Contributed by:

This document includes a set of comprehensive standards describing a national vision for the initial preparation of all teachers who teach mathematics. That is, in addition to early childhood and elementary school teachers who teach all disciplines, middle-grade teachers, and high school mathematics teachers, these standards are also directed toward special education teachers, teachers of emergent multilingual students, and all others who have responsibility for aspects of student learning in mathematics.

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STANDARDS FOR PREPARING TEACHERS OF

MATHEMATICS WRITING TEAM

Nadine Bezuk, San Diego State University, Chair

Jennifer M. Bay-Williams, University of Louisville, Leadership Team Member

Douglas H. Clements, University of Denver, Leadership Team Member

W. Gary Martin, Auburn University, Leadership Team Member

Julia Aguirre, University of Washington Tacoma

Timothy Boerst, University of Michigan

Elizabeth A. Burroughs, Montana State University

Ed Dickey, University of South Carolina

Rochelle Gutiérrez, University of Illinois at Urbana-Champaign

Elizabeth Hughes, University of Northern Iowa

DeAnn Huinker, University of Wisconsin — Milwaukee

Karen Karp, Johns Hopkins University

W. James Lewis*, University of Nebraska-Lincoln

Travis A. Olson, University of Nevada, Las Vegas

Randolph A. Philipp, San Diego State University

Nicole Rigelman, Portland State University

Marilyn E. Strutchens, Auburn University

Christine D. Thomas, Georgia State University

Dorothy Y. White, University of Georgia

W. James Lewis participated as a member of the Writing Team while serving at the National Science Foundation. Any

opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not

necessarily reflect the views of the National Science Foundation.

Thanks to the following people for their work in the design, production, and editing of this document:

Joe Champion, Boise State University; Website Director, Association of Mathematics Teacher Educators

Timothy Hendrix, Meredith College; Executive Director, Association of Mathematics Teacher Educators

Tony Nguyen, Meredith College; Graphic Designer and Webmaster, Association of Mathematics Teacher Educators

Bonnie Schappelle, San Diego State University; Research Assistant & Copy-Editor

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS i

MATHEMATICS WRITING TEAM

Nadine Bezuk, San Diego State University, Chair

Jennifer M. Bay-Williams, University of Louisville, Leadership Team Member

Douglas H. Clements, University of Denver, Leadership Team Member

W. Gary Martin, Auburn University, Leadership Team Member

Julia Aguirre, University of Washington Tacoma

Timothy Boerst, University of Michigan

Elizabeth A. Burroughs, Montana State University

Ed Dickey, University of South Carolina

Rochelle Gutiérrez, University of Illinois at Urbana-Champaign

Elizabeth Hughes, University of Northern Iowa

DeAnn Huinker, University of Wisconsin — Milwaukee

Karen Karp, Johns Hopkins University

W. James Lewis*, University of Nebraska-Lincoln

Travis A. Olson, University of Nevada, Las Vegas

Randolph A. Philipp, San Diego State University

Nicole Rigelman, Portland State University

Marilyn E. Strutchens, Auburn University

Christine D. Thomas, Georgia State University

Dorothy Y. White, University of Georgia

W. James Lewis participated as a member of the Writing Team while serving at the National Science Foundation. Any

opinion, findings, and conclusions or recommendations expressed in this material are those of the authors and do not

necessarily reflect the views of the National Science Foundation.

Thanks to the following people for their work in the design, production, and editing of this document:

Joe Champion, Boise State University; Website Director, Association of Mathematics Teacher Educators

Timothy Hendrix, Meredith College; Executive Director, Association of Mathematics Teacher Educators

Tony Nguyen, Meredith College; Graphic Designer and Webmaster, Association of Mathematics Teacher Educators

Bonnie Schappelle, San Diego State University; Research Assistant & Copy-Editor

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS i

4.
The work of preparing these standards was supported by a grant from the Brookhill Institute of Mathematics.

Standards for Preparing Teachers of Mathematics

Includes bibliographical references. (Portable Document Format)

ISBN 978-0-9986938

Recommended Citation Format

Association of Mathematics Teacher Educators. (2017). Standards for Preparing Teachers of

Mathematics. Available online at amte.net/standards.

Copying and Reprinting

© 2017 by the Association of Mathematics Teacher Educators. This electronic work is licensed under

a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make

fair use of the material. Permission is granted to quote brief passages from this publication in

reviews, provided the customary acknowledgment of the source is given.

Inquiries and requests can also be made by e-mail to amte-support@amte.net, or by mail to the

Association of Mathematics Teacher Educators, c/o Meredith College, 3800 Hillsborough Street,

Raleigh, NC 27607.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS ii

Standards for Preparing Teachers of Mathematics

Includes bibliographical references. (Portable Document Format)

ISBN 978-0-9986938

Recommended Citation Format

Association of Mathematics Teacher Educators. (2017). Standards for Preparing Teachers of

Mathematics. Available online at amte.net/standards.

Copying and Reprinting

© 2017 by the Association of Mathematics Teacher Educators. This electronic work is licensed under

a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.

Individual readers of this publication, and nonprofit libraries acting for them, are permitted to make

fair use of the material. Permission is granted to quote brief passages from this publication in

reviews, provided the customary acknowledgment of the source is given.

Inquiries and requests can also be made by e-mail to amte-support@amte.net, or by mail to the

Association of Mathematics Teacher Educators, c/o Meredith College, 3800 Hillsborough Street,

Raleigh, NC 27607.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS ii

5.
TABLES ............................................................................................................................................ vii

FIGURES ........................................................................................................................................... viii

VIGNETTES ............................................................................................................................................ ix

FOREWORD........................................................................................................................................... x

PREFACE ............................................................................................................................................ xii

Purpose..................................................................................................................................................................................xiv

Audience ................................................................................................................................................................................xiv

Organization of the Document ............................................................................................................................................xv

CHAPTER 1. INTRODUCTION ................................................................................................................ 1

Assumptions about Mathematics Teacher Preparation .................................................................................................... 1

A Well-Prepared Beginning Teacher of Mathematics ......................................................................................................... 3

CHAPTER 2. CANDIDATE KNOWLEDGE, SKILLS, AND DISPOSITIONS ................................................ 5

Organization of This Chapter ................................................................................................................................................ 5

What Should Well-Prepared Beginning Teachers of Mathematics Know and Be Able to Do, and What

Dispositions Should They Develop? ......................................................................................................................... 7

Standard C.1. Mathematics Concepts, Practices, and Curriculum ................................................................ 8

Standard C.2. Pedagogical Knowledge and Practices for Teaching Mathematics ..................................... 12

Standard C.3. Students as Learners of Mathematics ................................................................................... 18

Standard C.4. Social Contexts of Mathematics Teaching and Learning ..................................................... 21

Closing Remarks ................................................................................................................................................................... 24

CHAPTER 3. PROGRAM CHARACTERISTICS TO DEVELOP CANDIDATE KNOWLEDGE, SKILLS,

AND DISPOSITIONS ........................................................................................................... 25

Organization of this Chapter ............................................................................................................................................... 25

Standard P.1. Partnerships .............................................................................................................................. 27

Standard P.2. Opportunities to Learn Mathematics ..................................................................................... 29

Standard P.3. Opportunities to Learn to Teach Mathematics ..................................................................... 33

Standard P.4. Opportunities to Learn in Clinical Settings ............................................................................ 37

Standard P.5. Recruitment and Retention of Teacher Candidates ............................................................. 41

Closing Remarks ................................................................................................................................................................... 43

CHAPTER 4. ELABORATIONS OF THE STANDARDS FOR THE PREPARATION OF EARLY

CHILDHOOD TEACHERS OF MATHEMATICS ..................................................................... 45

Part 1. Elaborations of the Knowledge, Skills, and Dispositions Needed by Well-Prepared Beginning Early

Childhood Teachers of Mathematics ..................................................................................................................... 48

EC.1. Deep Understanding of Early Mathematics ......................................................................................... 48

EC.2. Positive Attitudes Toward Mathematics and Productive Dispositions Toward Teaching

Mathematics ...................................................................................................................................................... 53

EC.3. Mathematics Learning Trajectories: Paths for Excellence and Equity ............................................... 54

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS iii

FIGURES ........................................................................................................................................... viii

VIGNETTES ............................................................................................................................................ ix

FOREWORD........................................................................................................................................... x

PREFACE ............................................................................................................................................ xii

Purpose..................................................................................................................................................................................xiv

Audience ................................................................................................................................................................................xiv

Organization of the Document ............................................................................................................................................xv

CHAPTER 1. INTRODUCTION ................................................................................................................ 1

Assumptions about Mathematics Teacher Preparation .................................................................................................... 1

A Well-Prepared Beginning Teacher of Mathematics ......................................................................................................... 3

CHAPTER 2. CANDIDATE KNOWLEDGE, SKILLS, AND DISPOSITIONS ................................................ 5

Organization of This Chapter ................................................................................................................................................ 5

What Should Well-Prepared Beginning Teachers of Mathematics Know and Be Able to Do, and What

Dispositions Should They Develop? ......................................................................................................................... 7

Standard C.1. Mathematics Concepts, Practices, and Curriculum ................................................................ 8

Standard C.2. Pedagogical Knowledge and Practices for Teaching Mathematics ..................................... 12

Standard C.3. Students as Learners of Mathematics ................................................................................... 18

Standard C.4. Social Contexts of Mathematics Teaching and Learning ..................................................... 21

Closing Remarks ................................................................................................................................................................... 24

CHAPTER 3. PROGRAM CHARACTERISTICS TO DEVELOP CANDIDATE KNOWLEDGE, SKILLS,

AND DISPOSITIONS ........................................................................................................... 25

Organization of this Chapter ............................................................................................................................................... 25

Standard P.1. Partnerships .............................................................................................................................. 27

Standard P.2. Opportunities to Learn Mathematics ..................................................................................... 29

Standard P.3. Opportunities to Learn to Teach Mathematics ..................................................................... 33

Standard P.4. Opportunities to Learn in Clinical Settings ............................................................................ 37

Standard P.5. Recruitment and Retention of Teacher Candidates ............................................................. 41

Closing Remarks ................................................................................................................................................................... 43

CHAPTER 4. ELABORATIONS OF THE STANDARDS FOR THE PREPARATION OF EARLY

CHILDHOOD TEACHERS OF MATHEMATICS ..................................................................... 45

Part 1. Elaborations of the Knowledge, Skills, and Dispositions Needed by Well-Prepared Beginning Early

Childhood Teachers of Mathematics ..................................................................................................................... 48

EC.1. Deep Understanding of Early Mathematics ......................................................................................... 48

EC.2. Positive Attitudes Toward Mathematics and Productive Dispositions Toward Teaching

Mathematics ...................................................................................................................................................... 53

EC.3. Mathematics Learning Trajectories: Paths for Excellence and Equity ............................................... 54

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS iii

6.
EC.4. Tools, Tasks, and Talk as Essential Pedagogies for Meaningful Mathematics ................................. 58

EC.5 Understanding Young Children’s Mathematical Thinking Informs Teaching .................................... 61

EC.6. Collaboration With Families Enhances Children’s Mathematical Development .............................. 63

EC.7. Seeing Mathematics Through Children’s Eyes ..................................................................................... 64

EC.8. Creating Positive Early Childhood Learning Environments ................................................................ 66

Part 2. Elaborations of the Characteristics Needed by Effective Programs Preparing Early Childhood

Teachers of Mathematics ........................................................................................................................................ 68

EC.9. Mathematics Content Preparation of Early Childhood Teachers ...................................................... 69

EC.10. Mathematics Methods Experiences for Early Childhood Teachers ................................................. 70

EC.11. Clinical Experiences in Mathematics for Early Childhood Teachers ................................................ 71

Closing Remarks ................................................................................................................................................................... 72

CHAPTER 5. ELABORATIONS OF THE STANDARDS FOR THE PREPARATION OF UPPER

ELEMENTARY GRADES TEACHERS OF MATHEMATICS ..................................................... 73

Part 1. Elaborations of the Knowledge, Skills, and Dispositions Needed by Well-Prepared Beginning Teachers

of Mathematics in the Upper Elementary Grades ................................................................................................ 75

UE.1. Mathematics Concepts and Connections to Mathematical Practices ............................................... 75

UE.2. Pedagogical Knowledge and Teaching Practices ................................................................................. 82

UE.3. Tools to Build Student Understanding ................................................................................................. 84

UE.4. Assessment to Promote Learning and Improve Instruction .............................................................. 85

UE.5. Students’ Sense Making ......................................................................................................................... 86

UE.6. Ethical Advocates for Students .............................................................................................................. 87

Part 2. Elaborations of the Characteristics Needed by Effective Programs Preparing Mathematics Teachers

for Upper Elementary Grades ................................................................................................................................. 89

UE.7. Mathematical Content Preparation of Teachers of Mathematics in the Upper Elementary Grades

............................................................................................................................................................................ 89

UE.8. Mathematics Methods Coursework for Upper Elementary Teachers of Mathematics .................. 90

UE.9. Clinical Experiences for Upper Elementary Teachers of Mathematics ............................................. 91

Closing Remarks ................................................................................................................................................................... 92

CHAPTER 6. ELABORATIONS OF THE STANDARDS FOR THE PREPARATION OF MIDDLE LEVEL

TEACHERS OF MATHEMATICS ........................................................................................... 93

Part 1. Elaborations of the Knowledge, Skills, and Dispositions Needed by Well-Prepared Beginning Teachers

of Mathematics at the Middle Level ....................................................................................................................... 95

ML.1. Essential Understandings of Mathematics Concepts and Practices ................................................. 95

ML.2. Content Progressions for Middle Level Learners ............................................................................. 102

ML.3. Strategies to Support Early Adolescents ............................................................................................ 103

ML.4. Meaningful and Interdisciplinary Contexts ....................................................................................... 104

ML.5. Mathematical Practices of Middle Level Learners ............................................................................ 106

ML.6. Respond to the Needs of Early Adolescents ..................................................................................... 107

ML.7. Equitable Structures and Systems in Middle Schools ...................................................................... 109

Part 2. Elaborations of the Characteristics Needed by Effective Programs Preparing Middle level Teachers ......... 110

ML.8. Mathematics Content Preparation for Teachers of Mathematics at the Middle Level ................ 111

ML.9 Pedagogical Preparation for Middle Level Teachers of Mathematics ............................................. 113

ML.10. Clinical Experiences in Middle Level Settings .................................................................................. 114

Closing Remarks ................................................................................................................................................................. 116

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS iv

EC.5 Understanding Young Children’s Mathematical Thinking Informs Teaching .................................... 61

EC.6. Collaboration With Families Enhances Children’s Mathematical Development .............................. 63

EC.7. Seeing Mathematics Through Children’s Eyes ..................................................................................... 64

EC.8. Creating Positive Early Childhood Learning Environments ................................................................ 66

Part 2. Elaborations of the Characteristics Needed by Effective Programs Preparing Early Childhood

Teachers of Mathematics ........................................................................................................................................ 68

EC.9. Mathematics Content Preparation of Early Childhood Teachers ...................................................... 69

EC.10. Mathematics Methods Experiences for Early Childhood Teachers ................................................. 70

EC.11. Clinical Experiences in Mathematics for Early Childhood Teachers ................................................ 71

Closing Remarks ................................................................................................................................................................... 72

CHAPTER 5. ELABORATIONS OF THE STANDARDS FOR THE PREPARATION OF UPPER

ELEMENTARY GRADES TEACHERS OF MATHEMATICS ..................................................... 73

Part 1. Elaborations of the Knowledge, Skills, and Dispositions Needed by Well-Prepared Beginning Teachers

of Mathematics in the Upper Elementary Grades ................................................................................................ 75

UE.1. Mathematics Concepts and Connections to Mathematical Practices ............................................... 75

UE.2. Pedagogical Knowledge and Teaching Practices ................................................................................. 82

UE.3. Tools to Build Student Understanding ................................................................................................. 84

UE.4. Assessment to Promote Learning and Improve Instruction .............................................................. 85

UE.5. Students’ Sense Making ......................................................................................................................... 86

UE.6. Ethical Advocates for Students .............................................................................................................. 87

Part 2. Elaborations of the Characteristics Needed by Effective Programs Preparing Mathematics Teachers

for Upper Elementary Grades ................................................................................................................................. 89

UE.7. Mathematical Content Preparation of Teachers of Mathematics in the Upper Elementary Grades

............................................................................................................................................................................ 89

UE.8. Mathematics Methods Coursework for Upper Elementary Teachers of Mathematics .................. 90

UE.9. Clinical Experiences for Upper Elementary Teachers of Mathematics ............................................. 91

Closing Remarks ................................................................................................................................................................... 92

