The Role of Visual Representations in Learning Mathematics

1. The Role of Visual Representations in Learning Mathematics John Woodward Dean, School of Education University of Puget Sound Summer Assessment Institute August 3, 2012

2. Information Processing Psychology • How Do We Store Information? • How Do We Manipulate It? • What Mechanisms Enhance Thinking/ Problem Solving?

3. Information Processing Psychology Monitoring or Metacognition i m a g e s t e x t

4. The Traditional Multiplication Hierarchy 5x 3 35x 3 357x 3 357x 43

5. It Looks Like Multiplication 357x 43 1071 + 1428.. 15351 2 2 1 2 How many steps?

6. 43 589 The Symbols Scale Tips Heavily Toward Procedures What does all of this mean? 2201 -345 4x + 35 = 72 + x 589 x 73 5789 + 3577 .0009823 y = 3x + 1 9 10 7 2 7 3 +

7. Old Theories of Learning Practice Practice Show the concept or procedure Practice Practice Practice Practice

8. Better Theories of Learning Conceptual Demonstrations Visual Representations Discussions Return to Periodic Conceptual Demonstrations Controlledand Distributed Practice

9. The Common Core Calls for Understanding as Well as Procedures

10. Tools Manipulatives 100 10 1 Place Value or Number Coins Number Lines

11. Tools Fraction Bars Integer Cards

12. The Tasks 1/3 x 1/2 3 ) 102 3/4 = 9/12 as equivalent fractions 2/3 ÷ 1/2 .60 ÷ .20 1/3 + 1/4 4 - 3 = 4 + -3 = 1/3 - 1/4 4 - -3 =

13. Long Division 3 10 2 How would you explain the problem conceptually to students?

14. Hundreds Tens Ones 1 0 2

15. Hundreds Tens Ones 100 + 0 + 2 1 100 1

16. Hundreds Tens Ones 3 102 1 100 1

17. Hundreds Tens Ones 3 102 1 100 1

18. Hundreds Tens Ones 3 102 1 100 1

19. Hundreds Tens Ones 3 102 10 10 100 10 10 10 10 1 10 10 1 10 10

20. Hundreds Tens Ones 3 102 10 10 10 10 10 10 1 10 10 1 10 10

21. 3 102 1 10 1 3 Hundreds Tens Ones 10 10 10 10 10 10 10 10 10

22. 1 1 3102 1 1 1 1 1 1 1 10 1 1 1 3 Hundreds Tens Ones -9 1 10 10 10 10 10 10 10 10 10

23. 3102 1 1 1 1 1 1 1 1 1 1 3 Hundreds Tens Ones 1 1 -9 1 2 10 10 10 10 10 10 10 10 10

24. 3 102 3 4 Hundreds Tens Ones 1 10 1 10 -9 1 2 1 10 1 10 1 10 1 1 10 1 10 1 10 1 1 10 1

25. 3 102 3 4 Hundreds Tens Ones 1 10 1 10 -9 1 2 -1 2 1 10 1 10 1 10 1 1 10 1 10 1 10 1 1 10 1

26. 3102 3 4 Hundreds Tens Ones 1 10 1 10 -9 1 2 -1 2 0 1 10 1 10 1 10 1 1 10 1 10 1 10 1 1 10 1

