The Role of Representations in Fraction Addition and Subtraction Kathleen Cramer, Terry Wyberg, and Seth Leavitt Why do so many middle school stu- (NCTM 2000, p. 67). Concrete mod- dents find fraction addition and sub- els are critical forms of representation traction difficult despite the fact that and are needed to support students’ they have studied this topic since third understanding of, and operations with, or fourth grade? The Rational Number fractions. Other important repre- Project (RNP) (Cramer and Henry sentations include pictures, contexts, 2002; Cramer, Post, and delMas 2002) students’ language, and symbols. Trans- with support from the National Sci- lating among all these representations ence Foundation is currently engaged makes ideas meaningful to students. win a teaching experiment with sixth We have found the fraction circle graders from a large urban district in model to be the most powerful con- the Midwest to address this question. crete representation for fractions. The Kathleen Cramer, [email protected], Representation plays an important circle model helps build understanding and Terry Wyberg, [email protected], role when students are learning about of the part-whole model for fractions are colleagues at the University of Min- fractions. “Representations should be and the meaning of the relative size nesota. Their research interests focus on the teaching and learning of rational num- treated as essential elements in sup- of fractions. Fraction circles are also a bers. Seth Leavitt, Seth.Leavitt@mpls porting students’ understanding of powerful model for fraction addition .k12.mn.us, is a teacher at Field Middle mathematical concepts and relation- and subtraction. In this article, we School in Minneapolis. He is interested in ships; in communicating mathematical reflect on what we believe is involved improving mathematics instruction in kin- approaches, arguments, and under- in developing a deep understanding dergarten through high school classrooms. standings to one’s self and to others” of fraction addition and subtraction 490 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL ● Vol. 13, No. 8, April 2008 Copyright © 2008 The National Council of Teachers of Mathematics, Inc. www.nctm.org. All rights reserved. This material may not be copied or distributed electronically or in any other format without written permission from NCTM. based on our previous work as well as Fig. 1 Fraction circles our current teaching experiment. In our current work with two classrooms of sixth graders, students used fraction circles to review ideas involving mod- eling fractions, comparing fractions, finding equivalent fractions, and acting out fraction addition and subtraction concretely without connecting to sym- Black Yellow Brown Blue bols. Students then worked through 2 pieces 3 pieces 4 pieces a series of eight additional lessons to develop the meaning of common denominators for adding and subtract- ing fractions, with the goal of being able to add and subtract fractions symbolically. In the rest of this article, we share our reflections on the role The Role of Representations in that representations and connections Orange Pink Light Blue Gray among representations play in helping 5 pieces 6 pieces 7 pieces 8 pieces students understand fraction addition and subtraction procedures. Fraction Addition and Subtraction In our early teaching experiments, we used a variety of different models (fraction circles, chips, paper folding, Cuisenaire rods) to build meaning for fractions. The RNP has found that the fraction circle model is the most effec- White Purple Red Green tive representation for building mental 9 pieces 10 pieces 12 pieces 15 pieces images for fractions. These mental images support the ability to judge the because they are both one away from of fractions by relative size to as- relative size of fractions, which is an a whole, but fifths are bigger pieces sess students’ understanding of the essential skill in fraction understand- than sixths so there is a bigger piece numbers they were to operate on. ing (Cramer and Henry 2002). Figure missing.” Notice that the student’s lan- Consider this task: 1 shows the fraction circles that guage is strongly related to the fraction students used in the RNP teaching ex- circle model; she is picturing pieces of Order from smallest to largest this periments. We continue to use fraction a circle and seeing that both fractions set of fractions: 3/4, 1/10, 5/12, circles in the current study. Fraction are one fraction piece away from the 3/5, 14/15. circles support students’ construction whole. Her mental images are strong of the inverse relationship between