Cohort (I*C)
4
9.004
2.251
st
th
Note. 1 year =165, 5 year =143, Total sample=308.

F
10.994
1.322
1.291

η2
.129
.004
.017

p
.000
.251
.274

The univariate ANOVA revealed that the main effects were not qualified by an
2

interaction between institution and cohort, F (4, 308) = 1.291, p=.274, partial η = .017. Also,
2

there was a significant main effect of institution F (4, 308) = 10.994, p<.001, partial η = .129
2

but not for Cohort F (1, 308) = 1.322, p=.251, partial η =.004. These findings mean that the
effect of institution on the MPCK index did not depend on the cohort of the students.
Likewise, there was not a statistically significant difference between first-year and fifthyear cohorts on the MPCK index, but there was a statistically significant difference in the MPCK
index scores between the institutions. Further analysis is needed to find where those differences
were. Here it is important to notice that the standard deviations of the group means are large, as
reported in Table 5.21. This variation within each group and the small sample size in some
groups could be factors that explain why means are not statistically different.
Post Hoc Tests for Differences between Institutions
Table 5.23 shows significant differences involved in the comparison between institutions.
Table 5.23. Multiple Comparisons for MPCK Index
Institution
n
M
SD
A
B
C
D
A
15
3.73
1.33
1.13
n.s
1.29
C
62
2.60
1.30
-1.21
n.s
MPCK
D
42
3.81
1.60
1.37
index
E
131
2.44
1.34
F
58
3.12
1.04
Note. Differences are significant at p < .05 level. Non significant differences: n.s.
119

E
n.s
-0.52
0.69
-0.68
-

Institutions C, A, and E, in that order, had the greater means, and the remaining
institutions B and D, reported the lower means. Thus, the institutional mean of A is greater than
B and D, the institutional mean of C is greater than B, D, and E; and finally the institutional
mean of E is greater than B and D but smaller than C.
In sum, Peruvian future teachers demonstrated a low performance in mathematics
pedagogical content knowledge as measured by the MPCK index. Although their counterparts in
the international sample and Chilean sample also had difficulties in solving the items, scores
obtained by the Peruvian future teachers were lower, with the exception of the exit cohort of
institutions A and C who demonstrated similar performance to the international average. In spite
of this, results would imply that future teachers are exiting the program with deficiencies in
mathematics pedagogical content knowledge required to teach mathematics. These deficiencies
are likely to affect their ability to perform mathematics teaching in a way that allows their
students develop the three processes mandated by the mathematics curriculum (problem solving,
mathematical communication, and reasoning and demonstration).
Comparisons by institution showed that future teachers in institution A and C performed
better in the MPCK index. Their corresponding entry cohort scored higher than the other groups
in the same cohort and their exit cohort demonstrated a relative increase in the MPCK index.
These findings would indicate that these institutions are recruiting students with some potential
in mathematics knowledge and are providing their students with the opportunities to learn to
develop this domain of the teacher knowledge. The analysis on opportunity to learn showed for
example that these institutions plan to cover more topics and a greater percentage of their
students reported having studied mathematics topics required to teach mathematics; instead their

120

counterparts in others institutions (B and D), as already explained, showed some limitations in
this regard.

121

Research Question 4: Future Teachers’ Beliefs and Teacher Education Programs
Teacher education programs are the place where future teachers acquire dispositions for
mathematics teaching, and since programs obey policy mandates, it is expected that such
dispositions align with what the Ministry of Education recommends in the current reforms. At
the beginning of the program, future teachers might show beliefs that are opposite to or favorable
to the reform ideals, and this could depend on personal and schooling experiences. However, it is
anticipated that through their participation in the program, future teachers’ beliefs become more
aligned with the reform ideals. In the process of learning to teach mathematics this implies that
future teachers question views of mathematics in which the application of procedures and rules
are the core of mathematics learning and instead endorse more progressive views of
mathematics, in which questioning, exploring, and creativity are key elements for learning.
In this section I examine the beliefs of entry students and exit students across the
participating institutions. I start by providing some descriptive information on possible patterns
or differences, then I report on my use of MANOVA to see the effect of their cohort or
institution on the beliefs held by future teachers, and I identify the differences, as done before. I
also examine the correlations between belief scales and MPCK.
What Future Teachers Believe about Mathematics Teaching
Table 5.24 shows the means and standard deviations by institution and cohort for the five
beliefs scales examined in the survey. These scales, as explained in the methodology section,
examine beliefs about the nature of mathematics through “rules” and “inquiry”; beliefs about
learning mathematics through “following directions” and “active involvement” as scales aiming
to describe future teachers; and beliefs on students’ mathematical ability with “fixed ability.”
Likewise, due to their definition, some scales represent reform-oriented views of mathematics
122

(“inquiry” and “active involvement”), and others represent non-reform oriented views of
mathematics (“rules,” “following directions,” and “fixed ability”). This is an important point to
keep in mind in reading and interpreting the results.

Table 5.24. Means and Standard Deviation of Beliefs Scales
Scale

Cohort

Institutions
Total
A
B
C
D
E
1st Year
3.97
4.51
4.61
4.73
4.01
4.51
(1.38) (1.28)
(0.97)
(0.79)
(0.86) (0.96)
Rules
5th Year 4.42
4.68
4.50
4.51
4.39
4.53
(0.75) (0.58)
(0.64)
(0.71)
(0.75) (0.68)
Nature of
Mathematics
1st Year
4.50
4.41
4.98
4.66
5.22
4.79
(1.34) (1.19)
(0.94)
(0.83)
(0.58) (0.91)
Inquiry
5th Year 5.40
5.02
5.44
4.97
5.12
5.08
(0.71) (0.70)
(0.46)
(0.60)
(0.84) (0.68)
1st Year
3.00
3.64
3.52
3.85
2.48
3.46
Following
(1.01) (0.82)
(0.98)
(0.93)
(0.61) (1.01)
Directions
5th Year 2.77
3.33
2.78
3.31
2.84
3.15
(0.80) (0.73)
(0.89)
(0.78)
(0.77) (0.80)
Learning
mathematics
1st Year
4.83
4.76
5.16
4.96
5.14
5.01
Active
(0.54) (0.80)
(0.64)
(0.75)
(0.84) (0.76)
Involvement 5th Year 5.22
5.05
5.40
4.98
5.36
5.12
(0.55) (0.60)
(0.41)
(0.57)
(0.51) (0.57)
1st Year
2.93
2.97
3.06
3.29
2.39
3.01
Mathematics
(0.98) (0.86)
(1.07)
(0.91)
(0.57) (0.93)
as Fixed Ability
5th Year 2.48
3.00
2.90
2.99
2.74
2.91
(0.76) (0.92)
(0.68)
(0.75)
(0.82) (0.81)
Note. Standard deviations in parenthesis. Beliefs are measured with Likert scale 1= strongly
disagree, to 6 = strongly agree.

Regarding the nature of mathematics, Table 5.24 shows that first-year students favored
both views of mathematics, “rules” (M=4.51, SD=0.96) and “inquiry” (M=4.79, SD=0.91), with
the same intensity, although some differences are noticeable by institutions, which are analyzed
later. Fifth-year students also agreed mostly with both views of mathematics, with a slight
preference for beliefs that support “inquiry” (M=5.08, SD=0.68) over “rules” (M=4.53,
123

SD=0.68). Overall, these results imply that both views of the nature of mathematics are
compatible for future teachers rather than in opposition; in their mindset, there is a balance
between conceptual understanding and computational skills as features that should characterize
school mathematics. However, as demonstrated by the results, fifth-year students seemed to
support inquiry to a greater degree.
About beliefs on learning mathematics, future teachers endorsed beliefs that supported a
more constructivist view of mathematics, in which students have active participation; they
rejected views in which the center of instruction is the teacher, and this pattern was supported by
both entry and exit students. Table 5.24 shows that first-year students endorsed statements that
mathematics is “active involvement” (M=5.01, SD=0.76), and they showed less support for
learning mathematics through “following directions” (M=3.46, SD=1.01). In the same way, fifthyear students endorsed clearly statements that support learning mathematics as “active
involvement” (M=5.12, SD=0.57), and they disagreed with beliefs that emphasize learning by
memorization of procedures (M=3.15, SD=0.80). Likewise, comparisons of the totals show that
the degree of acceptance or rejection of these beliefs was stronger in the fifth-year cohort than in
the first-year; these results imply some effects of the teacher education program that will be
further examined later.
Finally, future teachers in both cohorts mostly disagreed with views of mathematics as a
“fixed ability”; the mean for the first-year students was 3.01 (SD=0.93), and the mean for fifthyear students was 2.91 (SD=0.81). This suggests that these future teachers rejected the notion
that mathematics is not for all students but only for those who have the natural ability; on the
contrary, these future teachers believe that all students are able to succeed in mathematics.

124

All in all, statements expressing beliefs that are most consistent with the reform
(“inquiry” and “active learning”) attracted greater support from future teachers, and statements
expressing beliefs that do not align with the reform purposes attracted both support and rejection.
Future teachers mostly endorsed views of mathematics as rules and procedures (“rules”), and
they showed more disagreement with views of learning mathematics by following teachers’
directions (“following directions”). Likewise, future teachers disagreed with ideas that support
the existence of a natural ability for learning mathematics (“fixed ability”).
The Relationship between Future Teachers’ Beliefs and Mathematics Pedagogical Content
Knowledge
According to several theorists, teachers’ beliefs are organized in systems that connect
them in ways that teachers can make sense of (Thompson, 1992). Theory also makes claims
about the relationship between future teachers’ beliefs and knowledge: having mathematical
pedagogical content knowledge that considers students’ thinking is related to a view of
mathematics that puts students in the center of the teaching-learning process and that puts
emphasis on the conceptual rather than on the procedural. In this section, I examine how entry
and exit students’ beliefs were related to the domains that have been explored in this study: the
nature of beliefs, mathematics learning, and mathematics, but also how they are related to their
results on the MPCK index. Table 5.25 shows the association between the variables.

125

Table 5.25. Intercorrelation for Beliefs Scales and MPCK Index by Cohort
Cohort

Rules

Inquiry

Rules
Inquiry
Teacher directions
Active involvement
Fixed ability
MPCK index

.49**
.47**
.27**
.27**
.08

.16
-.06
.50**
-.10
.25*

Teacher
directions
.36**
-.24**
-.00
.57**
-.10

Active
involvement
.14
.38**
-.06
-.06
.21*

Fixed
ability
.18*
-.20*
.53**
-.11
-.17*

MPCK
index
-.14
.17*
-.22*
.08
-.15
-

st

Note: Intercorrelations for 1 year cohort are below the diagonal (n= 137) and intercorrelations
th

for the 5 year are above the diagonal (n= 128). Correlations are significant at ** p < .01 and at
* p < .05

In the first year cohort, beliefs about mathematics as a set of rules, procedures, and facts
correlated significantly with the other scales, no matter if they were or were not reform oriented.
Thus, “rules” correlated positively with “inquiry” (.49) and “active involvement” (.27), which, as
explained before, are reform oriented, and with “teacher directions” (.47) and “fixed ability”
(.27), which support traditional views of mathematics. This suggests that future teachers entered
the teacher education program accepting several views of mathematics teaching, even though
they could be opposite or contradictory.
“Inquiry” and “active involvement,” both reform-oriented views, also had a medium
positive correlation (.50); this means that greater scores on beliefs that support a more dynamic
view of mathematics were related to views that put students in the center of mathematics
learning. Finally, “fixed ability” was positively correlated with “teacher directions” (.57), which
means that the stronger future teachers agreed that mathematics’ achievement depends on
students’ fixed ability, the more they endorsed beliefs about mathematics learning as heavily
teacher-centered.

