Fractions: A ProblemSolving Approach


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1 Fractions: A ProblemSolving Approach DeAnn Huinker, University of WisconsinMilwaukee (Submitted for publication in the Wisconsin Teacher of Mathematics, March 2010.) "What's a fraction?" Students, unfortunately, too often reply, "That's when you have a line with a number on top and a number on the bottom" rather than associating it with some quantity in everyday life that can be represented mathematically with symbols. The connection between realworld experiences and symbols is essential in enabling students with the power to make sense of fractions. Other conceptual knowledge connections needed by students to develop number sense and operation sense for fractions are shown in figure 1. These include connections among realworld experiences, concrete models and diagrams, oral language, and symbols. Mathematical Concept Symbolic Representation Concrete/Pictorial Representation RealWorld Representation Figure 1. Conceptual Knowledge Connections Students who are able to make these connections have demonstrated lasting ability to flexibly use their mathematical knowledge, both conceptual and procedural, to solve word problems. Towsley (1989) found that fourth grade students who had established these connections reasoned with fractions as quantities and not as two whole numbers when solving word problems. Other educators and researchers have also noted the importance of helping students make connections among various mathematical representations (Cramer & Bezuk 1991; Ellerbruch & Payne 1978; Lesh, Post, & Behr 1987). Students without connections among realworld experiences, concrete models and diagrams, oral language, and symbols lack the power to make sense of mathematics and do not see its usefulness in the world around them. They often have no choice but to rely on immature strategies such as key words or guessing in their attempts to solve problems. An instructional program for fractions that utilized a problemsolving approach as it emphasized these connections to develop fraction number sense and operation sense was implemented with a class of fifth grade students in a large urban school system. This article discusses the intended outcomes, the instructional sequence and activities, and the assessment of student learning. Intended Instructional Outcomes What fraction knowledge do we want fifth grade students to develop? By the end of the instructional program, it was hoped that students would: 1. Approach fraction word problems with confidence. 2. Use a variety of problem solving strategies to solve fraction word problems, such as acting out the situation with paper strips or by drawing diagrams, using invented symbolic procedures on paper and pencil, and visualizing actions on objects (mental logical reasoning). Fractions: A Problem Solving Approach (Huinker, 2010) 1
2 3. Demonstrate fraction number sense and operation sense by: (a) making connections among realworld, concrete/pictorial, oral language, and symbolic representations for all four operations; (b) seeing relationships among all four operations; (c) acquiring insight into the effects of an operation on a pair of fractions or a fraction and a whole number; and (d) realizing that a specific amount can have many names. 4. Communicate their fraction knowledge using oral, written, concrete, pictorial, and symbolic methods. Instructional Sequence and Activities The instructional activities encouraged students to investigate fractions through informal explorations. The experiences focused on the use of paper strips as concrete models and problem situations. The paper strips were used throughout all lessons. Too often manipulatives are used for a day or two and are then quickly put aside. This is not adequate time for students to construct understanding. The students were also encouraged to come up with their own methods for solving problems and to communicate their reasoning by sharing their approaches with each other. The instructional sequence for the initial fraction work and for addition and subtraction was modified from that presented by Ellerbruch and Payne (1978), Payne and Towsley (1990), and Payne, Towsley and Huinker (1990). Instructional ideas for multiplication and division were modified from Cramer and Bezuk (1991), Ott, Snook and Gibson (1991), Pothier and Sawado (1990) and Sweetland (1984). Oral Language and Comparisons. Each student was given a fraction kit which consisted of ten strips that measured 2 inches by 8.5 inches. These were their paper candy bars. Several days were spent folding the paper into fractions strips for halves, fourths, eighths, thirds, sixths, twelfths, fifths, and tenths. The other two strips remained wholes. The students discovered various ways to fold the strips into equalsized pieces and shared their strategies with each other. Throughout the days, the students were asked to show and compare many quantities with their fraction strips as they informally dealt with proper and improper fractions and mixed numbers. For example, Show me 1 fourth, 2 fourths, 3 fourths, 4 fourths, 5 fourths, 8 fourths. The students then worked in pairs to compare fractions presented as verbal or written oral language, such as "Which would you rather have 5 eighths of a candy bar or 5 sixths of a candy bar?" No fraction symbols were used for several days of instruction in order to establish a stronger connection between the concept and concrete representation. Each student would select one of the two fractions that were to be compared, show it with their fraction strips, and then check with their partner to see who had the larger fraction. Students found fractions that were larger than, smaller than, and equal to one whole candy bar. They also found that different fractions named the same amount, or were equivalent. When students were able to confidently use oral language to display and compare fractions and mixed numbers, it was time to introduce fraction symbols. Fraction Symbols and Comparisons. All students had previously worked with fraction symbols, but their use was deliberately delayed until students demonstrated some understanding of fraction quantities. Students used written oral names, such as 5 sixths in their written work and then the symbol, 5, was introduced and discussed as a short way to write it. The meaning of the top number was 6 interpreted as the number of parts and the bottom number as the size of the parts. Students now used fraction symbols along with oral language to display and compare fractions. It was interesting that some students continued to use written oral language rather than the symbols or used both. One student explained that she did this because the words were less confusing than the symbols. In this transition to using fraction symbols, students used their fraction strips to engage in the activity, "Who has the largest fraction?" This is version of highest card that uses a deck of fraction symbol Fractions: A Problem Solving Approach (Huinker, 2010) 2
