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PASCAL’S TRIANGLE AS ASSIMILATION PARADIGM: Standard Notation

PASCAL’S TRIANGLE AS ASSIMILATION PARADIGM: Standard Notation. Elizabeth B. Uptegrove Felician College uptegrovee@felician.edu. Background. Students first investigated combinatorics tasks The towers problem The pizza problem The binomial coefficients

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PASCAL’S TRIANGLE AS ASSIMILATION PARADIGM: Standard Notation

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  1. PASCAL’S TRIANGLE AS ASSIMILATION PARADIGM:Standard Notation Elizabeth B. Uptegrove Felician College uptegrovee@felician.edu

  2. Background Students first investigated combinatorics tasks The towers problem The pizza problem The binomial coefficients Students then learned standard notation

  3. Objectives Examine strategies that students used to generalize their understanding of counting problems Examine strategies that students used to make sense of the standard notation

  4. Theoretical Framework • Students should learn standard notation • Having a repertoire of personal representations can help • Revisiting problems helps students refine their personal representations

  5. Standard Notation • A standard notation provides a common language for communicating mathematically • Appropriate notation helps students recognize the important features of a mathematical problem

  6. Repertoires of Representations • Existing representations are used to deal with new mathematical ideas • But if existing representations are taxed by new questions, students refine the representations • Representations become more symbolic as students revisit problems • Representations become tools to deal with reorganizing and expanding understanding

  7. Research Questions • How do students develop an understanding of standard notation? • What is the role of personal representations?

  8. Data Sources • From long-term longitudinal study • Videotapes of a group of 5 students • Ankur, Brian, Jeff, Michael, and Romina • After-school problem-solving sessions (high school) • Individual task-based interviews (college) • Student work • Field notes

  9. Methodology • Summarize sessions • Code for critical events • Representations and notations • Sense-making strategies • Transcribe and verify

  10. Combinatorics Problems • Towers -- How many towers n cubes tall is it possible to build when there are two colors of cubes to choose from? • Pizzas -- How many pizzas is it possible to make when there are n different toppings to choose from?

  11. Combinatorics Notation • C(n,r) is the number of combinations of n things taken r at a time • C(n,r) gives the number of towers n cubes tall containing exactly r cubes of one color • C(n,r) gives the number of pizzas containing exactly r toppings when there are n toppings to choose from • C(n,r) gives the coefficient of the rth term of the expansion of (a+b)n • These numbers are found in Pascal’s Triangle

  12. Students’ Strategies • Early elementary: Build towers and draw pictures of pizzas • Later elementary: Tree diagrams, letter codes, organized lists • High school: Tables and numerical codes; binary coding; organization by cases

  13. Results • Students used their understanding of the pizza and towers problems to make sense of combinatorics notation and of the numbers in Pascal’s Triangle • Students used this understanding to make sense of a related combinatorics problem • Students regenerated or extended their work in interviews two or three years later

  14. Generating Pascal’s Identity • First explain a particular row of Pascal’s Triangle in terms of pizzas • Then explain a general row in terms of pizzas • First explain the addition rule in specific cases • Towers • Pizzas • Then explain the addition rule in the general case

  15. Pascal’s Identity(Student Version) • N choose X represents pizzas with X toppings when there are N toppings to choose from • N choose X+1 represents pizzas with X+1 toppings when there are N toppings to choose from • N+1 choose X+1 represents pizzas with X+1 toppings when there are N+1 toppings to choose from

  16. Pascal’s Identity(Student Explanation) • To the pizzas that have X toppings (selecting from N toppings), add the new topping • To the pizzas that have X+1 toppings (selecting from N toppings), do not add the new topping • This gives all the possible pizzas that have X+1 toppings, when there are N+1 toppings to choose from

  17. Taxicab Problem • Find the number of shortest paths from the origin (at the top left of a rectangular grid) to various points on the grid • The only allowed moves are to the right and down • C(n,r) gives the number of shortest paths from the origin to a point n segments away, containing exactly r moves to the right

  18. Taxicab ProblemDiagram

  19. Taxicab Problem(Student Strategies) • First connect the taxicab problem to the towers problem in specific cases • Then form the connection in the general case • Finally, connect to the pizza problem

  20. Romina Links the Taxicab Problem to “4 choose 3” and the Towers Problem

  21. Brian and Romina Make Some Connections

  22. Interview with Mike (2002)

  23. Interview (Mike) • Recall how to relate Pascal’s Triangle to pizzas and standard notation • Call the row r and the position in the row n • Write the equation

  24. Pizzas on The Triangle • The nth row is for the n-topping pizza problem • The first number in each row is the “no topping” pizza • The last number in each row in the “all toppings” pizza • The rest of the numbers are for 1, 2, … toppings • Michael: “We see like the physical connection”

  25. Pizzas and Pascal’s Identity

  26. Interview (Romina) • She explained standard notation in terms of towers, pizzas, and binary notation • She explained the addition rule in terms of towers, pizzas, and binary notation • She explained the taxicab problem in terms of towers

  27. Romina Links Pizzas, Towers, and Binary Notation

  28. Interview (Ankur) • He explained standard notation in terms of towers • He explained specific instance of the addition rule in terms of towers • He explained the general addition rule in terms of towers

  29. Ankur Explains Pascal’s Identity in Terms of Towers

  30. Conclusions • Students learned new mathematics by building on familiar powerful representations • Students built up abstract concepts by working on concrete problems • Students recognized the isomorphic relationship among three problems with different surface features • Their understanding appears durable

  31. Questions?

  32. The Towers Solution • For each cube in the tower, there are two choices: blue or red • For a tower 4 cubes tall, the number of towers is 2222 = 24 = 16 • For a tower n cubes tall, the number of towers is 2n.

  33. The Pizza Solution • There are two choices for each topping: on or off the pizza. When there are four toppings, there are 2222 = 24 = 16 possible pizzas • When there are n possible toppings, there are 2n possible pizzas

  34. Can You Explain theAddition Rule? (1998)

  35. Can You Explain theAddition Rule? Yes!

  36. Add a Blue to the Second 10

  37. Pascal’s Triangle

  38. Taxicab-Towers Isomorphism

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