PASCAL

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PASCALS TRIANGLE AS ASSIMILATION
PARADIGMStandard Notation

• Elizabeth B. Uptegrove
• Felician College
• uptegrovee_at_felician.edu

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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

• 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 (ab)n
• These numbers are found in Pascals 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
  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 specific cases
• Towers
• Pizzas
• 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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Romina Links the Taxicab Problem to 4 choose 3
and the Towers Problem
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Brian and Romina Make Some Connections
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Interview with Mike (2002)
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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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Pizzas on The Triangle

• 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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Pizzas and Pascals Identity
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Interview (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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Romina Links Pizzas, Towers, and Binary Notation
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Interview (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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Ankur Explains Pascals Identity 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

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Questions?
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The Towers Solution

• 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

• 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 theAddition Rule? (1998)
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Can You Explain theAddition Rule? Yes!
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Add a Blue to the Second 10
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Pascals Triangle
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(No Transcript)
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(No Transcript)
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Taxicab-Towers Isomorphism