the enough to allow her to be able to visu- After correctly ordering the fractions size of the denominator and the size of alize that the size of each piece differs. from smallest to largest, 1/10, 5/12, the fractional piece. Students readily The ability to compare frac- 3/5, 3/4, 14/15, the student reasoned see that the more a circle is partitioned tions with benchmarks will support as follows: into equal-sized parts, the smaller students’ skill in finding reason- each part becomes. When asked to able estimates to fraction addition 1/10 is the smallest because it is pick which fraction is larger, 3/10 or and subtraction problems. Students farthest from the whole. I know 5/12 3/20 (assuming the same-sized unit), should be able to determine if a frac- isn’t half of twelve yet, so it is under one student said, “Well, 10 has like, tion is close to 0, greater than or less 1/2. Then 3/5. It is two away from a they are bigger pieces than 20 because than 1/2, or close to 1. They should whole number. Then 3/4 and 14/15 there are less pieces.” When asked to also understand what constitutes one because both are missing one piece compare 5/6 with 4/5, the student whole. In our lessons, we consistently but a fourth is a lot bigger than a fif- said, “[The fraction] 4/5 is smaller posed questions about the ordering teenth, so it is missing a bigger piece. Vol. 13, No. 8, April 2008 ● MATHEMATICS TEACHING IN THE MIDDLE SCHOOL 491 The student understood that 1/10 tion skills. The following question comparing unit fractions. As you see was quite small. He recognized that was posed to the sixth graders within in the students’ answers, knowing that 5/12 is less than 1/2 and probably also individual interviews after two weeks 1/5 is less than 1/3 gave students the knew that 3/5 was greater than 1/2. of instruction: information needed to conclude that He compared two fractions close to 1 the total amount of flour must be less by accurately picturing the size of the Marty made two types of cookies. than 1 cup.) missing piece. Based on our experi- He used 1/5 cup of flour for one Here are some student responses, ences, we concluded: recipe and 2/3 cup of flour for the with additional questions placed in other. How much flour did he use in brackets: Before operating with fractions, students all? Is it greater than 1/2 cup or less need to understand what a fraction means. than 1/2 cup? Is the amount greater He used more than 1/2 cup because This involves understanding the part- than 1 cup or less than 1 cup? 2/3 is more than 1/2. He used less whole model for fractions and the ability to than 1 cup. Because 1/5 is less than judge the relative size of a fraction. (Although 1/5 cup is not as realis- 1/2. [Why would that be enough tic a measure as 2/3 cup of flour, we to know that the answer is less than The ability to order fractions plays used that value to encourage students 1?] Because 1/5 is 1 out of 5 and so an important role in students’ estima- to apply ordering ideas related to it’s smaller pieces. [What would you Fig. 2 Sixth-grade students’ thinking on homework assignments about estimation (a) (b) (c) (d) (e) (f) (g) (h) 492 MATHEMATICS TEACHING IN THE MIDDLE SCHOOL ● Vol. 13, No. 8, April 2008 need to add to 2/3 to equal 1 cup?] the student’s response in figure 2h. dation on which to build the concept 1/3; 1/3 is bigger than 1/5. Although there was no fraction circle of a common denominator for adding divided into hundredths, he knew that and subtracting fractions. It is already more than 1/2 because 2/3 hundredths were very small pieces, In planning the lessons for this lat- is greater than 1/2. It’s going to be less and he only had two of them, so he est teaching experiment, we consid- than 1 cup because 1/5 is not equal to explained that “2/100 is pretty much ered which representation might best 1/3 so it won’t cover that space. nothing.” demonstrate the common denomina- We were impressed with the depth tor procedure for adding and sub- More than 1/2 cup because 2/3 is of understanding that this group of tracting fractions. We looked at other already bigger than 1/2 so plus 1/5 sixth graders showed regarding frac- models, such as pattern blocks, chips, would be over 1/2. It’s less than 1 tion meaning, their ability to judge the and dot paper, which had been used cup. Because 2/3, 1/5 doesn’t fill up relative sizes of fractions using mul- by other NSF-supported curricula. the missing piece in 2/3. It just fills tiple strategies, and their use of order We looked closely at the positives and up a little pieces of it so it wouldn’t ideas to estimate fraction addition and negatives of these models using data equal a whole.