126

Regarding correlations between knowledge and beliefs, small positive correlations were
found for the MPCK index with the reform oriented beliefs: “inquiry” (.25) and “active
involvement” (.21), and negative correlation with “fixed ability” (-.17). These findings support
the hypothesis that better levels of mathematics pedagogical knowledge are associated with
dispositions that recognize students as active learners of mathematics.
In the fifth year cohort, future teachers seemed to be more sensitive to the different
conceptual orientations involved in the nature of mathematics and its learning; positive
correlations were found only between reform oriented beliefs or between non-reform oriented
beliefs. In the same way, negative correlations were found between reform and non-reform
oriented beliefs. Thus, regarding the beliefs that support traditional views of mathematics, the
“rules” scale correlated with “teacher directions” (.36), and with “fixed ability” (.18), and “fixed
ability” correlated with “teacher directions” (.53). On the side of reform oriented beliefs, “active
involvement” correlated with “inquiry” (.38). This differentiation in the meaning of beliefs is
confirmed with the negative correlations of the “inquiry” scale with “teacher directions” (-.24)
and with the “fixed ability” scale (-.20), which means that students who endorsed ideas of the
nature of mathematics as inquiry tended to rejected ideas that support learning mathematics as a
passive process in which students just have to follow directions, and they also rejected ideas that
students’ academic achievement depends on their natural ability.
Table 5.23 also shows some correlation between the MPCK index and beliefs for the exit
cohort. Thus as observed in the entry cohort, MPCK index correlated positively with “inquiry”
(.17) and negatively with “teacher directions” (-.22). The results agree with the hypothesis that
having more knowledge to teach mathematics is more associated with the reformed view of

127

mathematics, such as “inquiry.” Correspondingly, having more pedagogical knowledge is
negatively associated with traditional ideas about mathematics learning.
The Effect of Institution and Cohort in Future Teachers’ Beliefs
The descriptive results show some variations in the endorsement of beliefs by first year
and fifth year students, but it is not clear to what extent such variability of beliefs about the
nature of mathematics and learning were determined by cohort and institutions where the future
teachers were enrolled, and to what extent the variations by cohort and institution represent
significant differences. Two-way multivariate analysis of variance (MANOVA) was performed
to answer this question, by testing the hypothesis that the independent variables, cohort and
institution, had an effect on the five dependent variables and the beliefs scales, as a group.
The assumption of the homogeneity of the covariance matrices required for MANOVA
analysis was met, as assessed by Box’s M test (p = .065). Thus, the covariance matrices of the
dependent variables across all levels of the independent variables (cohort and institution) are
assumed to be different for the purposes of the MANOVA. Table 5.26 presents the results of the
multivariate test.

Table 5.26. Mutivariate Analysis of Variance for Beliefs Scales
Source
Institution (I)
Cohort (C)
IxC
st

Wilks' Λ
.745
.933
.929

Note. 1 year =137, 5

th

Df
20
5
20

Error
833.423
251.000
833.423

year =128, Total sample=265.

128

F
3.875
3.587
.935

p
.001
.001
.542

2

η
.071
.067
.018

Two-way MANOVA revealed no significant interaction between institution and cohort, F
2

(20, 833.423) =.935, p = .542, Wilks' Λ = .929, partial η =.018. This means that the effect of the
institutions on the dependent variables was not different between cohorts. Regarding the main
effects, the results show a significant multivariate main effect of institution on the set of five
2

beliefs scale as a group, F (20, 833.423) = 3.875, p = .001, Wilks' Λ = .745, partial η =.071. In
the same way, the results also show the significant effect of cohort on the set of the dependent
2

variables, Wilks' Λ = .933, F (5, 251) = 3.587, p = .001, partial η =.018.
Given the overall significance of the F test, the univariate main effects of the independent
variables on the five dependent variables were examined. Table 5.27 presents the results.

Table 5.27. Univariate Analysis of Variances for Beliefs Scales
Source

Dependent
Variable

Institution Rules
Inquiry
Following directions
Active involvement
Fixed ability
Cohort
Rules
Inquiry
Directions
Active involvement
Fixed ability

2

df

SS

MS

F

p

η

4
4
4
4
4
1
1
1
1
1

9.075
9.811
32. 473
5.612
13.368
.647
7.876
3.037
1.647
.461

2.269
2.453
8.118
1.403
3.342
.647
7.876
3.037
1.647
0.461

3.446
4.167
12.091
3.594
4.813
.982
13.380
4.523
4.219
0.665

.009
.003
.000
.007
.001
.323
.000
.034
.041
.416

.051
.061
.159
.053
.070
.004
.050
.017
.016
.003

Because there were five tests, one for each dependent variable, the Alpha level was
corrected to control for inflated type I error by dividing 0.05 by 5, and the p value to contrast is
0.01. The results show that significant univariate effects for institution were obtained for “rules,”
2

2

F (4, 255) = 3.446, p = .009, partial η =.051; “inquiry,” F (4, 255) = 4.167 p = .003, partial η ;
129

2

“following directions,” F (4, 255) = 12.091 p < .001, partial η =.159; “active involvement,” F
2

(4, 255) = 3.594 p = .007, partial η =.053; and “fixed ability” F (4, 255) = 4.813, p = .001,
2

partial η = .070. Overall, these results imply that for each dependent variable, the means of at
least two institutions were different. Such differences might be due to several reasons, which are
further examined in the discussion chapter.
Table 5.28 also shows that the cohorts had effects on the variation of one dependent
2

variable, but only for one scale, “inquiry” F (1, 255) = 13.380 p < .001, partial η =.050. The
results also show that there were no differences by cohort in the beliefs scales of “rules,”
“directions,” “active,” and “fixed ability,” at the level set after the Bonferroni correction (.01).
As observed in the preliminary analysis, students from both cohorts showed the same pattern of
support or rejection of beliefs, and although some variation was observed, these results confirm
than those means were not significantly different except in the case of “inquiry.” Significant
differences for this scale might suggest that staying in the program helps future teachers to define
their views on mathematics as a process of creativity and solving problems; however, this must
be read carefully, since although significant, the difference is small (0.29), and the means of both
cohorts remain around the same category of response “slightly agree”
Post-Hoc Tests for Differences between Institutions
Table 5.28 summarizes the multiple comparisons for the beliefs scales.

130

Table 5.28. Multiple Comparisons for Beliefs Scales
Institution n
M
SD
A
B
C
D
A
14
4.26
1.01
n.s
n.s
n.s
C
59
4.61
0.90
n.s
n.s
D
Rules
35
4.57
0.92
n.s
E
108 4.68
0.71
F
49
4.18
0.79
A
14
5.14
0.96
n.s
n.s
n.s
C
59
4.81
0.94
-0.36
n.s
D
Inquiry
35
5.17
0.87
0.31
E
108 4.86
0.71
F
49
5.21
0.61
A
14
2.84
0.83
-0.60
n.s
-0.76
C
59
3.43
0.78
n.s
n.s
D
Directions
35
3.20
0.94
-0.40
E
108 3.60
0.91
F
49
2.64
0.66
A
14
5.12
0.58
n.s
n.s
n.s
C
59
4.95
0.69
-0.28
n.s
Active
D
35
5.23
0.62
n.s
involvement
E
108 4.99
0.64
F
49
5.28
0.53
A
14
2.64
0.86
n.s
n.s
-0.54
C
59
2.95
0.87
n.s
n.s
Fixed
D
35
2.91
0.86
n.s
ability
E
108 3.18
0.86
F
49
2.52
0.70
Note. Differences are significant at p < .05 level. Non significant differences: n.s.
Scale

E
n.s
0.43
0.39
0.50
n.s
-0.40
n.s
-0.35
n.s
0.79
0.56
0.96
n.s
-0.33
n.s
-0.29
n.s
0.43
0.39
0.66
-

Regarding the “rules” scale, institution E had the lowest mean compared to the ones of
institutions B, C, and D. This does not mean that students from this institution supported this
view of mathematics less, since all institutional means were in the range of agree and slightly
agree. The “inquiry” scale shows that institutions C and E had the greater means when they were
compared with institutions B and D; in this case, more future teachers from C and E supported
131

this view, since their institutional means represent that they slightly agreed with these views of
mathematics, while their counterparts supported this view less. The scale of “directions” had
smaller means, which reflects that future teachers mostly disagreed with this view of
mathematics. Students from institution A and E disagreed more with this view, compared to
students from institutions B and D. Institution C had the third smaller mean, but enough to
differentiate it from institution D. The multiple comparisons for “active involvement” reveal
differences that were smaller, but were reported as significant; all are around the category
“slightly agree.” Again, the mean of institution E was greater than B and D. Also the mean of
institution C became different from institution B. Finally, the means for the scale of “fixed
ability” show, as reported previously, that future teachers disagreed with this kind of conception
of students’ ability to learn mathematics. The smaller means are the ones corresponding to
institutions A and E, which became different from institutions B, C, and D (especially institution
E).
The findings show that student teachers from both cohorts mostly endorsed beliefs that
are compatible with the reform ideals: they agreed with ideas that represent mathematics as a
process of inquiry and active involvement, and they disagreed with ideas that represent
mathematics learning as following directions and that consider mathematics as a fixed ability.
This system of beliefs could allow them to perform mathematics teaching as recommended by
the national curriculum design; however, a key aspect of mathematics is still held by future
teachers: mathematics is a set of rules and procedures. This belief seems to be reinforced through
the program since, the entry cohort agreed with this belief and the exit cohort showed stronger
levels of agreement. In spite of this trend, no evidence on differences by cohort on the belief
scales was found except for the scale of mathematics as an inquiry process.
132

The findings also show some consistency in how the scales varied by institutions.
Multiple comparisons allowed identifying that institution E holds the pattern of beliefs described
above, supporting the reform oriented beliefs and rejecting the non-reform beliefs, with the
exception of the “rules” scale. Its levels of agreement/disagreement (higher means in reform
oriented beliefs and lower means in non-reform oriented beliefs) were enough to show
differences with future teachers from institutions B and D (which, compared to other institutions,
consistently showed higher means for non-reform beliefs and lower means for the reform
oriented beliefs). This would suggest that some efforts are being made by institution E for their
future teachers to display these beliefs and that more work need to be done for institutions C and
E to nurture and strength reform oriented beliefs.
The intercorrelations between belief scales show that future teachers’ beliefs differ at
different points of teacher preparation. The exit cohort was more sensitive to the different views
of mathematics and its learning, while the entry cohort, held a more undifferentiated view of
mathematics, since they endorsed beliefs that could be considered opposite (i.e. “rules” and
“inquiry”). The results for the exit cohort implies that through the program they have developed
a better understanding of mathematics corresponding to the current pedagogies, which is
reflected in their significant differences for the scale of mathematics as a process of inquiry
(“inquiry”).

133

Research Question 5: The Opportunities to Learn Associated with Mathematics
Pedagogical Content Knowledge and Beliefs about Mathematics Teaching
In this section I examine the relationship between the variables analyzed in this study:
opportunities to learn, knowledge, and beliefs, in the exit cohort. As explained in the
methodology section, opportunity to learn was only explored for the exit cohort. The analysis
was done for the aggregate sample because the small sample size in each institution would not
allow identifying the relationships between the variables.
Course Topics, Mathematics Pedagogical Content Knowledge, and Beliefs for Mathematics
Teaching
Table 5.29 presents the correlations between the course work related to school
mathematics and mathematics education pedagogy with the expected outcomes of teacher
education programs: knowledge for mathematics teaching and reform oriented beliefs for
mathematics teaching.