CHAPTER 6. ELABORATIONS OF THE STANDARDS FOR THE PREPARATION OF MIDDLE LEVEL

TEACHERS OF MATHEMATICS ........................................................................................... 93

Part 1. Elaborations of the Knowledge, Skills, and Dispositions Needed by Well-Prepared Beginning Teachers

of Mathematics at the Middle Level ....................................................................................................................... 95

ML.1. Essential Understandings of Mathematics Concepts and Practices ................................................. 95

ML.2. Content Progressions for Middle Level Learners ............................................................................. 102

ML.3. Strategies to Support Early Adolescents ............................................................................................ 103

ML.4. Meaningful and Interdisciplinary Contexts ....................................................................................... 104

ML.5. Mathematical Practices of Middle Level Learners ............................................................................ 106

ML.6. Respond to the Needs of Early Adolescents ..................................................................................... 107

ML.7. Equitable Structures and Systems in Middle Schools ...................................................................... 109

Part 2. Elaborations of the Characteristics Needed by Effective Programs Preparing Middle level Teachers ......... 110

ML.8. Mathematics Content Preparation for Teachers of Mathematics at the Middle Level ................ 111

ML.9 Pedagogical Preparation for Middle Level Teachers of Mathematics ............................................. 113

ML.10. Clinical Experiences in Middle Level Settings .................................................................................. 114

Closing Remarks ................................................................................................................................................................. 116

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS iv

7.
CHAPTER 7. ELABORATIONS OF THE STANDARDS FOR THE PREPARATION OF HIGH SCHOOL

TEACHERS OF MATHEMATICS ......................................................................................... 117

Part 1. Elaborations of the Knowledge, Skills, and Dispositions Needed by Well-Prepared Beginning High

School Mathematics Teachers .............................................................................................................................. 120

HS.1. Essential Understandings of Mathematics Concepts and Practices in High School Mathematics

.......................................................................................................................................................................... 122

HS.2. Use of Tools and Technology to Teach High School Mathematics .................................................. 125

HS.3. Supporting Each and Every Student’s Opportunity to Learn Mathematics .................................... 127

HS.4. Cultivating Positive Mathematical Identities in Each and Every Student ........................................ 130

Part 2. Elaborations of the Characteristics Needed by Effective Programs Preparing High School

Mathematics Teachers ........................................................................................................................................... 133

HS.5. Effective Programs to Support Preparation of Teachers of Mathematics at the High School Level

.......................................................................................................................................................................... 133

HS.6. Partnerships to Support Preparation of Teachers of Mathematics at the High School Level...... 134

HS.7. Mathematical Content Preparation of Teachers of Mathematics at the High School Level......... 136

HS.8. Ethics and Values for Teaching Mathematics at the High School Level .......................................... 138

HS.9. Mathematics Methods Experiences for Teachers of Mathematics at the High School Level....... 141

HS.10. Clinical Experiences for Teachers of Mathematics at the High School Level ............................... 142

Closing Remarks ................................................................................................................................................................. 144

Potential Pathways for Preparing High School Mathematics Teachers ....................................................................... 145

CHAPTER 8. ASSESSING MATHEMATICS TEACHER PREPARATION ................................................. 147

Features of Effective Assessments Used in Mathematics Teacher Preparation ......................................................... 151

Recommendation AF.1. Focus on Mathematics Teaching in Assessments .............................................. 151

Recommendation AF.2. Promote Equity and Access in Assessments ...................................................... 152

Recommendation AF.3. Embody Openness in Assessments..................................................................... 152

Recommendation AF.4. Support Valid Inferences and Action Based on Assessments .......................... 152

Recommendation AF.5. Embody Coherence and Sustainability in Assessments.................................... 155

Assessing Quality of Mathematics Teacher Candidates ................................................................................................. 156

Recommendation AC.1. Assessment of Mathematical Knowledge Relevant to Teaching ..................... 156

Recommendation AC.2. Assessment of Mathematics Teaching Practice................................................. 157

Recommendation AC.3. Assessment of Dispositions ................................................................................. 157

Assessing Quality of Mathematics Teacher Preparation Programs .............................................................................. 158

Recommendation AP.1. Assessment of Stakeholder Engagement ........................................................... 158

Recommendation AP.2. Assessment of Program Curriculum and Instruction........................................ 159

Recommendation AP.3. Assessment of Effective Clinical Experiences ..................................................... 159

Recommendation AP.4. Assessment of Recruitment and Retention........................................................ 160

Closing Remarks ................................................................................................................................................................. 162

CHAPTER 9. ENACTING EFFECTIVE PREPARATION OF TEACHERS OF MATHEMATICS .................. 163

Improving the Preparation of Teachers of Mathematics ............................................................................................... 163

Process for Supporting Improvement of Mathematics Teacher Preparation ............................................................. 164

A Call to Action .................................................................................................................................................................... 166

Closing Remarks ................................................................................................................................................................. 167

REFERENCES ...................................................................................................................................... 169

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS v

TEACHERS OF MATHEMATICS ......................................................................................... 117

Part 1. Elaborations of the Knowledge, Skills, and Dispositions Needed by Well-Prepared Beginning High

School Mathematics Teachers .............................................................................................................................. 120

HS.1. Essential Understandings of Mathematics Concepts and Practices in High School Mathematics

.......................................................................................................................................................................... 122

HS.2. Use of Tools and Technology to Teach High School Mathematics .................................................. 125

HS.3. Supporting Each and Every Student’s Opportunity to Learn Mathematics .................................... 127

HS.4. Cultivating Positive Mathematical Identities in Each and Every Student ........................................ 130

Part 2. Elaborations of the Characteristics Needed by Effective Programs Preparing High School

Mathematics Teachers ........................................................................................................................................... 133

HS.5. Effective Programs to Support Preparation of Teachers of Mathematics at the High School Level

.......................................................................................................................................................................... 133

HS.6. Partnerships to Support Preparation of Teachers of Mathematics at the High School Level...... 134

HS.7. Mathematical Content Preparation of Teachers of Mathematics at the High School Level......... 136

HS.8. Ethics and Values for Teaching Mathematics at the High School Level .......................................... 138

HS.9. Mathematics Methods Experiences for Teachers of Mathematics at the High School Level....... 141

HS.10. Clinical Experiences for Teachers of Mathematics at the High School Level ............................... 142

Closing Remarks ................................................................................................................................................................. 144

Potential Pathways for Preparing High School Mathematics Teachers ....................................................................... 145

CHAPTER 8. ASSESSING MATHEMATICS TEACHER PREPARATION ................................................. 147

Features of Effective Assessments Used in Mathematics Teacher Preparation ......................................................... 151

Recommendation AF.1. Focus on Mathematics Teaching in Assessments .............................................. 151

Recommendation AF.2. Promote Equity and Access in Assessments ...................................................... 152

Recommendation AF.3. Embody Openness in Assessments..................................................................... 152

Recommendation AF.4. Support Valid Inferences and Action Based on Assessments .......................... 152

Recommendation AF.5. Embody Coherence and Sustainability in Assessments.................................... 155

Assessing Quality of Mathematics Teacher Candidates ................................................................................................. 156

Recommendation AC.1. Assessment of Mathematical Knowledge Relevant to Teaching ..................... 156

Recommendation AC.2. Assessment of Mathematics Teaching Practice................................................. 157

Recommendation AC.3. Assessment of Dispositions ................................................................................. 157

Assessing Quality of Mathematics Teacher Preparation Programs .............................................................................. 158

Recommendation AP.1. Assessment of Stakeholder Engagement ........................................................... 158

Recommendation AP.2. Assessment of Program Curriculum and Instruction........................................ 159

Recommendation AP.3. Assessment of Effective Clinical Experiences ..................................................... 159

Recommendation AP.4. Assessment of Recruitment and Retention........................................................ 160

Closing Remarks ................................................................................................................................................................. 162

CHAPTER 9. ENACTING EFFECTIVE PREPARATION OF TEACHERS OF MATHEMATICS .................. 163

Improving the Preparation of Teachers of Mathematics ............................................................................................... 163

Process for Supporting Improvement of Mathematics Teacher Preparation ............................................................. 164

A Call to Action .................................................................................................................................................................... 166

Closing Remarks ................................................................................................................................................................. 167

REFERENCES ...................................................................................................................................... 169

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS v

8.
STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS vi

9.
Table 0.1. Standards and Reports Related To Preparing Teachers of Mathematics..............................................................xii

Table 2.1. Standards and Related Indicators For Well-Prepared Beginning Teachers of Mathematics ............................... 6

Table 2.2. Mathematics Teaching Practices ............................................................................................................................... 15

Table 3.1. Standards and Related Indicators for Effective Programs for Preparing Beginning Teachers of

Mathematics ................................................................................................................................................................ 26

Table 3.2. Minimum Mathematics Content Preparation for Teacher Candidates ................................................................ 30

Table 4.1. Elaborations of Selected Candidate and Program Standards for Early Childhood Teachers of

Mathematics ................................................................................................................................................................ 46

Table 4.2. Connections to MET II (CBMS, 2012) Related to Number and Numeration in Early Childhood ......................... 50

Table 4.3. Connections to MET II (CBMS, 2012) Related to Operations and Algebraic Thinking in Early Childhood.......... 51

Table 4.4. Connections to MET II (CBMS, 2012) Related to Geometry, Measurement, and Data in Early Childhood ........ 53

Table 4.5. Research-based Developmental Learning Trajectories .......................................................................................... 55

Table 5.1. Elaborations of Candidate and Program Standards for Teachers of Mathematics for Upper Elementary

Grades .......................................................................................................................................................................... 74

Table 5.2. Connections to MET II (CBMS, 2012) Related to Multiplicative Structures in the Upper Elementary

Grades .......................................................................................................................................................................... 77

Table 5.3. Connections to MET II (CBMS, 2012) Related to Fractions and Decimals in the Upper Elementary Grades ..... 78

Table 5.4. Connections to MET II (CBMS, 2012) Related to Geometry and Measurement in the Upper Elementary

Grades .......................................................................................................................................................................... 80

Table 5.5. Connections to MET II (CBMS, 2012) Related to Algebraic Thinking in the Upper Elementary Grades ............. 81

Table 6.1. Elaborations of Candidate and Program Standards for Middle Level Teachers of Mathematics ...................... 94

Table 6.2. Connections to MET II (CBMS, 2012) Related to Ratios and Proportional Reasoning at the Middle Level ........ 96

Table 6.3. Connections to MET II (CBMS, 2012) Related to the Number System at the Middle Level ................................. 97

Table 6.4. Connections to MET II (CBMS, 2012) Related to Expressions, Equations, and Functions at the Middle

Level .............................................................................................................................................................................. 98

Table 6.5. Connections to MET II (CBMS, 2012) Related to Geometry at the Middle Level ................................................... 98

Table 6.6. Connections to MET II (CBMS, 2012) and SET (Franklin et al., 2015) Related to Statistics and Probability

at the Middle Level ...................................................................................................................................................... 99

Table 6.7. Interdisciplinary Modeling Task for the Middle Grades ....................................................................................... 105

Table 7.1. Elaborations of Candidate and Program Standards for High School Teachers of Mathematics ..................... 117

Table 8.1. Purposes for Assessment of Mathematics Teacher Preparation ........................................................................ 147

Table 8.2. Recommendations About Assessing Mathematics Teacher Preparation .......................................................... 149

Table 8.3. Relation of Assessment Recommendations to the Candidate Standards .......................................................... 150

Table 8.4. Relation of Assessment Recommendations to Program Standards ................................................................... 150

Table 8.5. Mathematics Teacher Preparation Quality: Attributes and Evidence ................................................................. 154

Table 9.1. Assumptions About Preparing Teachers of Mathematics .................................................................................... 163

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS vii

Table 2.1. Standards and Related Indicators For Well-Prepared Beginning Teachers of Mathematics ............................... 6

Table 2.2. Mathematics Teaching Practices ............................................................................................................................... 15

Table 3.1. Standards and Related Indicators for Effective Programs for Preparing Beginning Teachers of

Mathematics ................................................................................................................................................................ 26

Table 3.2. Minimum Mathematics Content Preparation for Teacher Candidates ................................................................ 30

Table 4.1. Elaborations of Selected Candidate and Program Standards for Early Childhood Teachers of

Mathematics ................................................................................................................................................................ 46

Table 4.2. Connections to MET II (CBMS, 2012) Related to Number and Numeration in Early Childhood ......................... 50

Table 4.3. Connections to MET II (CBMS, 2012) Related to Operations and Algebraic Thinking in Early Childhood.......... 51

Table 4.4. Connections to MET II (CBMS, 2012) Related to Geometry, Measurement, and Data in Early Childhood ........ 53

Table 4.5. Research-based Developmental Learning Trajectories .......................................................................................... 55

Table 5.1. Elaborations of Candidate and Program Standards for Teachers of Mathematics for Upper Elementary

Grades .......................................................................................................................................................................... 74

Table 5.2. Connections to MET II (CBMS, 2012) Related to Multiplicative Structures in the Upper Elementary

Grades .......................................................................................................................................................................... 77

Table 5.3. Connections to MET II (CBMS, 2012) Related to Fractions and Decimals in the Upper Elementary Grades ..... 78

Table 5.4. Connections to MET II (CBMS, 2012) Related to Geometry and Measurement in the Upper Elementary

Grades .......................................................................................................................................................................... 80

Table 5.5. Connections to MET II (CBMS, 2012) Related to Algebraic Thinking in the Upper Elementary Grades ............. 81

Table 6.1. Elaborations of Candidate and Program Standards for Middle Level Teachers of Mathematics ...................... 94

Table 6.2. Connections to MET II (CBMS, 2012) Related to Ratios and Proportional Reasoning at the Middle Level ........ 96

Table 6.3. Connections to MET II (CBMS, 2012) Related to the Number System at the Middle Level ................................. 97

Table 6.4. Connections to MET II (CBMS, 2012) Related to Expressions, Equations, and Functions at the Middle

Level .............................................................................................................................................................................. 98

Table 6.5. Connections to MET II (CBMS, 2012) Related to Geometry at the Middle Level ................................................... 98

Table 6.6. Connections to MET II (CBMS, 2012) and SET (Franklin et al., 2015) Related to Statistics and Probability

at the Middle Level ...................................................................................................................................................... 99

Table 6.7. Interdisciplinary Modeling Task for the Middle Grades ....................................................................................... 105

Table 7.1. Elaborations of Candidate and Program Standards for High School Teachers of Mathematics ..................... 117

Table 8.1. Purposes for Assessment of Mathematics Teacher Preparation ........................................................................ 147

Table 8.2. Recommendations About Assessing Mathematics Teacher Preparation .......................................................... 149

Table 8.3. Relation of Assessment Recommendations to the Candidate Standards .......................................................... 150

Table 8.4. Relation of Assessment Recommendations to Program Standards ................................................................... 150

Table 8.5. Mathematics Teacher Preparation Quality: Attributes and Evidence ................................................................. 154

Table 9.1. Assumptions About Preparing Teachers of Mathematics .................................................................................... 163

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS vii

10.
Figure 1.1. The teacher development continuum ..................................................................................................................... 3

1

Figure 2.1. Fraction-bar representation of the problem 3 ÷ . .................................................................................................. 8

5

Figure 2.2. Sample task for Pre-K–5 teacher candidates.......................................................................................................... 11

Figure 2.3. Sample task for middle level teacher candidates. ................................................................................................. 11

Figure 4.1. A learning trajectory for recognition of number and subitizing. ......................................................................... 57

Figure 9.1. The ongoing and cyclic nature of improving mathematics teacher preparation programs. .......................... 165

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS viii

1

Figure 2.1. Fraction-bar representation of the problem 3 ÷ . .................................................................................................. 8

5

Figure 2.2. Sample task for Pre-K–5 teacher candidates.......................................................................................................... 11

Figure 2.3. Sample task for middle level teacher candidates. ................................................................................................. 11

Figure 4.1. A learning trajectory for recognition of number and subitizing. ......................................................................... 57

Figure 9.1. The ongoing and cyclic nature of improving mathematics teacher preparation programs. .......................... 165

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS viii

11.
Vignette 4.1. Examining Memories of Learning Mathematics ................................................................................................. 54

Vignette 4.2. Building From What Children Understand Mathematically ............................................................................... 60

Vignette 4.3. Collaborative Sense Making of Children’s Mathematical Thinking ................................................................... 62

Vignette 4.4. Solving 21 + 32 ........................................................................................................................................................ 67

Vignette 5.1. Measuring Angles ................................................................................................................................................... 79

Vignette 5.2. Students’ Use of a Key-Words Strategy to Solve Word Problems ..................................................................... 83

Vignette 6.1. A Sequence of Course Activities Focused on Proportional Reasoning and Student Thinking ..................... 101

