Using Manipulatives to Construct Mathematical Meaning

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Using Manipulatives to Construct Mathematical
Meaning
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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
  childrens 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

• Whats 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 question1/2 ? 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 FractionsWhole 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 cant 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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Questions?Comments?Suggestions?