1. 1 Preparing for the NJ Math Assessments in the Middle Grades Dr. Eric Milou Rowan University Department of Mathematics milou@rowan.edu 856-256-4500 x3876

2. 2 Overview Conceptual vs. Procedural Debate –National Math Panel Number Sense & Computation Proficiency NJ mathematics assessments

3. 3 Rhetoric NY Times (5/15/06) In traditional math, children learn multiplication tables and specific techniques for calculating. In constructivist math, the process by which students explore the question can be more important than getting the right answer, and the early use of calculators is welcomed.

4. 4 NCTM Focal Points (9/12/06) September 12 Wall Street Journal article 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.

5. 5 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)

6. 6 Motivating Factors for Change Society’s hate for mathematics that is prevalent and acceptable –4 out of 10 adults hate mathematics* (twice as many people said they hated math as said that about any other subject) International test scores Industry concerns (no problem solving skills) National Council of Teachers of Mathematics (NCTM) Standards *2005 AP-AOL News poll

7. 7 Compute the following: 4 x 9 x 25 900 - 201 50 ÷ 1/2

8. 8 What’s “Typical?” in US

9. 9 Third International Math & Science Study (TIMSS) Procedures vs. Concepts

10. 10 Stated vs Developed

11. 11 Lesson Study 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

12. 12 We need a BALANCE Traditional text with conceptual supplement Conceptual text (EM, CMP, Core- Plus) with computational supplement

13. 13 Conceptual Understanding 24 ÷ 4 = 6 24 ÷ 3 = 8 24 ÷ 2 =12 24 ÷ 1 = 24 24 ÷ 1/2 = ??

14. 14 Fractions - Conceptually More than 1 or Less than 1 Explain your reasoning The F word

15. 15 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

16. 16 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

17. 17 Computational Balance 1000 ÷ 1.49 –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

18. 18 Computation is Important Engaging & Active Less passive worksheets Creative! More thinking & reasoning

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 Patterns 0 1 2 3 4 5 6 7 8 9

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 Example 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?

29. 29 Spinner Example BLUEORANGE 4 6 8 4 6 8 5 5 5 9 9 9

30. 30 Crossing the River 8 adults and 2 children need to cross a river and they have one small boat only available. The boat can hold ONLY: –One adult –One or two children How many one-way trips does it take for all 8 adults and 2 children to cross?

31. 31 2006 NJ Assessment Data NJASK3 6 non-calculator items (1/2 pt each) 21 MC - calculator allowed - 1 pt each 3 Open-ended - 3 pts each 14 out of 33 points is a passing score

32. 32 2006 NJ Assessment Data NJASK4 8 non-calculator items (1/2 pt each) 24 MC - calculator allowed - 1 pt each 5 Open-ended - 3 pts each 17.5 out of 43 points is a passing score

33. 33 2006 NJASK 5, 6, 7 NJASK5 JPM was 18/39 (46%) NJASK 6 JPM was 17/39 (44%) NJASK 7 JPM was 13/39 (33%) 10 pts per cluster (one cluster with 9 pts)

34. 34 2006 NJ Assessment Data GEPA All items allow a calculator 30 Multiple choice items - 1 pt each 6 Open-ended - 3 pts each 25 out of 48 points is a passing score

35. 35 2006 NJ Assessment Data HSPA All items allow a calculator 30 Multiple choice items - 1 pt each 6 Open-ended - 3 pts each 20.5 out of 48 points is a passing score

36. 36 Assessments Points by Cluster ClusterNJASK 3 NJASK 4 NJASK 5 NJASK 6 Number913109 Geometry8109 Algebra810 D/P/D810 Total334339 “200”1417.51817

37. 37 Assessments Points by Cluster ClusterNJASK7GEPAHSPA Number10127 Geometry912 Algebra101215 D/P/D101214 Total3948 “200”132520.5

38. 38 200 Score

39. 39 Implications & Inferences NJ Assessments are rigorous and conceptual NJ Math Standards are well aligned with NJ assessments Most districts have a well aligned curriculum –Then, what’s wrong?

40. 40 Algebra Placement Districts should not encourage all students to take Algebra I in grade 8; students should be taking Algebra I in grade 8 only if they are highly motivated, have a strong foundation in middle school mathematics, receive high grades in previous courses, intend to study calculus in high school, and only if the Algebra I courses are taught by teachers with mathematics certification.

41. 41 Fact #1 A

42. 42 Fact #2 B

43. 43 Fact #3 C

44. 44 Fact #4 D

45. 45 Fact #5 E

46. 46 Fact #6 F

47. 47 Fact #7 G

48. 48 Fact #8 H

49. 49 Fact #9 I

50. 50 What is this?

51. 51 What is this? F A C E

52. 52 What If? AB C DE F GH I

53. 53 Try Again

54. 54 Try Again D E C A D E

55. 55 What’s the Point? Isolated Facts –Less likely to retain information Connected Facts, Patterns, Fact in Context –More likely to retain information

56. 56 Characteristics of a good mathematics program CONCEPTUAL CONTEXTUAL CONSTUCTIVISM COMPUTATION TEST-PREP

57. 57 Thank You Dr. Eric Milou Rowan University milou@rowan.edu