# From Concrete Representations to Abstract Symbols by xnWgjn5

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```									   From Concrete
Representations
to Abstract Symbols
Elizabeth B. Uptegrove
Carolyn A. Maher
Rutgers University
BACKGROUND
• Students first investigated
– 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)
• 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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