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Using Manipulatives to Construct Mathematical Meaning. 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)

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theoretical framework
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)

NADE

instrumental and relational understanding
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
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

NADE

the importance of fractions
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
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

NADE

cuisenaire rods
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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students work on fractions
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
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
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
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 113
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
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

NADE

common denominator
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
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
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
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

NADE

subtracting fractions21
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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multiplying fractions
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
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
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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division problems
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
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
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
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
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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summary of results
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
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
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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