134

Table 5.29. Intercorrelations Between Course Topics, Knowledge, and Beliefs for Exit Student
Teachers
Course
topics
School mathematics

MPCK
index

Rules

Inquiry

Teacher
direction

Active
involvement

Fixed
ability

.18*

-.09

-.13

.01

-.04

.16

Measurement

.21

*

-.12

-.12

-.02

.09

-.02

Geometry

.12

.15

.20*

-.08

.17

-.06

Functions
Statistics
Calculus
Validation
Mathematics education
Pedagogy
Math foundation
Math context
Math thinking

.02
.12
-.08
.00

.00
-.16
.11
.03

.06
.02
.03
.06

.05
.01
.09
.04

.00
.14
.09
.15

.12
.07
-.04
-.18*

.02
.14
.24**

.03
.07
.01

.02
.15
.05

.03
.00
-.01

.17
.14
.14

0.06
0.03
-0.07

Math instruction
Teaching plans
Math teaching
Observation
Affective issues

.07
.03
.02
-.02
.09

-.07
.02
.01
.03
-.07

.03
.00
-.03
-.09
.18*

-.04
.05
.06
.06
-.09

.13
-.04
-.10
-.16
.08

0.12
0.07
0.11
0.15
-0.04

Numbers

Note: Correlations are significant at ** p <0.01 and at * p < .05. N= 123

Regarding the MPCK index, this variable was related to the fact of having studied
courses that cover “number” (.18) and “measurement” (.21). It also has a positive correlation
with “math thinking” (.24) which has to do with teaching about reasoning, argumentation, and
carrying procedures in mathematics.
The results show no association between belief scales (reform and non-reform oriented)
and course topics, except some few cases that could be considered spurious correlations. Thus,
taking “geometry” was associated with “inquiry” (.20). Also, having studied “affective issues” in
mathematics was related to the “inquiry” belief scale (.19).
135

Activities for Learning to Teach, Mathematical Pedagogical Content Knowledge, and
Beliefs about Mathematics Teaching
Contrary to expectations, only a few significant correlations were found between
opportunities to learn scale, the mathematics pedagogical content knowledge measured by the
MPCK index, and the belief scales as showed in Table 5.30.
Table 5.30. Intercorrelation Between Learning to Teach Scales, MPCK index, and Beliefs
Scales for Exit Student Teachers
Learning to Teach
MPCK Rules Inquiry Teacher
Active
Fixed
Scales
Index
direction involvement ability
Class participation
.06
.03
.03
.03
.06
.03
*
Class reading
.14
.06
.07
.19
.10
.04
Solving problem
.23*
-.07
.14
-.01
.30**
-.11
Instructional practice
.19*
-.07
.01
.01
.15
-.01
Instructional planning
.14
.00
.10
.01
.08
-.09
Assessment practice
.13
.05
.01
.04
.04
-.11
Reflection
.04
-.06
.00
.07
.05
-.04
Note: Correlations are significant at ** p < .01 and at * p < .05. N= 114

Regarding the learning to teach scales and knowledge, positive correlations were
identified between MPCK and “solving problem” (.23) and “instructional practice” (.19). The
greater a student’s exit score was, the more opportunities fifth-year students had in their program
to learn mathematics by problem solving activities, and to learn how to teach mathematics under
a solving problem approach.
In relation to learning to teach scales and beliefs, a positive correlation did exist between
“solving problems” and “active involvement” (.30). This intercorrelation suggests that students
who were more involved in problem solving activities endorsed more ideas that learning
mathematics involves inquiry in problem solving contexts.

136

Multiple regression analysis was planned to identify how the opportunities to learn in the
teacher preparation programs (independent variables) might shape the expected outcomes of
teacher education programs: knowledge and beliefs (dependent variables). Contrary to
expectations, only a few correlations were identified between these two set of variables.
The absence of meaningful correlations between school mathematics topics and beliefs
scales are expected because such opportunities for learning to teach are more focused on content
(studied and did not study) and the items did not ask about how those topics were taught, which
certainly could represent some influence on future teachers’ beliefs. However, expected
correlations between mathematics pedagogy topics and beliefs were not observed either with the
data available. This might have to do with the findings of the implemented and intended
curriculum regarding mathematics education pedagogy topics: most students were not exposed to
these topics because they were not covered by the programs. As discussed previously, topics
such as “mathematics foundations,” “context of mathematics education,” and “mathematics
thinking” can be useful to provide future teachers with opportunities to rethink the roles of
teacher and learner in mathematics teaching and to broaden their perspectives on the teaching
and learning of mathematics; therefore, these topics deserved the attention of the teacher
education programs to attempt any changes in the future teachers’ beliefs.
The absence of relationships between belief scales and activities for learning to teach
scales was also unexpected. Only one such relationship became visible: “solving problems” and
“active involvement,” which makes sense. Having opportunities to solve mathematical problems
and trying several strategies can help future teachers to think about mathematics as a process in
which getting the right answer is not the priority, but investigating different ways and strategies
to solve a problem is.
137

More associations were found between the MPCK index and the opportunities to learn
scales. The correlations between the MPCK index and the topics of “numbers,” and
“measurement” might have to do with the fact that the items that formed the index were related
to these topics. This association provided evidence on the relationship between these two
domains of knowledge: MCK and MPCK; a strong foundation on MPCK rests on a strong
foundation on MCK, therefore teacher education programs must provide future teachers with
both domains of knowledge.

138

CHAPTER 6
REFORMING TEACHER EDUCATION IN PERU: REFLECTION AND A VIEW TO THE
FUTURE
This study sought teacher education programs in Peru as places where future teachers are
expected to acquire the skills required for teaching school mathematics, according to the goals
proposed by the current national school curriculum. Given this premise and based on theories of
teacher learning, I examined the opportunity to learn delivered by teacher education programs to
their future teachers and the knowledge and beliefs resulting from those opportunities to learn in
order to see how these processes and outcomes can help future teachers to face the demands of
the school mathematics curriculum.
In relation to opportunities to learn, future teachers reported having been mostly exposed
to activities that allow them to gain practical knowledge for mathematics teaching; however, they
also reported not having been exposed to some important topics of school mathematics and
mathematics pedagogy that can strengthen their mathematics pedagogical content knowledge.
Syllabi analysis confirmed these vacuums in the curriculum of the teacher education programs.
Regarding knowledge, the results of this study showed that programs are having difficulties in
developing the knowledge for mathematics teaching which is necessary for future teachers to
connect mathematics content and pedagogy (An, 2004), and therefore necessary in developing
the processes (problem solving, reasoning and demonstration, and communication) and content
(number, geometry, and data) recommended in the national elementary curriculum. Regarding
beliefs, overall the results showed that future teachers endorsed beliefs related to mathematics
teaching that can help them to develop the kind of teaching encouraged by the reform, although
they still endorsed a core belief that describes the nature of mathematics as a set of rules and
139

