SCALE Middle School Math Forum

1. SCALE Middle School Math Forum December 11-12, 2005 Madison, Wisconsin Denver Public Schools Los Angeles Unified School District Madison Metropolitan School District Providence (RI) Public Schools California State University, Dominguez Hills California State University, Northridge University of Pittsburgh University of Wisconsin-Madison

2. Improving Middle School Student Math Proficiency What will make a difference?

3. (Teacher) In-service / Pre-service learning Curricula Standards / Assessment School / Community District / Government Teacher / Parent Classroom / Home Tonight we address middle school math teacher in-service learning Teacher In-service learning Student What are possible factors? For middle school math teachers: What you know you should understand

4. The views discussed tonight are informed primarily by: Our understanding of the relevant research literature Our understanding of mathematics 20 years of math PD work at CSUDH Five years of quilting/symmetry work in Madison The MMSD Math Masters courses The SCALE/QED Math Institutes

5. Math Explanation Structures Math Immersion Resources Math Immersion Concept paper Jackie Barab Matt Felton Eunice Krinsky Terry Millar

6. Now let us return to the…

7. Student Adding It Up: Helping Children Learn Mathematics* Mathematical Proficiency • conceptual understanding • procedural fluency • strategic competence • adaptive reasoning • productive disposition * National Research Council 2001

8. Adaptive Reasoning “Adaptive reasoning refers to the capacity to think logically about the relationships among concepts and situations. Such reasoning is correct and valid, stems from careful consideration of alternatives, and includes knowledge of how to justify the conclusions. In mathematics, adaptive reasoning is the glue that holds everything together, the lodestar that guides learning.” – Adding It Up

9. Teacher? Knowledge Understanding xxx

10. Hung-Hsi Wu: “The most difficult step in becoming a good teacher is to achieve a firm mastery of the mathematical contentknowledge. Without such a mastery, good pedagogy is impossible.” Liping Ma: “As I read this research, I kept thinking about the issue of teachers’ mathematical knowledge. Could it be that the “learning gap” was not limited to students? If so, there would be another explanation for U.S. students’ mathematical performance.” Tom Carpenter: “Teachers need flexible knowledge that they can adapt to their students and the demands of situations that arise in their classes. This kind of knowledge cannot be embedded in curriculum materials or scripted into instructional routines.”

11. Flavors of Knowledge Content Knowledge Pedagogical Knowledge Pedagogical Content Knowledge – Shulman Knowledgeof and about Mathematics – Ball Knowledge Packages – Ma Common and Specialized Content Knowledge – Ball

12. Knowledge vs Understanding Many people know that π is approximately 3.14 Fewer people understand that πis approximately 3.14 Typically neither middle school students nor teachers need to understand that πis approximately 3.14 But there are many things about π that they can understand

13. ) Circumference ( ) Diameter ( πis the ratio of the length of the circumferenceof any circle to the length of itsdiameter This is a definition and can be known and understood π= π=

14. ) ) Perimeter( Circum( Side( ) < ) Diameter( = 4 1 Middle school students can understand πis less than 4 & < π 4 = = =

15. Equilateral Triangle 6 2 Middle school students can understand πis greater than 3 > & = π = > = 3 =

16. Punch Line For middle school mathematics teachers: What youknowyou shouldunderstand. And this raises a question: What do you have when you have an understanding?

17. Perkins: What do you have when you have an understanding? An Explanation Structure “Itis a rich network of explanatory relationships that are encoded mentally in any of the many ways the mind has available – through words, images, cases in point, anecdotes, formal principles, and so on. This explanation structure is more than a memorized explanation: It is extensibleand revisable…” – Inside Understanding

18. Math Explanation Structures should help inform middle school math teacher (pre-service and in-service) learning through the use of networks of connected problems that we call Math Immersion Resources

19. Math Explanation Structures should haveconnective threadsthat mirror the cognitive contours of mathematics

20. Connective threads are themes that run through explanation structures connecting diverse elements. These connective threads can be developed through properly designed professional learning.

21. Example of connective threads in mathematics Similarity Sameness Equality Equivalence Congruence Isomorphism

22. Explanation structures and thus professional learning should have Extended Explanation Structure Explanation Structure both big ideas and connective threads.

23. Goal of examples this evening: Motivate a different long-term approach to middle school math teacher professional understanding that uses some math immersion resource-based professional development Tonight we will model this with immersion-likegames from a mathematical perspective involving adaptive reasoning and connective threads and building on variations of sameness

24. Cautionary Note! • These examples and the props are not themselves intended for middle school teacher mathematics professional development nor the classroom. • Rather, they are intended to provide a common conceptual space that is both unfamiliar and accessible to most of tonight’s diverse audience.

25. 2-Player Game:Sum of 15 Equipment: For each integer, 1 through 9 inclusive, one chip with that number 1 2 3 4 5 6 7 8 9 Play of the Game: Players alternate selecting chips from the above collection A win: A player whose set of chips has a subset of three chips whose numbers add to exactly 15 has a win.

