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.
3.
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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