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The Mathematical Education of Teachers II

W. James Lewis1

Sybilla Beckmann2

and Denise A. Spangler2

1University of Nebraska-Lincoln2University of Georgia

AMTE Jan 2013

Lewis, Beckmann, Spangler (UNL,UGA) METII 1 / 46
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The Mathematical Education of Teachers II

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Acknowledgements

We are grateful to all who have contributed to MET2, but we want to

express special thanks to:

Math for America for their financial support that made possible the

development, printing and dissemination of the MET2 report

The Brookhill Foundation and the National Science Foundation fortheir support of the 2010 and 2011 CBMS Forums on

Mathematics Education where participants provided valuable

comments and suggestions

The many teachers, mathematicians and mathematics educatorswho offered extensive and thoughtful criticism of an earlier draft of

MET2

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The MET2 Writing Team

Sybilla Beckmann, University of Georgia

Daniel Chazan, University of MarylandAl Cuoco, Education Development Center

Francis (Skip) Fennell, McDaniel College

Bradford Findell, The Ohio State University

Cathy Kessel, Mathematics Education Consultant

Karen King, National Council of Teachers of MathematicsW. James Lewis, University of Nebraska-Lincoln

William McCallum, University of Arizona

Ira Papick, University of Nebraska-Lincoln

Barbara Reys, University of Missouri

Ronald Rosier, Conference Board of the Mathematical Sciences

Richard Scheaffer, University of Florida

Denise A. Spangler, University of Georgia

Alan Tucker, State University of New York at Stony Brook

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Why MET2?

Much has changed in the past decade

NCTMs Curriculum Focal Points and Focus in High School

Mathematics

Foundations for Success, the 2008 National Mathematics Panel

Report

NSFs Math Science Partnerships and Noyce grants

Increased attention to the mathematical education of teachers

among mathematicians

Increasing common ground among mathematicians and

mathematics educators and

The Common Core State Standards for Mathematics

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Why MET2?

Address the professional development of mathematics teachers

Address math specialists, early childhood teachers, special

education teachers

Review and update the METRecommendations

Align mathematics teacher education with the Common Core

State Standards for Mathematics

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AMTE contributions to MET2

Writing team includes 9 AMTE members including 2 past

presidents

A 2010 survey of the AMTE membership provided valuableinformation about

1

how the 2001 MET Report was used by AMTE members,2 the influence of the MET Report,3 sections of the report they found most useful, and4 features or focus areas they recommend for additional emphasis in

the revised report

A 2012 AMTE Task Force report provided some of the most useful

feedback we received on the February 2012 public draft

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MET2 themes

There is intellectual substance in school mathematics.

Proficiency with school mathematics is necessary but not

sufficient mathematical knowledge for a teacher.

The mathematical knowledge needed for teaching differs from that

of other professions.

Mathematical knowledge for teaching can and should growthroughout a teachers career.

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MET2 Recommendation 1

Prospective teachers need mathematics courses that develop a solid

understanding of the mathematics they will teach.

The mathematical knowledge needed by teachers is substantial

yet quite different from that required in other mathematical

professions.

Prospective teachers need to understand the fundamental

principles that underlie school mathematics.

Coursework for prospective teachers should examine the

mathematics they will teach in depth, from a teachers perspective.

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Coursework that allows time to engage in reasoning, explaining, and

making sense of the math that prospective teachers will teach.Well-started beginning teachers need

Elementary teachers at least 12 hours on fundamental ideas of

elementary mathematics.

Middle grades (5-8) teachers at least 24 hours of mathematics

that includes 15 hours on fundamental ideas of school

mathematics appropriate for ML teachers.

High School teachers the equivalent of a major that includesthree courses with a primary focus on high school mathematics

from an advanced viewpoint.

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At each level, these recommendations include courses especially

designed for teachers.

The recommended statistics-probability courses need to be different

from the courses typically taken by STEM majors and from the

non-calculus-based statistics courses offered at many universities.