Vignette 6.2. Co-planning and Co-teaching to Support Every Student ................................................................................. 104

Vignette 6.3. Attending to Algebra Content and Language in a Classroom With Emergent Multilingual Students ......... 108

Vignette 7.1. Meaningful Algebraic Expressions...................................................................................................................... 121

Vignette 7.2. Providing Opportunities to Learn ....................................................................................................................... 128

Vignette 7.3. Mathematical Identity .......................................................................................................................................... 130

Vignette 7.4. A Student Teacher’s Revelation Related to Emergent Multilingual Students ................................................ 131

Vignette 7.5. The Importance of Coherent and Consistent Program Components ............................................................ 135

Vignette 7.6. Helping Candidates Consider Ethical Decisions They May Face ..................................................................... 140

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS ix

Vignette 4.2. Building From What Children Understand Mathematically ............................................................................... 60

Vignette 4.3. Collaborative Sense Making of Children’s Mathematical Thinking ................................................................... 62

Vignette 4.4. Solving 21 + 32 ........................................................................................................................................................ 67

Vignette 5.1. Measuring Angles ................................................................................................................................................... 79

Vignette 5.2. Students’ Use of a Key-Words Strategy to Solve Word Problems ..................................................................... 83

Vignette 6.1. A Sequence of Course Activities Focused on Proportional Reasoning and Student Thinking ..................... 101

Vignette 6.2. Co-planning and Co-teaching to Support Every Student ................................................................................. 104

Vignette 6.3. Attending to Algebra Content and Language in a Classroom With Emergent Multilingual Students ......... 108

Vignette 7.1. Meaningful Algebraic Expressions...................................................................................................................... 121

Vignette 7.2. Providing Opportunities to Learn ....................................................................................................................... 128

Vignette 7.3. Mathematical Identity .......................................................................................................................................... 130

Vignette 7.4. A Student Teacher’s Revelation Related to Emergent Multilingual Students ................................................ 131

Vignette 7.5. The Importance of Coherent and Consistent Program Components ............................................................ 135

Vignette 7.6. Helping Candidates Consider Ethical Decisions They May Face ..................................................................... 140

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS ix

12.
The Association of Mathematics Teacher Educators (AMTE), uniquely positioned as the lead organization for

mathematics teacher education in the United States, puts forth these standards as a national guide for the

preparation of prekindergarten through Grade 12 (Pre-K–12) teachers of mathematics. The mission of AMTE is

to promote the improvement of mathematics teacher education Pre-K–12 with stated goals focused on

effective mathematics teacher education programs and effective policies and practices related to mathematics

teacher education at all levels. Over the 25-year history of AMTE, the organization has produced three

standards documents: Principles to Guide the Design and Implementation of Doctoral Programs in Mathematics

Education (2002), designed for institutions of higher education to guide review, revision, or creation of doctoral

programs in mathematics education; Standards for Elementary Mathematics Specialists (first published 2009,

updated 2013), designed to define and advocate for effective preparation of mathematics specialists; and, now,

the Standards for the Preparation of Teachers of Mathematics (2017), created to address issues and challenges

facing teacher preparation and articulate a national and comprehensive vision for the initial preparation of

teachers of mathematics in Pre-K–12.

The AMTE Board of Directors’ decision to initiate the development of standards was ignited by Dr. Nadine Bezuk

of San Diego State University during her delivery of the Judith Jacobs Lecture (JJL) at the 2015 AMTE annual

conference. The JJL was established in 2003 to honor Dr. Judith E. Jacobs, one of the founding members of

AMTE. Since that time a renowned mathematics educator has been selected each year to deliver the JJL at the

annual conference. The focus of Dr. Bezuk’s lecture was the need for AMTE to provide leadership to the field by

developing standards for preparing teachers to teach mathematics. She argued, “While there exist a number of

documents that address various aspects of the initial preparation of mathematics teachers, there is no single,

definitive document addressing the initial preparation of mathematics teachers more globally.” Further, she

noted an absence, within the mathematics teacher education community, of a shared vision of what preparing

teachers entails and a limited understanding of the degree to which any such vision may be shared with other

stakeholders involved in the initial preparation of mathematics teachers. Unequivocally, the AMTE Board of

Directors was challenged by the lecture to develop and disseminate national standards, and as a result, the

AMTE Board of Directors made the decision to develop and disseminate Pre-K–12 standards for mathematics

teacher preparation.

In March 2015, the AMTE Board of Directors established a leadership team to develop the standards. The team

includes Douglas H. Clements of the University of Denver as lead developer for the early childhood grades;

Nadine Bezuk as chair and lead developer for upper elementary grades; Jennifer Bay-Williams of the University

of Louisville as lead developer for the middle level; and W. Gary Martin of Auburn University for high school.

The leadership team began by establishing criteria for the expertise required to write the standards. The

leadership team sought, as a collective body of knowledge across the members of the writing team, expertise

that included experiences in the preparation of mathematics teachers at the early childhood, upper

elementary, middle school, or high school level; the teaching of mathematics methods courses; the teaching of

mathematics coursework and development of mathematical knowledge for teaching; the supervision of clinical

placements and field experiences; the responsibility for recruiting and retaining students in mathematics

teacher education programs; pedagogy to support emergent multilingual learners; pedagogy to support special

needs students; advocacy for equity in mathematics teaching and learning; and the administration of

mathematics teacher education policy and change agency. With these criteria in mind, the leadership team

identified and invited mathematics educators and mathematicians to serve as members of the writing team.

The members of the writing team are listed in the document. The AMTE Board of Directors extends sincere

appreciation and gratitude to the members of the writing team. These individuals were dedicated and

committed to the significance of this work and the potential of the influence of these standards for improving

mathematics teaching.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS x

mathematics teacher education in the United States, puts forth these standards as a national guide for the

preparation of prekindergarten through Grade 12 (Pre-K–12) teachers of mathematics. The mission of AMTE is

to promote the improvement of mathematics teacher education Pre-K–12 with stated goals focused on

effective mathematics teacher education programs and effective policies and practices related to mathematics

teacher education at all levels. Over the 25-year history of AMTE, the organization has produced three

standards documents: Principles to Guide the Design and Implementation of Doctoral Programs in Mathematics

Education (2002), designed for institutions of higher education to guide review, revision, or creation of doctoral

programs in mathematics education; Standards for Elementary Mathematics Specialists (first published 2009,

updated 2013), designed to define and advocate for effective preparation of mathematics specialists; and, now,

the Standards for the Preparation of Teachers of Mathematics (2017), created to address issues and challenges

facing teacher preparation and articulate a national and comprehensive vision for the initial preparation of

teachers of mathematics in Pre-K–12.

The AMTE Board of Directors’ decision to initiate the development of standards was ignited by Dr. Nadine Bezuk

of San Diego State University during her delivery of the Judith Jacobs Lecture (JJL) at the 2015 AMTE annual

conference. The JJL was established in 2003 to honor Dr. Judith E. Jacobs, one of the founding members of

AMTE. Since that time a renowned mathematics educator has been selected each year to deliver the JJL at the

annual conference. The focus of Dr. Bezuk’s lecture was the need for AMTE to provide leadership to the field by

developing standards for preparing teachers to teach mathematics. She argued, “While there exist a number of

documents that address various aspects of the initial preparation of mathematics teachers, there is no single,

definitive document addressing the initial preparation of mathematics teachers more globally.” Further, she

noted an absence, within the mathematics teacher education community, of a shared vision of what preparing

teachers entails and a limited understanding of the degree to which any such vision may be shared with other

stakeholders involved in the initial preparation of mathematics teachers. Unequivocally, the AMTE Board of

Directors was challenged by the lecture to develop and disseminate national standards, and as a result, the

AMTE Board of Directors made the decision to develop and disseminate Pre-K–12 standards for mathematics

teacher preparation.

In March 2015, the AMTE Board of Directors established a leadership team to develop the standards. The team

includes Douglas H. Clements of the University of Denver as lead developer for the early childhood grades;

Nadine Bezuk as chair and lead developer for upper elementary grades; Jennifer Bay-Williams of the University

of Louisville as lead developer for the middle level; and W. Gary Martin of Auburn University for high school.

The leadership team began by establishing criteria for the expertise required to write the standards. The

leadership team sought, as a collective body of knowledge across the members of the writing team, expertise

that included experiences in the preparation of mathematics teachers at the early childhood, upper

elementary, middle school, or high school level; the teaching of mathematics methods courses; the teaching of

mathematics coursework and development of mathematical knowledge for teaching; the supervision of clinical

placements and field experiences; the responsibility for recruiting and retaining students in mathematics

teacher education programs; pedagogy to support emergent multilingual learners; pedagogy to support special

needs students; advocacy for equity in mathematics teaching and learning; and the administration of

mathematics teacher education policy and change agency. With these criteria in mind, the leadership team

identified and invited mathematics educators and mathematicians to serve as members of the writing team.

The members of the writing team are listed in the document. The AMTE Board of Directors extends sincere

appreciation and gratitude to the members of the writing team. These individuals were dedicated and

committed to the significance of this work and the potential of the influence of these standards for improving

mathematics teaching.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS x

13.
Another noteworthy aspect of the process was the attention given and time invested in a coherent and

comprehensive review of drafts. In the initial planning phase, the leadership team established and

implemented a review process to ensure that a broad range of stakeholders would have opportunities to

provide critical feedback throughout the development stages. As such, a group of mathematics educators was

identified and invited to serve on a review team to facilitate or assist in development of the document while the

writing group engaged in writing and refining chapters. The AMTE Board of Directors especially thanks the

review team for providing feedback on the drafts of the chapters on an ongoing basis. The following were

members of the review team:

Robert Q. Berry, III, University of Virginia; President-elect, National Council of Teachers of Mathematics

Francis (Skip) Fennell, McDaniel College; Past President, Association of Mathematics Teacher Educators;

Past President, National Council of Teachers of Mathematics; Vice Chair, Council for the Accreditation

of Educator Preparation Board of Directors

Matt Larson, Lincoln (Nebraska) Public Schools and President, National Council of Teachers of

Mathematics

Dale Oliver, Humboldt State University (California)

Margaret (Peg) Smith, University of Pittsburgh

Additionally, announcements and special requests for review of the draft standards went out to AMTE

members, other key stakeholders, and professional organizations representing mathematics education and the

mathematical sciences. The AMTE Board of Directors sincerely appreciates all individuals who provided

feedback on the draft document and professional organizations for assembling teams to provide feedback on

their behalf. We received feedback from the following organizations:

American Mathematical Association of Two-Year Colleges (AMATYC)

American Mathematical Society (AMS)

American Statistical Association (ASA)

Association for Middle Level Education (AMLE)

Association for Women in Mathematics (AWM)

Association of Mathematics Teacher Educators of Alabama (AMTEA)

Association of State Supervisors of Mathematics (ASSM)

Benjamin Banneker Association (BBA)

Council of Chief State School Officers (CCSSO)

Education Development Center (EDC)

Hoosier Association of Mathematics Teacher Educators (HAMTE)

Mathematical Association of America (MAA)

National Association of Mathematicians (NAM)

National Council of Supervisors of Mathematics (NCSM)

National Council of Teachers of Mathematics (NCTM)

Society for Industrial and Applied Mathematics (SIAM)

TODOS: Mathematics for All

The Standards for Preparing Teachers of Mathematics are aspirational, advocating for practices that support

candidates in becoming effective teachers of mathematics who guide student learning. These standards will

guide the improvement of individual teacher preparation programs and promote national dialogue and action

related to the preparation of teachers of mathematics.

Christine D. Thomas

AMTE President (2015–2017)

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS xi

comprehensive review of drafts. In the initial planning phase, the leadership team established and

implemented a review process to ensure that a broad range of stakeholders would have opportunities to

provide critical feedback throughout the development stages. As such, a group of mathematics educators was

identified and invited to serve on a review team to facilitate or assist in development of the document while the

writing group engaged in writing and refining chapters. The AMTE Board of Directors especially thanks the

review team for providing feedback on the drafts of the chapters on an ongoing basis. The following were

members of the review team:

Robert Q. Berry, III, University of Virginia; President-elect, National Council of Teachers of Mathematics

Francis (Skip) Fennell, McDaniel College; Past President, Association of Mathematics Teacher Educators;

Past President, National Council of Teachers of Mathematics; Vice Chair, Council for the Accreditation

of Educator Preparation Board of Directors

Matt Larson, Lincoln (Nebraska) Public Schools and President, National Council of Teachers of

Mathematics

Dale Oliver, Humboldt State University (California)

Margaret (Peg) Smith, University of Pittsburgh

Additionally, announcements and special requests for review of the draft standards went out to AMTE

members, other key stakeholders, and professional organizations representing mathematics education and the

mathematical sciences. The AMTE Board of Directors sincerely appreciates all individuals who provided

feedback on the draft document and professional organizations for assembling teams to provide feedback on

their behalf. We received feedback from the following organizations:

American Mathematical Association of Two-Year Colleges (AMATYC)

American Mathematical Society (AMS)

American Statistical Association (ASA)

Association for Middle Level Education (AMLE)

Association for Women in Mathematics (AWM)

Association of Mathematics Teacher Educators of Alabama (AMTEA)

Association of State Supervisors of Mathematics (ASSM)

Benjamin Banneker Association (BBA)

Council of Chief State School Officers (CCSSO)

Education Development Center (EDC)

Hoosier Association of Mathematics Teacher Educators (HAMTE)

Mathematical Association of America (MAA)

National Association of Mathematicians (NAM)

National Council of Supervisors of Mathematics (NCSM)

National Council of Teachers of Mathematics (NCTM)

Society for Industrial and Applied Mathematics (SIAM)

TODOS: Mathematics for All

The Standards for Preparing Teachers of Mathematics are aspirational, advocating for practices that support

candidates in becoming effective teachers of mathematics who guide student learning. These standards will

guide the improvement of individual teacher preparation programs and promote national dialogue and action

related to the preparation of teachers of mathematics.

Christine D. Thomas

AMTE President (2015–2017)

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS xi

14.
The future mathematical success of our nation’s children is largely dependent on the teachers of mathematics

they encounter from prekindergarten to Grade 12 (Pre-K–12). According to Tatto and Senk (2011), “If the quality

of education for every child is to be improved, the education of teachers needs to be taken seriously” (p. 134).

Those involved in preparing teachers of mathematics must ensure that all their candidates have the knowledge,

skills, and dispositions to provide all students access to meaningful experiences with mathematics.

The Association of Mathematics Teacher Educators (AMTE) is the largest U.S. professional organization devoted

to the preparation of teachers of mathematics. AMTE includes more than 1,000 members supporting preservice

teacher education and professional development of teachers of mathematics at all levels from Pre-K–12. AMTE

members include professors, researchers, teacher leaders, school-based and district mathematics supervisors

and coordinators, policymakers, graduate students, and others. The Standards described in this document

reflect AMTE’s leadership in shaping the preparation of Pre-K–12 teachers of mathematics, including clearly

articulated expectations for what well-prepared beginning mathematics teachers need to know and be able to do

upon completion of a certification or licensing program and the characteristics such programs must have to

support teachers' development.

Although the field continues to conduct research regarding effective practices for teacher preparation, we have

a growing research base describing what teaching practices affect student learning and student experiences in

mathematics classrooms. As an example, research indicates that focusing only on teachers' behaviors has a

smaller effect on teachers’ development and subsequently on their students’ learning than does focusing on

teachers’ knowledge of the subject, on the curriculum, or on how students learn the subject (Carpenter,

Fennema, Peterson, & Carey, 1988; Kennedy, 1998; Kwong et al., 2007; Philipp et al., 2007).

A number of recent documents address various aspects of the initial preparation of mathematics teachers.1

Table O.1 summarizes their focus. Although all these documents inform mathematics teacher preparation, no

single, comprehensive document addresses the initial preparation of mathematics teachers across Pre-K–12.

AMTE’s goal is for the standards in this document to provide a clear, comprehensive vision for initial

preparation of teachers of mathematics. Building on the documents in Table 0.1, we, in this document’s

standards, elaborate what beginning teachers of mathematics must know and be able to do as well as the

dispositions they must have to increase equity, access, and opportunities for the mathematical success of each

student. Given the challenges that teachers of mathematics face in preparing their students for future success,

mathematics teacher educators must be guided by a well-articulated vision to prepare teachers of mathematics

to meet those challenges. In this document, we take up that charge.

TABLE 0.1. STANDARDS AND REPORTS RELATED TO PREPARING TEACHERS OF MATHEMATICS

Standards and Reports Specific to Mathematics Teachers

The Mathematical Education of Teachers II (MET II)

The MET II (Conference Board of Mathematical Sciences [CBMS], 2012) addressed the mathematical content

knowledge well-prepared beginning teachers of mathematics should know at the elementary, middle and

high school levels.

