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PASCAL’S TRIANGLE AS ASSIMILATION PARADIGM: Standard Notation. Elizabeth B. Uptegrove Felician College [email protected] 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 l.jpg

PASCAL’S TRIANGLE AS ASSIMILATION PARADIGM:Standard Notation

Elizabeth B. Uptegrove

Felician College

[email protected]


Background l.jpg
Background

Students first investigated combinatorics tasks

The towers problem

The pizza problem

The binomial coefficients

Students then learned standard notation


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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


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Theoretical Framework

  • Students should learn standard notation

  • Having a repertoire of personal representations can help

  • Revisiting problems helps students refine their personal representations


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Standard Notation

  • A standard notation provides a common language for communicating mathematically

  • Appropriate notation helps students recognize the important features of a mathematical problem


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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


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Research Questions

  • How do students develop an understanding of standard notation?

  • What is the role of personal representations?


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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


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Methodology

  • Summarize sessions

  • Code for critical events

    • Representations and notations

    • Sense-making strategies

  • Transcribe and verify


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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?


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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


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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


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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


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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


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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


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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


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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



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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





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Interview Towers Problem(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


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Pizzas on The Triangle Towers Problem

  • 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”



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Interview Towers Problem(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



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Interview Towers Problem(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



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Conclusions Towers Problem

  • 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


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Questions? Towers Problem


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The Towers Solution Towers Problem

  • 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.


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The Pizza Solution Towers Problem

  • 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


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Can You Explain the Towers ProblemAddition Rule? (1998)


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Can You Explain the Towers ProblemAddition Rule? Yes!



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Pascal’s Triangle Towers Problem



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