Using Manipulatives to Construct

					            Using Manipulatives to
            Construct Mathematical
                   Meaning
Suzanne Reynolds           Elizabeth Uptegrove
St. Thomas Aquinas College Felician College
sreynold@stac.edu          uptegrovee@felician.edu


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             Theoretical Framework
• Understanding can be instrumental
  (procedural) or relational (conceptual)
  – Skemp, 1976
• Manipulatives can help elementary
  students make sense of fractions
  – Steencken (2002); Reynolds (2005)
• When presented with rich mathematical
  experiences, college students can move
  beyond procedural understanding
  – Glass and Maher (2002)
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   Instrumental and Relational
         Understanding
• Instrumental (procedural) understanding
  – Knowing what to do (but not why)
  – Example: Dividing fractions
        • Invert the divisor and multiply
• Relational (conceptual) understanding
  – Knowing both what to do and why
  – Example: Dividing fractions
        • See results from “Ribbons and Bows”




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          Rationale for This Study
• Research has shown that Cuisenaire rods
  have helped elementary students make
  sense of fractions
• Our students have often been
  unsuccessful in performing basic
  operations on fractions
   – They know the procedures but not the reasons
     for the procedures
   – Hence, they often misremember the procedures
   – They are unable to recognize when an answer
     does not make sense
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 The Importance of Fractions
• Fractions are important in many areas of
  higher-level mathematics
    – Rate
    – Proportionality
    – Algebra
• When students develop conceptual
  understanding of fractions, they become
  more confident in their general
  mathematical ability
    – They can become less intimidated by other
      mathematical topics

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Our Students’ Characteristics
• Most students:
   – Relied on rules which were sometimes
     imperfectly recalled
   – Did not relate fraction problems to real
     situations
   – Did not recognize unreasonable answers
• College students made mistakes similar to
  children’s mistakes:
   – Adding numerators and denominators
   – Cross multiplying
   – Multiplying whole numbers and fractions
     separately

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           Cuisenaire® Rods
• Developed by Georges Cuisenaire (Belgian
  educator) in the 50s
• Focus is on the length of the rod, which is
  related to color
• The rods are versatile
   – There are no markings requiring specific
     divisions (e.g. 10ths)
   – A rod can be used to represent any rational
     number



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          Cuisenaire Rods




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 Students’ Work on Fractions
• Representing and comparing fractions
• Adding and subtracting fractions
• Multiplying fractions
  – Whole number · fraction
  – Mixed number · fraction
  – Fraction · fraction
• Dividing fractions
  – Whole number ÷ fraction
  – Fraction ÷ fraction
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              Representing and
             Comparing Fractions
• Exploring relationship among the
  rods, including fractional
  relationships
• Assigning fraction names to the rods
• Using the rods to compare fractions


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          Representing Fractions

• Assign the number name 1 to the
  orange rod
• What are the number names for all
  the other rods?




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            If the orange rod is 1…
• Working with the model:
  – The white rod is 1/10 because 10 whites = 1
    orange
  – The red rod is 1/5 because 5 reds = 1 orange
  – The yellow rod is 1/2 because 2 yellows = 1
    orange




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            If the orange rod is 1…
• Extrapolating from the model:
  – The concept of equivalent fractions emerges
  – Red = 2 whites = 2/10, lt. green = 3/10, purple
    = 4/10, … blue = 9/10




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           Comparing Fractions
• The question
  – Which is larger, 2/3 or 3/4?
  – By how much?
  – Demonstrate using a model
• The process
  – Assign the number name 1 to a selected
    rod or train of rods
  – Find rods that represent 2/3 and 3/4
  – Find the number name of the rod(s) that
    represent the difference

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  Which is larger, 2/3 or 3/4?
       By how much?




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          Common Denominator

• Comparisons can lead naturally to
  the concept of common denominator.
• Can students use the model to
  discern the meaning of common
  denominator?
• Usually, we have to tell them, or at
  least provide hints.


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            Finding Common
          Denominator Via Model
• The train representing 1 is 12 white
  rods long; 1 = 12/12
• The green rod representing 1/4 is 3
  white rods in length; 1/4 = 3/12
• The purple rod representing 1/3 is 4
  white rods in length; 1/3 = 4/12
• The difference is 1 white rod = 1/12

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            Subtracting Fractions
• Comparisons lead to the concept of
  difference (subtraction)
• But some students have a great deal of
  difficulty with word problems related to
  fraction minus fraction
  – Possibly, they never developed the concept
    of fraction as number (not operator)
  – We are still searching for ways to help
    students understand these operations


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 The Chocolate Bar Problem
• I had a chocolate bar. I gave 1/2 of
  the bar to Jason and 1/3 of the bar to
  John. What fraction of the chocolate
  bar did I have left?
• Use Cuisenaire rods to model your
  answer



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    A Chocolate Bar Solution




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            Subtracting Fractions
• What’s the difference between these
  two problems?
   – The problem we assigned
           • I have 1/2 of a cookie. I give 1/3 of a cookie to
             Bob. What fraction of a cookie do I have left?
   – The problem some students answered
           • I have 1/2 of a cookie. I give 1/3 of what I have
             to Bob. What fraction of what I started with do I
             have left?


