The Role of Representations in and 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 . 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 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 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 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 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 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 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. subtraction. We strongly believe that we collected during another fraction the fraction circle model gave students study. We found that students had Students used 1/2 as a benchmark, the mental imagery for fractions that difficulty simply representing frac- compared unit fractions, and deter- supported their understanding. This tions with the pattern blocks and were mined amounts to add to 2/3 to make understanding provided a strong foun- unable to use them to add fractions. a whole. Students were able to do this because they constructed strong Fig. 3 Students’ errors with chip and dot paper models mental images for fractions. Based on student responses to estimation tasks like this one, we concluded:

Estimation and visualization are im- portant. These abilities will help students monitor their work when finding exact answers.

Figure 2 shows examples of sixth graders’ thinking on estimation tasks drawn from homework assigned before students were asked to find exact answers to fraction addition and subtraction problems. Students were asked if the solution given in each problem was a reasonable estimate. Notice that students coordinated several ordering ideas to judge if the answers to addition and subtraction tasks were reasonable. Their work with fraction circles helped them xxx judge reasonableness, because they were able to visualize the relative sizes of the fractions and combine them by figure 3 mathtypedot spacing mentally adding on or taking away. Students were able to visualize almost any fraction, regardless of whether 1 2 3 + = they had actually seen that fraction 4 3 7 with the fraction circle model. Note dot spacing

Vol. 13, No. 8, April 2008 ● Mathematics Teaching in the Middle School 493 Students’ reaction to the chip and dot amount shown on the black fraction in these examples. For some students, paper model was interesting in that circle. One student suggested that to recording with symbols meant using they used it to reinforce the incorrect get the exact answer, you need to have numbers. For others, it meant writing strategy of adding numerators and all the pieces the color of the smallest in words. denominators (see fig. 3). Notice that piece. We concretely exchanged the Students had many opportuni- the students did not use a common yellow piece (1/2) for 3 pinks (3/6). ties to connect their actions with the unit to show each fraction. Instead Students’ previous work with equiva- fraction circles to pictures of fraction they used a unit equal to the number lence ideas based on the fraction circle circles to symbols. Students’ writ- of parts in each denominator. They model supported this step of finding ten explanations were often messy, added the number of parts shaded and common denominators. Students which sometimes led to errors. After the total number of parts to calculate easily saw that 4/6 of the whole circle much discussion about, and guid- 3/7 as the answer. was covered, which became an “A ha!” ance from, Principles and Standards for Students who tried to model ad- moment for some. At this point, it School Mathematics (NCTM 2000), we dition with chips or dot paper did was tempting to jump right into the concluded that students’ own record- not realize the need to express both procedure but we held off. Finding ing systems would be more meaning- fractions as equivalent using the same a common denominator is critical to ful than a teacher directed one: “It is number of chips in the unit or the understanding how to add and subtract important to encourage students to same number of square units in the with symbols. To truly grasp this idea, represent their ideas in ways that make dot paper model. In other words, they a majority of students need extended sense to them, even if their first repre- did not realize the need to express periods of time with the fraction sentations are not conventional ones” both fractions as equivalent with a circles, determining for themselves the (NCTM 2000, p. 67). For example, common denominator. When consid- need to change the different pieces on many sixth-grade students used arrows ering what the students did with the the circle to those that are of the same to show the change from the given chips and dot paper to try to model color. Students solved many problems fraction to an equivalent form. This fraction addition, we realized that using the fraction circles to add and seems to be a reasonable way to show nothing is obvious in the model itself subtract before they made the connec- what is being done with the procedure. to show the need for finding common tion between exchanging circle pieces Figure 5 shows a few examples of how denominators. To add two fractions and finding common denominators students recorded their fraction addi- accurately using chips and dot paper, symbolically. We concluded: tion and subtraction. We concluded: students need to know already that they have to find a common denomi- Students need to experience acting out Making connections between concrete nator and realize how the denomina- addition and subtraction concretely with actions and symbols is an important tor relates to the final answer. an appropriate model before operating part of understanding. Students should We believe that students would with symbols. be encouraged to find their own way of understand common denominators if recording with symbols. they constructed for themselves the Moving from a concrete rep- need to find common denominators. resentation to a symbolic one is At some point, sixth-grade stu- The fraction circle model made this not as simple as it seems. Written dents need experiences with fractions need more obvious than other models language and verbal language play (denominators less than 20) at the we examined. Therefore, in our current important roles in helping students symbolic level. Students need to be teaching experiment, we began fraction translate from concrete to symbolic able to imagine fraction circles to help addition and subtraction with frac- representation. Students need time solve problems, but they must actu- tion circles and estimation. We asked to describe what they do with their ally calculate with symbols (numbers). students to show 1/2 (a yellow piece) concrete models before they are We found that even though almost plus 1/6 (a pink piece) on their frac- able to use symbols to meaningfully all students knew what to do to add tion circle unit (the black circle). We record their work. Figure 4 illustrates and subtract fractions symbolically, asked, “How much of the whole circle how sixth-grade students use written their ability to do so easily without was covered?” The discussion led to an language to describe what they did the concrete model depended on how estimate that the answer was obviously with the fraction circles. The idea of quickly they were able to recall their greater than 1/2 but less than 1. Stu- exchanging equivalent fractions to facts. Students who still counted on dents had difficulty naming the exact find common denominators is shown their fingers were at a disadvantage

