From Concrete Representations to Abstract Symbols

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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.

				
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