procedures. Likewise, the data revealed no interaction effect between cohort and institution in
the outcome variables (beliefs and knowledge). Finally, comparison of institutions showed that
some institutions (A, C, and E) were consistent in showing evidence that are providing
opportunities to learn to their student teachers and helping them to nurture reform oriented
beliefs; however, these institutions still need to enhance the mathematics pedagogical content
knowledge of their student teachers.
Overall, data from this investigation suggest that teacher education programs are not
meeting future teachers’ needs to teach mathematics as the reform requires. The development of
skills, such as problem solving, reasoning and demonstration, and mathematical communication
for the learning of mathematics content indicated in the school curriculum requires strong
mathematics pedagogical content knowledge but future teachers showed difficulties in this
regard. The reform also requires a dynamic view of mathematics learning; however, the marked
endorsement of mathematics as rules and procedures potentially can lead teachers to set aside
any effort to pose strategies for problem solving, although they did show endorsement of other
reform oriented beliefs. All this raises questions about the process of learning to teach
mathematics that takes place within institutions to drive knowledge and beliefs needed for the
reform.
In this chapter I discuss why teacher education programs may be failing to fulfill the
expectations that the current reform poses for preparing teachers with the knowledge and beliefs
they need to enact the reform curriculum. Following a top-down organization, I start by
discussing how the reform considers these knowledge and beliefs throughout their mandates,
how knowledge and beliefs are being taken into account within teacher education programs, and
how knowledge and beliefs are being addressed in the process of learning to teach. Implications
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for policy and practice are discussed at each of these layers when they are pertinent. Finally,
some reflections on directions for future research and limitations of the study are also discussed.
Knowledge and Beliefs in the National Context of the Reform
Teacher education programs do not work in a vacuum but within a national system where
policies define their purposes and processes. Thus, if the reform requires that programs prepare
teachers to be able to enact the practices described in the national curriculum design, and if
research has proved the indispensable character of the knowledge and beliefs for the success of
the reform, then policy should have clear mandates related to the desired teachers knowledge and
beliefs for teaching the school curriculum. This is not, however, observed in the Peruvian
context, and it seems that reformers are not aware of the importance of these two elements for
learning to teach mathematics.
As explained in Chapter 2, the Ministry of Education has included as one of its main
policies the establishment of the accreditation and evaluation of teacher education programs in
order to assure the quality of teacher preparation. While quality teaching involves knowledge and
dispositions as broadly discussed by the literature, it is striking to find that the model of
accreditation is not emphatic in addressing these two key elements among their criteria. The
models for certification and accreditation in effect fail to include a cognitive model for how
future teachers are expected to acquire the knowledge needed for teaching (Ball et al., 2001;
Shulman, 1986) and instead they only consider structural aspects of the teacher preparation
programs. As shown in Table 2.6 of Chapter 2, the model for accreditation includes many
factors, and among them only one is related to the teaching learning process in the program.
Learning evaluation is part of this factor, and institutions are requested to assess the knowledge,
skills, and attitudes posed in their professional profile for future teachers. The model leaves to
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the institution the responsibility to decide their evaluation and the rigor of their strategies and
instruments. Thus, if the institution is not aware of the importance of content knowledge and
pedagogical content knowledge, these crucial domains for teacher performance might not be
assessed. Instead, the institution may consider other elements for evaluation which, while
important, do not have direct influence on teacher practices. The implications of this are further
developed in the following section.
Another indicator that policies on teacher education programs are overlooking the
importance of knowledge and beliefs is the missing alignment between the curriculum of teacher
education programs and the national school curriculum as part of the accreditation model.
Including this criterion would indicate that policy makers seek to assure that future teachers
develop knowledge and beliefs for teaching the reform curriculum, and to assure that future
teachers are exposed to the content and processes involved in all subject matters of the school
curriculum so that student teachers can teach those subjects. By contrast, the standards related to
the study plan address issues of credits, hours, and course sequence.
Although the implementation of the accreditation policy was not the focus of this study,
the literature revised and the documents analyzed show that policy makers and practitioners are
not paying full attention to pedagogical content knowledge and beliefs for the preparation of
future teachers, since they are not being considered in their real dimensions and importance. This
situation demands an important change in direction if the reform’s aims are to be fulfilled.
Currently a number of strategies are in place that may address this concern and produce the
desired change. One of them, the development of learning standards for basic education, beyond
helping to unpack the processes and content posed in the national school curriculum, can help
teacher education programs to align their curriculum with the school curriculum, and can help
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policy makers of accreditation to be aware of the competences that student teachers have to
develop if their programs are to be accredited.
In this line, the findings of this study can provide important feedback to policy makers in
relation to the accreditation requested by the Ministry of Education. Including criteria on the
alignment between the national school curriculum and the teacher education curriculum as part
of the quality model could help to assure that future teachers graduate from their teacher
preparation programs with the knowledge needed to teach the reformed curriculum.
Knowledge and Beliefs in the Teacher Education Programs
Teacher education also takes place within a cultural context in which institutions define
their philosophy and discourses on teaching based on the ones proposed by the reform. The
assumption is that the programs that are aware of the reform guidelines and that have adopted
them in their discourse and philosophy are likely to prepare teachers under that approach (Tatto,
1996). The reformed school curriculum requires that teacher education programs are aware of
what these guidelines demand, and that they consider both knowledge and beliefs as part of their
discourses, visions, and goals. However, if this awareness is not found at the top level among
those who design policy, as discussed above, then it could be expected that teacher education
programs’ philosophy does not capture fully what is needed for future teachers to implement the
curriculum successfully.
The document analysis presented in Chapter 5 showed that teacher education programs
have incorporated the principles of socio-constructivism in their discourse somehow; they have
included them in the description of their curriculum and of their profile of future teachers, which
indicates that these programs are familiar with this pedagogical orientation. Constructivism as a
way of learning was introduced in Peru in the curriculum reform of the nineties, and it was used
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in the professional development of in-service teachers; therefore, it is expected that teacher
education programs have adopted it as part of their philosophy.
Regarding the guidelines for the mathematics curriculum, the findings showed that the
curriculum for mathematics education of three institutions (A, C, and E) had some alignment
with problem solving. From this, it should be expected that these ideals are being implemented in
the preparation of future teachers, and that they are reflected in future teachers’ knowledge and
beliefs. Conversely, the findings indicated a gap between the institutional discourse and the skills
future teachers need to implement the reformed curriculum. Even institutions that are consistent
in their discourse on solving problem, through all their documentation, have graduates who
showed deficient levels of mathematics pedagogical content knowledge and also endorsed
traditional beliefs about the nature of mathematics.
Those findings drive the question on to what extent the institutions of teacher preparation
recognize the importance of assuring that future teachers develop knowledge and beliefs during
their participation in the program. The document analysis found that although the documents
consider knowledge as part of their profile, this key factor is only a small part of a bigger
discourse, which includes other aspects that define teachers’ and future teachers’ roles, such as
students’ personal development, education for life, and commitment with the community. In this
bigger discourse, the focus on students’ learning as the teachers’ main responsibility seems to get
lost. This can explain why the key aspects of learning to teach, knowledge and beliefs, are
weakly addressed or are not properly emphasized in the teacher profile of the institutions.
In addition to the institutional philosophy, human resources also constitute an important
part of the context of teacher education. It is important to consider what teacher educators and
future teachers bring to the teacher education programs. Having teacher educators endowed with
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the appropriate skills and dispositions will help future teachers to develop, in turn, the skills and
dispositions (or the knowledge and beliefs) required for teaching the school curriculum. In the
same way, having future teachers with strong knowledge base on mathematics content can
facilitate the acquisition of more rigorous mathematics content knowledge and mathematics
pedagogical content knowledge. The process of learning to teach will be hindered, however, if
teacher preparation programs have to supply student teachers with basic mathematics knowledge
which should have been developed in school before entering the program.
This study did not focus on teacher educators, and that is an important line for future
research; it focused instead on student teachers, and the findings about their performance in the
MPCK index uncovered deficiencies in first year students as well as fifth year students. Thus, it
is possible that future teachers are entering the program with deficiencies in mathematics skills
that the programs cannot remedy totally, especially considering that future elementary teachers
are prepared to be generalist teachers and the courses of mathematics that these student teachers
receive as part of their preparation are just a few. Likewise, although teacher education programs
have admission tests to select candidates, passing the test does not assure that future teachers
have strong mathematical knowledge. As explained in Chapter 2, universities in Peru mostly
require low scores on the admission test to be accepted in their programs of teacher education
compared to the higher standards on the admission test set by other programs in the same
institutions.
That said, it is important that teacher education programs consider potential limitations in
future teachers’ prior mathematical knowledge in order to pose strategies that help their students
to overcome them and to develop the knowledge required for teaching the mathematics
curriculum. The potential deficiencies of teacher candidates also pose implications for policy.
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The future teachers surveyed in this study showed their good intentions and motivations for
teaching as the basis for their decision to pursue a teaching career; however, this is not enough to
become a teacher able to enhance students’ learning. Rigorous filters to recruit future teachers, as
is done in other countries, could help to select teacher candidates with strong subject matter
knowledge (Tatto et al., 2012).
Knowledge and Beliefs in the Process of Learning to Teach
As mentioned before, this study assumed that in order to assure the development of the
knowledge and beliefs necessary for teaching mathematics, teacher education programs must
provide future teachers with thoughtful and meaningful learning experiences. These learning
experiences have to do with course topics, learning activities, and opportunities for reflection on
teaching.
The design of the study, which included students in the initial and the final stage of their
programs of teacher education, allowed performing analysis by cohort; in this way, changes in
beliefs or gains in knowledge could be observed. Consequently, any significant variation in the
two variables between the two cohorts favoring the fifth-year cohort could indicate that the
programs are having some influence in future teachers due to the learning opportunities they
have been exposed.
Regarding variation on knowledge, the findings showed no interaction effect between
institution and cohort and no effect of the cohort in the results on the MCPK index, but it showed
the effect of the institution on the MPCK index. The no interaction effect questions the influence
of the programs in the development of future teachers’ knowledge. Specifically it questions the
quality of the opportunities to learn that future teachers had during their preparation as teachers.
The no significant differences between cohorts for the whole sample also support this concern.
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The significant differences by institution suggests two possibilities: variation in the MPCK index
is due to the student teachers who are being recruited by the institutions, or the variation is due to
the curriculum, strategies, and strategies allocated by the institutions to support the learning to
teach process. More sophisticated analysis, such as hierarchical linear modeling might be needed
to test these hypotheses and to identify better the effect of the institutions. However, based on the
data of this study it is possible to pose some conjectures.
About variations in the MPCK index due to the kind of students attracted by each
institution, demographics of future teachers enrolled in institution A and institution C, which got
the greater means in the index, showed that these groups of students have particular
characteristics. Institution A is a private university in Peru; therefore, their students mostly
belong to middle class; this was confirmed with the higher levels of parental education found in
this institution. Belonging to an economically privileged group means more opportunities and
resources for basic education; and therefore a better starting point for the teacher learning
process. For its part, institution C is one of the most prestigious public universities; it attracts
many candidates every year and the admission test represents a strong competition among
students to reach a vacancy. Future teachers from this institution mostly belong to lower socio
economic groups compared to their peers in institution A but the filter set by the admission test
somehow helps to select the best candidates for teacher education. Although we do not know
about the kinds of knowledge and skills that are tested in these evaluations, having rigorous
preparation to be admitted in these institutions can help future teachers to revise their
mathematical knowledge and to be in better shape to initiate the program of teacher education.
Regarding the variation of MPCK index due to the opportunities to learn provided by the
institutions, the differences in opportunities to learn and MPCK index across the institutions
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support this hypothesis. Multiple comparisons showed that institutions B and D consistently got
smaller means in the scales of opportunities to learn; in addition their future teachers reported the
lower percentages in the questions on the implemented curriculum for school mathematics and
mathematics education pedagogy. Instead the remaining institutions (A, C, and E) were
consistent in showing positive results for the variables of opportunity to learn, in both the
intended and the implemented curriculum; coincidentally future teachers from these institutions
demonstrated the better performance in the MPCK items. All this evidence suggests that the
mathematical pedagogical content knowledge that future teachers can acquire and develop will
depend on the institution in which they are enrolled.
The positive correlation, found in the aggregate sample, between the MPCK index on the
scales of solving problems (the opportunity to participate in mathematics classes as a problem
solver), instructional practice (the opportunity to learn about how to teach problem solving), and
on the fact of being exposed to the topics of numbers, measurement, and mathematics thinking
demonstrate the importance of these learning opportunities for the development of mathematics
pedagogical content knowledge. The existence of these associations, however, does not deny the
limited mathematical pedagogical content knowledge of Peruvian future teachers; instead they
call for more attention to the learning experiences future teachers are exposed in their programs.
If student teachers are entering programs with preexisting deficiencies in content knowledge to
teach mathematics, then they need more opportunities to learn. Chapter 2, on the Peruvian
context, showed the deficiencies in mathematics learning of students, as demonstrated in national
evaluations of students’ academic achievement. Teacher education programs must be aware of
these deficiencies that potentially affect teacher candidates in order to, as already discussed,
address the actions needed to remedy them since future teachers cannot learn how to teach
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mathematics pedagogy on a weak mathematical content knowledge basis, especially considering
that the reform demands the implementation of a more sophisticated curriculum.
Opportunities to learn have to do not only with the amount of topics or pedagogical
strategies future teachers are exposed, but also the quality in the deliverance of such topics and
pedagogical strategies. The evidence about the intended and implemented curriculum found in
this study showed some flaws in the curriculum that can hinder the acquisition of mathematics
pedagogical content knowledge. Contrary to what was observed in the study, future teachers
need more pedagogical and foundational resources to be able to tackle the mathematical
processes and content mandated in the mathematics curriculum. Alignment between the topics
on content knowledge and the topics on mathematical pedagogical content knowledge can also
help to assure better opportunities to learn. As important as exposing future teachers to all
mathematical content included in the school curriculum is exposing them to opportunities to
learn how to teach such content addressing the three processes stated in the mathematics
curriculum: problem solving, reasoning and demonstration, and mathematical communication.
Regarding future teachers’ beliefs, if beliefs about the nature of mathematics and about
the mathematics learning of entry students are different from the ones supported by the reform,
then this could represent a challenge for teacher education programs, since programs would have
to define strategies to transform such beliefs during their participation in the program. On the
other hand, if entry students endorse beliefs supported by the reform, then teacher education
programs should work to strengthen these beliefs so that future teachers tend to teach
mathematics according to these beliefs despite the pressures they could face in their first years of
teaching. In both cases, the teacher education program must provide learning experiences so that
future teachers, at the end of their preparation, endorse reform-oriented beliefs.
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In this study, first-year students and fifth-year students endorsed the beliefs required for
teaching the reformed mathematics curriculum and at the same time they endorsed beliefs about
mathematics as rules and procedures which are contrary to the reform ideals. This particular set
of beliefs for mathematics teaching is the core of the teachers’ beliefs system, and therefore it
has more strength than other beliefs to remain unchanged. This is not favorable for the purposes
of the reform because endorsing these kinds of beliefs could make future teachers step back to
traditional approaches of teaching, especially if future teachers do not have a conceptual
understanding of mathematics and its pedagogy. It is remarkable that first-year students start the
program endorsing beliefs that are compatible with the reform; here it can be hypothesized that
first-year students bring to the program these dispositions because of their personal experiences
during schooling. As explained before, ideas that recognize student-centered pedagogies have
been part of the discourse used by policy and practitioners since the nineties; so it is likely that
students have had the chance to experience some aspects of that pedagogy and, in consequence,
endorse those views of teaching. If this were the case, teacher education programs must reinforce
these dispositions and encourage reflective thinking among their students, so they can examine in
depth other views of mathematics that are opposed.
In relation to the variation of beliefs by institution or cohort, no interaction effect
between these two variables could be identified. Likewise, the cohort effect was identified only
for the scale of “inquiry,” which considers mathematics as a process of solving problems. This
means that at the end of their preparation future teachers support a more dynamic view of
mathematics teaching and learning than their counterparts in the first year. Despite the fact that
the data did not show significant differences by cohort for the other scales, some trace of change
of beliefs toward the end of the program was observed in the aggregate data. The different
150

