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Using Manipulatives to Construct Mathematical Meaning Suzanne Reynolds Elizabeth Uptegrove St. Thomas Aquinas College Felician College sreynold@stac.edu uptegrovee@felician.edu 2/28/08 NADE 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) 2/28/08 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” 2/28/08 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 2/28/08 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 2/28/08 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 2/28/08 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 2/28/08 NADE 7 Cuisenaire Rods 2/28/08 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 2/28/08 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 2/28/08 NADE 10 Representing Fractions • Assign the number name 1 to the orange rod • What are the number names for all the other rods? 2/28/08 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 2/28/08 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 2/28/08 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 2/28/08 NADE 14 Which is larger, 2/3 or 3/4? By how much? 2/28/08 NADE 15 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. 2/28/08 NADE 16 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 2/28/08 NADE 17 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 2/28/08 NADE 18 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 2/28/08 NADE 19 A Chocolate Bar Solution 2/28/08 NADE 20 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? 2/28/08 NADE 21 Models for 1/2 1/3 2/28/08 NADE 22 Answering the question 1/2 1/3 2/28/08 NADE 23 Multiplying Fractions • Whole number times mixed number • Mixed number times fraction • Mixed number operations help develop notion of the distributive rule 2/28/08 NADE 24 Multiplying Fractions Whole Number · Mixed Number • Example: Use the rods to model 3 times 2 1/3 2/28/08 NADE 25 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 2/28/08 NADE 26 Multiplication -- Model 1 2/28/08 NADE 27 Multiplication -- Model 2 2/28/08 NADE 28 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 2/28/08 NADE 29 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 2/28/08 NADE 30 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 2/28/08 NADE 31 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 2/28/08 NADE 32 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 2/28/08 NADE 33 Ribbons and Bows Illustrations 2/28/08 NADE 34 Models for 6 Divided by 2 2/28/08 NADE 35 Model for 6 divided by 1/2 2/28/08 NADE 36 Model for 1 3/4 1/2 2/28/08 NADE 37 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 2/28/08 NADE 38 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? 2/28/08 NADE 39 Future Directions • Consider students’ learning styles • The meaning of – Fraction as number – Common denominator • Check for retention – At a later time – In other situations 2/28/08 NADE 40 REFERENC ES Davis, R. (1984). Learning Mathematics: The Cognitive Science Approach to Mathematics Education. Norwood, NJ: Ablex Publishing. Davis, R. (1992). Understanding Ō Understanding.Õ Journal of Mathematical Behavior 11, 225-241. Glass, B. & Maher, C. (2002). Comparing represe ntations and reasoning in young children with two- year college students. In A. Cockburn & E. Nardi (Eds.), Proceedings of the 26th Annual Meeting for the International Group for the Psychology of Mathematics Education, 3 (pp. 1-8). Norwich, UK. Kamii, C., Lewis, B. & Kirkland, L. (2001). Manipulatives: When are they useful? Journal of Mathematical Behavior, 20, 21-31. Lamon, S. (1999). Teaching Fractions and Ratios for Understanding. Mahwah, NJ: Lawrenc e Erlbaum Associates. Reynolds, S. (2005). A study of fourth-grade students' explorations into comparing fractions. (Doctoral dissertation, Rutgers, The State University of New Jersey, 2005) Dissertation Abstracts International, 66/04.p.1305. AAT3171003 Reynolds, S. & Uptegrove, E. (2006). Using manipulatives to teach students in college developmental math classes about fractions. In J. Novotn‡ ,H. Moraov‡ , M. Kr‡ tk‡ , & N. Stehlikov‡ (Eds.), Proceedings of the 30th Conference of the International Group for the Psychology of Mathematics Education (Vol. 1, p. 319). Prague, Czech Republic. Skemp, R. (2002). Instrumental understanding and relational understanding. In D. Tall & M. Thomas (Eds.), Intelligence, Learning and Understanding in Mathematics: A Tribute to Richard Skemp (pp. 131-150). Flaxton, Australia: PostPressed. (Original work published 1976) Steencken, E. (2002). Explorations to Build Meaning about Fractions. New Brunswick, NJ: Robert B. Davis Institute for Learning, Rutgers The State University of New Jersey 2/28/08 NADE 41 Questions? Comments? Suggestions? 2/28/08 NADE 42