For the purposes of this document, mathematics teacher preparation includes preparation to teach statistics,

following common practice. However, we recognize that statistics and statistics education, although related to

mathematics and mathematics education, are distinct. This document will use the term mathematics to

encompass mathematics and statistics; when the distinction between mathematics and statistics is important

to emphasize, statistics will be identified separately.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS xii

they encounter from prekindergarten to Grade 12 (Pre-K–12). According to Tatto and Senk (2011), “If the quality

of education for every child is to be improved, the education of teachers needs to be taken seriously” (p. 134).

Those involved in preparing teachers of mathematics must ensure that all their candidates have the knowledge,

skills, and dispositions to provide all students access to meaningful experiences with mathematics.

The Association of Mathematics Teacher Educators (AMTE) is the largest U.S. professional organization devoted

to the preparation of teachers of mathematics. AMTE includes more than 1,000 members supporting preservice

teacher education and professional development of teachers of mathematics at all levels from Pre-K–12. AMTE

members include professors, researchers, teacher leaders, school-based and district mathematics supervisors

and coordinators, policymakers, graduate students, and others. The Standards described in this document

reflect AMTE’s leadership in shaping the preparation of Pre-K–12 teachers of mathematics, including clearly

articulated expectations for what well-prepared beginning mathematics teachers need to know and be able to do

upon completion of a certification or licensing program and the characteristics such programs must have to

support teachers' development.

Although the field continues to conduct research regarding effective practices for teacher preparation, we have

a growing research base describing what teaching practices affect student learning and student experiences in

mathematics classrooms. As an example, research indicates that focusing only on teachers' behaviors has a

smaller effect on teachers’ development and subsequently on their students’ learning than does focusing on

teachers’ knowledge of the subject, on the curriculum, or on how students learn the subject (Carpenter,

Fennema, Peterson, & Carey, 1988; Kennedy, 1998; Kwong et al., 2007; Philipp et al., 2007).

A number of recent documents address various aspects of the initial preparation of mathematics teachers.1

Table O.1 summarizes their focus. Although all these documents inform mathematics teacher preparation, no

single, comprehensive document addresses the initial preparation of mathematics teachers across Pre-K–12.

AMTE’s goal is for the standards in this document to provide a clear, comprehensive vision for initial

preparation of teachers of mathematics. Building on the documents in Table 0.1, we, in this document’s

standards, elaborate what beginning teachers of mathematics must know and be able to do as well as the

dispositions they must have to increase equity, access, and opportunities for the mathematical success of each

student. Given the challenges that teachers of mathematics face in preparing their students for future success,

mathematics teacher educators must be guided by a well-articulated vision to prepare teachers of mathematics

to meet those challenges. In this document, we take up that charge.

TABLE 0.1. STANDARDS AND REPORTS RELATED TO PREPARING TEACHERS OF MATHEMATICS

Standards and Reports Specific to Mathematics Teachers

The Mathematical Education of Teachers II (MET II)

The MET II (Conference Board of Mathematical Sciences [CBMS], 2012) addressed the mathematical content

knowledge well-prepared beginning teachers of mathematics should know at the elementary, middle and

high school levels.

For the purposes of this document, mathematics teacher preparation includes preparation to teach statistics,

following common practice. However, we recognize that statistics and statistics education, although related to

mathematics and mathematics education, are distinct. This document will use the term mathematics to

encompass mathematics and statistics; when the distinction between mathematics and statistics is important

to emphasize, statistics will be identified separately.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS xii

15.
National Council of Teachers of Mathematics’ Council for the Association of Educator Preparation (CAEP) Standards

The National Council of Teachers of Mathematics (NCTM) CAEP Standards (NCTM & CAEP, 2012a, 2012b)

described what effective preservice teachers of secondary mathematics should know and be able to do,

informing program reviews for middle and high school mathematics programs.

Statistical Education of Teachers (SET)

SET (Franklin et al., 2015) describes the statistical content knowledge well-prepared beginning teachers of

mathematics should know at the elementary, middle and high school levels.

Teacher Education and Development Study in Mathematics (TEDS-M)

In the TEDS-M, researchers examined and discussed findings and challenges related to the mathematics

education of future primary, middle, and high school teachers (Tatto & Senk, 2011).

Standards Not Specific to Mathematics That Also Apply to Teachers of Mathematics

Association for Middle Level Education (AMLE) Middle Level Teacher Preparation Standards with Rubrics and

Supporting Explanations

AMLE (2012) describes five standards, including one on content, which addresses content, standards for

students, and the interdisciplinary nature of knowledge.

Council for Exceptional Children (CEC) Initial Preparation Standards

The CEC (2012) standards require that beginning professionals understand and use mathematics concepts

to individualize learning for students.

Council for the Accreditation of Educator Preparation (CAEP) Accreditation Standards

The CAEP (2013) describes candidate and program expectations that define quality programs. Among these

is the expectation that candidates demonstrate content and pedagogical knowledge in their content

domains.

National Association for the Education of Young Children (NAEYC) Standards for Initial and Advanced Early

Childhood professional Preparation Programs

The NAEYC (2010) professional standards describe the importance of knowing mathematics and teaching it

in ways that promote sense making and nurture positive development.

Standards for Experienced Teachers of Mathematics

Mathematics Specialists

In the Standards for Elementary Mathematics Specialists (2013), AMTE outlined “particular knowledge, skills,

and dispositions” needed by elementary mathematics specialists who “teach and support others who teach

mathematics at the elementary level.” (p. iv)

Interstate Teacher Assessment and Support Consortium (InTASC) Model Core Teaching Standards

The InTASC Model Core Teaching Standards (Council of Chief State School Officers [CCSSO], 2013) are used in

states, school districts, professional organizations, and teacher education programs to support teachers.

National Board for Professional Teaching Standards (NBPTS) Standards

The NBPTS standards recognize accomplished teachers and included certifications for early childhood and

elementary school generalists (2012a, 2012b), and middle school and high school mathematics teachers

(2010).

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS xiii

The National Council of Teachers of Mathematics (NCTM) CAEP Standards (NCTM & CAEP, 2012a, 2012b)

described what effective preservice teachers of secondary mathematics should know and be able to do,

informing program reviews for middle and high school mathematics programs.

Statistical Education of Teachers (SET)

SET (Franklin et al., 2015) describes the statistical content knowledge well-prepared beginning teachers of

mathematics should know at the elementary, middle and high school levels.

Teacher Education and Development Study in Mathematics (TEDS-M)

In the TEDS-M, researchers examined and discussed findings and challenges related to the mathematics

education of future primary, middle, and high school teachers (Tatto & Senk, 2011).

Standards Not Specific to Mathematics That Also Apply to Teachers of Mathematics

Association for Middle Level Education (AMLE) Middle Level Teacher Preparation Standards with Rubrics and

Supporting Explanations

AMLE (2012) describes five standards, including one on content, which addresses content, standards for

students, and the interdisciplinary nature of knowledge.

Council for Exceptional Children (CEC) Initial Preparation Standards

The CEC (2012) standards require that beginning professionals understand and use mathematics concepts

to individualize learning for students.

Council for the Accreditation of Educator Preparation (CAEP) Accreditation Standards

The CAEP (2013) describes candidate and program expectations that define quality programs. Among these

is the expectation that candidates demonstrate content and pedagogical knowledge in their content

domains.

National Association for the Education of Young Children (NAEYC) Standards for Initial and Advanced Early

Childhood professional Preparation Programs

The NAEYC (2010) professional standards describe the importance of knowing mathematics and teaching it

in ways that promote sense making and nurture positive development.

Standards for Experienced Teachers of Mathematics

Mathematics Specialists

In the Standards for Elementary Mathematics Specialists (2013), AMTE outlined “particular knowledge, skills,

and dispositions” needed by elementary mathematics specialists who “teach and support others who teach

mathematics at the elementary level.” (p. iv)

Interstate Teacher Assessment and Support Consortium (InTASC) Model Core Teaching Standards

The InTASC Model Core Teaching Standards (Council of Chief State School Officers [CCSSO], 2013) are used in

states, school districts, professional organizations, and teacher education programs to support teachers.

National Board for Professional Teaching Standards (NBPTS) Standards

The NBPTS standards recognize accomplished teachers and included certifications for early childhood and

elementary school generalists (2012a, 2012b), and middle school and high school mathematics teachers

(2010).

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS xiii

16.
This document includes a set of comprehensive standards describing a national vision for the initial preparation

of all teachers Pre-K–12 who teach mathematics. That is, in addition to early childhood and elementary school

teachers who teach all disciplines, middle grade teachers, and high school mathematics teachers, these

standards are also directed toward special education teachers, teachers of emergent multilingual students, and

all others who have responsibility for aspects of student learning in mathematics.

These standards are intended to

guide the improvement of individual teacher preparation programs,

inform the accreditation process of such programs,

influence policies related to preparation of teachers of mathematics, and

promote national dialogue and action related to preparation of teachers of mathematics.

These standards are aspirational, advocating for mathematics teacher preparation practices that support

candidates in becoming high-quality teachers who are ethical advocates for children and effectively guide

student learning aligned with research and best practices, rather than describing minimum levels of

competency needed by beginning teachers. The standards are intended both to build on existing research

about mathematics teacher preparation and existing standards and to motivate researchers to investigate

areas that are less well understood.

The audience for these standards includes all those involved in mathematics teacher preparation, including

faculty and others involved in the initial preparation of mathematics teachers; classroom teachers and other

Pre-K–12 school personnel who support student teachers and field placements; coordinators of mathematics

teacher preparation programs; deans, provosts, and other program administrators who make decisions

regarding content and funding of mathematics teacher preparation programs; CAEP, the largest accreditor of

teacher education programs in the United States as well as state licensure or credentialing

agencies/organizations; NCTM, the professional association responsible for setting standards for educator-

preparation programs for preservice, middle, and high school mathematics; and other organizations, including

specialized professional associations (e.g., NAEYC, CEC) and agencies focused on and involved in the

preparation of mathematics teachers.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS xiv

of all teachers Pre-K–12 who teach mathematics. That is, in addition to early childhood and elementary school

teachers who teach all disciplines, middle grade teachers, and high school mathematics teachers, these

standards are also directed toward special education teachers, teachers of emergent multilingual students, and

all others who have responsibility for aspects of student learning in mathematics.

These standards are intended to

guide the improvement of individual teacher preparation programs,

inform the accreditation process of such programs,

influence policies related to preparation of teachers of mathematics, and

promote national dialogue and action related to preparation of teachers of mathematics.

These standards are aspirational, advocating for mathematics teacher preparation practices that support

candidates in becoming high-quality teachers who are ethical advocates for children and effectively guide

student learning aligned with research and best practices, rather than describing minimum levels of

competency needed by beginning teachers. The standards are intended both to build on existing research

about mathematics teacher preparation and existing standards and to motivate researchers to investigate

areas that are less well understood.

The audience for these standards includes all those involved in mathematics teacher preparation, including

faculty and others involved in the initial preparation of mathematics teachers; classroom teachers and other

Pre-K–12 school personnel who support student teachers and field placements; coordinators of mathematics

teacher preparation programs; deans, provosts, and other program administrators who make decisions

regarding content and funding of mathematics teacher preparation programs; CAEP, the largest accreditor of

teacher education programs in the United States as well as state licensure or credentialing

agencies/organizations; NCTM, the professional association responsible for setting standards for educator-

preparation programs for preservice, middle, and high school mathematics; and other organizations, including

specialized professional associations (e.g., NAEYC, CEC) and agencies focused on and involved in the

preparation of mathematics teachers.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS xiv

17.
ORGANIZATION OF THE DOCUMENT

The AMTE Standards for Preparing Teachers of Mathematics are organized in the following way:

Chapter 1 describes the overall framework, including a set of assumptions that underlie the

recommendations made throughout the document.

Chapter 2 provides standards for the professional knowledge, skills, and dispositions that well-

prepared beginning teachers of mathematics need to possess related to content, teaching, learners

and learning, and the social context of mathematics education. Each standard includes indicators

describing what attainment of that standard by candidates entails.

Chapter 3 describes standards for mathematics teacher preparation programs designed to develop the

knowledge, skills, and dispositions of their teacher candidates described in Chapter 2. Again, each

standard includes indicators of what attainment of that standard by a program entails.

Chapters 4 through 7 provide specific elaborations of the standards in Chapters 2 and 3 to relate them

to the specific needs for preparation of teachers of mathematics at different levels of instruction and

discuss their alignment with other standards. These grade-bands include Prekindergarten to Grade 2

(early childhood), Grades 3 through 5 (upper elementary), Grades 6 through 8 (middle level), and

Grades 9 through 12 (high school).

Chapter 8 provides recommendations for effectively assessing candidates and programs in meeting the

standards and elaborations.

Chapter 9 provides advice on how the vision of this document can be attained, including action steps

that those involved in mathematics teacher preparation.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS xv

The AMTE Standards for Preparing Teachers of Mathematics are organized in the following way:

Chapter 1 describes the overall framework, including a set of assumptions that underlie the

recommendations made throughout the document.

Chapter 2 provides standards for the professional knowledge, skills, and dispositions that well-

prepared beginning teachers of mathematics need to possess related to content, teaching, learners

and learning, and the social context of mathematics education. Each standard includes indicators

describing what attainment of that standard by candidates entails.

Chapter 3 describes standards for mathematics teacher preparation programs designed to develop the

knowledge, skills, and dispositions of their teacher candidates described in Chapter 2. Again, each

standard includes indicators of what attainment of that standard by a program entails.

Chapters 4 through 7 provide specific elaborations of the standards in Chapters 2 and 3 to relate them

to the specific needs for preparation of teachers of mathematics at different levels of instruction and

discuss their alignment with other standards. These grade-bands include Prekindergarten to Grade 2

(early childhood), Grades 3 through 5 (upper elementary), Grades 6 through 8 (middle level), and

Grades 9 through 12 (high school).

Chapter 8 provides recommendations for effectively assessing candidates and programs in meeting the

standards and elaborations.

Chapter 9 provides advice on how the vision of this document can be attained, including action steps

that those involved in mathematics teacher preparation.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS xv

18.
CHAPTER 1. INTRODUCTION

As a professional community, mathematics teacher educators have begun to define, research, and refine the

characteristics of effective teachers of mathematics and, in particular, the professional proficiencies of a well-

prepared beginning teacher of mathematics. This document describes a set of proficiencies for well-prepared

beginners and for programs preparing mathematics teachers. Although these proficiencies are grounded in

available research, in many areas that research is not yet sufficient to determine the specific knowledge, skills,

and dispositions that will enable beginning teachers to be highly effective in their first years of teaching. Hence,

the standards presented in this document are intended to engage the mathematics teacher education

community in continued research and discussion about what candidates must learn during their initial

preparation as teachers of mathematics.

ASSUMPTIONS ABOUT MATHEMATICS TEACHER PREPARATION

The Standards for Preparing Teachers of Mathematics are centered on five foundational assumptions about

mathematics teacher preparation. These assumptions reflect the emerging consensus of those involved in

mathematics teacher preparation in response to the needs of both their teacher candidates and the students

those candidates will teach. These assumptions underlie the standards presented in Chapters 2 and 3 as well as

the grade-band elaborations in Chapters 4 through 7.

Assumption #1. Ensuring the success of each and every learner requires a deep,

integrated focus on equity in every program that prepares teachers of mathematics.

Over the past decades, the need for a central focus on issues related to equity in mathematics

education has become clear in reflecting on the uneven performance of students by various

demographic factors (AMTE, 2015; NCTM, 2000, 2014a, 2014b). Although equity, diversity, and social-

justice issues need to be specifically addressed as standards, they must also be embedded within all

the standards. Addressing these issues solely within the context of “equity standards” might be

misinterpreted to imply that these issues are not important within the other standards; conversely, if

they are not directly addressed in standards addressing equity, their centrality to the mission of

mathematics teacher preparation can be overlooked. Thus, we believe that equity must be both

addressed in its own right and embedded within every standard. Every standard must be built on the

premise that it applies to each and every student, recognizing that equity requires acknowledging the

particular context, needs, and capabilities of each and every learner rather than providing identical

opportunities to students.

Assumption #2. Teaching mathematics effectively requires career-long learning.

