Rational numbers such as fractions “...are among the most complex and important mathematical ideas that children encounter before they reach secondary school” (Behr, Wachsmuth, Post, & Lesh, 1984, p. 323)—a very powerful statement about a topic often dreaded by students regardless of their disability status. Educators at all grade levels recurrently experience students’ difficulties to understand and work with fractions.
Because of these difficulties, students often desire to avoid fractions, despite the evident need for facility with this part of mathematical knowledge, which is ever present in life. Whether students have an interest in cooking, working in the yard, decorating their bedroom, planning the amount of time needed to get the bus or walk to school, developing a budget to save for a desired purchase, or making change, they need fractions.
With the wide availability of programmable calculators that include the favored fraction “button,” students often ask why is it necessary to know how to compute fractions by hand, as long as they know the correct key(s) to press on their calculator? Certainly, calculators can compute fractions quickly and, at times, more accurately than students, but to properly input fractions into a calculator, a student must know what information to enter, the order in which to enter it, how to interpret the output, and how to estimate whether the answer generated by a calculator is reasonable. Basic knowledge and ability to work with numerical quantities such as fractions is commonly referred to as number sense.
Number sense relates to an individual’s fluidity and flexibility with numbers. It is defined as having a sense of what numbers mean and being able to mentally manipulate mathematical concepts and quantities to make comparisons in the immediate environment (Gersten & Chard, 1999). If students do not have a solid understanding of what a fraction is or what its components represent, it can be expected that they would have great difficulty in manipulating fractions in calculations and problem solving both in and out of the classroom. In effect, number sense relating to fractions needs to be firmly established for conceptual understanding, mathematical reasoning, and effective problem solving with fractions to develop.
Few studies have examined students’ understanding of fractions and effective instructional methods for teaching students fractions. One such study, The Rational Number Project, consisted of a series of activities designed to assess students’ quantitative understanding and ability to use fractions in estimation, computation, and problem solving. Results from studies on students’ approach to fractions highlighted their initial reliance on manipulatives to aid them in computation (Behr et al., 1985; Behr et al., 1984; Post et al., 1986; Post et al., 1985). As students establish a solid foundation of knowledge and understanding of fractions their dependence on manipulatives lessened to verification purposes only (Post et al., 1985). However, the progression away from the use of manipulatives as students develop greater fluency in translating among different representations of fractions aids in their understanding and ease of application of fractions (Post et al., 1985). Thus, the use of manipulatives in instruction should also decrease once concepts are introduced and understanding is established as a way to facilitate greater automaticity.
Direct instruction in fractions was explored in an early study conducted by Perkins and Cullinan (1985) that involved third grade students who had no previous instruction in fractions, but did have prerequisite skills of basic addition and subtraction. Three interrelated factors were emphasized in this exploration of direct instruction: instructional design, presentation techniques, and organization of instruction. Instructional design components focused on teaching strategies by identifying and teaching all preskills and components of the strategies so students would be able to solve various types of fraction problems. Presentation techniques involved elements designed to aid students in maintaining high levels of attention-to-task and learning skills to mastery. Organization of instruction focused on matching the student to the curriculum and efficient use of classroom time and resources. Findings from the study indicate that students demonstrate an improvement in their problem solving skills once direct instruction was implemented. After instruction was completed, students were able to maintain their problem solving skills at levels comparable to their performance towards the end of the study. Besides maintaining their problem solving skills, students were also able to transfer their skills to problem types for which they had not received direct instruction. Direct instruction then proved effective in teaching students problem-solving skills with fractions, and facilitated their generalization and maintenance of the skills during and after instruction. Thus, the effectiveness of direction instruction in teaching fractions is clearly supported for early elementary students who have had no previous instruction in fractions.
In light of the above-mentioned information, effective instruction in fractions should begin with determining the level of knowledge related to fractions students already possess. Instruction should incorporate manipulatives of different shapes, sizes, and types to represent fractional quantities presented in various symbolic forms in both conceptual and computational applications. As students develop competence and flexibility in solving computational problems, contextual problem solving examples can be presented to facilitate students’ success in applying their knowledge of fractions to meaningful situations. Yet the assumption should not be made that students will be able to correctly apply computation skills with fractions to comparable problem solving situations. Once a conceptual base in fractions is firmly established, and students demonstrate competency and fluency with computations, problem solving applications should be integrated and used in instruction to help students develop competency in working with fractions in real life situations.
Behr, M. J., Wachsmuth, I., & Post, T. R. (1985). Construct a sum: A measure of children’s understanding of fraction size. Journal for Research in Mathematics Education, 16(2), 120-131.
Behr, M. J., Wachsmuth, I., Post, T. R., & Lesh, R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal for Research in Mathematics Education, 15(5), 323-341.
Bottge, B. A. (1999). Effects of contextualized math instruction on problem solving of average and below-average achieving students. The Journal of Special Education, 33(2), 81-92.
Bottge, B. A., & Hasselbring, T. S. (1993). A comparison of two approaches for teaching complex, authentic mathematics problems to adolescents in remedial math classes. Exceptional Children, 59(6), 556-566.
Gersten, R., & Chard, D. (1999). Number sense: Rethinking arithmetic instruction for students with mathematical disabilities. The Journal of Special Education, 33(1), 18-28.
Perkins, V., & Cullinan, D. (1985). Effects of direct instruction intervention for fraction skills. Education and Treatment of Children, 8(1), 41-50.
Post, T. R., Behr, M. J., & Lesh, R. (1986). Research-based observations about children’s learning of rational number concepts. Focus on Learning Problems in Mathematics, 8(1), 39-47.
Post, T. R., Wachsmuth, I., Lesh, R., & Behr, M. J. (1985). Order and equivalence of rational numbers: A cognitive analysis. Journal for Research in Mathematics Education, 16(1), 18-36.
Teresa Foley is an Instructor of Mathematics at Asnuntuck Community College in Enfield, Connecticut.