1. Basic Facts Understanding and Automaticity John SanGiovanni http://jsangiovanni.hcpss.wikispaces.net

2. Let’s Take a Test…

3. Traditionally, how have we taught (or learned) basic facts?

4. They don’t know their facts.

6. Drill on arithmetic facts does not necessarily lead to recall……. Drill must be preceded by sound instruction. 1935 - Brownell and Chazal, 1935 1935

7. Has our approach to teaching basic facts met the needs of all of our students?

8. Has our approach met the needs of ANY of our students? Understanding Number sense Has it aligned with our ideas of good teaching?

9. Why hasn’t memorization worked?

10. Turn to a “shoulder buddy” and identify all of the skills needed for this everyday task.

12. In math… 2 + 4 ½ + ¾ 2y + 4y 0.2 + 0.4 62 + 34

13. Have we confused memorization with AUTOMATICITY?

14. Processing of new information makes heavy use of working memory. As skills are repeated, the brain recognizes the information and can process it more quickly and with less effort. Automaticity reduces the load of working memory by as much as 90% (Schneider, 2003)

15. Computation in the “real world” is done mentally 84.6% of the time. - Northcote and McIntosh 1999 With this in mind…

16. Solve the next in problem mentally.

17. 49 + 27

21. Does memorization contribute to understanding?

22. Does knowing your facts mean you understand?

23. If our approach that uses memorization doesn’t help kids develop number sense and/or computational fluency? When do they develop it?

24. If we memorize 6 + 8… Will it help with 56 + 38?

25. If I understand… Multiplying by 5 is the same as Multiplying by ½ of 10 Multiplying by 5 is the same as Multiplying by ½ of 10 4 x 5 is the same as ½ of 4 x 10 I know 68 x 5

26. If I understand… Multiplying by 9 is the same as Multiplying by 1 group less than 10 Multiplying by 9 is the same as Multiplying by 1 group less than 10 8 x 9 is the same as (8 x 10) – (8 x 1) 8 x 9 is the same as (8 x 10) – (8 x 1) I know 37 x 9

27. New Tricks for an Old Dog or Old Tricks for a New Dog Foundational Understanding Instruction in Context Intentional Practice Independent Practice Assessment

28. What +/-/x/÷ means.. Commutative property Associative property * What +/-/x/÷ means.. Commutative property Associative property * Foundational Understanding Picture Removed

31. Pictures Removed

33. Patterns and The Beauty of Mathematics

34. Develop Understanding +1/+2 (counting on) 38 + 1 74 + 2

35. Develop Understanding +1/+2 (counting on) +0 198 + 0 0 + 56

36. Develop Understanding +1/+2 (counting on) +0 +10 10 + 23 45 + 10

37. Develop Understanding +1/+2 (counting on) +0 +10 Doubles 33 + 33 45 + 45

38. Develop Understanding +1/+2 (counting on) +0 +10 Doubles Make ten 4 + 66 53 + 7

39. Develop Understanding +1/+2 (counting on) +0 +10 Doubles Make ten +8/+9 (Using Ten) 38 + 7 59 + 4

40. Develop Understanding +1/+2 (counting on) +0 +10 Doubles Make ten +8/+9 (Using Ten) Using Doubles 33 + 34 45 + 46

41. Develop Understanding +1/+2 (counting on) +0 +10 Doubles Make ten +8/+9 (Using Ten) Using Doubles Using Knowns

42. Develop Understanding x 2 (doubles) 12 x 2 210 x 2

43. Develop Understanding x 2 (doubles) x 10 12 x 10

44. Develop Understanding x 2 (doubles) x 10 x 5 (1/2 of x10) 33 x 5 (33 x 10) ÷ 2

45. Develop Understanding x 2 (doubles) x 10 (double x5) x 5 x 1 14 x 1

46. Develop Understanding x 2 (doubles) x 10 (double x5) x 5 x 1 x 0

47. Develop Understanding x 2 (doubles) x 10 (double x5) x 5 x 1 x 0 x 3 (x2 + x1) 6 x 3 (12 + 6) 14 x 3 (28 + 14)

48. Develop Understanding x 2 (doubles) x 10 (double x5) x 5 x 1 x 0 x 3 (x2 + x1) x 4 (x2)(x2) 7 x 4 (7 x 2) x 2 23 x 4 (23 x 2) x 2

49. Develop Understanding x 2 (doubles) x 10 (double x5) x 5 x 1 x 0 x 3 (x2 + x1) x 4 (x2)(x2) x 6 (x3)(x2) 7 x 6 (7 x 3) x 2 42 x 6 (42 x 3) x 2

50. Develop Understanding x 2 (doubles) x 10 (double x5) x 5 x 1 x 0 x 3 (x2 + x1) x 4 (x2)(x2) x 6 (x3)(x2) x 8 (x2)(x2)(x2) 8 x 5 (5 x 2) x 2 x 2 31 x 8 (31 x 2) x 2 x 2

51. Develop Understanding x 2 (doubles) x 10 (double x5) x 5 x 1 x 0 x 3 (x2 + x1) x 4 (x2)(x2) x 6 (x3)(x2) x 8 (x2)(x2)(x2) x 9 (x10) – (x9) 6 x 9 (6 x 10) – (6 x 1) 18 x 9 (180) - 18

52. Develop Understanding x 2 (doubles) x 10 (double x5) x 5 x 1 x 0 x 3 (x2 + x1) x 4 (x2)(x2) x 6 (x3)(x2) x 8 (x2)(x2)(x2) x 9 (x10) – (x9) x 7 ---- just one

53. Good Mathematics Teaching Context Problem Solving Practice Assessment Picture Removed

54. Draw 10 meatballs on one of your paper plates. Draw 9 meatballs on your other paper plate.

55. Picture Removed

57. Intentional Practice Picture Removed

58. Intentional Practice 1.Fold a piece of paper in half and then in half again. 1.Label your columns a number, double it, double it again, double it again. 1.Write some numbers in the first column and complete the other columns.

59. Intentional Practice

60. Engaging Practice: Games Picture Removed

61. Engaging Practice: Games

63. Assessment What is our purpose? Picture Removed

64. Traditional Tests Negative impact They don’t teach anything Can reinforce inefficient strategies Avoid overuse Picture Removed

68. Data from Observation Picture Removed

69. Student Progress

70. Positive Attitudes Recognize progress Provide support Reasonable expectations Picture Removed

71. All students are able to master the basic facts-including children with learning disabilities. Children simply need to construct efficient MENTAL tools that will help them. - Van de Walle, 2005

72. But we have to: teach them in context with problems. practice them in an intentional and engaging way. assess them in a fair way that promotes growth. believe that they can.