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From Concrete Representations to Abstract Symbols

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Taxicab Problem ... First connect the taxicab problem to the towers problem in specific cases. ... Explain taxicab problem in terms of towers. Interview (Ankur) ... – PowerPoint PPT presentation

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Title: 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 (ab)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
    Pascals 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 Pascals Identity
  • First explain a particular row of Pascals
    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
Pascals Identity(Student Version)
  • N choose X represents pizzas with X toppings when
    there are N toppings to choose from.
  • N choose X1 represents pizzas with X1 toppings
    when there are N toppings to choose from.
  • N1 choose X1 represents pizzas with X1
    toppings when there are N1 toppings to choose
    from.

16
Pascals Identity(Student Explanation)
  • To the pizzas that have X toppings (selecting
    from N toppings), add the new topping.
  • To the pizzas that have X1 toppings (selecting
    from N toppings), do not add the new topping.
  • This gives all the possible pizzas that have X1
    toppings, when there are N1 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 Pascals 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.
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