27. Hundreds Tens Ones 3 1 0 2 3 4 1 10 1 10 -9 1 2 -1 2 0 1 10 1 10 1 10 1 1 10 1 10 1 10 1 1 10 1

28. 3 5 3 7 3 7 3 7 3 7 2 5 2 5 2 5 2 5 2 5 + + - ÷ x The Case of Fractions

29. Give Lots of Practice to Those who Struggle 2/3 + 3/5 = 2/3 + 3/5 = 2/3 + 3/5 = 2/3 + 3/5 = 2/3 + 3/5 = 3/4 - 1/2 = 3/4 - 1/2 = 3/4 - 1/2 = 3/4 - 1/2 = 3/4 - 1/2 = 3/5 x 1/6 = 3/5 x 1/6 = 3/5 x 1/6 = 3/5 x 1/6 = 3/5 x 1/6 = 4/9 ÷ 1/2 = 4/9 ÷ 1/2 = 4/9 ÷ 1/2 = 4/9 ÷ 1/2 = 4/9 ÷ 1/2 = 2/3 + 3/5 = 2/3 + 3/5 = 2/3 + 3/5 = 2/3 + 3/5 = 2/3 + 3/5 = 3/4 - 1/2 = 3/4 - 1/2 = 3/4 - 1/2 = 3/4 - 1/2 = 3/4 - 1/2 = 3/5 x 1/6 = 3/5 x 1/6 = 3/5 x 1/6 = 3/5 x 1/6 = 3/5 x 1/6 = 4/9 ÷ 1/2 = 4/9 ÷ 1/2 = 4/9 ÷ 1/2 = 4/9 ÷ 1/2 = 4/9 ÷ 1/2 = 3/5 x 1/6 = 3/5 x 1/6 = 3/5 x 1/6 = 3/5 x 1/6 = 3/5 x 1/6 = 4/9 ÷ 1/2 = 4/9 ÷ 1/2 = 4/9 ÷ 1/2 = 4/9 ÷ 1/2 = 4/9 ÷ 1/2 =

30. Why Operations on Fractions Are So Difficult • Students are used to the logic of whole number counting • Fractional numbers are a big change • Operations on fractions require students to think differently • Addition and subtraction of fractions require one kind of thinking • Multiplication and division require another kind of thinking • Contrasting operations on whole numbers with operations on fractions can help students see the difference

31. Counting with Whole Numbers Counting with Whole Numbers is Familiar and Predictable ... ... 0 1 2 3 98 99 100

32. Counting with Whole Numbers Even When We Skip Count, the Structure is Predictable and Familiar ... ... 0 1 2 3 4 5 6 98 99 100 101 102

33. The “Logic” Whole Number Addition Whole Numbers as a Point of Contrast 0 1 2 3 4 5 6 7 8 9 3 + 4 = 7 Students just assume the unit of 1 when they think addition.

34. Counting with Fractions Counting with Fractional Numbers is not Necessarily Familiar or Predictable ? 0 1/3 1

35. The Logic of Adding and Subtracting Fractions 1 3 1 + 4 ? 1 3 1 4 We can combine the quantities, but what do we get?

36. Students Need to Think about the Part/Wholes 1 4 0 1 0 1 1 3 The parts don’t line up 0 1

37. Common Fair Share Parts Solves the Problem 1 3 4 12 3 12 1 4

38. Work around Common Units Solves the Problem 4 12 7 12 3 12 Now we can see how common units are combined

39. The Same Issue Applies to Subtraction 1 3 1 4 - What do we call what is left when we find the difference?

40. Start with Subtraction of Fractions We Need Those Fair Shares in Order to be Exact = 4 12 3 - 12 1 12 Now it is easier to see that we are removing 3/12s

41. Multiplication of Fractions Multiplication of Fractions: A Guiding Question When you multiply two numbers, the product is usually larger than either of the two factors. 3 x 4 = 12 When you multiply two proper fractions, the product is usually smaller. Why? 1/3 x 1/2 = 1/6

42. Let’s Think about Whole Number Multiplication 3 x 4 = 12 = 3 groups of 4 cubes = 12 cubes

43. An Area Model of Multiplication 3 x 4 Begin with an area representation 4 units

44. An Area Model of Multiplication 3 x 4 Begin with an area representation 4 units 3 units

45. An Area Model of Multiplication 3 x 4 = 12 4 units 3 units

46. An Area Model of Multiplication ½ x 4 4 units Begin with an area representation

47. An Area Model of Multiplication ½ x 4 Begin with an area representation 4 units 1/2 units

48. An Area Model of Multiplication ½ x 4 4 Begin with an area representation 1/2

49. An Area Model of Multiplication ½ x 4 4 4 red units 1/2

50. An Area Model of Multiplication ½ x 4 4 ½ of the 4 red shown in stripes 1/2