1. 1 NCSM Annual Meeting March 2007 Dr. Eric Milou Rowan University Department of Mathematics milou@rowan.edu 856-256-4500 x3876 Conceptual Understanding and Basic Skill: Discussion Points for Teachers and to Parents

2. 2 Overview National News in Mathematics Education Conceptual vs. Procedural Debate Number Sense & Computation Proficiency

3. 3 NCTM Focal Points (9/12/06) Not Back to Basics at All Press articles did not represent the substance or intent of the focal points. The focal points are not about the basics; they are about important foundational topics. NCTM has always supported learning the basics. Students should learn and be able to recall basic facts and become computationally fluent, but such knowledge and skills should be acquired with understanding.

4. 4 Education Week 11/1/06 We cannot afford to waste time on polarization. What is important is that we pragmatically address critical target areas to improve mathematics education. We cannot be distracted from our primary mission—to match tactical initiatives in other, newly technological societies that are snatching our competitive advantage in innovation—while we bicker over modest differences in approach. (Jere Confrey)

5. 5 Compute the following: 4 x 9 x 25 How many ounces are in a gallon? 50 ÷ 1/2

6. 6 Peanuts

7. 7 What’s “Typical?” in US

8. 8 Third International Math & Science Study (TIMSS) Procedures vs. Concepts

9. 9 Stated vs Developed

10. 10 Math Lessons Demonstrates a procedure Assigns similar problems to students as exercises Homework assignment Presents a problem without first demonstrating how to solve it Individual or group problem solving Compare and discuss multiple solution methods Summary, exercises and homework assignment

11. 11 We need a BALANCE Pedagogical Balance Content Balance –Conceptual Understanding –Algorithmic Proficiency These are NOT Dichotomous

12. 12 Conceptual Understanding 24 ÷ 4 = 6 24 ÷ 3 = 8 24 ÷ 2 =12 24 ÷ 1 = 24 24 ÷ 1/2 = ??

13. 13 Fractions - Conceptually More than 1 or Less than 1 Explain your reasoning The F word

14. 14 Which is larger? 2/3 + 3/4 + 4/5 + 5/6 OR 4 12.5 x 45 OR 4.5 x 125 1/3 + 2/4 + 2/4 + 5/11 OR 2

15. 15 Where’s the Point? 2.43 x 5.1 = 12393 4.85 x 4.954 = 240269 21.25 x 1.08 = 2295 1.25 x 64 = 80 4.688 x 1.355 = 635224 46.88 x 1.355 = 635224 4.688 x 135.5 = 635224 46.88 x 13.55 = 635224

16. 16 Computational Balance 1000 ÷ 1.49 = 671.1409396 = Torture! Big Macs Sell for $1.49, how many Big Macs can I buy for $10.00? –1 is $1.50 –2 are $3 –4 are $6 –6 are $9 Mental Mathematics is a vital skill

17. 17 Computation is Important Engaging & Active Less passive worksheets More thinking & reasoning

18. 18 Numbers Are Everywhere

19. 19 Name That Number - Computational Practice Target #: 6 3 8 17 1 3

20. 20 Active Computation Fifty (1, 2, 3, 4, 5, 6 and addition) Buzz (3) Product Game Wipe Out Software: Math Arena

21. 21 Multiplication Wheels 0 1 2 3 4 5 6 7 8 9 4 Facts

22. 22 Conceptual & Contextual 8+ 7 = ? How do we teach this? xxx x xx x x x x x x x x x x x

23. 23 17 - 8 = 1 7 - 8 / / 0 17 2 7 8 --> --> 10 --> --> --> --> --> --> --> 17

24. 24 1000 - 279 = ? 279+1 = 280+ 20 = 300+700 = 1000

25. 25 Multiplication 13 x 17 = ? 1 3 x 1 7 1 2 9 031 2 2 1 ------- 10 7 10 3 1 0 0 3 0 7 0 2 1 221

26. 26 Conceptual approach leads to ? x 7 x3x3 x2x2 3x 7x 21 Algebra: (x + 3) (x + 7) =

27. 27 Contextual Problem Solving Not more traditional word problems Placing mathematical lessons into settings Giving students a reason to learn the skill Motivating students

28. 28 Contextual Problem Solving ≠ More Word Problems

29. 29 Example 1: Sneakers SECOND purchase Nike ConverseReebok Nike Converse Reebok FIRSTFIRST

30. 30 100 Students –50 1st time Nike buyers –30 1st time Converse buyers –20 1st time Reebok buyers How many would buy Nike the second time? 50 x.4 + 30 x.2 + 20 x.1 = 28

31. 31 Example 2: Game Show You must select one spinner. Both spinners above will be spun once. The spinner with the higher number showing wins $1,000,000 for that person. Which spinner will you select?

32. 32 Spinner Example BLUEORANGE 4 6 8 4 6 8 5 5 5 9 9 9

33. 33 Math Essays?

34. 34 Fact #1 A

35. 35 Fact #2 B

36. 36 Fact #3 C

37. 37 Fact #4 D

38. 38 Fact #5 E

39. 39 Fact #6 F

40. 40 Fact #7 G

41. 41 Fact #8 H

42. 42 Fact #9 I

43. 43 What is this?

44. 44 What is this? F A C E

45. 45 What If? AB C DE F GH I

46. 46 Try Again

47. 47 Try Again D E C A D E

48. 48 What’s the Point? Isolated Facts –Less likely to retain information Connected Facts, Patterns, Facts in Context –More likely to retain information

49. 49 Characteristics of a good mathematics program CONCEPTUAL CONTEXTUAL CONSTRUCTIVISM COMPUTATION

50. 50 Thank You Dr. Eric Milou Rowan University milou@rowan.edu