STUDENTS BUILDING MATHEMATICAL CONNECTIONS THROUGH COMMUNICATION

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STUDENTS BUILDING MATHEMATICAL CONNECTIONS
THROUGH COMMUNICATION

• Elizabeth B. Uptegrove
• uptegrovee_at_felician.edu
• Carolyn A. Maher
• cmaher_at_rci.rutgers.edu
• Rutgers University
• Graduate School of Education

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Research Question
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Theoretical Framework

• Maher (1998)
• Communicating their ideas helps students to
  develop and consolidate mathematical thinking
• Justifying their thinking helps students develop
  mathematical reasoning skills
• Sfard (2001)
• Students learn to think mathematically by
  participating in discourse about ideas arguing,
  asking questions, and anticipating feedback

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Background

• Longitudinal study (Maher, 2005)
• Public school students were followed from first
  grade through college
• After-school problem-investigation session
  involving four students during the sophomore year
  of high school (March 1998)

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Data Sources and Analysis

• Data
• Videotapes (two cameras)
• Student work
• Analysis
• Transcripts verified and reviewed
• Events selected for analysis
• Student work on Pascals Triangle
• Mathematical ideas described and analyzed

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Problems Investigated

• Pizza How many pizzas is it possible to make
  when there are n toppings to choose from?
• Towers How many towers n cubes tall can be built
  when choosing from white and blue cubes?

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Students Earlier Findings

• The answer to both problems is 2n
• The answers to the towers problem can be
  enumerated by the numbers in row n of Pascals
  Triangle
• C(n,r) gives the number of towers with exactly r
  blue cubes when selecting from blue and white
  cubes
• They are not sure how the pizza problem fits with
  Pascals Triangle

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Results

• First, students communicated their ideas about
  relationships
• Between towers and Pascals Triangle and the
  binomial expansion
• Between pizza problem and Pascals Triangle
• Students provided support for these ideas
• They answered questions
• They responded to arguments
• Then they described how the towers and pizza
  problems are related to each other

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Episode 1 Towers

• Students represented (ab)2 by towers two cubes
  tall
• Students related those towers to row 2 of
  Pascals Triangle

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Episode 1 Towers
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Episode 1 Towers(continued)

• Building towers can be related to expanding the
  binomial (ab)
• Adding a blue cube is like multiplying by a
• Adding a white cube is like multiplying by b

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Episode 2 Pizzas
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Episode 2 Pizzas

• Students try to explain how the different
  two-topping pizzas can be found in row 2 of
  Pascals Triangle
• Row 2 is 1 2 1
• 1 plain pizza
• 2 one pepperoni, one pepper
• 1 both toppings
• But students do not identify this relationship
  instead Ankur proposes
• 1 pepperoni
• 2 both (counted twice)
• 1 peppers
• ? plain

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Episode 3 Connecting Towers Pizzas
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Episode 3 Connecting Towers Pizzas

• Michaels insight
• Blue cube topping present
• White cube topping not present
• The others built on this insight

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Analysis UsingConversational Turns

• Conversational turn units are tied sequences of
  utterances that constitute speakers turn at talk
  and at holding the floor (Powell, 2003)
• Used to measure students participation in
  discourse
• Coded for types of discourse
• Asking questions or expressing uncertainty
• Answering questions
• Disagreeing
• Making connections
• Expressing understanding

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Students Participationin Discourse
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Types of Discourse inConversational Turns
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Summary

• Episodes 1 and 2
• Students communicated their ideas about Pascals
  Triangle and its relationship to the towers and
  pizza problems
• Students supported their ideas through
  demonstrations and by offering specific examples
• Episode 3
• Students built on Michaels critical insight
• Students described a new connection the
  isomorphism between the towers and pizza problems

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Conclusions

• The framework for developing a new connection was
  provided by earlier discussions about Pascals
  Triangle
• Communicating about and supporting those earlier
  ideas helped students make this connection
  between two problems of equivalent structure

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Questions?
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Pascals Triangle
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Pascals Triangle
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Pascals Triangle