Representation: Getting at the heart of mathematical understandingPrepared for the National Council of Teachers of Mathematics

2011 Annual Meeting Indianapolis IN

Thursday, April 14, 2011

Beth Schefelker, Mathematics Teaching Specialist, MPS

Melissa Hedges, Mathematics Teaching Specialist, MTSD

WALT and Success Criteria Today we are learning to…

Understand and use modes of representation for teaching and learning mathematics To show student understanding To support teaching for understanding To aid in differentiation of math lessons

We will know we will have achieved this when we… view student mathematical understanding through the lens

of representation. reflect on our use of the modes of representation for

teaching and learning mathematics

Turn and Talk

What does it mean to know how to compare fractions?

What does it mean to understand how to compare fractions?

What’s the difference?

Exploring Difference

1. Independently study the fraction pairs. No talking!

2. Decide…Which is larger? About by how much?

3. Estimate the difference.

4. Challenge: Share the difference as a unit fraction.

(A unit fraction has a one in the numerator

e.g.: ½, ⅓, ¼).

No common denominators please!

No pencils! No talking!

Fraction Pairs

⅔ or ¼

⅜ or ¾

5/3 or 7/4

Supporting your thinking…

Tap into your first “life line.” Turn to your partner and share you thinking.

Select your second “life line.” Still no common denominatorsContinue with your task of comparing the fractions

Alright – now reach for your third “life line.”Still no common denominators.Continue with your task.

Thinking about thinkingWhich is larger? About by how much?

2/3 or 1/4

3/8 or 3/4

5/3 or 7/4

As you reasoned through this task:

What mathematical struggles ensued? What insights emerged?

Without pencils? (mental image and oral language)

With fraction strips? (manipulative representation)

With pencils? (pictures or symbolic representation)

Building Understanding

In what ways did the various representations: help build and clarify understanding? support your ability to communicate

mathematically?

As children move between and among these representations for concepts, there is a better chance of a concept being formed correctly and understood more deeply.

Manipulativemodels

Pictures

Real-worldsituations

Oral/Writtenlanguage

Written symbols

Modes of representation of a mathematical idea

Lesh, Post & Behr (1987)

Process Standard: RepresentationA Scaffold for Learning

What Practices did we tap into as we compared fractions and estimated differences?

Connections to Common Core Standards 4th Extend understanding of fraction

equivalence and ordering (4.NF.1; 4.NF.2)

3rd Develop understanding of fractions as numbers (3.NF.3a; 3.NF.3d)

2nd Reason with shapes and their attributes (2.G.3)

Looking at student thinking…

How do the representations: help students clarify thinking and make

sense of the task? surface misconceptions and

understandings?

Student A Student B

Student C Student D

Supporting differentiation

...students who use concrete materials develop more precise and more comprehensive mental representations, often show more motivation and on-task behavior, understand mathematical ideas, and better apply these ideas to life situations. The Access Center – Improving Outcomes for All

Students K-8

Connecting back to instruction…How might modes of representation help you further differentiate?

When learners are able to represent a problem or mathematical situation in a way that is meaningful to them, the problem becomes more accessible.

Concrete → Representational → Abstract

Instructional Approach

Thank you for coming!

This powerpoint presentation will be posted on the

Milwaukee Mathematics Partnership website

mmp.uwm.edu