patterns of correlations between the beliefs scales examined in this study reported by the two
cohorts suggest that the fifth-year students had some learning experiences during their
participation in the program that allowed them to be more sensitive to the several approach
related to mathematics teaching and learning than the first-year students. However, as shown in
the earlier chapter, just a few associations were found between the beliefs scales and course
topics. The MT21 study (Schmidt, Blömeke et al., 2011) also found that belief scales did not
have much relation with course work, but with the specific learning experiences as found in this
study: the more future teachers are engaged in activities as problem solvers, the more there is
endorsement of the beliefs that support mathematics learning requires students be involved
actively in solving problem activities. Regarding the institutional effect, as in the case of MPCK
index, findings showed that institutions can make a difference in shaping future teachers’ beliefs;
however as explained before the lack of correlations between the beliefs and opportunities to
learn, as measured in this study, did not help to hypothesize how such differences are being
produced. More research on specific learning experiences included in the teacher education
curriculum for mathematics education need to be explored to better understand the institutional
effect on future teachers’ beliefs.
The results about the correlation between the beliefs scales and the MPCK index are also
worthy of attention. The MPCK index was associated positively with reform-oriented beliefs,
and negatively associated with non-reform oriented beliefs which imply that having specialized
knowledge for mathematics teaching can provide future teachers with more insights to think
about mathematics and its teaching in a more dynamic way and vice versa. The findings of the
TEDS-M study also found similar results about the correlation between knowledge and beliefs;
their correlations between these variables were low but significant (Tatto et al., 2012).
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Descriptive and multiple comparisons for these two variables support such association; thus it
was possible to observe that institutions with low results in the MPCK index (institutions B and
D) got higher means on non-reform oriented beliefs (“rules,” “teacher direction,” and “fixed
ability”). All these evidence on the relationship between beliefs and knowledge measures build
on the argument about the importance of addressing both knowledge and beliefs during teacher
preparation.
Having the certificate that proves the completion of the program does not mean that
future teachers are ready to face the demands of the mathematics school curriculum. The low
results in the MPCK index and the endorsement of beliefs supporting mathematics as a set of
rules and procedures would indicate that future teachers are likely to perform the traditional
mathematics teaching that the reform is just trying to change, that is, the same kind of teaching
that is behind the poor performance of Peruvian students in mathematics. The results of this
study indicate that the enhancement of mathematics education for elementary future teachers is
an urgent task. Such task involves joint and concerted effort from institutions that plan policies
and institutions that execute them. So far, the Ministry of Education has set some guidelines to
improve teacher preparation, but the mechanisms that can help this purpose (such as evaluation,
learning standards, selection mechanisms, etc.) are not there yet, or are not operating. So, at this
point, the efforts of the Ministry of Education to respond to the problems of teacher preparation
seem to be symbolic. On their part, teacher education programs seem not to be aware of the
particularities of the reformed school curriculum and are failing to arrange optimal conditions for
the process of learning to teach mathematics. Closing this gap is necessary to reach the ultimate
goal which is the enhancement of students’ academic achievement in mathematics.

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Directions for Future Research
This study paves the road for further research on teacher education programs in Peru and
their responsiveness to the current educational reforms in mathematics teaching. The themes
examined in this study are very much needed to provide feedback on the policies on teacher
preparation being implemented as part of the National Educational Project and others that could
be launched to support the guidelines established by this policy. The study design is useful to
develop a model for a larger research project that could be carried out with more rigorous
methodology and instruments. Having two cross-sectional cohorts to compare the outcomes
resulting from years of preparation allows posing conclusions about the influence of the
program, although longitudinal study could provide more certain information since it would help
to control any bias of the cohort effect.
Sampling design might consider teacher preparation corresponding with nesting data
(future teachers within programs) to select participants and institutions; likewise, some criteria of
randomness should be introduced to have the chance of generalizing the results to a broad
spectrum of teacher preparation. The tentative study should include other routes of certification
as well. The current study focused mostly on teacher education programs in universities, but it is
important to examine the opportunities to learn as well as the knowledge and beliefs of student
teachers in superior pedagogical institutes (ISPs). This kind of institution also contributes to the
teaching force and its curriculum is designed by the Ministry of Education; so it would be
important to know how successful these programs are at infusing the reform oriented beliefs and
developing the mathematical knowledge for teaching among future teachers.
Likewise, having a larger sample size could allow using complex measures of
mathematics content knowledge and pedagogical content knowledge, which mostly require
153

multiples forms to cover all domains of their variables. Further study also should include
information related to teacher educators who teach mathematics-related courses, which can help
us to understand better how the processes of learning to teach are taking place and how beliefs
and knowledge are contributing in the outcomes expected in future teachers. In addition,
program coordinators also could participate in the study to provide detailed information on the
philosophy of the institution related to the preparation of teachers.
Future research should also examine the attention teacher education programs give to
general pedagogy and practicum and how these two components of the curriculum contribute to
the development of knowledge for mathematics teaching and the acquisition of reform-oriented
beliefs. TEDS-M findings have found that curriculum for teacher education include many
courses on general pedagogy at the expenses of courses that could help future teachers to gain
specialized knowledge for teaching mathematics (Tatto et al., 2012). Research has also found
that practical experiences help future teachers in the development of mathematics knowledge for
teaching (Youngs & Qian, 2013). Collecting data on this regard could help to have a better
picture of the opportunities to learn in teacher education programs.
Another line of research is related to the relationship between future teachers’ beliefs and
mathematics pedagogical content knowledge for teaching. Endorsing reform-oriented beliefs is
not enough to assure good practice if future teachers’ knowledge for mathematics teaching is not
solid. Besides, including measures of beliefs about mathematics as problem solving,
mathematical communication, and reasoning and demonstration could help to analyze the
alignment between the school mathematics curriculum and the preparation of future teachers.

154

Limitations
The study does have limitations. Its sample size represents a limitation to generalize
results to other institutions of teacher preparation in Peru, although in theory these institutions
represent the best case scenarios for teacher education in Peru. Likewise, the small number of
participants in some institutions might affect the variability in the variables measured and hinder
their possibilities for multiple comparisons. Regarding the cohort, the ideal scenario to make
claims about the effectiveness of the program would be with a longitudinal design in which
future teachers are surveyed when they are entering and exiting the program. Using different
groups can be problematic, since a cohort effect can influence the results (i.e. entry students’
particular characteristics). Another limitation has to do with the lack of institutional information
that could strengthen the findings; documents were limited to provide information about the
philosophy of the programs and their goals for their graduated. Finally, the MPCK index also
had limitations to represent the whole spectrum of the construct of mathematics pedagogical
content knowledge: items that formed it were too hard or too easy to answer; consequently,
although it is only a proxy of the construct, it could have limitations to discriminate the future
teachers’ skills related to this knowledge domain. Despite these constrains, the findings still can
provide promising directions for future research. In any case, it should be noted that there are no
previous studies in Peru that answer the research questions posed by this study; thus, the results
can be helpful to propose hypotheses that can be tested later by using the rigorous procedures
used by TEDS-M.

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APPENDICES

156

APPENDIX A
15

Future Teacher Questionnaire

GENERAL BACKGROUND
1. How old are you?
_________ years
______________________________________________________________________________
2. What is your gender?
A. Female
B. Male

1
2

______________________________________________________________________________
3. About how many books are there in your <parents’ or guardian’s> home?
(Do not count magazines, newspapers, or your school books.)
Check one box
A.
B.
C.
D.
E.

None or few (0-10 books)
Enough to fill one shelf (11-25 books)
Enough to fill one bookcase (26-100 books)
Enough to fill two bookcases (101-200 books)
Enough to fill three or more bookcases (more than
200 books)

15

1
2
3
4
5

Items are reprinted with the authorization of Teresa Tatto, principal investigator of the TEDSM study.
157

______________________________________________________________________________
4. Do you have any of these items at your <parents’ or guardian’s> home?
Check one box in each row.
Yes
No
A.
B.
C.
D.
E.
F.
G.
H.
I.
J.
K.

Calculator
Computer (excluding TV/video game computers)
Study desk/table for your use
Dictionary
Encyclopedia (as a book or CD)
Playstation, Game Cube, Xbox or other TV/Video game system
DVD player
Three or more cars, small trucks or sport utility vehicles
<Country specific>
<Country specific>
<Country specific>

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

1

2

______________________________________________________________________________
5. What is the highest level of education completed by your mother (or stepmother or female
guardian)?
Check one box
A.
B.
C.
D.
E.
F.
G.
H.
I.

<ISCED 1> primary
<ISCED 2> lower secondary
<ISCED 3> upper secondary
<ISCED 4B> post-secondary non-tertiary
<ISCED 5B> practical training
<ISCED 5A> first degree
<ISCED 5A> second degree
Beyond <ISCED 5A>, first degree
I don’t know

158

1
2
3
4
5
6
7
8
9

______________________________________________________________________________
6. What is the highest level of education completed by your father (or stepfather or male
guardian)?
Check one box
A. <ISCED 1> primary
1
B. <ISCED 2> lower secondary
2
C. <ISCED 3> upper secondary
3
D. <ISCED 4B> post-secondary non-tertiary
4
E. <ISCED 5B> practical training
5
F. <ISCED 5A> first degree
6
G. <ISCED 5A> second degree
7
H. Beyond <ISCED 5A>, first degree
8
I.
I don’t know
9
______________________________________________________________________________
7. How often do you speak <language of test> at home?
Check one box
A. Always
1
B. Almost always
2
C. Sometimes
3
D. Never
4
_____________________________________________________________________________
8a. What was the highest <year/grade> level at which you studied mathematics in <secondary
school>?
Check one box
Highest level
completed
A.
B.
C.
D.
E.
F.
G.

<Year 12> (Advanced level)
<Year 12>
<Year 11>
<Year 10>
Below <Year 10>
<Country specific>
<Country specific>

1
2
3
4
5
6
7

159

______________________________________________________________________________
8b. What is the most advanced mathematics <course> that you took in <secondary school>?
Check one box
Most advanced
course
A. <Country specific>
1
B. <Country specific>
2
C. <Country specific>
3
D. <Country specific>
4
E. <Country specific>
5
F. <Country specific>
6
______________________________________________________________________________
9. In secondary school, what was the usual level of <marks/grades> that you received?
Check one box
A. Always at the top of my year level
1
B. Usually near the top of my year level
2
C. Generally above average for my year level
3
D. Generally about average for my year level
4
E. Generally below average for my year level
5
______________________________________________________________________________
10. Prior to commencing your teacher education program, did you have another career?
[For the purposes of this question, a career is having a paid job that you regarded as likely to
form your life’s work.]
A.
B.

Yes
No

1
2

160

______________________________________________________________________________
11. To what extent does each of the following identify your reasons for becoming a teacher?
A minor
reason
A
significa
nt major
A reason
reason

Not a
reason

Check one box in each row

A.
B.
C.
D.
E.
F.
G.
H.
I.

I was always a good student in school.
1
2
3
4
I am attracted by the availability of teaching
1
2
3
4
I love mathematics.
1
2
3
4
I believe that I have a talent for teaching.
1
2
3
4
I like working with young people.
1
2
3
4
I am attracted by teacher salaries.
1
2
3
4
I want to have an influence on the next
1
2
3
4
I see teaching as a challenging job.
1
2
3
4
I seek the long-term security associated with being
1
2
3
4
a teacher.
______________________________________________________________________________
12. Did any of the following circumstances hinder your studies during your teacher preparation
program?
Check one box
Yes
No
A. Had family responsibilities that made it difficult to do my best
1
2
B. Had to borrow money
1
2
C. Had to work at a job
1
2
______________________________________________________________________________
13. How do you see your future in teaching?
Check one box
A.
B.
C.
D.
E.

