Ways to Think About Content-Based Professional Development: Adapting Existing Models and Materials to Meet Diverse Need - PowerPoint PPT Presentation

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  1. Ways to Think About Content-Based Professional Development:Adapting Existing Models and Materials to Meet Diverse Needs Steve Benson Mark Driscoll E. Paul Goldenberg sbenson@edc.orgmdriscoll@edc.orgpgoldenberg@edc.org Education Development Center Newton, Massachusetts

  2. What is Content-Based Professional Development (CBPD)? • Traditional model: Undergraduate or graduate level mathematics course taught within a Department of Mathematics, usually not specifically designed for preservice or inservice teachers • Three different models/vehicles: • Curriculum implementation • Analysis of student thinking • Connecting higher-level and school mathematics

  3. Plan for this session • Introduction • Difference of two squares • Illustrating the models through the activity: • Curriculum Implementation • Analysis of Student Thinking • Connecting higher-level and school mathematics • Discussion

  4. Learning by Doing, for Teachers Algebraic ideas from arithmeticNCSM — Anaheim, 2005 From a new comprehensive K-5 NSF program by EDC and Harcourt School Publishers A project of With support from www2.edc.org/mathworkshop

  5. To serve the teacher… To work, PD must improve life for the teacher. It must start where teachers are and acknowledge what they do know as well as what they might not know. • Connect directly with their practice • Capture time they already have • If it’s curriculum, it must Serve the children • Capture the adult’s curiosity • Meet the needs of the job of teaching tests • Allow easy start without independent PD

  6. …what could be less sexy than memorizing 4th grade multiplication facts?

  7. Just the facts • Start by knowing 4  4, 5  5, 6  6, 7  7, … • Have most others and easily work out what they don’t have memorized. • Goal now is to consolidate!

  8. What helps kids memorize multiplication facts? Something memorable!

  9. Surprise What is 6  6?

  10. Surprise 3635 What is 6  6? What is 5  7?

  11. Surprise What is 7  7?

  12. Surprise 4948 What is 7  7? What is 6  8?

  13. Surprise What is 8  8?

  14. Surprise 6463 What is 8  8? What is 7  9?

  15. Surprise What is 9  9?

  16. Surprise 8180 What is 9  9? What is 8  10?

  17. Is this always true?

  18. Is this always true? Is this number times itself always one more than

  19. Is this always true? Is this number times itself always one more than the product of these two numbers?

  20. But why does it work?!

  21. One way to look at it 5  5

  22. One way to look at it Removing a column leaves 5  4

  23. One way to look at it Replacing as a row leaves 6  4 with one left over.

  24. One way to look at it Removing the leftover leaves 6  4 showing that it is one less than 5 5.

  25. A second look Don’t bother counting! A square array.

  26. A second look Removing a column leavesit narrower by 1.

  27. A second look Replacing as a row leaves it narrower by 1 and taller by 1 (with 1 left over).

  28. A second look Removing the leftover shows that the new array contains one less dot than the square.

  29. What’s the gain? • An aid for remembering 6  8 or 7  9 • 7  7 = 49 • 6  8 = 48 • (6  8) = (7  7) - 1 Direct benefit!

  30. What’s the gain? • An aid for remembering 6  8 or 7  9 • 7  7 = 49 • 6  8 = 48 • (6  8) = (7  7) – 1 • A practical tool for (some) calculations • A hint at a BIG IDEA lurking Investment in the future!

  31. Further Investigation • In the process of taking this idea further, the children get more multiplication practice. • Is there a pattern that lets us use 7  7…

  32. Further Investigation • In the process of taking this idea further, the children get more multiplication practice. • Is there a pattern that lets us use 7  7 to derive 5  9?

  33. Experiment a moment Find a pattern that shows how 7  7 relates to 5  9…

  34. Experiment a moment …or how 8  8 relates to 6  10…

  35. Experiment a moment …or how 9  9 relates to 7  11…

  36. (7 – 1) (7 + 1) = 7  7– 1 • Or use 9 as an example • (9 – 1)  (9 + 1) = 9  9 – 1 • 8  10 = 81 – 1 n n – 1 n + 1

  37. (7 – 2) (7 + 2) = 7  7– 4 • Or use 8 as an example • (8 – 2)  (8 + 2) = 8  8 – 4 • 6  10 = 64 – 4 n n – 2 n + 2

  38. (7 – 3) (7 + 3) = 7  7– 9 • Or use 10 as an example • (10 – 3)  (10 + 3) = 10  10 – 9 • 7  13 = 100 – 9 n n – 3 n + 3

  39. Where does this lead?

  40. Where does this lead? To do… 53 47

  41. Where does this lead? To do… …I think… 53 3 more than 50 47

  42. Where does this lead? To do… …I think… 53 3 more than 50 47 3 less than 50 • 50  50 (well, 5  5 and …) … 2500 • Minus 3  3 – 9

  43. Where does this lead? To do… …I think… 53 3 more than 50 47 3 less than 50 • 50  50 (well, 5  5 and …) … 2500 • Minus 3  3 – 9 • 2491

  44. Why does it work? 50 53 47 3

  45. Thanks! Contact Information E. Paul Goldenberg pgoldenberg@edc.org www2.edc.org/mathworkshop

  46. Bye! Thanks! Contact Information E. Paul Goldenberg pgoldenberg@edc.org www2.edc.org/mathworkshop

  47. Fostering Algebraic Thinking Toolkit • Mathematics PD resources designed to help teachers in grades 6-10 learn to identify, describe, and foster algebraic thinking in their students. • Helping teachers to understand students' thinking through the analysis of different kinds of data, such as written student work, student transcripts, and classroom observation. • Instructional implications are also considered, from the perspective of an understanding of how algebraic thinking develops. • 54 hours of professional development, divided into four modules, each focused on a different type of classroom data.

  48. Toolkit Habits of Mind Framework • Doing-Undoing • Building Rules to Represent Functions • Abstracting from Computation

  49. Doing-Undoing The Doing: Divide 31 by 5. Divide 31 by 7. The Undoing: What numbers leave a remainder of 1 when divided by 5 and a remainder of 3 when divided by 7?

  50. A Toolkit Premise Examining classroom artifacts—e.g., student work--can afford fresh insights into mathematical content