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# Using Manipulatives to Construct Mathematical Meaning

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1. 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

2. 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” NADE

3. 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

4. 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 NADE

5. 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

6. 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 NADE

8. 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 NADE

9. Representing and Comparing Fractions • Exploring relationship among the rods, including fractional relationships • Assigning fraction names to the rods • Using the rods to compare fractions NADE

10. Representing Fractions • Assign the number name 1 to the orange rod • What are the number names for all the other rods? NADE

11. 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 NADE

12. 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 NADE

13. 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

14. 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. NADE

15. 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 NADE

16. 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 NADE

17. 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

18. 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? NADE

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

20. Multiplying FractionsWhole Number · Mixed Number • Example: Use the rods to model 3 times 2 1/3 NADE

21. 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 NADE

22. 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 NADE

23. 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 NADE

24. 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 NADE

25. 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 NADE

26. 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 NADE

27. 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 NADE

28. 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? NADE

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