From Concrete Representations to Abstract Symbols

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From Concrete Representationsto Abstract Symbols

• Elizabeth B. Uptegrove
• Carolyn A. Maher
• Rutgers University
• Graduate School of Education

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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

• Videotapes
• 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 (ab)n.

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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 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.

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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.

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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.

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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.

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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 ProblemDiagram
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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 (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.

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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.

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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.

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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.