3 cards. In another activity students were challenged to find fractions that were larger than, smaller than, and equivalent to one whole candy bar and to write about their discoveries. One purpose of this activity was to have students consider fractions such as 3 thirds or 3 3 and 8 eighths or 8. It is rare that students 8 have discussions or even see fractions in which the top number and the bottom number are the same. The understanding that this is equivalent to one whole is important for students to discuss. This means they have enough of the same sized parts to make a whole candy bar. It is also rare for students to explore fractions greater than one whole during initial fraction instruction. This is unfortunate as it leads to misconceptions. Building from the oral language and concrete experiences, students can readily understand and explain that a fraction such as 8 means that you have 8 of the samesized parts, that you 5 have more than enough to make a whole candy bar, in fact you have enough to make one whole candy bar and then still have 3 fifths of another candy bar. To extend the ideas to fraction symbols not restricted to their fraction strips, students went hunting for equivalent fractions. First they found all the names for one half among their fraction strips and then students invented many more, such as , 80,, and. Next the students went hunting for other names for 1 third, 2 thirds, 1 fourth, and 3 fourths. Another activity which focused on equivalent fractions, as well as informally explored addition, challenged students to work in pairs to find and record different ways to build one whole. This was a favorite exploration of the students. Paula and Brandy worked together and find the ways shown in Figure 2. Figure 2. Finding Ways to Make One Whole Fractions: A Problem Solving Approach (Huinker, 2010) 3
4 Addition and Subtraction. More structured work with addition and subtraction began by posing word problems for the students to act out with their fraction strips. Both addition and subtraction problems with like fractions and related fractions were investigated on the first day. Beginning only with like fractions is a source of student misconceptions and interference with later learning. Students should experience like and related fractions interchangeably from the first day of fraction instruction. Then they learn that you do not have different rules but rather like fractions are only a special case of adding fractions. For example, "Let's pretend that Shawn had 1 half of a candy bar and then Latarre gave him 3 fourths of another candy bar. How much would Shawn have then?" Some students lined up the fractions, compared them with a whole, and realized that the answer was 1 whole candy bar and 1 fourth of another. Other students reasoned that 1 half is the same as 2 fourths, so they traded in the half for fourths which resulted in an answer of 5 fourths. Whereas other students said that they could not solve the problem because the pieces were not the same size. After listening to the thinking and reasoning of their classmates, these students gained insight and were eager to try the next problem. Number sentences using written oral language were used to record the results of each problem at this initial stage of development in adding and subtracting fractions. For example, the problem was written on the board as 1 half + 3 fourths. Then after doing some trading it was rewritten as 2 fourths + 3 fourths = 5 fourths = 1 whole and 1 fourth. This use of oral language promotes thinking of fractions as quantities and discourages students from merely adding numerators and denominators. When students were successful with trading in or renaming fractions to get same size pieces, a connection was made from the written oral language to the fraction symbols. The students began using a cross out technique as shown in Figure 3. To show that they had made a trade or renamed a fraction, they merely crossed it out and wrote the equivalent fraction next to it. Some of the students had remembered how they had been shown to write fraction addition problems vertically in the previous grade. The class talked about this and it was unanimous that it makes much more sense to just write the equation horizontally and to cross out the fraction when you rename it. Figure 3. Adding Fractions by Renaming Now that the students were familiar with fraction symbols and problem situations, they were asked to pose their own word problems as shown in Figure 3. Often times they were given a specific number sentence for which they were to pose a problem. This strengthens the connection from symbols to real life experiences. Once they had posed a problem, they would find the solution by using their fraction strips. These problems often became the basis for further exploration by other students in future lessons. The problems were also compiled into a fraction word problem book. Multiplication and Division. The instructional unit also included several lessons focused on the exploration of multiplication and division of fractions through problem situations. Word problems were Fractions: A Problem Solving Approach (Huinker, 2010) 4
