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Solving the Problem:What Mathematics Do Teachers Need to Know?Hyman Bass

Forum on Mathematical Competencies in Higher Education Bogotá, Colombia • November 22, 2006

Deborah Ball, Mark Thames, Heather Hill. Jennifer Lewis, Ed Wall, Laurie Sleep, Kara Suzuka, Imani Goffney, Deborah Zopf, Seán Delaney, Tim Boerst, Raven McCrory, Geoffrey Phelps, Helen Siedel, Andreas Stylianides, . . .

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49X 25

3

4925x

405108

1485

(a) 4925x

100225

325

(b) 4925x

125025

1275

(c)

What mathematical steps produced each of these answers?

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What mathematics do teachers need to know?

• Where the question came from (for me and for the field) and how it has evolved

• Why the question matters• Different ways to answer the question

• How different answers inform practice and policy

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What is the “problem”?

Teachers’ knowledge of mathematics

and their ability to use it in practice

The quality of mathematics

teaching and learning

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Some approaches to solving the problem, and the unresolved question• Require more mathematics for certification– More mathematics courses– A major (or minor) in mathematics– Mathematics test

• Recruit mathematically trained people into teaching– Engineers, accountants, mathematicians, …

• Fund mathematically focused professional development But what kind of mathematical knowledge,

skill,and reasoning is needed in teaching, and how can it be developed?

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Working on a practice-based theory of mathematical knowledge for teaching (MKT)

• Why we set out to do this• How we have developed a laboratory that intertwines practice, theory development, and empirical testing

• Investigating three components of the laboratory

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Elements of our approach1. Study instruction, and identify the

mathematical work of teaching2. Analyze what mathematical knowledge is

needed to do that work effectively, and how it must be understood to be useful for the work

3. Develop, test, and refine measures of MKT using multiple methods as a means to evaluate professional education, investigate effects on students’ learning, and improve theory

4. Develop and evaluate approaches to helping teachers learn mathematical knowledge for teaching

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A laboratory for studying the mathematical knowledge needed for teaching

Teaching elementary students Developing measures of

mathematical knowledge Teaching teachers

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Teaching elementary school mathematics

Data archive of one complete year in a third grade class

Deliberate design of laboratory classes (PCMI, others)

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Using our own teaching and others’ to study MKT• Identify tasks of teaching that demand mathematical skill and reasoning

• Analyze the sort of mathematical knowledge, skill, and sensibilities needed

• Experiment with trying to teach particular ideas and learn what is involved mathematically in practice

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Developing measures of mathematical knowledge for teaching

1 ÷34

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Write a story or a situation or

make a diagram for which

is the mathematical formulation.

Student A Student B Student C

x32

55 x

32

55 x

3255

+17

25

5+1770

50 1

2550

875+

16

0000

875

875

Which of the students is using a method that would work to multiply any two whole numbers?

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Pedagogical Content Knowledge

Common ContentKnowledge (CCK)

Specialized

Content Knowledge (SCK)

Knowledge of Content and Students (KCS)

Knowledge of Content and Teaching (KCT)

Subject Matter Knowledge

Knowledge at the

mathematical horizon

Knowledge of

curriculum

Shulman’s original categorization(1985) compared with ours

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Using the development of measures to refine theory of mathematical knowledge for teaching

Which of these can be interpreted as a representation of 3/4?

How many 4’s are there in 3?

a)

b)

c)

d)

e)

f)

g)

i)

18 crayons out of a box of 24

.75

I want to share 3 bottles of soda equally among 4 people. How much will each person get?

h)0 1 2 3

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What does this item measure?

A. .5 7 .01 11.4

B. .60 2.53 3.12 .45

C. .6 4.25 .565 2.5

D. These lists are all equally good for assessing whether students understand how to order decimal numbers.

Which of the following lists would be best for assessingwhether your students understand decimal ordering?

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Try to write a specialized content knowledge item.Ms. Lin looks up the definition of “even number” in several textbooks and reference books. Which of the following definitions is both mathematically correct and accessible below the middle school level?

a)b)c)d)

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Validating our measuresHow do we interpret teachers’ performance on

our questions?

1. Their score reflects their mathematical thinking– Cognitive interviews

2. Higher scores mean higher-quality mathematics instruction– Videotape validation study

3. Scores reflect common and specialized knowledge of content– Mathematician and non-teacher interviews

4. Higher scores related to improved student learning– Study of Instructional Improvement student

gains analysis

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Using measures work to learn about MKT• Testing our conceptual ideas• Investigating the categories and challenging them empirically

• Validating the theory• Getting new ideas for theory development

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What have we learned and where are we going?• Teaching mathematics presents special kinds of

mathematical demands.• Mathematical knowledge for teaching is both more than

knowledge, more elaborated, and different from that needed for other mathematically-intensive tasks or occupations.

• Centrality of mathematical interpretation, analysis, and production, as well as explanation and representation

• Study and develop theory about the mathematical reasoning, argument, and language special to using mathematics in and for teaching

• The promise and need to integrate work on MKT with work on attending to equity

• Develop knowledge of what people who teach mathematics to teachers need to know