From Concrete Representations to Abstract Symbols

1. From Concrete Representationsto Abstract Symbols Elizabeth B. Uptegrove Carolyn A. Maher Rutgers University Graduate School of Education

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 • Videotapes • 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.

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 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 a specific case. • 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. 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.

21. Interview (Romina) • Explain standard notation in terms of towers, pizzas, and binary notation. • Explain addition rule in terms of towers, pizzas, and binary notation. • Explain taxicab problem in terms of towers.

22. Interview (Ankur) • Explain standard notation in terms of towers. • Explain specific instance of addition rule in terms of towers. • Explain general addition rule in terms of towers.

23. 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.