# Presentation: 037089

## What Mathematics Knowledge is Needed for Teaching Mathematics?

- Deborah Loewenberg Ball
- University of Michigan

**Archived Information**

## Multiply:  What is the correct answer?

**Notes:**
Ask people to calculate this problem using whatever method they typically use.

## What is Necessary Mathematics Knowledge for Teaching?

- _Examine teaching_, and identify the mathematical work that teaching entails
- Analyze _what_ mathematical knowledge –– topics and skills –– is needed to do that  work
- Analyze also the qualities of that knowledge –– _how_ it must be understood and known to be serviceable for the work

## Evaluating and interpreting common incorrect student answers

**What is the misunderstanding**** ****here?**

## Explaining mathematical concepts and procedures

**Why do we “move over” before writing 70?**

## Evaluating and appraising different methods

**Which of these students is using a method that**
**could be used to multiply any two whole numbers?**

## Using representations to model and

**Using representations to model and**
**reason about procedures**

## Qualities of Mathematics Knowledge for Effective Instruction

- Respectful of the integrity of the discipline
- Able to be extended and opened up for learners –– “unpacked”
- Justified, reasoned
- Connected within and across domains, building on earlier ideas and anticipating more advanced topics
- Organized psychologically as well as logically

## Using definitions

**Textbook definition**:  A closed flat two-dimensional shape whose sides are formed by line segments.

**Which shapes are polygons?**

## Definition: A simple closed plane curve formed by straight line segments. 

-  But –– what’s a mathematically acceptable definition of “polygon” that is usable by fifth graders?

## Looking Back:  What Mathematics Do Teachers Have to Do?

## Looking Back:  What Mathematics Do Teachers Have to Do?

- Design mathematically accurate explanations that are comprehensible and useful for students
- Use mathematically appropriate and comprehensible definitions
- Represent ideas carefully, mapping between a physical or graphical model, the symbolic notation, and the operation or process
- Interpret and make mathematical and pedagogical judgments about students’ questions, solutions, problems,and insights (both predictable and unusual)
- Be able to respond productively to students’ mathematical questions and curiosities
- Make judgments about the mathematical quality of instructional materials and modify as necessary
- Be able to pose good mathematical questions and problems that are productive for students’ learning
- Assess students’ mathematics learning and take next steps

## Looking Back:  What Mathematics Do Teachers Need to Know to Do Those Things?

## Topics and ideas that are fundamental to the school curriculum–– and beyond 

Tools and skills for reasoning about mathematical claims, ideas, representations, and solutions; and sensibility about what constitutes adequate proof

Fluency and care with mathematical language and notation

Familiarity with applications of mathematics

- Tools and skills for reasoning about mathematical claims, ideas, representations, and solutions; and sensibility about what constitutes adequate proof
- Fluency and care with mathematical language and notation
- Familiarity with applications of mathematics

- Looking Back:  What Mathematics Do Teachers Need to **Know** to Do Those Things?

*In general:*

## Looking Back:  What Mathematics Do (Elementary) Teachers Need to Know?

- Concepts of number and place value notation
- Operations
- Number theory and number systems
- Common algorithms and how and why they work
- Concepts and tools of algebra
- Geometric concepts and reasoning
- Concepts and tools of statistics and probability

- Representing and connecting representations (e.g., symbols, graphs, geometric models)
- Mathematical language and definitions
- Mathematical reasoning and justification
- Good sense about mathematical precision
- Mathematical curiosity and interest

- What provides mathematical leverage?

## Conclusion:Knowing Mathematics for Teaching ––Three Core Principles 

- Teachers need to know the same things and be able to do the same things as any adult, and _much more_.
- Knowledge needed for teaching is _different_ from what is needed for other occupations or professions where mathematics is used (e.g., physics, mathematics, carpentry, tailoring, business).
- Knowledge needed for teaching must be _usable_ for the mathematical work that _teachers_ have to do.