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Concepts vs Skills: A Bogus Dichotomy

Concepts vs Skills: A Bogus Dichotomy. NCTM Regional Meeting Richmond, VA October 2007 Dr. Eric Milou Rowan University Department of Mathematics milou@rowan.edu 856-256-4500 x3876. Overview. National News in Mathematics Education Conceptual vs. Procedural Debate

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Concepts vs Skills: A Bogus Dichotomy

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  1. Concepts vs Skills: A Bogus Dichotomy NCTM Regional Meeting Richmond, VA October 2007 Dr. Eric Milou Rowan University Department of Mathematics milou@rowan.edu 856-256-4500 x3876

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

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

  6. Peanuts

  7. What’s “Typical?” in US

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

  9. Stated vs Developed

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

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

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

  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. 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. 1.49 1000 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. Computation is Important • Engaging & Active • Less passive worksheets • More thinking & reasoning

  18. Numbers Are Everywhere

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

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

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

  22. 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

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

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

  25. 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

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

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

  28. Contextual Problem Solving ≠ More Word Problems

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

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

  33. Math Essays?

  34. Fact #1 A

  35. Fact #2 B

  36. Fact #3 C

  37. Fact #4 D

  38. Fact #5 E

  39. Fact #6 F

  40. Fact #7 G

  41. Fact #8 H

  42. Fact #9 I

  43. What is this?

  44. What is this? F A C E

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

  46. Try Again

  47. Try Again D E C A D E

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

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

  50. Thank You Dr. Eric Milou Rowan University milou@rowan.edu http://www.rowan.edu/colleges/las/departments/math/facultystaff/milou/eric.html

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