Representation: Getting at the heart of mathematical understanding

1. Representation: Getting at the heart of mathematical understanding Wisconsin Mathematics Council Green Lake Annual Conference Thursday, May 6, 2010 Sarah Burzynski, Math Teacher Leader, Longfellow School MPS Melissa Hedges, Math Teaching Specialist, MPS The Milwaukee Mathematics Partnership (MMP), an initiative of the Milwaukee Partnership Academy (MPA), is supported with funding from the National Science Foundation.

2. 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

3. 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?

4. Exploring Difference 1. Turn over one card at a time. On your own study the fractions you see. No talking! 2. Decide…Which is larger? About by how much? 3. Estimate the difference. 4. Be prepared to share the estimated 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!

5. Supporting your thinking… Tap into your first “life line.” • Turn to your partner and share your thinking. • Still no pencils! • 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…still estimating! • Continue with your task.

6. 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 & oral language) • With pencils? (pictorial or symbolic representation) • With fraction strips? (manipulative representation)

7. Building Understanding • In what ways did the various representations help build and clarify understanding? • In what ways did the representations support your ability to communicate mathematically?

8. Modes of representation of a mathematical idea Pictures 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. Written symbols Manipulative models Real-world situations Oral/Written language Lesh, Post & Behr (1987)

9. Process Standard: RepresentationA Scaffold for Learning • When learners are able to represent a problem or mathematical situation in a way that is meaningful to them, the problem becomes more accessible. (NCTM, 2000) • 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. (NCTM, 2000)

10. References National Research Council. (2001). Adding it up. Mathematics Learning Study Committee, Center for Education, Division of Behavioral Sciences and Education, National Research Council. Washington, DC: National Academy Press. National Research Council. (2002). Helping Children Learn Mathematics. Mathematics Learning Study Committee, J. Kilpatrick & J. Swafford, Editors. Center for Education, Division of Behavioral Sciences and Education. Washington, DC: National Academy Press. Wisconsin Department of Public Instruction. (1998). Wisconsin’s model academic standards for mathematics. Madison, WI: Author.

11. Looking at student thinking… In reviewing the student work, what are you noticing? What does the representation being used by the student tell us about his/her understanding of the mathematics? How do the children use representations to clarify thinking and make sense of the task? How does their use of representations surface misconceptions?

12. Student A Student B

13. Student C Student D

14. Wisconsin State Framework Descriptors Mathematical Processes Grade K5 – Grade 8 Mathematical Processes: Students will effectively use mathematical knowledge, skills and strategies related to reasoning, communication, connections, representations, and problem solving.

15. Learner communicates mathematically Descriptors, such as but not limited to: • Communicate mathematical ideas and reasoning using the vocabulary of mathematics in a variety of ways e.g., using words, numbers, symbols, pictures, charts, tables, diagrams, graphs, and models. • Connect mathematics to the real world, as well as within mathematics. • Create and use representations to organize, record, and communicate mathematical ideas. • Solve and analyze routine and non-routine problems.

16. Connecting back to instruction… • What ideas surfaced today that reinforce what you already do during your math class? • What ideas surfaced this morning that you might pay more attention to during your math class? • How might modes of representation help you differentiate your instruction?

17. 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