26. 1 8 7 2 3 5 2 4 6 5 4 6 1 7 A win! 2-Player Game: Sum of 15 Example 1 2 3 4 5 6 7 8 9 Play Player 1 Player 2

27. 2-Player Game:Sum of 150 Equipment: For the multiples of ten, 10 through 90 inclusive, one chip with that number 10 20 30 40 50 60 70 80 90 Play of the Game: Players alternate selecting chips from the above collection A win: A player whose set of chips has a subset of three chips whose numbers add to exactly 150 has a win.

28. 10 1 20 2 30 3 4 40 5 50 6 60 7 70 80 8 90 9 20 50 80 2 5 8 Is the sum of 150 game the same as the sum of 15 game? The elements of 150 The correspondence The elements of 15 Math “Talk” An isomorphism is a correspondence between the elements of two structures that is structure preserving 20 + 50+ 80 10*(2 + 5+ 8) 10*15 150 15 2 + 5+ 8

29. The rest of this evening will be spent on Tic-Tac-Toe and variations How many “different” first moves are there? We will investigate other variations of “sameness” within a Tic-Tac-Toe framework.

30. Tic-Tac-Dud-V = only vertical wins Tic-Tac-Dud-H = only horizontal wins Variations on a Theme How interesting can Tic-Tac-Toe be??

31. Tic-Tac-Dull = vertical and horizontal wins Tic-Tac-Toe = include diagonals

32. Tic-Tac-Cylinder

33. Tic-Tac-Donut

34. Tic-Tac-Mobius

35. Math Immersion-type investigations Using tic-tac-X games • Investigations: • Beginning • Intermediate • Advanced • Really advanced!

36. Math Immersion-type investigations For Tic-Tac-X, X = Dud-V, Dud-H, Dull, Toe, Cylinder, Donut, Mobius: Beginning Investigations: • How many different wins are possible? • For each element (square) of the game, how many wins can that element be in? • Can the players cooperate so that each has an edge-through-edge win (for example, a horizontal win)? • Can the players cooperate so that each has a corner-through-corner win (for example, a diagonal win)?

37. Math Immersion-type investigations For Tic-Tac-X, X = Dud-V, Dud-H, Dull, Toe, Cylinder, Donut, Mobius: Intermediate Investigations: 5. Does either player have a winning strategy? 6. If the players cooperate, is a draw always possible? 7. Is it possible for two different elements (squares) to be in more than one win?

38. Math Immersion-type investigations For Tic-Tac-X, X = Dud-V, Dud-H, Dull, Toe, Cylinder, Donut, Mobius: Advanced Investigations: 8. How many final positions are there? 9. Which pairs of games are isomorphic? 10. If game A is isomorphic to game B, and game B is isomorphic to game C, then is it always the case that game A is isomorphic to game C? 11. Is the sum of 15 game isomorphic to tic-tac-toe?

39. Math Immersion-type investigations Really Advanced investigations: 12. How many different Tic-Tac-X games are there satisfying: • Can be played on a 3x3 tick-tac-toe surface • The collection of wins is closed under the symmetries • of the square • c. There is a topological space in which the 3x3 surface • can be embedded so that each win is a geodesic

40. Stereolithography Apparatus at the Milwaukee School of Engineering

41. Stereolithography 3D Systems, Inc., Valencia, CA 3D graphing calculator! Entire operation is driven by systems of (parametric) equations: Input: mathematics; Output: solid shape.

42. Stereolithography 1- laser 2- mirror 3- positioning mechanism 4- liquid polymer with photoinitiator 5- part Math in: Reality out:

43. 1 5 9 1 6 8 2 4 9 2 5 8 2 6 7 3 4 8 7 3 5 4 5 6 2-Player Game: Sum of 15 Analysis All possible wins: How many wins include 1? How many wins include 2?

44. 1 5 9 1 6 8 2 4 9 2 5 8 2 6 7 1 2 3 4 5 6 7 8 9 3 4 8 7 3 5 4 5 6 the structure is the collection of possible wins Tic-Tac-Toe: Elements are the spaces for marks the structure is the collection of possible wins Sum of 15: Elements are the chips

45. 2 9 4 7 5 3 6 1 8 Sum of 15 elements Tic-Tac-Toe elements 1 2 3 4 Isomorphism 5 6 7 The correspondence is structure preserving: Under this correspondence, every win in sum of 15 corresponds to a win in tic-tac-toe, and visa versa 8 9

46. Mathematical Proficiency Students ADAPTIVE REASONING Immersion-based PD Math Explanation Structures Teachers (Extensible and Revisable) Principles of Learning Mathematics Pedagogy

47. Schools of Education Schools Districts Math departments Psych departments Conclusion? Develop teacher math explanation structures by Developing math immersion resource-based PD by Developing new kinds of partnerships Explanation structure Immersion resource New partnerships