Experienced teachers reflecting on their first year of teaching mathematics have frequently described

how much more they can now accomplish, given their current level of teaching competence and

understanding of the mathematics and students they are teaching. Teachers improve through

reflective experience and through intentional efforts to seek additional knowledge. They use that

knowledge to build their understanding of the mathematics they teach and to support their

improvement in supporting students’ learning of mathematics. This process must begin during their

initial preparation and continue throughout their careers. Knowing that candidates will complete

teacher preparation programs without the expertise they will later develop focuses attention on

priorities for beginning teachers. Those priorities become the knowledge, skills, and dispositions of a

well-prepared beginner.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 1

As a professional community, mathematics teacher educators have begun to define, research, and refine the

characteristics of effective teachers of mathematics and, in particular, the professional proficiencies of a well-

prepared beginning teacher of mathematics. This document describes a set of proficiencies for well-prepared

beginners and for programs preparing mathematics teachers. Although these proficiencies are grounded in

available research, in many areas that research is not yet sufficient to determine the specific knowledge, skills,

and dispositions that will enable beginning teachers to be highly effective in their first years of teaching. Hence,

the standards presented in this document are intended to engage the mathematics teacher education

community in continued research and discussion about what candidates must learn during their initial

preparation as teachers of mathematics.

ASSUMPTIONS ABOUT MATHEMATICS TEACHER PREPARATION

The Standards for Preparing Teachers of Mathematics are centered on five foundational assumptions about

mathematics teacher preparation. These assumptions reflect the emerging consensus of those involved in

mathematics teacher preparation in response to the needs of both their teacher candidates and the students

those candidates will teach. These assumptions underlie the standards presented in Chapters 2 and 3 as well as

the grade-band elaborations in Chapters 4 through 7.

Assumption #1. Ensuring the success of each and every learner requires a deep,

integrated focus on equity in every program that prepares teachers of mathematics.

Over the past decades, the need for a central focus on issues related to equity in mathematics

education has become clear in reflecting on the uneven performance of students by various

demographic factors (AMTE, 2015; NCTM, 2000, 2014a, 2014b). Although equity, diversity, and social-

justice issues need to be specifically addressed as standards, they must also be embedded within all

the standards. Addressing these issues solely within the context of “equity standards” might be

misinterpreted to imply that these issues are not important within the other standards; conversely, if

they are not directly addressed in standards addressing equity, their centrality to the mission of

mathematics teacher preparation can be overlooked. Thus, we believe that equity must be both

addressed in its own right and embedded within every standard. Every standard must be built on the

premise that it applies to each and every student, recognizing that equity requires acknowledging the

particular context, needs, and capabilities of each and every learner rather than providing identical

opportunities to students.

Assumption #2. Teaching mathematics effectively requires career-long learning.

Experienced teachers reflecting on their first year of teaching mathematics have frequently described

how much more they can now accomplish, given their current level of teaching competence and

understanding of the mathematics and students they are teaching. Teachers improve through

reflective experience and through intentional efforts to seek additional knowledge. They use that

knowledge to build their understanding of the mathematics they teach and to support their

improvement in supporting students’ learning of mathematics. This process must begin during their

initial preparation and continue throughout their careers. Knowing that candidates will complete

teacher preparation programs without the expertise they will later develop focuses attention on

priorities for beginning teachers. Those priorities become the knowledge, skills, and dispositions of a

well-prepared beginner.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 1

19.
Assumption #3. Learning to teach mathematics requires a central focus on

Teaching is often approached as a general craft that is independent of the content being taught.

Effective mathematics teaching, however, requires not just general pedagogical skills but also content-

specific knowledge, skills, and dispositions. To support student learning and develop positive

dispositions toward mathematics, mathematics teachers at every level of instruction need deep and

flexible knowledge of the mathematics they teach, of how students think about and learn

mathematics, of instructional approaches that support mathematical learning, and of the societal

context in which mathematics is taught and used in everyday life to effectively support student

learning of, and positive dispositions toward, mathematics.

Assumption #4. Multiple stakeholders must be responsible for and invested in

preparing teachers of mathematics.

Preparing teachers to teach in ways that ensure that each and every student learns important

mathematics requires the concerted effort of everyone who holds a stake in students’ future

successes. Mathematics teacher educators and mathematicians; other teacher educators; program

and school administrators; classroom teachers, including special education teachers; families and

communities; policymakers and others in the educational system all play critical roles. When these

groups send mixed messages about how mathematics is best taught and learned, beginning teachers

receive incomplete and fragmented visions of how to enact effective mathematics learning

environments for their students. Successful mathematics teacher preparation requires a shared vision

of mathematics learning outcomes for students, of effective mathematics learning environments, and

of the kinds of experiences that best support a mathematics teacher’s continuing growth and

development. Moreover, stakeholders must feel both included in the development of that vision and

accountable for enacting that vision.

Assumption #5. Those involved in mathematics teacher preparation must be

committed to improving their effectiveness in preparing future teachers of

Mathematics teacher preparation program structures differ widely, as do the needs and backgrounds

of teacher candidates. Additionally, mathematics teacher preparation occurs within a range of

contexts; in the United States, hundreds of institutions as well as online and school district programs

prepare teachers of mathematics, and each one is unique. Thus, program personnel need to discern

how existing research might apply to their context and how they can respond to issues not yet

addressed by research. Further, they must assess the relationship between their unique program and

its effectiveness, sharing relevant findings with the broader mathematics teacher preparation

community (e.g., through publications and presentations at conferences).

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 2

Teaching is often approached as a general craft that is independent of the content being taught.

Effective mathematics teaching, however, requires not just general pedagogical skills but also content-

specific knowledge, skills, and dispositions. To support student learning and develop positive

dispositions toward mathematics, mathematics teachers at every level of instruction need deep and

flexible knowledge of the mathematics they teach, of how students think about and learn

mathematics, of instructional approaches that support mathematical learning, and of the societal

context in which mathematics is taught and used in everyday life to effectively support student

learning of, and positive dispositions toward, mathematics.

Assumption #4. Multiple stakeholders must be responsible for and invested in

preparing teachers of mathematics.

Preparing teachers to teach in ways that ensure that each and every student learns important

mathematics requires the concerted effort of everyone who holds a stake in students’ future

successes. Mathematics teacher educators and mathematicians; other teacher educators; program

and school administrators; classroom teachers, including special education teachers; families and

communities; policymakers and others in the educational system all play critical roles. When these

groups send mixed messages about how mathematics is best taught and learned, beginning teachers

receive incomplete and fragmented visions of how to enact effective mathematics learning

environments for their students. Successful mathematics teacher preparation requires a shared vision

of mathematics learning outcomes for students, of effective mathematics learning environments, and

of the kinds of experiences that best support a mathematics teacher’s continuing growth and

development. Moreover, stakeholders must feel both included in the development of that vision and

accountable for enacting that vision.

Assumption #5. Those involved in mathematics teacher preparation must be

committed to improving their effectiveness in preparing future teachers of

Mathematics teacher preparation program structures differ widely, as do the needs and backgrounds

of teacher candidates. Additionally, mathematics teacher preparation occurs within a range of

contexts; in the United States, hundreds of institutions as well as online and school district programs

prepare teachers of mathematics, and each one is unique. Thus, program personnel need to discern

how existing research might apply to their context and how they can respond to issues not yet

addressed by research. Further, they must assess the relationship between their unique program and

its effectiveness, sharing relevant findings with the broader mathematics teacher preparation

community (e.g., through publications and presentations at conferences).

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 2

20.
A WELL-PREPARED BEGINNING TEACHER OF MATHEMATICS

As stated in Assumption 2, the development of teachers’ content and teaching knowledge, skills, and

dispositions occurs over a career-long trajectory. For example, InTASC developed learning progressions to

describe “a coherent continuum of expectations for teachers from beginning through accomplished practice”

(Council of Chief State School Officers, 2013, p. 6). Figure 1.1 depicts the career-long continuum of teacher

Figure 1.1. The teacher development continuum.

Note. Adapted from Developing the Analytic Framework: A Tool for Supporting Innovation and Quality Design in the

Preparation and Development of Science and Mathematics Teachers (p. 9) by C. R. Coble, 2012. Washington, DC:

Association of Public and Land-grant Universities. Copyright 2012 by APLU.

The standards in this document address primarily the initial preparation phase of the trajectory depicted in

Figure 1.1, with some attention to the recruitment of teacher candidates. Chapter 2 provides clear expectations,

based on the current knowledge base and national recommendations, for what a well-prepared beginning

teacher of mathematics needs to know and be able to do as well as productive dispositions they need to

develop, while Chapter 3 describes what programs need to provide to enable candidates to meet these

expectations. Well-prepared beginning teachers of mathematics must be committed to supporting the

mathematical success of each and every student, and with proper support from the mathematics teacher

education community, they will continue to become more effective throughout their careers.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 3

As stated in Assumption 2, the development of teachers’ content and teaching knowledge, skills, and

dispositions occurs over a career-long trajectory. For example, InTASC developed learning progressions to

describe “a coherent continuum of expectations for teachers from beginning through accomplished practice”

(Council of Chief State School Officers, 2013, p. 6). Figure 1.1 depicts the career-long continuum of teacher

Figure 1.1. The teacher development continuum.

Note. Adapted from Developing the Analytic Framework: A Tool for Supporting Innovation and Quality Design in the

Preparation and Development of Science and Mathematics Teachers (p. 9) by C. R. Coble, 2012. Washington, DC:

Association of Public and Land-grant Universities. Copyright 2012 by APLU.

The standards in this document address primarily the initial preparation phase of the trajectory depicted in

Figure 1.1, with some attention to the recruitment of teacher candidates. Chapter 2 provides clear expectations,

based on the current knowledge base and national recommendations, for what a well-prepared beginning

teacher of mathematics needs to know and be able to do as well as productive dispositions they need to

develop, while Chapter 3 describes what programs need to provide to enable candidates to meet these

expectations. Well-prepared beginning teachers of mathematics must be committed to supporting the

mathematical success of each and every student, and with proper support from the mathematics teacher

education community, they will continue to become more effective throughout their careers.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 3

21.
STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 4

22.
CHAPTER 2. CANDIDATE KNOWLEDGE, SKILLS, AND

Teaching is a complex enterprise, and teaching mathematics is particularly demanding. Thus, that initial

preparation focused on teachers’ knowledge of the subject, on the curriculum, or on how students learn the

subject is more effective than preparation focused on teachers' specific behaviors is not surprising (cf. Ball &

Forzani, 2011; Philipp et al., 2007). As described in Accreditation Standards and Evidence: Aspirations for Educator

Preparation (Council for the Accreditation of Educator Preparation (CAEP), 2013), teacher candidates must learn

“critical concepts and principles of their discipline and, by completion, are able to use discipline-specific

practices flexibly” (p. 2). This chapter includes standards and indicators to describe the specific knowledge, skills,

and dispositions that well-prepared mathematics teacher candidates at all levels will know and be able to do

upon completion of an initial preparation program. Additional expectations specific to a particular grade-band

are provided in Chapters 4–7 as elaborations. We refer to those who are starting their careers after completion

of a teacher preparation program as “well-prepared beginning teachers of mathematics” (“well-prepared

beginners” for short).

ORGANIZATION OF THIS CHAPTER

This chapter includes four equally important and interrelated standards that describe the knowledge, skills, and

dispositions that well-prepared beginners need to acquire. The first standard, “Knowledge of Mathematics

Concepts, Practices, and Curriculum,” describes disciplinary knowledge involved in the teaching of mathematics.

The second, “Knowledge and Pedagogical Practices for Teaching Mathematics,” describes research-based

practices or strategies for effective mathematics teaching. The third, “Knowledge of Students as Learners of

Mathematics,” describes what teachers need to know about their students’ mathematical knowledge, skills,

representations, and dispositions, for both individual students and groups of students. The final standard in

this chapter, “Social Contexts of Mathematics Teaching and Learning,” describes the knowledge and

dispositions beginning teachers must have related to the social, historical, and institutional contexts of

mathematics that affect teaching and learning, themes also woven into the first three standards.

As indicated in Table 2.1, each standard includes specific indicators, along with accompanying explanations.

These standards and indicators apply to all well-prepared beginning teachers of mathematics from

prekindergarten through high school.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 5

Teaching is a complex enterprise, and teaching mathematics is particularly demanding. Thus, that initial

preparation focused on teachers’ knowledge of the subject, on the curriculum, or on how students learn the

subject is more effective than preparation focused on teachers' specific behaviors is not surprising (cf. Ball &

Forzani, 2011; Philipp et al., 2007). As described in Accreditation Standards and Evidence: Aspirations for Educator

Preparation (Council for the Accreditation of Educator Preparation (CAEP), 2013), teacher candidates must learn

“critical concepts and principles of their discipline and, by completion, are able to use discipline-specific

practices flexibly” (p. 2). This chapter includes standards and indicators to describe the specific knowledge, skills,

and dispositions that well-prepared mathematics teacher candidates at all levels will know and be able to do

upon completion of an initial preparation program. Additional expectations specific to a particular grade-band

are provided in Chapters 4–7 as elaborations. We refer to those who are starting their careers after completion

of a teacher preparation program as “well-prepared beginning teachers of mathematics” (“well-prepared

beginners” for short).

ORGANIZATION OF THIS CHAPTER

This chapter includes four equally important and interrelated standards that describe the knowledge, skills, and

dispositions that well-prepared beginners need to acquire. The first standard, “Knowledge of Mathematics

Concepts, Practices, and Curriculum,” describes disciplinary knowledge involved in the teaching of mathematics.

The second, “Knowledge and Pedagogical Practices for Teaching Mathematics,” describes research-based

practices or strategies for effective mathematics teaching. The third, “Knowledge of Students as Learners of

Mathematics,” describes what teachers need to know about their students’ mathematical knowledge, skills,

representations, and dispositions, for both individual students and groups of students. The final standard in

this chapter, “Social Contexts of Mathematics Teaching and Learning,” describes the knowledge and

dispositions beginning teachers must have related to the social, historical, and institutional contexts of

mathematics that affect teaching and learning, themes also woven into the first three standards.

As indicated in Table 2.1, each standard includes specific indicators, along with accompanying explanations.

These standards and indicators apply to all well-prepared beginning teachers of mathematics from

prekindergarten through high school.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 5

23.
TABLE 2.1. STANDARDS AND RELATED INDICATORS FOR WELL-PREPARED BEGINNING TEACHERS OF

STANDARD RELATED INDICATORS

C.1. Mathematics Concepts, Practices, and Curriculum

Well-prepared beginning teachers of mathematics C.1.1. Know Relevant Mathematical Content

possess robust knowledge of mathematical and C.1.2. Demonstrate Mathematical Practices and

statistical concepts that underlie what they encounter Processes

in teaching. They engage in appropriate mathematical

C.1.3. Exhibit Productive Mathematical Dispositions

and statistical practices and support their students in

doing the same. They can read, analyze, and discuss C.1.4. Analyze the Mathematical Content of

curriculum, assessment, and standards documents as Curriculum

well as students’ mathematical productions. C.1.5. Analyze Mathematical Thinking

C.1.6. Use Mathematical Tools and Technology

C.2. Pedagogical Knowledge and Practices for Teaching Mathematics

Well-prepared beginning teachers of mathematics C.2.1. Promote Equitable Teaching

have foundations of pedagogical knowledge, effective C.2.2. Plan for Effective Instruction

and equitable mathematics teaching practices, and

C.2.3. Implement Effective Instruction

positive and productive dispositions toward teaching

mathematics to support students’ sense making, C.2.4. Analyze Teaching Practice

understanding, and reasoning. C.2.5. Enhance Teaching Through Collaboration With

Colleagues, Families, and Community

Members

C.3. Students as Learners of Mathematics

Well-prepared beginning teachers of mathematics C.3.1. Anticipate and Attend to Students’ Thinking

have foundational understandings of students’ About Mathematics Content

mathematical knowledge, skills, and dispositions. C.3.2. Understand and Recognize Students’

They also know how these understandings can Engagement in Mathematical Practices

contribute to effective teaching and are committed to

C.3.3. Anticipate and Attend to Students’

expanding and deepening their knowledge of

Mathematical Dispositions

students as learners of mathematics.