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Throughout their careers, teachers need opportunities for continued

professional growth in their mathematical knowledge.Satisfying the minimum requirements for initial certification to

teach mathematics does not ensure that even outstanding future

teachers have the knowledge of mathematics, of teaching, and of

students that is possessed by successful experienced teachers.

The need for professional development takes on increased

importance due to the wide adoption of the CCSS.

A reasonable goal for initial certification at the secondary level isto create beginning teachers who are able to teach competently a

portion of the high school curriculum.

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All courses and professional development experiences for

mathematics teachers should develop the habits of mind of amathematical thinker and problem-solver, such as reasoning and

explaining, modeling, seeing structure, and generalizing.

Courses should also use the flexible, interactive styles of teachingthat will enable teachers to develop these habits of mind in their

students.

To help their students achieve the CCSS Standards for

Mathematical Practice, teachers must not only understand thepractices of the discipline but also how these practices can occur

in school mathematics and be acquired by students.

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MET2 Recommendations 1 4

Apply to

early childhood and elementary-level generalist teachers

middle grades and high school teachers who teach mathematicsclasses

teachers of special needs students, ELL, other special groups,

when those teachers have direct responsibility for teachingmathematics.


MET2 R d i 5
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Teacher education must be recognized as an important part of

mathematics/statistics departments missions and should beundertaken in collaboration with mathematics educators.

More mathematics/statistics faculty need to become deeply

involved in professional development for teachers and becomeinvolved with local schools and districts. National and regional

efforts are needed to help prepare mathematics/statistics faculty

to contribute effectively to teacher education.

Mathematics/statistics departments must provide graduate levelcourses designed to meet the professional needs of PreK-12

mathematics teachers.


MET2 R d ti 6
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Mathematics education, including the mathematical education of

teachers, can be greatly strengthened by the growth of a mathematics

education community that includes mathematicians as one of many

constituencies committed to working together to improve mathematics

instruction at all levels and to raise professional standards for teaching.


El t T h
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Elementary Teachers

. . . this report recommends that before beginning to teach, an

elementary teacher should study in depth, and from a teachers

perspective, the vast majority of K5 mathematics, its connections to

prekindergarten mathematics, and its connections to grades 68mathematics. By itself, this expectation is not sufficient to guarantee

high quality teaching. . . . However, there is no substitute: a strong

understanding of the mathematics a teacher will teach is necessary for

good teaching.


El t T h
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Elementary TeachersEssential Grades K 5 Ideas for Teachers

Operations and Algebraic Thinking (Kindergarten Grade 5).

The different types of problems solved by addition, subtraction,

multiplication, and division, and meanings of the operations

illustrated by these problem types.

Teaching-learning paths for single-digit addition and associated

subtraction and single-digit multiplication and associated division,

including the use of properties of operations (i.e., the field axioms).

Recognizing the foundations of algebra in elementary

mathematics, including understanding the equal sign as meaningthe same amount as rather than a calculate the answer symbol.


Ch 4: Elementary Teachers
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Ch 4: Elementary TeachersThe Common Core State Standards and the mathematics that elementary teachers

should study

Operations and Algebraic Thinking (Kindergarten Grade 5 5).

Illustrative activities:1 Recognize that commutativity for multiplication is not obvious and

use arrays to explain why multiplication is commutative.

MP 3, 5.


The commutative property of multiplication
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Why should multiplication be commutative? This is not obvious!

3 5

5 3


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So why is A B equal to B A???

The temptation is just to check a bunch of examples, but is there some

underlying reason why it always works?


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So why is A B equal to B A???

The temptation is just to check a bunch of examples, but is there someunderlying reason why it always works?