I expect it to be my lifetime career.
It could possibly be my lifetime career.
It is something I can do until I find the career that I really want.
I will probably not seek employment as a teacher.
I don’t know.

161

1
2
3
4
5

BELIEFS ABOUT THE NATURE OF MATHEMATICS
1. To what extent do you agree or disagree with the following beliefs about the nature of
mathematics?

A. Mathematics is a collection of rules and procedures
that prescribe how to solve a problem
B. Mathematics involves the remembering and
application of definitions, formulas, mathematical
facts and procedures
C. Mathematics involves creativity and new ideas.
D. In mathematics many things can be discovered and
tried out by oneself.
E. When solving mathematical tasks you need to
know the correct procedure else you would be lost.
F. If you engage in mathematical tasks, you can
discover new things (e.g., connections, rules,
concepts).
G. Fundamental to mathematics is its logical rigor and
preciseness.
H. Mathematical problems can be solved correctly in
many ways.
I. Many aspects of mathematics have practical
relevance.
J. Mathematics helps solve everyday problems and
tasks.
K. To do mathematics requires much practice, correct
application of routines, and problem solving
strategies.
L. Mathematics means learning, remembering and
applying.

162

Strongly
agree

Slightly
disagree
Slightly
agree
Agree

Disagree

Strongly
disagree

Check one box in each row

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

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5

6

1

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3

4

5

6

1

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5

6

1

2

3

4

5

6

1

2

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4

5

6

1

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5

6

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5

6

1

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5

6

1

2

3

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5

6

BELIEFS ABOUT LEARNING MATHEMATICS
2. From your perspective, to what extent would you agree or disagree with each of the following
statements about learning mathematics?

163

Strongly
agree

Slightly
agree

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

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4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

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5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

Agree

Slightly
disagree

A. The best way to do well in mathematics is to
memorize all the formulas.
B. The best way to do well in mathematics is to
memorize all the formulas.
C. It doesn’t really matter if you understand a
mathematical problem, if you can get the right
answer.
D. To be good in mathematics you must be able to
solve problems quickly.
E. Pupils learn mathematics best by attending to the
teacher’s explanations.
F. When pupils are working on mathematical
problems, more emphasis should be put on getting
the correct answer than on the process followed.
G. In addition to getting a right answer in
mathematics, it is important to understand why the
answer is correct.
H. Teachers should allow pupils to figure out their
own ways to solve mathematical problems.
I. Non-standard procedures should be discouraged
because they can interfere with learning the correct
procedure.
J. Hands-on mathematics experiences aren’t worth the
time and expense.
K. Time used to investigate why a solution to a
mathematical problem works is time well spent.
L. Pupils can figure out a way to solve mathematical
problems without a teacher’s help.
M Teachers should encourage pupils to find their own
.
solutions to mathematical problems even if they are
inefficient.
N. It is helpful for pupils to discuss different ways to
solve particular problems.

Disagree

Strongly
disagree

Check one box in each row

BELIEFS ABOUT MATHEMATICS ACHIEVEMENT
3. To what extent do you agree or disagree with each of the following statements about pupil
achievement in <primary/lower secondary> mathematics?

A. Since older pupils can reason abstractly, the use of
hands-on models and other visual aids becomes less
necessary
B. To be good at mathematics you need to have a kind
of “mathematical mind”.
C. Mathematics is a subject in which natural ability
matters a lot more than effort.
D. Only the more able pupils can participate in multistep problem solving activities.
E. In general, boys tend to be naturally better at
mathematics than girls.
F. Mathematical ability is something that remains
relatively fixed throughout a person’s life.
G. Some people are good at mathematics and some
aren’t.
H. Some ethnic groups are better at mathematics than
others.

164

Strongly
agree

Agree

Slightly
disagree
Slightly
agree

Disagree

Strongly
disagree

Check one box in each row

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

BELIEFS ABOUT PREPAREDNESS FOR TEACHING MATHEMATICS
4. Please indicate the extent to which you think your teacher education program has prepared you
to do the following when you start your teaching career.

A. Communicate ideas and information about
mathematics clearly to pupils
B. Establish appropriate learning goals in mathematics
for pupils
C. Set up mathematics learning activities to help
pupils achieve learning goals
D. Use questions to promote higher order thinking in
mathematics
E. Use computers and ICT to aid in teaching
mathematics
F. Challenge pupils to engage in critical thinking
about mathematics
G. Establish a supportive environment for learning
mathematics
H. Use assessment to give effective feedback to pupils
about their mathematics learning
I. Provide parents with useful information about your
pupils’ progress in mathematics
J. Develop assessment tasks that promote learning in
mathematics
K. Incorporate effective classroom management
strategies into your teaching of mathematics
L. Have a positive influence on difficult or
unmotivated pupils
M Work collaboratively with other teachers

165

Strongly
agree

Agree

Slightly
disagree
Slightly
agree

Disagree

Strongly
disagree

Check one box in each row

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

4

5

6

1

2

3

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5

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1

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1

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3

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6

1

2

3

4

5

6

BELIEFS ABOUT PROGRAM EFFECTIVENESS
5. To what extent do you agree or disagree with the following statements?
The instructors who teach mathematics-related <courses> in your current teacher
preparation program:
Strongly
agree

Agree

Slightly
disagree
Slightly
agree

Disagree

Strongly
disagree

Check one box in each row

A. Model good teaching practices in their teaching
1
2
3
4
5
6
B. Draw on and use research relevant to the content of
1
2
3
4
5
6
their <courses>
C. Model evaluation and reflection on their own
1
2
3
4
5
6
teaching
D. Value the learning and experiences you had prior to
1
2
3
4
5
6
starting the program
E. Value the learning and experiences you had in your
1
2
3
4
5
6
field experience and or practicum
F. Value the learning and experiences you had in your
1
2
3
4
5
6
teacher preparation program
______________________________________________________________________________
6. Overall, how effective do you believe your pre-service teacher education program was in
preparing you to teach mathematics?
Check one box
A.
B.
C.
D.

Very ineffective
Ineffective
Effective
Very effective

1
2
3
4

166

KNOWLEDGE
1. A <Grade 1> teacher asks her students to solve the following four story problems, in any way
they like, including using materials if they wish.
Problem 1: [Jose] has 3 packets of stickers. There are 6 stickers in each pack. How many
stickers does [Jose] have altogether?
Problem 2: [Jorgen] had 5 fish in his tank. He was given 7 more for his birthday. How many
fish did he have then?
Problem 3: [John] had some toy cars. He lost 7 toy cars. Now he has 4 cars left. How many
toy cars did [John] have before he lost any?
Problem 4: [Marcy] had 13 balloons. 5 balloons popped. How many balloons did she
have left?
The teacher notices that two of the problems are more difficult for her children than the other
two. Identify the TWO problems which are likely to be more DIFFICULT to solve for <Grade
1> children.

Problem _______ and Problem _______

167

______________________________________________________________________________
2. The following problem was given to children in <primary> school.
The graph shows the number of pens, pencils, rulers, and erasers sold by a store in a week.

The names of the items are missing from the graph. Pens were the item most often sold. Fewer
erasers than any other item were sold. More pencils than rulers were sold.
(a) How many pencils were sold?
Check one box
A.
B.
C.
D.

40
80
120
140

1
2
3
4

(b) Some <primary> students would experience difficulty with a problem of this type. What is
the main difficulty you would expect? Explain clearly with reference to the problem.

168

3. When teaching children about length measurement for the first time, Mrs. [Ho] prefers to
begin by having the children measure the width of their book using paper clips, then again using
pencils. Give TWO reasons she could have for preferring to do this rather than simply teaching
the children how to use a ruler?

4. [Jeremy] notices that when he enters 0.2 × 6 into a calculator his answer is smaller than 6, and
when he enters 6 ÷ 0.2 he gets a number greater than 6. He is puzzled by this, and asks his
teacher for a new calculator!
(a) What is [Jeremy’s] most likely misconception?

(b) Draw a visual representation that the teacher could use to model 0.2 × 6 to help [Jeremy]
understand WHY the answer is what it is?

169

OPPORTUNITY TO LEARN
UNIVERSITY OR TERTIARY LEVEL MATHEMATICS
1. Consider the following topics in university level mathematics. Please indicate whether you
have ever studied each topic.
Check one box in each row.
Studied
Not
studied
A. Foundations of Geometry or Axiomatic Geometry (e.g., Euclidean
1
2
axioms)
B. Analytic/Coordinate Geometry (e.g., equations of lines, curves,
conic sections, rigid transformations or isometrics)
C. Non-Euclidean Geometry (e.g., geometry on a sphere)
D. Differential Geometry (e.g., sets that are manifolds, curvature of
curves, and surfaces)
E. Topology
F. Linear Algebra (e.g., vector spaces, matrices, dimensions,
eigenvalues, eigenvectors)
G. Set Theory
H. Abstract Algebra (e.g., group theory, field theory, ring theory,
ideals)
I. Number Theory (e.g., divisibility, prime numbers, structuring
integers)
J. Beginning Calculus Topics (e.g., limits, series, sequences)
K. Calculus (e.g., derivatives and integrals)
L. Multivariate Calculus (e.g., partial derivatives, multiple integrals)
M Advanced Calculus or Real Analysis or Measure Theory
N. Differential Equations (e.g., ordinary differential equations and
partial differential equations)
O. Theory of Real Functions, Theory of Complex Functions or
Functional Analysis
P. Discrete Mathematics, Graph theory, Game theory, Combinatorics
or Boolean Algebra
Q. Probability
R. Theoretical or Applied Statistics
S. Mathematical Logic (e.g., truth tables, symbolic logic, propositional
logic, set theory, binary operations)

170

1

2

1

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1

2

1

2

1

2

SCHOOL LEVEL MATHEMATICS
2. Consider the following list of mathematics topics that are often taught at the <primary> or
<secondary> school level. Please indicate whether you have studied each topic as part of your
current teacher preparation program.
Check one box in each row.
Studied
A.

B.

C.

D.
E.
F.
G.

Numbers (e.g., whole numbers, fractions, decimals, integer,
rational, and real numbers; number concepts; number theory;
estimation; ratio and proportionality)
Measurement (e.g., measurement units; computations and
properties of length, perimeter, area, and volume; estimation
and error)
Geometry (e.g., 1-D and 2-D coordinate geometry, Euclidean
geometry, transformational geometry, congruence and
similarity, constructions with straightedge and compass, 3-D
geometry, vector geometry)
Functions, Relations, and Equations (e.g., algebra,
trigonometry, analytic geometry)
Data Representation, Probability, and Statistics
Calculus (e.g., infinite processes, change, differentiation,
integration)
Validation, Structuring, and Abstracting (e.g., Boolean
algebra, mathematical induction, logical connectives, sets,
groups, fields, linear space, isomorphism, homomorphism)

Not
studied

1

2

1

2

1

2

1

2

1

2

1

2

1

2

______________________________________________________________________________
3. In your teacher preparation program, at what level is emphasis given to learning mathematics?
Check one box in each row.
Yes
A.
B.
C.

