Representation: Getting at the heart of mathematical understanding Prepared 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 denominators  Continue 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 thinking Which 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. Manipulative models Pictures Real-world situations Oral/Written language Written symbols Modes of representation of a mathematical idea Lesh, Post & Behr (1987)‏

Process Standard: Representation A Scaffold for Learning When students gain access to mathematical representations and the ideas they represent, they have a set of tools to significantly expand their capacity to think and communicate mathematically. PSSM (2001).

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

Connections to Common Core Standards 4 th Extend understanding of fraction equivalence and ordering (4.NF.1; 4.NF.2) 3 rd Develop understanding of fractions as numbers (3.NF.3a; 3.NF.3d) 2 nd 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 CStudent 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