C.4. Social Contexts of Mathematics Teaching and Learning

Well-prepared beginning teachers of mathematics C.4.1. Provide Access and Advancement

realize that the social, historical, and institutional C.4.2. Cultivate Positive Mathematical Identities

contexts of mathematics affect teaching and learning

C.4.3. Draw on Students’ Mathematical Strengths

and know about and are committed to their critical

roles as advocates for each and every student. C.4.4. Understand Power and Privilege in the History

of Mathematics Education

C.4.5. Enact Ethical Practice for Advocacy

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 6

STANDARD RELATED INDICATORS

C.1. Mathematics Concepts, Practices, and Curriculum

Well-prepared beginning teachers of mathematics C.1.1. Know Relevant Mathematical Content

possess robust knowledge of mathematical and C.1.2. Demonstrate Mathematical Practices and

statistical concepts that underlie what they encounter Processes

in teaching. They engage in appropriate mathematical

C.1.3. Exhibit Productive Mathematical Dispositions

and statistical practices and support their students in

doing the same. They can read, analyze, and discuss C.1.4. Analyze the Mathematical Content of

curriculum, assessment, and standards documents as Curriculum

well as students’ mathematical productions. C.1.5. Analyze Mathematical Thinking

C.1.6. Use Mathematical Tools and Technology

C.2. Pedagogical Knowledge and Practices for Teaching Mathematics

Well-prepared beginning teachers of mathematics C.2.1. Promote Equitable Teaching

have foundations of pedagogical knowledge, effective C.2.2. Plan for Effective Instruction

and equitable mathematics teaching practices, and

C.2.3. Implement Effective Instruction

positive and productive dispositions toward teaching

mathematics to support students’ sense making, C.2.4. Analyze Teaching Practice

understanding, and reasoning. C.2.5. Enhance Teaching Through Collaboration With

Colleagues, Families, and Community

Members

C.3. Students as Learners of Mathematics

Well-prepared beginning teachers of mathematics C.3.1. Anticipate and Attend to Students’ Thinking

have foundational understandings of students’ About Mathematics Content

mathematical knowledge, skills, and dispositions. C.3.2. Understand and Recognize Students’

They also know how these understandings can Engagement in Mathematical Practices

contribute to effective teaching and are committed to

C.3.3. Anticipate and Attend to Students’

expanding and deepening their knowledge of

Mathematical Dispositions

students as learners of mathematics.

C.4. Social Contexts of Mathematics Teaching and Learning

Well-prepared beginning teachers of mathematics C.4.1. Provide Access and Advancement

realize that the social, historical, and institutional C.4.2. Cultivate Positive Mathematical Identities

contexts of mathematics affect teaching and learning

C.4.3. Draw on Students’ Mathematical Strengths

and know about and are committed to their critical

roles as advocates for each and every student. C.4.4. Understand Power and Privilege in the History

of Mathematics Education

C.4.5. Enact Ethical Practice for Advocacy

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 6

24.
WHAT SHOULD WELL-PREPARED BEGINNING TEACHERS OF

MATHEMATICS KNOW AND BE ABLE TO DO, AND WHAT

DISPOSITIONS SHOULD THEY DEVELOP?

The guiding question for this chapter is “Recognizing that learning to teach is an ongoing process over many

years, what are reasonable expectations for the most important knowledge, skills, and dispositions that

beginning teachers of mathematics must possess to be effective?” Answering this question is difficult because

some aspects of teaching will not be well learned initially, even though they may be critically important to

student learning. What a beginner knows is also a significant equity issue because students with the greatest

needs are often taught by teachers with the least experience (Kalogrides, Loeb, & Béteille, 2012; Oakes, 2008).

Mathematics teachers, from the very beginning of their careers, must robustly understand the mathematical

content knowledge for the age groups or grades they may teach, along with the content taught to the age

groups preceding and following those they teach—and in a different and deeper way than is often presented in

textbooks, curriculum documents, or standards. Such knowledge affects their students’ learning (e.g., Hill,

Rowan, & Ball, 2005; National Mathematics Advisory Panel, 2008).

Well-prepared beginners must be ready to teach each and every student in their first classrooms. Although

pedagogical skills develop over time, beginners must have an initial repertoire of effective and equitable

teaching strategies; for example, in selecting tasks, orchestrating classroom discussions, building on prior

knowledge, and connecting conceptual understanding and procedural fluency (NCTM, 2014a). All teachers,

including well-prepared beginners, must hold positive dispositions about mathematics and mathematics

learning, such as the notions that mathematics can and must be understood, and that each and every student

can develop mathematical proficiency, along with a commitment to imbue their students with similar beliefs

and dispositions.

To teach effectively, one must hold knowledge of learners and learning, both general pedagogical knowledge

and knowledge specific to the learning and teaching of mathematics. Understanding mathematical learners

includes knowing about their backgrounds, interests, strengths, and personalities as well as knowing how

students think about and learn mathematics, including possible misconceptions and creative pathways they may

take in learning (Ball & Forzani, 2011; Clements & Sarama, 2014; Sztajn, Confrey, Wilson, & Edgington, 2012).

Well-prepared beginners must understand—at least at an initial level—how to assess the understandings and

competencies of their students and use this knowledge to plan and modify instruction using research-based

instructional strategies (e.g., Ball & Forzani, 2011; Shulman, 1986).

Mathematics teaching and learning are influenced by social, historical, and institutional contexts. Beginning

teachers must be aware of learners’ social, cultural, and linguistic resources; know learners’ histories; and

recognize how power relationships affect students’ mathematical identities, access, and advancement in

mathematics (e.g., Gutiérrez, 2013b; Martin, 2015; Strutchens et al., 2012; Wager, 2012). For example, classroom

dynamics and social interactions strongly influence students’ emerging mathematical identities, which in turn

affect the students’ learning opportunities. In short, well-prepared beginners must be ethical advocates for

every student.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 7

MATHEMATICS KNOW AND BE ABLE TO DO, AND WHAT

DISPOSITIONS SHOULD THEY DEVELOP?

The guiding question for this chapter is “Recognizing that learning to teach is an ongoing process over many

years, what are reasonable expectations for the most important knowledge, skills, and dispositions that

beginning teachers of mathematics must possess to be effective?” Answering this question is difficult because

some aspects of teaching will not be well learned initially, even though they may be critically important to

student learning. What a beginner knows is also a significant equity issue because students with the greatest

needs are often taught by teachers with the least experience (Kalogrides, Loeb, & Béteille, 2012; Oakes, 2008).

Mathematics teachers, from the very beginning of their careers, must robustly understand the mathematical

content knowledge for the age groups or grades they may teach, along with the content taught to the age

groups preceding and following those they teach—and in a different and deeper way than is often presented in

textbooks, curriculum documents, or standards. Such knowledge affects their students’ learning (e.g., Hill,

Rowan, & Ball, 2005; National Mathematics Advisory Panel, 2008).

Well-prepared beginners must be ready to teach each and every student in their first classrooms. Although

pedagogical skills develop over time, beginners must have an initial repertoire of effective and equitable

teaching strategies; for example, in selecting tasks, orchestrating classroom discussions, building on prior

knowledge, and connecting conceptual understanding and procedural fluency (NCTM, 2014a). All teachers,

including well-prepared beginners, must hold positive dispositions about mathematics and mathematics

learning, such as the notions that mathematics can and must be understood, and that each and every student

can develop mathematical proficiency, along with a commitment to imbue their students with similar beliefs

and dispositions.

To teach effectively, one must hold knowledge of learners and learning, both general pedagogical knowledge

and knowledge specific to the learning and teaching of mathematics. Understanding mathematical learners

includes knowing about their backgrounds, interests, strengths, and personalities as well as knowing how

students think about and learn mathematics, including possible misconceptions and creative pathways they may

take in learning (Ball & Forzani, 2011; Clements & Sarama, 2014; Sztajn, Confrey, Wilson, & Edgington, 2012).

Well-prepared beginners must understand—at least at an initial level—how to assess the understandings and

competencies of their students and use this knowledge to plan and modify instruction using research-based

instructional strategies (e.g., Ball & Forzani, 2011; Shulman, 1986).

Mathematics teaching and learning are influenced by social, historical, and institutional contexts. Beginning

teachers must be aware of learners’ social, cultural, and linguistic resources; know learners’ histories; and

recognize how power relationships affect students’ mathematical identities, access, and advancement in

mathematics (e.g., Gutiérrez, 2013b; Martin, 2015; Strutchens et al., 2012; Wager, 2012). For example, classroom

dynamics and social interactions strongly influence students’ emerging mathematical identities, which in turn

affect the students’ learning opportunities. In short, well-prepared beginners must be ethical advocates for

every student.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 7

25.
Well-prepared beginning teachers of mathematics possess

STANDARD C.1. robust knowledge of mathematical and statistical concepts

KNOWLEDGE OF that underlie what they encounter in teaching. They engage

in appropriate mathematical and statistical practices and

MATHEMATICS FOR support their students in doing the same. They can read,

TEACHING analyze, and discuss curriculum, assessment, and standards

documents as well as students’ mathematical productions.

Having a robust knowledge of the mathematics content being taught is foundational to the success of a well-

prepared beginning teacher of mathematics. Well-prepared beginners can read, analyze, and discuss

curriculum, assessment, and standards documents as well as students’ mathematical productions. Without that

foundation, they will be unable to support their students’ learning of mathematics. As addressed in the preface,

in this document the term mathematics encompasses mathematics and statistics, because school mathematics

teachers are responsible for instruction in both content areas. For cases in which the distinction between

mathematics and statistics is important to emphasize, statistics is identified separately.

INDICATOR C.1.1. KNOW RELEVANT MATHEMATICAL CONTENT

Well-prepared beginning teachers of mathematics have solid and flexible knowledge of core

mathematical concepts and procedures they will teach, along with knowledge both beyond what

they will teach and foundational to those core concepts and procedures.

Well-prepared beginning teachers of mathematics understand and solve problems in more than one way,

explain the meanings of key concepts, and explain the mathematical rationales underlying key procedures. For

1

example, a well-prepared beginner for Upper Elementary Grades recognizes that simplifying 3 ÷ 5 indicates the

question “How many fifths are in 3?” Using a visual diagram as in Figure 2.1 and considering that one whole is

comprised of five fifths leads one to realize that the answer will be 3 × 5 or 15.

1

5

↓

𝟏

Figure 2.1. Fraction-bar representation of the problem 3 ÷ .

𝟓

This result can be generalized so that students recognize that dividing by any unit fraction is equivalent to

multiplying by the denominator. Thus, this procedure can be built on a solid and flexible understanding of

underlying mathematics. (See Chapters 4 through 7 for additional examples of the specific content for well-

prepared beginners at each grade-band.)

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 8

STANDARD C.1. robust knowledge of mathematical and statistical concepts

KNOWLEDGE OF that underlie what they encounter in teaching. They engage

in appropriate mathematical and statistical practices and

MATHEMATICS FOR support their students in doing the same. They can read,

TEACHING analyze, and discuss curriculum, assessment, and standards

documents as well as students’ mathematical productions.

Having a robust knowledge of the mathematics content being taught is foundational to the success of a well-

prepared beginning teacher of mathematics. Well-prepared beginners can read, analyze, and discuss

curriculum, assessment, and standards documents as well as students’ mathematical productions. Without that

foundation, they will be unable to support their students’ learning of mathematics. As addressed in the preface,

in this document the term mathematics encompasses mathematics and statistics, because school mathematics

teachers are responsible for instruction in both content areas. For cases in which the distinction between

mathematics and statistics is important to emphasize, statistics is identified separately.

INDICATOR C.1.1. KNOW RELEVANT MATHEMATICAL CONTENT

Well-prepared beginning teachers of mathematics have solid and flexible knowledge of core

mathematical concepts and procedures they will teach, along with knowledge both beyond what

they will teach and foundational to those core concepts and procedures.

Well-prepared beginning teachers of mathematics understand and solve problems in more than one way,

explain the meanings of key concepts, and explain the mathematical rationales underlying key procedures. For

1

example, a well-prepared beginner for Upper Elementary Grades recognizes that simplifying 3 ÷ 5 indicates the

question “How many fifths are in 3?” Using a visual diagram as in Figure 2.1 and considering that one whole is

comprised of five fifths leads one to realize that the answer will be 3 × 5 or 15.

1

5

↓

𝟏

Figure 2.1. Fraction-bar representation of the problem 3 ÷ .

𝟓

This result can be generalized so that students recognize that dividing by any unit fraction is equivalent to

multiplying by the denominator. Thus, this procedure can be built on a solid and flexible understanding of

underlying mathematics. (See Chapters 4 through 7 for additional examples of the specific content for well-

prepared beginners at each grade-band.)

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 8

26.
Indicator C.1.2. Demonstrate Mathematical Practices and Processes

Well-prepared beginning teachers of mathematics have solid and flexible knowledge of

mathematical processes and practices, recognizing that these are tools used to solve problems and

communicate ideas.

The mathematical knowledge of well-prepared beginning teachers of mathematics includes ability to use

mathematical and statistical processes and practices (NCTM, 2000; NGA & CCSSO, 2010; Shaughnessy, Chance,

& Kranendonk, 2009) to solve problems. They use mathematical language with care and precision. These

teachers can explain their mathematical thinking using grade-appropriate concepts, procedures, and language,

including grade-appropriate definitions and interpretations for key mathematical concepts. They can apply their

mathematical knowledge to real-world situations by using mathematical modeling to solve problems

appropriate for the grade levels and the students they will teach. They are able to effectively use

representations and technological tools appropriate for the mathematics content they will teach. They regard

doing mathematics as a sense-making activity that promotes perseverance, problem posing, and problem

solving. In short, they exemplify the mathematical thinking that will be expected of their students.

Well-prepared beginners recognize processes and practices when they emerge in their mathematical thinking

and highlight these actions and behaviors when they observe them in others. Over time, beginning teachers

can (a) better distinguish intricacies among the various processes and practices, determining those that are at

the crux of a mathematical investigation and (b) see the interrelationships among the processes and practices.

Well-prepared beginners understand that mathematics is a human endeavor that is practiced in and out of

school, across many facets of life. They know that mathematics has a history and includes contributions from

people with different genders and cultural, linguistic, religious, and racial/ethnic backgrounds. Mathematics is

based on constructed conventions and agreements about the meanings of words and symbols, and these

conventions vary. Well-prepared beginners are aware that algorithms considered as standard in the United

States differ from algorithms used in other countries and that some alternative algorithms have different,

desirable properties that make them worth knowing. This idea is elaborated in later standards and indicators of

this chapter as well as in Chapters 4–7.

Indicator C.1.3. Exhibit Productive Mathematical Dispositions

Well-prepared beginning teachers of mathematics expect mathematics to be sensible, useful, and

worthwhile for themselves and others, and they believe that all people are capable of thinking

mathematically and are able to solve sophisticated mathematical problems with effort.

Well-prepared beginning teachers of mathematics know that one's success in mathematics depends on a

productive disposition toward the subject and on hard work (National Research Council [NRC], 2001a). They

believe that requisite characteristics of high-quality teaching of mathematics include a commitment to sense

making in mathematical thinking, teaching, and learning and to developing habits of mind, including curiosity,

imagination, inventiveness, risk-taking, and persistence. For example, when faced with a challenge to common

practice or to their current understandings or beliefs, well-prepared beginning teachers have the intellectual

courage and mathematical disposition to not reject the challenge but to investigate the proposed idea, applying

their own critical thinking and using all available resources.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 9

Well-prepared beginning teachers of mathematics have solid and flexible knowledge of

mathematical processes and practices, recognizing that these are tools used to solve problems and

communicate ideas.

The mathematical knowledge of well-prepared beginning teachers of mathematics includes ability to use

mathematical and statistical processes and practices (NCTM, 2000; NGA & CCSSO, 2010; Shaughnessy, Chance,

& Kranendonk, 2009) to solve problems. They use mathematical language with care and precision. These

teachers can explain their mathematical thinking using grade-appropriate concepts, procedures, and language,

including grade-appropriate definitions and interpretations for key mathematical concepts. They can apply their

mathematical knowledge to real-world situations by using mathematical modeling to solve problems

appropriate for the grade levels and the students they will teach. They are able to effectively use

representations and technological tools appropriate for the mathematics content they will teach. They regard

doing mathematics as a sense-making activity that promotes perseverance, problem posing, and problem

solving. In short, they exemplify the mathematical thinking that will be expected of their students.

Well-prepared beginners recognize processes and practices when they emerge in their mathematical thinking

and highlight these actions and behaviors when they observe them in others. Over time, beginning teachers

can (a) better distinguish intricacies among the various processes and practices, determining those that are at

the crux of a mathematical investigation and (b) see the interrelationships among the processes and practices.

Well-prepared beginners understand that mathematics is a human endeavor that is practiced in and out of

school, across many facets of life. They know that mathematics has a history and includes contributions from

people with different genders and cultural, linguistic, religious, and racial/ethnic backgrounds. Mathematics is

based on constructed conventions and agreements about the meanings of words and symbols, and these

conventions vary. Well-prepared beginners are aware that algorithms considered as standard in the United

States differ from algorithms used in other countries and that some alternative algorithms have different,

desirable properties that make them worth knowing. This idea is elaborated in later standards and indicators of

this chapter as well as in Chapters 4–7.

Indicator C.1.3. Exhibit Productive Mathematical Dispositions

Well-prepared beginning teachers of mathematics expect mathematics to be sensible, useful, and

worthwhile for themselves and others, and they believe that all people are capable of thinking

mathematically and are able to solve sophisticated mathematical problems with effort.