Learning mathematics at the level of the school curriculum
Learning school mathematics topics at a deeper more
conceptual level than the school curriculum
Learning mathematics beyond the school curriculum with no
direct relation to the school curriculum

171

No

1

2

1

2

1

2

MATHEMATICS EDUCATION/<PEDAGOGY>
4. Consider the following list of mathematics education/<pedagogy> topics. Please indicate
whether you have studied each topic as part of your current teacher preparation program.
Check one box in each row.
Studied
Not
studied
A.

B.

C.

D.

E.

F.
G.
H.

Foundations of Mathematics (e.g., mathematics and
philosophy, mathematics epistemology, history of
mathematics)
Context of Mathematics Education (e.g., role of mathematics
in society, gender/ethnic aspects of mathematics
achievement)
Development of Mathematics Ability and Thinking (e.g.,
theories of mathematics ability and thinking; developing
mathematical concepts; reasoning, argumentation, and
proving; abstracting and generalizing; carrying out
procedures and algorithms; application; modeling)
Mathematics Instruction (e.g., representation of mathematics
content and concepts, teaching methods, analysis of
mathematical problems and solutions, problem posing
strategies, teacher-pupil interaction)
Developing Teaching Plans (e.g., selection and sequencing
the mathematics content, studying and selecting textbooks
and instructional materials)
Mathematics Teaching: Observation, Analysis and
Reflection
Mathematics Standards and Curriculum
Affective Issues in Mathematics (e.g., beliefs, attitudes,
mathematics anxiety)

172

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2

1

2

1

2

1

2

______________________________________________________________________________
5. In the mathematics education<pedagogy/teaching methods> courses that you have taken or
are currently taking in your teacher preparation program, how frequently did you do any of the
following?

Never
L.
M.
N.
O.

173

Often

G.
H.
I.
J.
K.

Listen to a lecture
Ask questions during class time
Participate in a whole class discussion
Make presentations to the rest of the class
Teach a class session using methods of my own choice
Teach a class session using methods demonstrated by the
instructor
Work together in groups during class
Read about research on mathematics
Read about research on mathematics education
Read about research on teaching and learning
Analyze examples of teaching (e.g., film, video, transcript of
lesson)
Write mathematical proofs
Solve problems in applied mathematics
Solve a given mathematics problem using multiple strategies
Use computers or calculators to solve mathematics problems

Occasionally

A.
B.
C.
D.
E.
F.

Rarely

Check one box in each row.

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

______________________________________________________________________________
6. In your current teacher preparation program, how frequently did you engage in activities that
gave you the opportunity to learn how to do the following?

Never
174

Occasionally

Often

A. Accommodate a wide range of abilities in each lesson
B. Analyze and use national or state standards or frameworks for
school mathematics
C. Analyze pupil assessment data to learn how to assess more
effectively
D. Assess higher−level goals (e.g., problem-solving, critical
thinking)
E. Assess low−level objectives (factual knowledge, routine
procedures and so forth)
F. Build on pupils’ existing mathematics knowledge and thinking
skills
G. Create learning experiences that make the central concepts of
subject matter meaningful to pupils
H. Create projects that motivate all pupils to participate
I. Deal with learning difficulties so that specific pupil outcomes
are accomplished
J. Develop games or puzzles that provide instructional activities
at a high interest level
K. Develop instructional materials that build on pupils’
experiences, interests and abilities
L. Explore how to apply mathematics to real-world problems

Rarely

Check one box in each row.

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

6. (continued.)
In your current teacher preparation program, how frequently did you engage in activities that
gave you the opportunity to learn how to do the following?

Never
O.
P.
Q.
R.
S.
T.

U.
V.
W.
X.
Y.
Z.

175

Often

N.

Explore how to use manipulative (concrete) materials or
physical models to solve mathematics problems
Explore mathematics as the source for real-world
problems
Give useful and timely feedback to pupils about their
learning
Help pupils learn how to assess their own learning
Learn how to explore multiple solution strategies with
pupils
Learn how to show why a mathematics procedure works
Locate suitable curriculum materials and teaching
resources
Make distinctions between procedural and conceptual
knowledge when teaching mathematics concepts and
operations to pupils
Use assessment to give effective feedback to parents or
guardians
Use assessment to give feedback to pupils about their
learning
Use classroom assessments to guide your decisions about
what and how to teach
Use pupils’ misconceptions to plan instruction
Use standardized assessments to guide your decisions
about what and how to teach
Integrate mathematical ideas from across areas of
mathematics

Occasionally

M.

Rarely

Check one box in each row.

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

EDUCATION AND <PEDAGOGY>
7. Consider the following topics in education and <pedagogy>. Please indicate whether you
have studied each topic as part of your current teacher preparation program.
Check one box in each row.
Studied
Not
studied
A. History of Education and Educational Systems (e.g.,
historical development of the national system, development
of international systems)
B. Philosophy of Education (e.g., ethics, values, theory of
knowledge, legal issues)
C. Sociology of Education (e.g., purpose and function of
education in society, organization of current educational
systems, education and social conditions, diversity,
educational reform)
D. Educational Psychology (e.g., motivational theory, child
development, learning theory)
E. Theories of Schooling (e.g., goals of schooling, teacher’s
role, curriculum theory and development, didactic/teaching
models, teacher-pupil relations, school administration and
leadership)
F. Methods of Educational Research (e.g., read, interpret and
use education research; theory and practice of action
research)
G. Assessment and Measurement: Theory and Practice
H. Knowledge of Teaching (e.g., knowing how to teach pupils
of different backgrounds, use resources to support instruction,
manage classrooms, communicate with parents)

176

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2

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2

1

2

1

2

1

2

1

2

1

2

1

2

TEACHING FOR DIVERSITY AND REFLECTION ON PRACTICE
8. In your teacher preparation program, how often did you have the opportunity to learn to do the
following?

Never
J.

Develop strategies to identify your learning needs

B.
C.
D.
E.
F.
G.
H.

177

Often

I.

Develop specific strategies for teaching students with
behavioral and emotional problems
Develop specific strategies and curriculum for teaching pupils
with learning disabilities
Develop specific strategies and curriculum for teaching
gifted pupils
Develop specific strategies and curriculum for teaching pupils
from diverse cultural backgrounds
Accommodate the needs of pupils with physical disabilities in
your classroom
Work with children from poor or disadvantaged backgrounds
Use teaching standards and codes of conduct to reflect on your
teaching
Develop strategies to reflect upon the effectiveness of your
teaching
Develop strategies to reflect upon your professional knowledge

Occasionally

A.

Rarely

Check one box in each row.

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

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4

1

2

3

4

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2

3

4

1

2

3

4

1

2

3

4

______________________________________________________________________________
9. In your teacher preparation program, how often did you have the opportunity to learn to do the
following?

Never
D.
E.
F.
G.
H.
I.
J.
K.
L.

178

Often

C.

Study stages of child development and learning
Develop research projects to test teaching strategies for pupils
of diverse abilities
Consider the relationship between education, social justice and
democracy
Observe teachers modeling new teaching practices
Develop and test new teaching practices
Set appropriately challenging learning expectations for pupils
Learn how to use findings from research to improve knowledge
and practice
Connect learning across subject areas
Study ethical standards and codes of conduct expected of
teachers
Create methods to enhance pupils’ confidence and self-esteem
Identify opportunities for changing existing schooling practices
Identify appropriate resources needed for teaching

Occasionally

A.
B.

Rarely

Check one box in each row.

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

SCHOOL EXPERIENCE AND THE PRACTICUM
The questions in this section focus on the school experience part of your teacher education
program.
10. Did you spend any time in a <primary or secondary school> on <field experience and/or
practicum> as part of your teacher preparation program?
1

Yes

2

No

If you checked ‘Yes’, please continue the rest of the survey.
If you checked ‘No’, please go to QUESTION 15.
______________________________________________________________________________
11. For what proportion of this time were you temporarily in charge of teaching the class (as
opposed to observation, assistance, individual tutoring, etc.)?
Check one box.
A.
Less than ¼ of the time
1
B.
¼ or more, but less than ½
2
C.
½ or more, but less than ¾
3
D.
¾ or more
4
______________________________________________________________________________
12. For about how much of the time in the <field experience/practicum>, was one of your
assigned <mentors/instructors/supervisors> present in the same room as you?

A.
B.
C.
D.

Check one box.
1
2
3
4

Less than ¼ of the time
¼ or more, but less than ½
½ or more, but less than ¾
¾ or more

179

______________________________________________________________________________
13. During the school experience part of your program, how often were you required to do each
of the following?

Never

Rarely

Occasionally

Often

Check one box in each row.

A.

Observe models of the teaching strategies you were learning
in your <courses>

1

2

3

4

B.

Practice theories for teaching mathematics that you were
learning in your <courses>
Complete assessment tasks that asked you to show how you
were applying ideas you were learning in your <courses>

1

2

3

4

1

2

3

4

Receive feedback about how well you had implemented
teaching strategies you were learning in your <courses>
Collect and analyze evidence about pupil learning as a result
of your teaching methods

1

2

3

4

1

2

3

4

Test out findings from educational research about
difficulties pupils have in learning in your <courses>
Develop strategies to reflect upon your professional
knowledge

1

2

3

4

1

2

3

4

Demonstrate that you could apply the teaching methods you
were learning in your <courses>

1

2

3

4

C.
D.
E.
F.
G.
H.

180

________________________________________________________________________________
14. To what extent do you agree or disagree with the following statements about the <field
experience and/or practicum> you had in your teacher preparation program?

Slightly
agree

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

The feedback I received from my <supervising teacher/mentor/
instructors> helped me to improve my understanding of pupils.

1

2

3

4

G The feedback I received from my <supervising teacher/mentor/
instructors> helped me improve my teaching methods.

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

A I had a clear understanding of what my school-based <supervising
teacher/mentor/instructors> expected of me as a teacher in order to
pass the <field experiences/ practicum>.
B My school-based <supervising teacher/mentor/instructors> valued
the ideas and approaches I brought from my <university/college>
teacher education program.
C My school-based <supervising teacher/mentor/instructors> used
criteria/standards provided by my <university/college> when
reviewing my lessons with me.
D I learned the same criteria or standards for good teaching in my
<courses> and in my <field experiences /practicum>.
E In my <field experience / practicum> I had to demonstrate to my
supervising teacher that I could teach according to the same
criteria/standards used in my <university/college> <courses>.
F

H The feedback I received from my <supervising teacher/mentor/
instructors> helped me improve my understanding of the
curriculum.
I The feedback I received from my <supervising teacher/mentor/
instructors> helped me improve my knowledge of mathematics
content.
J The methods of teaching I used in my <field experiences/
practicum> were quite different from the methods I was learning
in my <university/college> <courses>.
K The regular supervising teacher in my <field experiences/
practicum> classroom taught in ways that were quite different
from the methods I was learning in my <university/college>
<courses>.
181

Agree

Slightly
disagree

Disagree

Check one box in each row.

COHERENCE OF YOUR TEACHER EDUCATION PROGRAM
15. Consider all of the <courses> in the program including subject matter <courses> (e.g.,
mathematics), mathematics <pedagogy> <courses>, and general education <pedagogy>
<courses>. Please indicate the extent to which you agree or disagree with the following
statements.

B.
C.
D.
E.

F.