494 Mathematics Teaching in the Middle School ● Vol. 13, No. 8, April 2008 Fig. 4 Connecting actions with circles and pictures to symbols Fig. 5 Students’ recording systems

Fig. 6 Student’s description of using a number line to add two fractions

when working only with symbols. We rely on students being able to under- found the following: stand the need for a common denomi- nator to be able to use these models. Students need easy recall of their multi- For example, to add 3/4 + 1/5 with plication and facts. chips, the student needs to under- stand that he or she has to represent Our past experience observing each fraction using the same number students using chips and dot paper of chips. Determining that number to add and subtract fractions and our is equivalent to finding the common new work employing a number line denominator. Modeling each fraction to add and subtract fractions led us to with that number of chips corresponds consider the power of connecting their to finding equivalent fractions with the newly learned procedure for adding common denominator. and subtracting fractions to a different In our current work, we used the model as a way to reinforce the pro- number line model for addition and cedure. As mentioned earlier, some of subtraction after students learned the models found in current textbooks to find a common denominator. We

Vol. 13, No. 8, April 2008 ● Mathematics Teaching in the Middle School 495 found that although it is an effective equivalent amount on the number are able to estimate answers to frac- model for reinforcing the procedure line is supported by a student’s ability tion addition and subtraction tasks for adding and subtracting fractions, to find equivalent fractions symboli- and judge reasonableness of answers it is less so for the introduction of cally. Finding the total amount on the to fraction operation tasks because the concept of common denomina- number line is the final step involving they have strong mental representa- tors. Once the idea of a common adding fractions now that each are tions for these numbers and are able denominator was developed with shown using common denominators. to manipulate them mentally. In our fraction circles, the number line was We believe the following: latest teaching experiment, we devel- introduced to reinforce it. Students oped lessons to help students build used their understanding of common Connecting the procedure to a new repre- meaning for finding exact answers denominators to make sense of the sentation may be an effective strategy to to fraction addition and subtraction number line as a model for adding reinforce the procedure. problems using common denomi- and subtracting fractions. Consider nators. We found that the fraction the student’s work in figure 6. Stu- SUMMARY circles vividly demonstrate the need dents were given a page of eight The concrete models we choose to for finding common denominators number lines partitioned differently: help students build meaning for frac- when adding and subtracting fractions halves, thirds, fourths, sixths, eighths, tions and operations are important. and that the fraction circles show the ninths, tenths, and twelfths. They In our work, we have determined steps to exchanging given fractions were asked to show how to add 2/3 that the fraction circle model sup- with equivalent ones with common and 2/9 on a number line. The deci- ports students’ understanding of the denominators. We will continue to sion as to which number line to use part-whole model for fractions and study how fraction circles and other to add the two fractions corresponds provides students with mental repre- models contribute to students’ mas- to finding the common denomina- sentations that enable them to judge tery of this difficult procedure. tor. Identifying each fraction as an the relative size of fractions. Students Bibliography Cramer, Kathleen, and Apryl Henry. “Using Manipulative Models to Build Number Sense for Addition and Frac- tions.” In Making Sense of Fractions, , and Proportions, 2002 Yearbook of the National Council of Teachers of Mathematics (NCTM), edited by Bonnie Litwiller and George Bright, pp. 41–48. Reston, VA: NCTM, 2002. Cramer, Kathleen, Thomas R. Post, and Robert C. delMas. “Initial Fraction Learning by Fourth- and Fifth-Grade Students: A Comparison of the Ef- fects of Using Commercial Curricula with the Effects of Using the Project Curriculum.” Journal for Research in 33 (March 2002): 111–44. National Council of Teachers of Math- ematics (NCTM). Principles and Stan- dards for School Mathematics. Reston, VA: 2000. Post, Thomas R., and Kathleen Cramer, “Children’s Strategies in Ordering Rational Numbers.” Teacher 35 (October 1987): 33–35. l

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