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          Models for 1/2  1/3




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     Answering the question
           1/2  1/3




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          Multiplying Fractions
• Whole number times mixed number
• Mixed number times fraction
• Mixed number operations help
   develop notion of the distributive rule




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    Multiplying Fractions
Whole Number · Mixed Number
 • Example: Use
   the rods to
   model 3 times
   2 1/3




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            Multiplication: Mixed
           Number Times Fraction
• Use Cuisenaire rods to show 1 3/4 • 1/2
• Model 1: Make a model of 1 3/4 and find
  a rod that is half that length
• Model 2: Take half of 1 and half of 3/4
  – Illustrates the distributive rule
  – 1/2 (1 + 3/4) = 1/2 · 1 + 1/2 · 3/4




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      Multiplication -- Model 1




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          Multiplication -- Model 2




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           Division Problems
• Problems to develop the meaning of the
  division algorithm
   – Ribbons and bows
• Problems to show the difference between
  dividing by n and dividing by 1/n
   – What is 6 divided by 2?
   – What is 6 divided by 1/2?
• A problem to show the difference between
  multiplying by 1/n and dividing by 1/n
   – What is 1 3/4 divided by 1/2?
   – Compare to earlier multiplication problem


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              Ribbons and Bows
•    Short ribbons are 1 yard long
•    Middle-size ribbons are 2 yards long
•    Long ribbons are 3 yards long
•    Bows can be unit fractions in length
      – 1/2, 1/3, 1/4, 1/5 of a yard long
• Bows can be multiples of unit
  fractions in length
      – 2/3, 3/4 of a yard long


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How Many Bows? (Unit Fractions)
 • A short ribbon (1 yard long) makes:
    – 2 bows that are 1/2 yard long
    – 3 bows that are 1/3 yard long
    – n bows that are 1/n yards long
 • A middle-size ribbon (2 yards long) makes:
    – 4 bows that are 1/2 yard long
    – 2n bows that are 1/n yards long
 • A ribbon that is m yards long makes:
    – n · m bows that are 1/n yards long



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            How Many Bows?
           (Nonunit Fractions)
• If the ribbon is 2 yards long and the bow is
  1/3 of a yard long, you can make 2 · 3 = 6
  bows
• What if the bow is 2/3 of a yard long?
   – If the bow is twice as long, you can make half as
     many: 6  2 = 3 bows
• If the ribbon is n yards long, and the bow is
  2/3 of a yard long…
   – 3n gives the number of bows that are 1/3 of a
     yard long
   – If the bow is twice as long, you can make half as
     many: 3n  2 = number of bows
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              How Many Bows?
               (General Rule)
• n = Length of the ribbon
• k / m = Size of the bow
• n · m = How many bows of size 1/m
• Divide n · m by k to get the number of
  bows of size k/m
• Symbolically:
      – Number of bows = nm/k
• In words:
      – Invert and multiply
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Ribbons and Bows Illustrations




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    Models for 6 Divided by 2




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    Model for 6 divided by 1/2




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          Model for 1 3/4  1/2




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              Summary of Results
• Some students found the Cuisenaire rods
  useful
  – They used rods to visualize problems
  – They used rods to determine the reasonableness
    of their answers
  – They used rods to make sense of algorithms
           • But relating the rods to the symbols remained an issue
• Other students resisted using them
  – They preferred to practice computational fluency
  – They were not interested in making sense of the
    algorithms
  – They resisted using tools designed for children
           • Models are for those who can’t figure out the answer the
             “right” way

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                     Conclusions
• Cuisenaire rods can be helpful in some cases
  – We found them to be useful in assessing student
    comprehension
  – Models helped expose student thinking
  – The rods can help some students make sense of the
    standard algorithms
• It takes time and patience to achieve results
  – Overcoming some students’ resistance can be an
    issue
  – Some students might not find the rods useful
       • Different learning styles?



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           Future Directions
• Consider students’ learning styles
• The meaning of
   – Fraction as number
   – Common denominator
• Check for retention
   – At a later time
   – In other situations


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                                           REFERENC ES

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