Each stage of the program seemed to be planned to meet
the main needs I had at that stage of my preparation.
Later <courses> in the program built on what was taught
in earlier <courses> in the program.
The program was organized in a way that covered what
I needed to learn to become an effective teacher.
The <courses> seemed to follow a logical sequence of
development in terms of content and topics.
Each of my <courses> was clearly designed to prepare
me to meet a common set of explicit standard
expectations for beginning teachers.
There were clear links between most of the <courses> in
my teacher education program.

182

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

1

2

3

4

Agree

Slightly
agree

A.

Slightly
disagree

Disagree

Check one box in each row.

APPENDIX B
Coding for MPCK items

16

Table B1. Item “Jose”
Code
20
10

11

70

79

99

Response

Item:

MFC505

Correct Response
Problem 1 and Problem 3 (or Problem 3 and Problem 1)
Partially Correct Response
Problem 1 only correct (with or without Problems 2 and 4)
Examples:
• Problem 1 and Problem 2 (or 2 and 1)
• Problem 1 and Problem 4 (or 4 and 1)
 Problem 1 andcorrect (with or without Problems 2 and 4)
Problem 3 only Problem (blank)
Examples:
• Problem 3 and Problem 2 (or 2 and 3)
• Problem 3 and Problem 4 (or 4 and 3)

Problem 3 and Problem (blank)
Incorrect Response
At least one problem selected but neither Problem 1 nor Problem 3.
Examples:
• Problem 2 and Problem 4 (or 4 and 2)
• Problem 2 and Problem (blank)

Other incorrect (including Problem out, erased, stray(blank) illegible, or
crossed 4 and Problem marks,
off task)
Non-response
Blank

16

The release items and their coding system are available in IEA (2012). These tables are
reprinted with the authorization of Teresa Tatto, principal investigator of the TEDS-M study.
183

Table B2. Item “Difficulty”
Code
20

10

11

12

79

99

Response

Item:

MFC502B

Correct Response
Responses that refer to reading and comprehension difficulties related
to the complexity of the language used in the question with reasons
and/or references to specific examples.
Examples:
• The language used is quite challenging. Example, “fewer than
any other” and “more pencils than rulers”.
• Students would be challenged by the difficulty/complexity of the
wording in the question such as ‘most often’ ‘fewer’. There is a
considerable load on their ‘higher order’ skills as they are required
to organize, interpret and relate back to the graph.
• The items described in the text are listed in a different order to
the bars on the graph creating logistic or sequencing challenges.
Partially Correct Response
Less detailed responses that recognize that the language is likely to be a
difficulty for children but without reasons or examples.
Examples:
• They would have trouble with the language used in the question.
• Reading and comprehending the text would be difficult for many
children.
• There is a considerable amount of information to read, organize,
sequence and relate to the graph.
A statement describing difficulties attributable to the graph rather than
the text.
Examples:
• They would have trouble reading the graph.
• The names are missing from the graph and they wouldn’t have
experienced this before.
A statement attributing difficulties to the level of problem-solving or
analysis required without explaining how/why.
Examples:
• They would have trouble analyzing the information in the problem.
• The problem requires problem-solving strategies and they would
have trouble with that.
Incorrect Response
Incorrect (including crossed out, erased, stray marks, illegible, or off
task)
Non-response
Blank
184

Table B3. Item “Paper clip”
Code

Response

Item:

MFC513

Note: Significant and acceptable reasons
Reason 1: (Understanding of what measurement is) Using familiar/different units
enables understanding of what measurement is, that any object/unit can be used to
measure, that the scale on a ruler is just the repetition of a basic unit.
Reason 2: (Need for standard units) Use of non-standard units can, by creating
uncertainty about length, show the need for standard/formal units and possibly
create opportunities to discuss the (historical) development of measurement.
Reason 3: (Choosing most appropriate unit) Using objects of different lengths
helps children learn how to decide which unit/object is the most appropriate to
measure a given length.
Correct Response
20
Responses that give any TWO of the three significant and acceptable
reasons noted above.
10

11

Partially Correct Response
Responses that give Reason 1 only: (Understanding of what
measurement is)
Examples:
• Using familiar objects to measure enables young students to focus just
on the idea of measurement before they have to deal with formal units
and the skill of using a ruler.
• Using everyday objects to measure shows that anything can be used to
measure and makes measurement easier to understand because there is
no abstract scale to read.
Responses that give Reason 2 only: (Need for standard units)
Examples:
• Using non-standard units of different length to measure gives differing
numbers of units for the same length and shows that we need standard
units.
• Using different units like paper clips and pencils to measure means that
students will get different answers for the same length and through
discussion about what measurement is can come to realize the need for
a common unit and more formal system of measurement.

185

Table B3 (cont’d)
12

70

71

79

99

Responses that give Reason 3 only: (Choosing most appropriate unit)
Examples:
• The teacher wants the students to see that they should think about which
unit is most appropriate for different lengths. Pencils would be more
efficient for larger lengths than paperclips, for example. Paperclips would
better for shorter lengths. Paces would be better for very long lengths.
• This would show that long lengths are best measured with large units
(pencils) and short lengths are best measured by small units (paper clips).
Incorrect Response
Responses that focus on motivation, enjoyment, etc.
Examples:
• Using concrete materials is more fun, motivating, interesting and
engaging.
• It is not as boring for the students if the teacher uses a variety of methods
and aids
• The teacher knows that the students will enjoy their work more if
they can use hands-on materials
Responses that focus on other unrelated or insignificant aspects.
Examples:
• Using familiar objects such as pencils encourages estimation skills.
• The teacher wants to encourage creativity by getting students to
measure with paper clips and pencils.
 So that her children will crossed out, erased, stray marks, illegible, or
Other incorrect (including know how to measure with paperclips and
pencils.
off task)
Non-response
Blank

186

Table B4. Item “Jaime”
Code
20

10

11

70

79

99

Response

Item:

MFC208A

Correct Response
Responses that suggest the misconception is that multiplication always
gives a larger answer and that division always gives a smaller answer.
Example:
• He thinks that when you multiply the answer should be larger and
when you divide the answer should be smaller.
Partially Correct Response
Responses that suggest the misconception is that multiplication always
gives a larger answer or that division always gives a smaller answer but
not both.
Examples:
• He thinks that when you multiply the answer should be larger
than either/both numbers.
 He thinks that suggest that Jeremy considers 0.2 as a whole number.
Responses that division should give an answer that is smaller than the
numbers
Example: you started with.
• He thinks he is multiplying and dividing by 2 rather than by 0.2.
Incorrect Response
Responses relating to understanding of decimal numbers, decimal
multiplication/division or use of a calculator.
Example:
• He doesn’t understand decimal multiplication (or division).
• He doesn’t know how to use his calculator.
• Mathematical operations.
• The decimal point.
Other incorrect (including crossed out, erased, stray marks, illegible, or
off task)
Non-response
Blank

187

17

Table B5. Item “Diagram”.
Code
20

10

11

12

70

71

79

99

Response

Item:

MFC208B

Correct Response
A suitable visual representation that clearly shows why 0.2 × 6 is 1.2.
Example:
• 6 lots of 0.2 making it clear that 5 lots of 0.2 = 1, probably with
some annotation. See Pictures 1, 2, 3 and 4 below.
Partially Correct Response
A visual representation that shows 6 lots of 0.2 but does NOT make it
clear how this equals 1.2. Accept 0.2 shown as one-fifth or as twotenths.
Example: See Picture 5 below.
A visual representation that shows how 5 lots of 0.2 make a whole
but does NOT make it clear how 6 lots of 0.2 equals 1.2
Example: See Picture 6 below.
A visual representation of an equation 0.2 × 6 = 1.2 without showing
why it is true.
Example: See Picture 7 below.
• 0.2 + 0.2 + 0.2 + 0.2 + 0.2 + 0.2 = 1.2
Incorrect Response
A visual representation showing 6 lots of 0.2 without showing what
0.2 is or how 5 lots of 0.2 equals 1.
Example: See Picture 8 below.
An example in words suggesting counting in lots of 0.2.
Example:
• “Count 6 lot’s of 0.2 as follows: 0.2, 0.4, 0.6, 0.8, 1.0, 1.2”
Note: This is a good teaching strategy but is not a visual representation.
Other incorrect (including crossed out, erased, stray marks, illegible, or
off task)
Example: An equation or written calculation of the form 0.2 × 6 = 1.2
Non-response
Blank

17

Pictures with examples of correct, partially correct, and incorrect responses for this item are
available in IEA (2012b).
188

APPENDIX C
Future Teachers’ Performance in MPCK Items by Cohort and Institution
Table C1. Future Teachers’ Performance for Item “Jose” ( MFC505) by Cohort and Institution
Full
credit

Partial
credit

No
credit

A

80.0

20.0

0.0

C

57.1

38.1

4.8

D

85.7

14.3

0.0

E

67.1

24.7

8.2

F

80.0

17.1

2.9

Total

72.2

22.8

4.9

A

80.0

20.0

0.0

C

61.0

29.3

9.8

D

84.6

15.4

0.0

E

59.6

26.9

13.5

F

82.6

8.7

8.7

Total

68.5

22.8

8.7

Cohort Institution

1st
Year

5th
Year

189

Table C2. Future Teachers’ Performance for Item “Difficulty” (MFC502B) by Cohort and
Institution.
Full
credit

Partial
credit

No
credit

A

0.0

60.0

40.0

C

0.0

71.4

28.6

D

0.0

86.2

13.8

E

1.3

62.7

36.0

F

2.9

91.4

5.7

Total

1.2

73.9

24.8

A

0.0

100.0

0.0

C

0.0

48.8

51.2

D

0.0

92.3

7.7

E

0.0

53.6

46.4

F

0.0

82.6

17.4

Total

0.0

63.6

36.4

Cohort Institution

1st
Year

5th
Year

190

Table C3. Future Teachers’ Performance for Item “Paper clip” (MFC513) by Cohort and
Institution.
Cohort Institution

Full
credit

Partial
credit

No
credit

A

80.0

0.0

14.3

85.7

D

0.0

17.2

82.8

E

0.0

6.7

93.3

F

0.0

11.4

88.6

Total

0.0

10.9

89.1

A

0.0

70.0

30.0

C
5th
Year

20.0

C
1st
Year

0.0

0.0

19.5

80.5

D

0.0

46.2

53.8

E

0.0

16.1

83.9

F

0.0

26.1

73.9

Total

0.0

25.2

74.8

191

Table C4. Future Teachers’ Performance for Item “Jeremy” (MFC208A) by Cohort and
Institution.
Cohort Institution

Full
credit

Partial
credit

Not
credit

A

80.0

9.5

14.3

76.2

D

27.6

10.3

62.1

E

9.3

8.0

82.7

F

2.9

17.1

80.0

Total

11.5

10.9

77.6

A

20.0

10.0

70.0

C
5th
Year

0.0

C
1st
Year

20.0

14.6

4.9

80.5

D

23.1

0.0

76.9

E

3.6

1.8

94.6

F

8.7

4.3

87.0

Total

10.5

3.5

86.0

192

Table C5. Future Teachers’ Performance for Item “Diagram” ( MFC208B) by Cohort and
Institution.
Cohort Institution

Full
credit

Partial
credit

No
credit

A

100.0

0.0

0.0

100.0

D

3.4

10.3

86.2

E

1.3

0.0

98.7

F

0.0

2.9

97.1

Total

1.2

2.4

96.4

A

0.0

10.0

90.0

B
5th
Year

0.0

C
1st
Year

0.0

0.0

0.0

100.0

C

15.4

15.4

69.2

D

7.1

0.0

7.1

E

0.0

8.7

91.3

F

4.2

3.5

92.3

Total

0.0

0.0

100.0

193

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194

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