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How Do I Balance Computational and Conceptual Understanding?

How Do I Balance Computational and Conceptual Understanding?. Dr. Eric Milou Rowan University Department of Mathematics milou@rowan.edu 856-256-4500 x3876. Overview. National News in Mathematics Education NJ mathematics assessments Conceptual vs. Procedural Debate

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How Do I Balance Computational and Conceptual Understanding?

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  1. How Do I Balance Computational and Conceptual Understanding? Dr. Eric Milou Rowan University Department of Mathematics milou@rowan.edu 856-256-4500 x3876

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

  3. NCTM Focal Points (9/12/06) • Not Back to Basics at All • 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.

  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. 2006 State Testing Results

  6. 2006 NJASK 5, 6, 7 • 30 MC and 3 Open Ended: 39 points • Calculator allowed on ALL questions • 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) • Sample Items at: http://www.nj.gov/njded/assessment/ms/5-7/

  7. 2006 NJ GEPA Data • 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

  8. Assessments Points by Cluster

  9. Assessments Points by Cluster

  10. 200 Score

  11. 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?

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

  13. What’s “Typical?” in US

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

  15. Stated vs Developed

  16. 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

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

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

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

  20. 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

  21. 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

  22. 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

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

  24. Name That Number - 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 • Wipe Out • Software: Math Arena

  26. 0 9 1 8 2 7 3 6 4 5 Patterns

  27. 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

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

  29. 1000 - 279 = ? 279 +1 = 280 + 20 = 300 +700 = 1000

  30. 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

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

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

  33. 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?

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

  35. 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?

  36. Fact #1 A

  37. Fact #2 B

  38. Fact #3 C

  39. Fact #4 D

  40. Fact #5 E

  41. Fact #6 F

  42. Fact #7 G

  43. Fact #8 H

  44. Fact #9 I

  45. What is this?

  46. What is this? F A C E

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

  48. Try Again

  49. Try Again D E C A D E

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

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