Well-prepared beginning teachers of mathematics know that one's success in mathematics depends on a

productive disposition toward the subject and on hard work (National Research Council [NRC], 2001a). They

believe that requisite characteristics of high-quality teaching of mathematics include a commitment to sense

making in mathematical thinking, teaching, and learning and to developing habits of mind, including curiosity,

imagination, inventiveness, risk-taking, and persistence. For example, when faced with a challenge to common

practice or to their current understandings or beliefs, well-prepared beginning teachers have the intellectual

courage and mathematical disposition to not reject the challenge but to investigate the proposed idea, applying

their own critical thinking and using all available resources.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 9

27.
Indicator C.1.4. Analyze the Mathematical Content of Curriculum

Well-prepared beginning teachers of mathematics read, analyze, interpret, and enact mathematics

curricula, content trajectories, standards documents, and assessment frameworks for the grades

in which they are being prepared to teach.

The alignment of standards, instructional materials, and assessment is critical in designing a cohesive, well-

articulated curriculum. Well-prepared beginning teachers of mathematics are aware that the mathematics they

teach is based on a variety of, often nested, documents. They know that connections exist among standards,

curriculum documents, instructional materials, and assessment frameworks and have dispositions and

commitment to analyze these guides to inform their teaching. They have the content preparation and the

dispositions to analyze instructional resources, including those provided by textbook publishers and those

available from sources online, to determine whether these resources fully address the content expectations

described in standards and curriculum documents. When the materials fall short of the standards or

expectations, well-prepared beginners are able to decide whether to replace or adapt the materials to better

address the content and process expectations.

Well-prepared beginners realize that in addition to the curriculum and standards that they are accountable to

teach, other resources can support their efforts to design rigorous, coherent mathematics instruction. Such

resources include learning or standards progressions (cf. Generating Increased Science and Mathematics

Opportunities, 2012; Institute for Mathematics and Education, n.d.) that describe relationships among

standards within and across grades. Note that in addition to content progressions, other types of progressions

to be considered are developmental progressions or learning trajectories. Well-prepared beginners understand

the content within these materials and can discuss them with colleagues, administrators, and families of their

students in ways that make sense to these audiences.

Through analyzing available resources, well-prepared beginners are able to make decisions about the

sequencing and time required to teach the content in depth as well as to make important connections among

the mathematics taught in the grades or units before and after what they are teaching.

Indicator C.1.5. Analyze Mathematical Thinking

Well-prepared beginning teachers of mathematics analyze different approaches to mathematical

work and respond appropriately.

Well-prepared beginning teachers of mathematics analyze both written and oral mathematical productions

related to key mathematical ideas and look for and identify sensible mathematical reasoning, even when that

reasoning may be atypical or different from their own. Well-prepared beginners value varied approaches to

solving a problem, recognizing that engaging in mathematics is more than finding an answer. They make

mathematical connections among these approaches to clarify underlying mathematical concepts. Well-

prepared beginners recognize the importance of context and applications in uses of mathematics and statistics.

They make connections across disciplines in ways that illuminate mathematical ideas.

The tasks in Figures 2.2 and 2.3, which might be used in a mathematics methods or mathematics content

course for teachers, exemplify the level of mathematical analysis expected of well-prepared beginners.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 10

Well-prepared beginning teachers of mathematics read, analyze, interpret, and enact mathematics

curricula, content trajectories, standards documents, and assessment frameworks for the grades

in which they are being prepared to teach.

The alignment of standards, instructional materials, and assessment is critical in designing a cohesive, well-

articulated curriculum. Well-prepared beginning teachers of mathematics are aware that the mathematics they

teach is based on a variety of, often nested, documents. They know that connections exist among standards,

curriculum documents, instructional materials, and assessment frameworks and have dispositions and

commitment to analyze these guides to inform their teaching. They have the content preparation and the

dispositions to analyze instructional resources, including those provided by textbook publishers and those

available from sources online, to determine whether these resources fully address the content expectations

described in standards and curriculum documents. When the materials fall short of the standards or

expectations, well-prepared beginners are able to decide whether to replace or adapt the materials to better

address the content and process expectations.

Well-prepared beginners realize that in addition to the curriculum and standards that they are accountable to

teach, other resources can support their efforts to design rigorous, coherent mathematics instruction. Such

resources include learning or standards progressions (cf. Generating Increased Science and Mathematics

Opportunities, 2012; Institute for Mathematics and Education, n.d.) that describe relationships among

standards within and across grades. Note that in addition to content progressions, other types of progressions

to be considered are developmental progressions or learning trajectories. Well-prepared beginners understand

the content within these materials and can discuss them with colleagues, administrators, and families of their

students in ways that make sense to these audiences.

Through analyzing available resources, well-prepared beginners are able to make decisions about the

sequencing and time required to teach the content in depth as well as to make important connections among

the mathematics taught in the grades or units before and after what they are teaching.

Indicator C.1.5. Analyze Mathematical Thinking

Well-prepared beginning teachers of mathematics analyze different approaches to mathematical

work and respond appropriately.

Well-prepared beginning teachers of mathematics analyze both written and oral mathematical productions

related to key mathematical ideas and look for and identify sensible mathematical reasoning, even when that

reasoning may be atypical or different from their own. Well-prepared beginners value varied approaches to

solving a problem, recognizing that engaging in mathematics is more than finding an answer. They make

mathematical connections among these approaches to clarify underlying mathematical concepts. Well-

prepared beginners recognize the importance of context and applications in uses of mathematics and statistics.

They make connections across disciplines in ways that illuminate mathematical ideas.

The tasks in Figures 2.2 and 2.3, which might be used in a mathematics methods or mathematics content

course for teachers, exemplify the level of mathematical analysis expected of well-prepared beginners.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 10

28.
Students are given the following prompt:

How many pencils will our classroom need to last through the school year?

Propose a strategy to answer this question.

Four student responses follow:

1) We should vote on whether we have enough.

2) We should survey teachers about how many they have used in the past.

3) We should determine how many words each pencil can write and estimate how many words our

class will write in a year.

4) We should collect data for a week before trying to answer the question.

What would you ask to further each student’s approach to developing a model to answer this question?

Figure 2.2. Sample task for Pre-K–5 teacher candidates.

Students are given the following prompt:

One number is 3 times another number, and their sum is 30. What are the two numbers?

Four student responses follow:

1) 30 ÷ 3 = 10, so 10 and 30.

2) 30 ÷ 4 = 7.5, so 7.5 and 22.5.

3) Half of 30 is 15. Half of 15 is 7.5, so go up and down 7.5 from 15.

4) x + 3x = 30, so x = 7.5.

What question would you ask to clarify each student’s thinking?

Figure 2.3. Sample task for middle level teacher candidates.

Indicator C.1.6. Use Mathematical Tools and Technology

Well-prepared beginning teachers of mathematics are proficient with tools and technology

designed to support mathematical reasoning and sense making, both in doing mathematics

themselves and in supporting student learning of mathematics.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 11

How many pencils will our classroom need to last through the school year?

Propose a strategy to answer this question.

Four student responses follow:

1) We should vote on whether we have enough.

2) We should survey teachers about how many they have used in the past.

3) We should determine how many words each pencil can write and estimate how many words our

class will write in a year.

4) We should collect data for a week before trying to answer the question.

What would you ask to further each student’s approach to developing a model to answer this question?

Figure 2.2. Sample task for Pre-K–5 teacher candidates.

Students are given the following prompt:

One number is 3 times another number, and their sum is 30. What are the two numbers?

Four student responses follow:

1) 30 ÷ 3 = 10, so 10 and 30.

2) 30 ÷ 4 = 7.5, so 7.5 and 22.5.

3) Half of 30 is 15. Half of 15 is 7.5, so go up and down 7.5 from 15.

4) x + 3x = 30, so x = 7.5.

What question would you ask to clarify each student’s thinking?

Figure 2.3. Sample task for middle level teacher candidates.

Indicator C.1.6. Use Mathematical Tools and Technology

Well-prepared beginning teachers of mathematics are proficient with tools and technology

designed to support mathematical reasoning and sense making, both in doing mathematics

themselves and in supporting student learning of mathematics.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 11

29.
Well-prepared beginning teachers of mathematics are proficient in using both digital tools and physical

manipulatives for solving mathematical problems and as a means of enhancing or illuminating mathematical

and statistical concepts. Well-prepared beginners know when and how to use physical manipulatives to explore

mathematical and statistical ideas and to build conceptual understanding of these. Furthermore, they are

prepared to use “mathematical action technologies” (cf. NCTM, 2014a, p. 79), powerful tools that will be a part

of the lives of the students they teach. They know that physical and digital simulations are critical for

understanding key statistical concepts. They are familiar with the use of virtual manipulatives, interactive

electronic depictions of physical manipulatives, and know how these can support sophisticated explorations of

mathematical concepts (Moyer-Packenham, Niezgoda, & Stanley, 2005).

Well-prepared beginners “make sound decisions about when such tools enhance teaching and learning,

recognizing both the insights to be gained and possible limitations of such tools” (NCTM, 2012, p. 3). Not every

tool, whether electronic or physical, is appropriate in every situation, and different tools may provide different

insights into a context. Well-prepared beginners recognize the fast rate at which technologies emerge and are

committed to staying abreast of new tools, analyzing their potential and limitations for students’ mathematics

Well-prepared beginning teachers of mathematics have

STANDARD C.2. foundations of pedagogical knowledge, effective and

PEDAGOGICAL equitable mathematics teaching practices, and positive and

productive dispositions toward teaching mathematics to

KNOWLEDGE AND support students’ sense making, understanding, and

PRACTICES FOR reasoning.

TEACHING

MATHEMATICS

Teaching mathematics effectively involves significant knowledge, skills, and dispositions. Well-prepared

beginning teachers of mathematics draw upon their knowledge of mathematics content, processes, and

curriculum and blend it with effective and equitable mathematics teaching practices. They begin teaching with a

solid foundation of pedagogical content knowledge. That is, they have studied teaching approaches unique to

the subject matter of mathematics, such as knowledge of how to structure and represent mathematics content

(e.g., fractions, ratios, algebra) in ways that make it accessible to students; knowledge of the common

conceptions, misconceptions, and difficulties that students encounter when learning mathematics; and

knowledge of specific teaching strategies to assess and address students’ learning needs in mathematics

(Shulman, 1986). Well-prepared beginners realize that the teaching of mathematics has its own nuances and

complexities specific to the discipline and are ready to start using and expanding their established foundations

of pedagogical content knowledge to guide their instructional actions in planning for and interacting with each

and every student in mathematics classrooms.

Well-prepared beginners take seriously their responsibilities as new teachers to develop each student’s ability

to engage in mathematical sense making, understanding, and reasoning. Toward that end, they present

positive attitudes toward the subject of mathematics as a discipline when interacting with students, families,

colleagues, and community members. They also demonstrate productive dispositions toward teaching

mathematics, including engagement in collaborative communities of practice and ongoing learning as

professional educators. Well-prepared beginners know that they will continue to develop and refine their

pedagogies and practices, and, as with all complex endeavors, their skills and effectiveness in teaching

mathematics will improve with time, experience, and deliberate reflection.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 12

manipulatives for solving mathematical problems and as a means of enhancing or illuminating mathematical

and statistical concepts. Well-prepared beginners know when and how to use physical manipulatives to explore

mathematical and statistical ideas and to build conceptual understanding of these. Furthermore, they are

prepared to use “mathematical action technologies” (cf. NCTM, 2014a, p. 79), powerful tools that will be a part

of the lives of the students they teach. They know that physical and digital simulations are critical for

understanding key statistical concepts. They are familiar with the use of virtual manipulatives, interactive

electronic depictions of physical manipulatives, and know how these can support sophisticated explorations of

mathematical concepts (Moyer-Packenham, Niezgoda, & Stanley, 2005).

Well-prepared beginners “make sound decisions about when such tools enhance teaching and learning,

recognizing both the insights to be gained and possible limitations of such tools” (NCTM, 2012, p. 3). Not every

tool, whether electronic or physical, is appropriate in every situation, and different tools may provide different

insights into a context. Well-prepared beginners recognize the fast rate at which technologies emerge and are

committed to staying abreast of new tools, analyzing their potential and limitations for students’ mathematics

Well-prepared beginning teachers of mathematics have

STANDARD C.2. foundations of pedagogical knowledge, effective and

PEDAGOGICAL equitable mathematics teaching practices, and positive and

productive dispositions toward teaching mathematics to

KNOWLEDGE AND support students’ sense making, understanding, and

PRACTICES FOR reasoning.

TEACHING

MATHEMATICS

Teaching mathematics effectively involves significant knowledge, skills, and dispositions. Well-prepared

beginning teachers of mathematics draw upon their knowledge of mathematics content, processes, and

curriculum and blend it with effective and equitable mathematics teaching practices. They begin teaching with a

solid foundation of pedagogical content knowledge. That is, they have studied teaching approaches unique to

the subject matter of mathematics, such as knowledge of how to structure and represent mathematics content

(e.g., fractions, ratios, algebra) in ways that make it accessible to students; knowledge of the common

conceptions, misconceptions, and difficulties that students encounter when learning mathematics; and

knowledge of specific teaching strategies to assess and address students’ learning needs in mathematics

(Shulman, 1986). Well-prepared beginners realize that the teaching of mathematics has its own nuances and

complexities specific to the discipline and are ready to start using and expanding their established foundations

of pedagogical content knowledge to guide their instructional actions in planning for and interacting with each

and every student in mathematics classrooms.

Well-prepared beginners take seriously their responsibilities as new teachers to develop each student’s ability

to engage in mathematical sense making, understanding, and reasoning. Toward that end, they present

positive attitudes toward the subject of mathematics as a discipline when interacting with students, families,

colleagues, and community members. They also demonstrate productive dispositions toward teaching

mathematics, including engagement in collaborative communities of practice and ongoing learning as

professional educators. Well-prepared beginners know that they will continue to develop and refine their

pedagogies and practices, and, as with all complex endeavors, their skills and effectiveness in teaching

mathematics will improve with time, experience, and deliberate reflection.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 12

30.
Indicator C.2.1. Promote Equitable Teaching

Well-prepared beginning teachers of mathematics structure learning opportunities and use teaching

practices that provide access, support, and challenge in learning rigorous mathematics to advance the

learning of every student.

Teaching for access and equity is evidenced as well-prepared beginning teachers view their roles as developing

robust and powerful mathematical identities in their students, demonstrating commitment to view each and

every student as a capable and unique learner of mathematics. Well-prepared beginning teachers embrace and

build on students’ current mathematical ideas and on students’ ways of knowing and learning, including

attending to each student’s culture, race/ethnicity, language, gender, socioeconomic status, cognitive and

physical abilities, and personal interests. They also attend to developing students’ identities and agency so that

students can see mathematics as components of their cultures and see themselves in the mathematics. For

instance, well-prepared beginners often connect mathematics to students’ everyday experiences or ask

students to tell stories because they know that making mathematics relevant to students’ lives can make the

learning experience more meaningful and raise achievement (Turner, Celedón-Pattichis, Marshall, & Tennison,

2009). Ensuring equitable mathematics learning outcomes for each student is an essential goal and a significant

challenge. Achieving this goal requires (a) clear and coherent mathematical goals for students’ learning, (b)

expectations for the collective work of students in the classroom, (c) effective methods of supporting the

learning of mathematics by each student, and (d) provision of appropriate tools and resources targeted to

students' specific needs.

Teaching with a commitment to access and equity entails striving to reach each student whose life is affected by

what occurs in the mathematics classroom. Well-prepared beginning teachers plan for and use an equity-based

pedagogy (AMTE, 2015) by structuring learning opportunities to provide access, support, and challenge in

learning mathematics. This practice includes considering students’ individual needs, cultural experiences, and

interests as well as prior mathematical knowledge when selecting tasks and planning for mathematics

instruction (Leonard, Brooks, Barnes-Johnson, & Berry, 2010). For example, what everyday, informal language

might support or hinder the specialized use of language in mathematics? What are students’ prior experiences

with specific mathematical representations or strategies? What scaffolds are needed to support students with

special needs? What assessments can help identify the strengths and weaknesses of students who are in need

of interventions? What topics, activities, places, books, movies, television shows, and video games are currently

popular with students and could be incorporated into tasks or problem-solving opportunities?

Well-prepared beginning teachers of mathematics strongly believe that each and every student can learn

mathematics with understanding, and they take conscious and intentional actions to build students’ agency as

mathematical learners (AMTE, 2015; Gutiérrez, 2009). That is, the teachers believe in each student’s ability to

make sense of mathematical tasks and situations, to engage in mathematical discourse, and to judge the

validity of solutions. For example, the beginning teacher envisions a classroom community in which students

present ideas, challenge one another, and construct meaning together; varied mathematical strengths are

valued and celebrated. Well-prepared beginners understand the importance of communicating the relevance of

mathematics—specifically, that students can use mathematics to address problems and issues in their homes

and communities.