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3 5

5 3


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3 5

5 3


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3 5

5 3


Elementary Teachers
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Elementary TeachersThe preparation and professional development of elementary teachers

The mathematics of elementary school is full of deep and interesting

ideas, which can be studied repeatedly, with increasing depth and

attention to detail and nuance. Therefore, whereas prospectiveteachers will undertake an initial study of elementary mathematics

from a teachers perspective in their preparation program, practicing

teachers will benefit from delving more deeply into the very same

topics.


Elementary Teachers
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Elementary TeachersThe preparation and professional development of elementary teachers

Programs for Prospective Teachers

Professional Development for Practicing Teachers

Challenges in the Education of Elementary Teachers

Elementary Mathematics Specialists

Refers to AMTEs 2009 standards for EMS

Early Childhood Teachers

Teachers of Special Populations


Middle Grades Teachers
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Middle Grades TeachersEssential Grades 6 8 Ideas for Teachers

Ratio and Proportional Relationships (Grades 67).

Illustrative activities:

2. Compare and contrast different ways to find values in proportional

relationships and in inversely proportional relationships. Forexample, explain why linear interpolation can be used with

proportional relationships but not with inversely proportional

relationships.

MP 3, 4, 7.


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Middle Grades TeachersEssential Grades 6 8 Ideas for Teachers

Miles

Miles

Gallons

Gallons

4

100

8

200

12

300

16

400

20

500

0

0

100

4

200

8

300

12

400

16

500

20

+ 100

+ 4

+ 100

+ 4

+ 100

+ 4

+ 100

+ 4


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dd e G ades eac e sEssential Grades 6 8 Ideas for Teachers

100 200 25

1

Miles:

Gallons: 4 8

300

12

400

100 200 300 400

16

4 8 12 16

375

15

-

-

0

0

Miles

Gallons

13 14 15

325350

375


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Essential Grades 6 8 Ideas for Teachers

Suppose that 2 people take 8 hours to mow 5 acres of grass.

(Assume all the people work at the same steady pace.)

1 2 4 8People:

Acres: 5 10 202.5

2

2

2

2

2

2

1 2 4 8People:

Hours: 8 4 216

2

2

2

2

2

2

Proportional relationship Inverselyproportional relationship


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Essential Grades 6 8 Ideas for Teachers

Suppose that 2 people take 8 hours to mow 5 acres of grass.

(Assume all the people work at the same steady pace.)

3

3

1 2 3 4People:

Acres: 5 102.5 7.5

3

3

1 2 3 4People:

Hours: 8 416 5.33

# acres# people = 2.5 # hours # people = 16

Proportional relationship Inverselyproportional relationship


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The Preparation and Professional Development of Middle Grades Teachers

. . . prospective and practicing middle grades teachers need to be

aware of representations, be they drawings, tape diagrams, number

lines, or physical models, used in the earlier grades and how thoserepresentations may lend themselves to establishing and extending

mathematical ideas into the middle grades.


High School Teachers
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gIntroduction

A double discontinuity:

Felix Klein:

The young university student [was] confronted with problems that didnot suggest . . . the things with which he had been concerned at school.

When, after finishing his course of study, he became a teacher . . . he

was scarcely able to discern any connection between his task and his

university mathematics.



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Introduction

Not just forward looking, but also connecting back:

the mathematical topics in courses for prospective high school

teachers and in professional development for practicing teachers

should be tailored to the work of teaching, examining connections

between middle grades and high school mathematics as well as those

between high school and college.


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Introduction

The need for opportunities to engage in mathematical practices and

develop mathematical habits of mind:

teachers need opportunities for the full range of mathematicalexperience themselves: struggling with hard problems, discovering

their own solutions, reasoning mathematically, modeling with

mathematics, and developing mathematical habits of mind.



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Introduction

Outline:

1 Essentials in the mathematical preparation of high school

teachers.

2 Important additional mathematics content that can be learned in

undergraduate electives or in professional development programs

for practicing teachers.

