Conceptual Understanding and Basic Skills: Discussion Points for Teachers and Parents

1. Conceptual Understanding and Basic Skills: Discussion Points for Teachers and Parents NCSM 2008 Dr. Eric Milou Rowan University Department of Mathematics milou@rowan.edu 856-256-4500 x3876

2. Overview • National Math Panel Recommendations • Conceptual vs. Procedural Debate • Number Sense & Computation Proficiency

3. National Math Panel (NMP) • A focused, coherent progression of mathematics learning, with an emphasis on proficiency with key topics, should become the norm in elementary and middle school mathematics curricula. Any approach that continually revisits topics year after year without closure is to be avoided.

4. NMP • Instructional practice should be informed by high-quality research, when available, and by the best professional judgment and experience of accomplished classroom teachers. High-quality research does not support the contention that instruction should be either entirely “student centered” or “teacher directed.”

5. NMP • A major goal for K–8 mathematics education should be proficiency with fractions (including decimals, percents, and negative fractions).

6. Response (Gary Stager) • The Report of the National Mathematics Advisory Panel does not dispute that teachers spend lots of time teaching fractions. The report merely urges that teachers do even more of the same while hoping for a different result. A definition of insanity comes to mind. • It would be bad enough if wasted time was the only consequence of the fanatical fraction focus, but too many students get the idea that they can’t do math. This damages their inclination towards learning other forms of mathematics. Given the importance of mathematics and the widespread mathphobia sweeping the land, students can ill afford to a diminution in their self-image as capable mathematicians.

7. Response (Peanuts)

8. 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)

9. Compute the following: 4 x 9 x 25 How many ounces are in a gallon? 4 x 4 ÷ 4 x 4 30 ÷ 3/4

10. What’s “Typical?” in US

11. Third International Math & Science Study (TIMSS) Proceduresvs. Concepts

12. Stated vs Developed

13. We need a BALANCE • Balance • Direct Instruction • Constructivism • Balance • Conceptual Understanding • Algorithmic Proficiency • These are NOT Dichotomous

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

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

16. 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

17. Conceptual Fraction task • Kim’s teacher asked her class to design a flag using four colors, dividing a square into parts, and to color the parts as follows: • 1/2 is colored red • 1/4 is colored blue • 1/8 is colored green • Any other part is to be left white

18. Flags

19. Harder Task • A chocolate bar is separated into several equal pieces. • If Laura eats 1/4 of the pieces; and • Paul eats 1/2 of the remaining pieces; • There are six pieces left over • Into how many pieces was the original bar divided?

20. Chocolate Bar P A U L LAURA 16 pieces

21. 1.49 1000 Decimals • 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

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

23. Numbers Are Everywhere

24. Computational Practice Target #: 6 3 3 17 1 8

25. Active Computation • Fifty (1, 2, 3, 4, 5, 6 and addition) • Buzz (3) • Product Game

26. Conceptual & Contextual • 8 + 7 = ? • How do we teach this? x x x x x x x x x x x x x x x x x

27. 8 + 7 = ? 8 + 7 = ? 5 2 10 + 5 = 15

28. 17 - 8 = 0 17 / / 1 7 - 8 2 7 8 --> --> 10 --> --> --> --> --> --> --> 17

29. How Many Circles? 50

30. 1000 - 279 = ? 279 +1 = 280 + 20 = 300 +700 = 1000

31. 1000 - 279 = ? 1000 - 1 = 999 999 -278 721

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

33. Conceptual approach leads to ? • Algebra: (x + 3) (x + 7) = x 7 x 3 x2 7x 3x 21

34. Contextual Problem Solving ≠ More Word Problems

35. Example 1: Sneakers SECOND purchase Nike Converse Reebok F I R S T Nike Converse Reebok

36. N C R N C R 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

37. Example 2: Drug Steady State • 10 mg Zrytec daily • 50% of Zrytec is eliminated daily • What % is in the body • after 7 days • 10 days • n days?

38. 10mg Daily of Zrytec 10 1 15 2 17.5 3 18.75 4 5 19.375 6 19.6875

39. Fact #1 A

40. Fact #2 B

41. Fact #3 C

42. Fact #4 D

43. Fact #5 E

44. Fact #6 F

45. Fact #7 G

46. Fact #8 H

47. Fact #9 I

48. What is this?

49. What is this? F A C E

50. What If? C A B F D E I G H