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From Concrete Representations to Abstract Symbols Elizabeth B. Uptegrove Carolyn A. Maher Rutgers University Graduate School of Education BACKGROUND • Students first investigated combinatorics tasks. – The towers problem – The pizza problem – The binomial coefficients • Students then learned standard notation. 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. Theoretical Framework • Students should learn standard notation. • Having a repertoire of personal representations can help. • Revisiting problems helps students refine their personal representations. Standard Notation • A standard notation provides a common language for communicating mathematically. • Appropriate notation helps students recognize the important features of a mathematical problem. 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. Research Questions • How do students develop an understanding of standard notation? • What is the role of personal representations? Data Sources • Videotapes – After-school problem-solving sessions (high school) – Individual task-based interviews (college) • Student work • Field notes Methodology • Summarize sessions • Code for critical events – Representations and notations – Sense-making strategies • Transcribe and verify 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? Combinatorics Notation n n C(n, r) Cr n Cr r • 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. 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. 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. 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. Pascal’s Identity (Student Version) N N N 1 X X 1 X 1 • 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. 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. 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. Taxicab Problem Diagram 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. Interview (Mike) r r 1 r 1 1 n n n • 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. 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. Interview (Ankur) 3 3 4 1 2 2 • 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. 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.