3 Essential mathematical experiences for practicing teachers.



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Essentials in Mathematical Preparation

Courses taken by a variety of undergraduate majors

Single- and multi-variable calculus

Introduction fo linear algebra

Statistics and probability



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Essentials in Mathematical Preparation

Courses intended for all mathematics majors

Number theoryA comparison of arithmetic in Z and Z/nZ helps teachers understandthe importance of the lack of zero divisors when teaching the factor to

solve techniques for quadratic and higher-degree equations.

For example, how would one add words to turn these equations into a

coherent logical argument?

x2 5x + 6 = 0

(x 3)(x 2) = 0

x 3 = 0, x 2 = 0

x = 3, x = 2

Does the argument work over Z/6Z?



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Courses designed primarily for prospective teachers

Geometry and transformations. The approach to geometry in the

Common Core State Standards replaces the initial phases of axiomatic

Euclidean geometry. In the latter, the triangle congruence and

similarity criteria are derived from axioms. The Common Core, on theother hand, uses a treatment based on translations, rotations,

reflections, and dilations, whose basic angle and distance preserving

properties are taken as axiomatic. It is essential that teachers see a

detailed exposition of this development.


High School TeachersI Addi i l M h i

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Important Additional Mathematics

It is impossible to learn all the mathematics one will use in any

mathematical profession, including teaching, in four years of college.Therefore teachers will need opportunities to learn further topics

throughout their careers.


High School TeachersE ti l E i f P ti i T h
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Essential Experiences for Practicing Teachers

Many teachers prepared before the era of the CCSS will needopportunities to study content that they have not previously taught,

particularly in the areas of statistics and probability.

. . . teachers need experiences that renew and strengthen their interest

in and love for mathematics, help them represent mathematics as a

living discipline to their students by exemplifying mathematical

practices, figure out how to pose tasks to students that highlight the

essential ideas under consideration, to listen to and understand

students ideas, and to respond to those ideas and point out flaws instudents arguments.


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Sample undergraduate mathematics sequences:

Short sequence

33 semester-hoursLong sequence

42 semester-hours


Can the MET2 vision be realized, and what will it take?
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Academic culture is slow to change, but it is changing. Here is some

evidence:

1993 (20 years ago) The Joint Math Meetings had 7 talks aboutthe mathematical education of teachers including:

Thoughts on the mathematical preparation of

teachers. Alan Schoenfeld

Rethinking the mathematical education of teachers:What do we know and what do we need to figure

out? Glenda Lappan

2001 (12 years ago) The JMM had 33 talks about the

mathematical education of teachers. Many were related

to the draft recommendations of The MathematicalEducation of Teachers

2013 There were at least 70 talks and posters related to the

mathematical education of teachers.



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The Mathematical Sciences Research Institute (MSRI) has an

Educational Advisory Committee chaired by Deborah Ball and for 10years it has sponsored an annual workshop on Critical Issues in

Mathematics Education. Past workshops include:

2005 Mathematical Knowledge for Teaching K-8

2007 Teaching Teachers Mathematics

2011 The Mathematical Education of Teachers

2012 Teacher Education in view of the Common Core

And a related workshop

2009 Using Partnerships to Strengthen Elementary

Mathematics Teacher



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In AMTE, the profession has an organization focused on the

mathematical education of teachers

1991 In November, AMTE began with a meeting of 15 people

2001 185 people attended the 2001 AMTE national conference;

AMTE has 670 members

. . . we must be more active at every level.

Think about getting involved. . . . Let people know

about the importance of mathematics teacher

education and AMTE every day.

Skip Fennel, AMTE President, AMTE News, March

2001

2013 581 registered for the 2013 national conference; AMTE

has 942 members


Discussion

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Your thoughts and comments?

How can we engage our colleagues in discussions about

implementing MET2 recommendations?

How can we work towards implementing MET2 recommendationsat our own institutions?

How can we work to change policies so they become aligned with

MET2 recommendations?

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