1. Old Computations, New Representations Lynn T. Goldsmith Nina Shteingold nshteingold@edc.org Lynn T. Goldsmith Nina Shteingold nshteingold@edc.org

2. © EDC. Inc., ThinkMath! 2007 http://www2.edc.org/thinkmath/

3. © EDC. Inc., ThinkMath! 2007 Plan of the presentation: ThinkMath: examples of using different representations in teaching addition, subtraction, multiplication, and division. Discussion: - how does using a variety of representations help to build computational fluency? - how does using a variety of representations help to de-bug a concept? ThinkMath: examples of using different representations in teaching addition, subtraction, multiplication, and division. Discussion: - how does using a variety of representations help to build computational fluency? - how does using a variety of representations help to de-bug a concept?

4. © EDC. Inc., ThinkMath! 2007 (Some of) The Problems that Teachers Experience: Different students have different learning styles Different students learn with different pace Without computational fluency students cannot progress to fully comprehend related concepts Flows in conceptual understanding are frequent There is just not enough time! Different students have different learning styles Different students learn with different pace Without computational fluency students cannot progress to fully comprehend related concepts Flows in conceptual understanding are frequent There is just not enough time!

5. © EDC. Inc., ThinkMath! 2007 One Way of Solving These Problems: Using Multiple Representations

6. © EDC. Inc., ThinkMath! 2007 Example of Addition and Subtraction

7. © EDC. Inc., ThinkMath! 2007 From representing number as a quantity and as a position…

8. © EDC. Inc., ThinkMath! 2007 … to representing addition and subtraction both as a change in the position on the number line…

9. © EDC. Inc., ThinkMath! 2007 … and as a change in quantity.

10. © EDC. Inc., ThinkMath! 2007 What are some of characteristics of the number line representation of addition and subtraction?

11. © EDC. Inc., ThinkMath! 2007 Observing patterns

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13. Numbers grow… Students do not have to use the number line to complete the task, but they can if they need.

14. © EDC. Inc., ThinkMath! 2007 The level of abstraction grows. Students rely more and more on their internal representation.

15. © EDC. Inc., ThinkMath! 2007 Cross Number Puzzles

16. © EDC. Inc., ThinkMath! 2007 6 small counters, 4 large counters

17. © EDC. Inc., ThinkMath! 2007 7 blue counters, 3 gray counters

18. © EDC. Inc., ThinkMath! 2007 Does not matter how you count counters, small and then large, or blue and then gray, you’ll always have the total of 10.

19. © EDC. Inc., ThinkMath! 2007 Cross Number Puzzles Underline “any order, any grouping” property of addition and subtraction

20. © EDC. Inc., ThinkMath! 2007 Interplay of different representations: numbers are represented by “sticks” (each worth10) and “dots” (each worth 1); addition is represented by a part of a Cross Number Puzzle.

21. © EDC. Inc., ThinkMath! 2007 Moving towards addition algorithm

22. © EDC. Inc., ThinkMath! 2007 Adding money is a very good concrete representation of addition

23. © EDC. Inc., ThinkMath! 2007 Using place value to add and subtract: 1.Same amount on both sides of a thick line; 2.Only multiples of 10 in one of the columns.

24. © EDC. Inc., ThinkMath! 2007 It works with more than 2-digit Numbers too. And with more than 2 numbers

25. © EDC. Inc., ThinkMath! 2007 Multiplication and Division Representation Repeated jumps on a number line Counting objects in equal groups Counting North-South and East-West roads and intersections Counting lines in one direction, lines in another direction, and intersections Counting combinations Counting dots in an array Counting rows, columns, and blocks Calculation “area” Repeated jumps on a number line Counting objects in equal groups Counting North-South and East-West roads and intersections Counting lines in one direction, lines in another direction, and intersections Counting combinations Counting dots in an array Counting rows, columns, and blocks Calculation “area”

26. © EDC. Inc., ThinkMath! 2007 Repeated jumps

27. © EDC. Inc., ThinkMath! 2007 Groups of the same size

28. © EDC. Inc., ThinkMath! 2007 Combinations

29. © EDC. Inc., ThinkMath! 2007 Combinations of letters (and digits)

30. © EDC. Inc., ThinkMath! 2007 Lines and intersections This representation Is good for showing commutative property of multiplication as well as for showing what multiplying by 0 means.

31. © EDC. Inc., ThinkMath! 2007 Underlying distributive property

32. © EDC. Inc., ThinkMath! 2007 Underlying distributive property - on a more complex level

33. © EDC. Inc., ThinkMath! 2007 Array representation of multiplication

34. © EDC. Inc., ThinkMath! 2007 One cannot just count any more!

35. © EDC. Inc., ThinkMath! 2007 And then to area. This representation is well expandable to include multiplication of fractions.

36. © EDC. Inc., ThinkMath! 2007 Interplay of array and Cross Number Puzzle

37. © EDC. Inc., ThinkMath! 2007 How multiplication and division are related Notice how standard notation for division is being introduced (lower part of the page).

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39. Connections: Multiplication and division sentences are used to describe different situations (representations); earlier number sentences were introduced as their Records.

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42. How does using a variety of representations help to build computational fluency: Allows for students’ different learning styles Allows for different pace Helps to increase practice in computation yet to avoid boredom ? Allows for students’ different learning styles Allows for different pace Helps to increase practice in computation yet to avoid boredom ?

43. © EDC. Inc., ThinkMath! 2007 How does using a variety of representations help to de-bug a concept: A representation underlines some properties of a concept but obscures others.