5 posed for students to act out with their fraction strips, and then they shared their approaches and reasoning with each other. After a word problem was solved, the mathematical operation was identified and symbols were used to write a number sentence to record the relationships among the numbers in the problem. For example, it the story involved combining equal parts then a multiplication equation was written. If the story involved taking an amount and separating it into equal parts, then a division equation was written. Students also drew diagrams as a written record of their solutions as shown in Figure 4. If It is important to note that all problems were presented to the students in context. The increased accessibility to the mathematical ideas in that students could use their fraction strips to act out the situations. The symbols just became a written record of the story context and the solution. Figure 4. Student Solutions to Multiplication and Division Problem Situations Students interchangeably investigated multiplication of a whole number and a fraction, such as 3 x 2 3 ; division of a whole number by a fraction, ; division of a fraction by a fraction ; and division of a mixed number by a whole number, Students were also asked to pose their own 2 multiplication and division word problems which were compiled in a fraction word problem book that was put into the class library. Assessment of Student Learning Journals. Each student had a Fraction Journal which was written in daily. The students were sometimes asked to compare fractions and explain their reasoning in writing or with diagrams in their journals. The students pose their own word problem for a specific number sentence and wrote it in their journal, as well as drew diagrams or used symbols to show and explain how to solve the problem. At other times the students were just asked to write down something that they had learned that day. The journal allowed for ongoing assessment of student learning throughout the instructional program, as well as selfassessment by the students. For example, one day Mike wrote, I don't understand the mixed fraction and it s confusing me. So could you help me? Interviews. The students were individually interviewed before and after the instructional program. They were asked to compare fractions, perform fraction computations, and solve fraction word problems. All but one student initially compared fractions based on the size of the individual numbers. They thought 1 6 was larger than 1 2 because six was larger than two. Only one of the 28 students initially recalled that fractions needed a common denominator before adding or subtracting them. All the other students stated that the answer to was 4 6 and the answer to was 4 5. The students were allowed to use paper circles or paper strips in the initial interview, but only a few choose to do so with futile attempts. Fractions: A Problem Solving Approach (Huinker, 2010) 5
6 During the followup interviews, all students dealt with the fraction symbols as representing some quantity and did not merely manipulate the symbols without meaning. Students either used the fraction strips or drew diagrams to help them solve the problems or they reasoned through the solving of the problem by mentally visualizing the fractions. All students solved the comparison problems correctly, almost all successfully solved the addition and subtraction word problems, and many students also solved the multiplication and division problems. Paperandpencil Test. The students were given a paperandpencil pretest and posttest. These test results further substantiated the results of the interviews. Before instruction, almost all students added or subtracted both numerators and denominators and compared fractions based on the individual numbers. Whereas after instruction, only a few students resorted to meaningless symbol manipulations. This class of fifth grade students were immersed in fraction instruction for almost four weeks for about an hour each day. The results show that a focus on problem solving and making connections among representations build a strong and solid foundation for making sense of fractions. Conclusions These students developed intuitive quantitative understandings of fractions which they used to solve problems, as well as to pose problems. The role of contexts were essential for student learning. Problem contexts were used each and every day of instruction from initial concept development through division of fractions. The challenge of students posing their own word problems and selecting their own contexts was also a regular and essential part of instruction and learning. Students can experience success with fractions if they are given opportunities to investigate and explore connections among realworld experiences, concrete models and diagrams, oral language, and symbols, to communicate their findings, and to construct their understandings. Many of the students commented, I learned that fractions are easy. The two students who summed it up best were Wallace who stated, I learned that fractions are the best because it's important to use fractions all of our life and Shannon who said, I learned that you can do mostly anything with fractions. Resources Cramer, K., & Bezuk, N. (1991). Multiplication of fractions: Teaching for understanding. Arithmetic Teacher 39, Ellerbruch, L. W., & Payne, J. N. (1978). A Teaching sequence from initial fraction concepts through the addition of unlike fractions. In M. Suydam (Ed.), Developing computational skills (pp ). Reston, VA: National Council of Teachers of Mathematics. Lesh, R., Post, T., & Behr, M. (1987). Representations and translations among representations in mathematics learning and problem solving. In C. Janvier (Ed.), Problems of Representation in the Teaching and Learning of Mathematics (pp ). Hillsdale, NJ: Lawrence Erlbaum. Ott, J. M., Snook, D. L., & Gibson, D. L. (1991). Understanding partitive division of fractions. Arithmetic Teacher 39, Payne, J. N., & Towsley, A. E. (1990). Implications of NCTM's standards for teaching fractions and decimals. Arithmetic Teacher 37, Payne, J. N., Towsley, A. E., & Huinker, D. M. (1990). Fractions and decimals. In J. N. Payne (Ed.), Mathematics for the Young Child (pp ). Reston, VA: National Council of Teachers of Mathematics. Pothier, Y., & Sawado, D. (1990). Partitioning: An approach to fractions. Arithmetic Teacher 38, Sweetland, R. (1984). Understanding multiplication of fractions. Arithmetic Teacher 32, Towsley, A. (1989). The use of conceptual and procedural knowledge in the learning of concepts and multiplication of fractions in grades 4 and 5. Doctoral dissertation, University of Michigan. Fractions: A Problem Solving Approach (Huinker, 2010) 6
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