Additionally, well-prepared beginners intentionally foster growth mindsets among students about learning

mathematics and persistently counter manifestations of fixed mindsets (e.g., that some people are good at

mathematics and others are not). This practice includes public praise for contributions, use of applicable

strategies, and perseverance (Boaler, 2016; Dweck, 2008). For example, well-prepared beginners acknowledge

mistakes as critical for learning and help students view mistakes as important in the learning process and for

engaging in mathematics.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 13

Well-prepared beginning teachers of mathematics structure learning opportunities and use teaching

practices that provide access, support, and challenge in learning rigorous mathematics to advance the

learning of every student.

Teaching for access and equity is evidenced as well-prepared beginning teachers view their roles as developing

robust and powerful mathematical identities in their students, demonstrating commitment to view each and

every student as a capable and unique learner of mathematics. Well-prepared beginning teachers embrace and

build on students’ current mathematical ideas and on students’ ways of knowing and learning, including

attending to each student’s culture, race/ethnicity, language, gender, socioeconomic status, cognitive and

physical abilities, and personal interests. They also attend to developing students’ identities and agency so that

students can see mathematics as components of their cultures and see themselves in the mathematics. For

instance, well-prepared beginners often connect mathematics to students’ everyday experiences or ask

students to tell stories because they know that making mathematics relevant to students’ lives can make the

learning experience more meaningful and raise achievement (Turner, Celedón-Pattichis, Marshall, & Tennison,

2009). Ensuring equitable mathematics learning outcomes for each student is an essential goal and a significant

challenge. Achieving this goal requires (a) clear and coherent mathematical goals for students’ learning, (b)

expectations for the collective work of students in the classroom, (c) effective methods of supporting the

learning of mathematics by each student, and (d) provision of appropriate tools and resources targeted to

students' specific needs.

Teaching with a commitment to access and equity entails striving to reach each student whose life is affected by

what occurs in the mathematics classroom. Well-prepared beginning teachers plan for and use an equity-based

pedagogy (AMTE, 2015) by structuring learning opportunities to provide access, support, and challenge in

learning mathematics. This practice includes considering students’ individual needs, cultural experiences, and

interests as well as prior mathematical knowledge when selecting tasks and planning for mathematics

instruction (Leonard, Brooks, Barnes-Johnson, & Berry, 2010). For example, what everyday, informal language

might support or hinder the specialized use of language in mathematics? What are students’ prior experiences

with specific mathematical representations or strategies? What scaffolds are needed to support students with

special needs? What assessments can help identify the strengths and weaknesses of students who are in need

of interventions? What topics, activities, places, books, movies, television shows, and video games are currently

popular with students and could be incorporated into tasks or problem-solving opportunities?

Well-prepared beginning teachers of mathematics strongly believe that each and every student can learn

mathematics with understanding, and they take conscious and intentional actions to build students’ agency as

mathematical learners (AMTE, 2015; Gutiérrez, 2009). That is, the teachers believe in each student’s ability to

make sense of mathematical tasks and situations, to engage in mathematical discourse, and to judge the

validity of solutions. For example, the beginning teacher envisions a classroom community in which students

present ideas, challenge one another, and construct meaning together; varied mathematical strengths are

valued and celebrated. Well-prepared beginners understand the importance of communicating the relevance of

mathematics—specifically, that students can use mathematics to address problems and issues in their homes

and communities.

Additionally, well-prepared beginners intentionally foster growth mindsets among students about learning

mathematics and persistently counter manifestations of fixed mindsets (e.g., that some people are good at

mathematics and others are not). This practice includes public praise for contributions, use of applicable

strategies, and perseverance (Boaler, 2016; Dweck, 2008). For example, well-prepared beginners acknowledge

mistakes as critical for learning and help students view mistakes as important in the learning process and for

engaging in mathematics.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 13

31.
Indicator C.2.2. Plan for Effective Instruction

Well-prepared beginning teachers of mathematics attend to a multitude of factors to design

mathematical learning opportunities for students, including content, students' learning needs, students’

strengths, task selection, and the results of formative and summative assessments.

Careful and detailed planning is necessary to design lessons that build on students’ mathematical thinking

while supporting development of key mathematical ideas. Well-prepared beginners realize that planning takes

substantial amounts of time. They recognize the importance of having clear understandings of the mathematics

content and mathematics learning goals for each unit and lesson as well as how these particular goals fit within

a developmental progression of student learning (Daro, Mosher, & Corcoran, 2011). Well-prepared beginners

are able to articulate and clarify mathematics learning goals during the planning process, knowing that the

goals are the starting point that “sets the stage for everything else” (Hiebert, Morris, Berk, & Jansen, 2007, p. 57).

To provide effective instruction, one considers the prior knowledge and experiences students bring to a lesson

and how the task will be set up or launched in meaningful contexts to ensure that each and every student has

access to the content and contexts (Jackson, Garrison, Wilson, Gibbons, & Shahan, 2013). Therefore, well-

prepared beginners strive to design classroom environments in which students have opportunities to

communicate their thinking, listen to the thinking of others, connect mathematics to a variety of contexts, and

make connections across mathematical ideas and subject areas. They plan purposeful and meaningful

questions to probe student thinking, make the mathematics visible for discussion, and encourage reflection

and justification (NCTM, 2014a).

In effective mathematics planning, teachers select meaningful tasks to motivate student learning, develop new

mathematical knowledge, and build connections between conceptual and procedural understanding. Well-

prepared beginners understand that providing students opportunities to think, reason, and solve problems

requires cognitively challenging mathematical tasks (Stein, Smith, Henningsen, & Silver, 2009). Additionally, well-

prepared beginners engage students on a regular basis with mathematical tasks that promote reasoning and

problem solving, provide multiple entry points, have high ceilings to offer challenges, and support varied

solution strategies (NCTM, 2014a). One can identify such tasks only by considering the ways in which students

might solve them. Therefore, well-prepared beginners analyze tasks and lessons, anticipating students’

approaches and responses (Gravemeijer, 2004; Stigler & Hiebert, 1999). They understand that anticipating

students’ responses involves considering both the array of strategies, both conventional and unconventional,

that students might use to solve the task, and the ways those strategies relate to the mathematical concepts,

representations, and procedures that students are learning (Stein, Engle, Smith, & Hughes, 2008).

Well-prepared beginners attend to the needs of their students in their planning of lessons and units. That is, in

their lesson planning, the beginners incorporate inclusive and equity-based teaching practices. Formative

assessment is an integral aspect of effective instruction; therefore, lesson planning must include a plan for

monitoring and assessing student understanding (Black & Wiliam, 1998). Well-prepared beginners have

repertoires of strategies to elicit evidence of students’ progress toward the intended mathematics learning

goals, such as being able to use observation checklists, interviews, writing prompts, exit tickets, quizzes, and

tests. They realize that although they have anticipated student responses, the evidence from these

assessments may require a departure from the planned lesson or may affect subsequent lessons within a unit

of study. They also recognize their responsibilities to develop interventions to support the students who are not

reaching the objectives of the grade-level instruction. Knowing the importance of monitoring and attending to

the progress of students who are struggling to learn, the well-prepared beginning teacher uses results from

formative assessments to design targeted instruction presented outside the regular grade-level mathematics

session.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 14

Well-prepared beginning teachers of mathematics attend to a multitude of factors to design

mathematical learning opportunities for students, including content, students' learning needs, students’

strengths, task selection, and the results of formative and summative assessments.

Careful and detailed planning is necessary to design lessons that build on students’ mathematical thinking

while supporting development of key mathematical ideas. Well-prepared beginners realize that planning takes

substantial amounts of time. They recognize the importance of having clear understandings of the mathematics

content and mathematics learning goals for each unit and lesson as well as how these particular goals fit within

a developmental progression of student learning (Daro, Mosher, & Corcoran, 2011). Well-prepared beginners

are able to articulate and clarify mathematics learning goals during the planning process, knowing that the

goals are the starting point that “sets the stage for everything else” (Hiebert, Morris, Berk, & Jansen, 2007, p. 57).

To provide effective instruction, one considers the prior knowledge and experiences students bring to a lesson

and how the task will be set up or launched in meaningful contexts to ensure that each and every student has

access to the content and contexts (Jackson, Garrison, Wilson, Gibbons, & Shahan, 2013). Therefore, well-

prepared beginners strive to design classroom environments in which students have opportunities to

communicate their thinking, listen to the thinking of others, connect mathematics to a variety of contexts, and

make connections across mathematical ideas and subject areas. They plan purposeful and meaningful

questions to probe student thinking, make the mathematics visible for discussion, and encourage reflection

and justification (NCTM, 2014a).

In effective mathematics planning, teachers select meaningful tasks to motivate student learning, develop new

mathematical knowledge, and build connections between conceptual and procedural understanding. Well-

prepared beginners understand that providing students opportunities to think, reason, and solve problems

requires cognitively challenging mathematical tasks (Stein, Smith, Henningsen, & Silver, 2009). Additionally, well-

prepared beginners engage students on a regular basis with mathematical tasks that promote reasoning and

problem solving, provide multiple entry points, have high ceilings to offer challenges, and support varied

solution strategies (NCTM, 2014a). One can identify such tasks only by considering the ways in which students

might solve them. Therefore, well-prepared beginners analyze tasks and lessons, anticipating students’

approaches and responses (Gravemeijer, 2004; Stigler & Hiebert, 1999). They understand that anticipating

students’ responses involves considering both the array of strategies, both conventional and unconventional,

that students might use to solve the task, and the ways those strategies relate to the mathematical concepts,

representations, and procedures that students are learning (Stein, Engle, Smith, & Hughes, 2008).

Well-prepared beginners attend to the needs of their students in their planning of lessons and units. That is, in

their lesson planning, the beginners incorporate inclusive and equity-based teaching practices. Formative

assessment is an integral aspect of effective instruction; therefore, lesson planning must include a plan for

monitoring and assessing student understanding (Black & Wiliam, 1998). Well-prepared beginners have

repertoires of strategies to elicit evidence of students’ progress toward the intended mathematics learning

goals, such as being able to use observation checklists, interviews, writing prompts, exit tickets, quizzes, and

tests. They realize that although they have anticipated student responses, the evidence from these

assessments may require a departure from the planned lesson or may affect subsequent lessons within a unit

of study. They also recognize their responsibilities to develop interventions to support the students who are not

reaching the objectives of the grade-level instruction. Knowing the importance of monitoring and attending to

the progress of students who are struggling to learn, the well-prepared beginning teacher uses results from

formative assessments to design targeted instruction presented outside the regular grade-level mathematics

session.

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 14

32.
Indicator C.2.3. Implement Effective Instruction

Well-prepared beginning teachers of mathematics use a core set of pedagogical practices that are

effective for developing students’ meaningful learning of mathematics.

Teachers must not only understand the mathematics they are expected to teach (Ball, Thames, & Phelps, 2008)

and understand how students learn that mathematics (Fuson, Kalchman, & Bransford, 2005), they must be

skilled in using content-focused instructional pedagogies to advance the mathematics learning of each and

every student (Forzani, 2014). Well-prepared beginning teachers of mathematics have begun to develop skillful

use of a core set of effective teaching practices, such as those described in Principles to Actions (NCTM, 2014a)

and listed in Table 2.2 below.

TABLE 2.2. MATHEMATICS TEACHING PRACTICES (NCTM, 2014a)

Establish mathematics goals to focus learning. Effective teaching of mathematics establishes clear goals for the

mathematics that students are learning, situates goals within learning progressions, and uses the goals to

guide instructional decisions.

Implement tasks that promote reasoning and problem solving. Effective teaching of mathematics engages

students in solving and discussing tasks that promote mathematical reasoning and problem solving and

allow multiple entry points and varied solution strategies.

Use and connect mathematical representations. Effective teaching of mathematics engages students in making

connections among mathematical representations to deepen understanding of mathematics concepts and

procedures and as tools for problem solving.

Facilitate meaningful mathematical discourse. Effective teaching of mathematics facilitates discourse among

students to build shared understanding of mathematical ideas by analyzing and comparing student

approaches and arguments.

Pose purposeful questions. Effective teaching of mathematics uses purposeful questions to assess and advance

students’ reasoning and sense making about important mathematical ideas and relationships.

Build procedural fluency from conceptual understanding. Effective teaching of mathematics builds fluency with

procedures on a foundation of conceptual understanding so that students, over time, become skillful in

using procedures flexibly as they solve contextual and mathematical problems.

Support productive struggle in learning mathematics. Effective teaching of mathematics consistently provides

students, individually and collectively, with opportunities and supports to engage in productive struggle as

they grapple with mathematical ideas and relationships.

Elicit and use evidence of student thinking. Effective teaching of mathematics uses evidence of student thinking

to assess progress toward mathematical understanding and to adjust instruction continually in ways that

support and extend learning.

Note. Reprinted from Principles to Actions: Ensuring Mathematical Success for All (p. 10) by the National Council of

Teachers of Mathematics, 2014. Reston, VA: NCTM. Copyright 2014 by National Council of Teachers of

Mathematics. Used with permission.

Well-prepared beginners enter classrooms with commitment to, and initial skills for, enacting effective

mathematics instruction. They can identify mathematics learning goals for lessons, articulate how those goals

relate to the selected tasks, and use the goals to guide their instructional decisions throughout a lesson. They

can distinguish between high-level and low-level tasks on the basis of the cognitive demand for their students.

They are growing in their abilities to implement high-level tasks that promote reasoning and problem solving

with students and to engage students in meaningful mathematical discourse on those tasks, without lowering

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 15

Well-prepared beginning teachers of mathematics use a core set of pedagogical practices that are

effective for developing students’ meaningful learning of mathematics.

Teachers must not only understand the mathematics they are expected to teach (Ball, Thames, & Phelps, 2008)

and understand how students learn that mathematics (Fuson, Kalchman, & Bransford, 2005), they must be

skilled in using content-focused instructional pedagogies to advance the mathematics learning of each and

every student (Forzani, 2014). Well-prepared beginning teachers of mathematics have begun to develop skillful

use of a core set of effective teaching practices, such as those described in Principles to Actions (NCTM, 2014a)

and listed in Table 2.2 below.

TABLE 2.2. MATHEMATICS TEACHING PRACTICES (NCTM, 2014a)

Establish mathematics goals to focus learning. Effective teaching of mathematics establishes clear goals for the

mathematics that students are learning, situates goals within learning progressions, and uses the goals to

guide instructional decisions.

Implement tasks that promote reasoning and problem solving. Effective teaching of mathematics engages

students in solving and discussing tasks that promote mathematical reasoning and problem solving and

allow multiple entry points and varied solution strategies.

Use and connect mathematical representations. Effective teaching of mathematics engages students in making

connections among mathematical representations to deepen understanding of mathematics concepts and

procedures and as tools for problem solving.

Facilitate meaningful mathematical discourse. Effective teaching of mathematics facilitates discourse among

students to build shared understanding of mathematical ideas by analyzing and comparing student

approaches and arguments.

Pose purposeful questions. Effective teaching of mathematics uses purposeful questions to assess and advance

students’ reasoning and sense making about important mathematical ideas and relationships.

Build procedural fluency from conceptual understanding. Effective teaching of mathematics builds fluency with

procedures on a foundation of conceptual understanding so that students, over time, become skillful in

using procedures flexibly as they solve contextual and mathematical problems.

Support productive struggle in learning mathematics. Effective teaching of mathematics consistently provides

students, individually and collectively, with opportunities and supports to engage in productive struggle as

they grapple with mathematical ideas and relationships.

Elicit and use evidence of student thinking. Effective teaching of mathematics uses evidence of student thinking

to assess progress toward mathematical understanding and to adjust instruction continually in ways that

support and extend learning.

Note. Reprinted from Principles to Actions: Ensuring Mathematical Success for All (p. 10) by the National Council of

Teachers of Mathematics, 2014. Reston, VA: NCTM. Copyright 2014 by National Council of Teachers of

Mathematics. Used with permission.

Well-prepared beginners enter classrooms with commitment to, and initial skills for, enacting effective

mathematics instruction. They can identify mathematics learning goals for lessons, articulate how those goals

relate to the selected tasks, and use the goals to guide their instructional decisions throughout a lesson. They

can distinguish between high-level and low-level tasks on the basis of the cognitive demand for their students.

They are growing in their abilities to implement high-level tasks that promote reasoning and problem solving

with students and to engage students in meaningful mathematical discourse on those tasks, without lowering

STANDARDS FOR PREPARING TEACHERS OF MATHEMATICS 15