

One
Point of View


Fractions
and Other Rational Numbers


Thomas
R. Post, University of Minnesota




National and international assessments results indicate significant student difficulties in learning about rational numbers. In 1979 only 24% of the nation’s 13 year olds could estimate the sum of 12/13 and 7/8 given the possibilities 1,2,19,21 and I don’t know. Fiftyfive percent selected either 19 or 21 as the estimated sum! Why do such difficulties occur? Surely part of the answer resides in the fact that rational number instruction is poorly conceptualized and implemented by the various textbook series. Most approaches do not deal significantly with any type of manipulative material even though many excellent ones exist, (Fraction circles, cuisenaire rods, chips, and number lines are examples.) Students are expected to formulate fraction related concepts primarily on the basis of their interaction with the printed page which, by definition, can only include pictorial models (iconic representations) and symbols. This is a very severe misunderstanding of the nature of human learning. It is further exacerbated by the fact that most texts focus on operations with fractions rather than fundamental concepts, such as partitioning, and order and equivalence. Thus students very early can be found generating long lists of equivalent fractions, adding, subtracting, multiplying, dividing, without having the foggiest idea of what they are doing. Even well conceived fraction instruction involves more complex forms of numerical relationships than previously encountered. I am speaking here of the transition from additive to multiplicative structures, a truly major transition in mathematical domains. Before rational number, virtually every type of numerical problem situation could be viewed as a variation on the counting theme. With the beginning of serious attention to multiplicative relationships, students find that the additive baggage which has served them so well in the past becomes no longer fruitful, i.e., 1/2 + 1/3 2/5. Children for the first time must employ condition and relativistic kinds of thought processes and begin to identify new types of relationships between numbers, i.e., 1/3 is less than 1/2 even though 3 is greater than two, but 3/7 is greater than 2/7 because 3 is greater than 2. Multiplicative patterns work, but not additive ones. For example, 2/3 = 4/6 but 2/3 3/4. There are new rules for addition and subtraction, but the old rules for multiplication seemingly continue to work. Rational numbers include a variety of interpretations other than the part whole model. Decimals, ratios, operators, number lines, and indicated divisions are all considered rational number subconstructs. Ideally as students progress through the curricula, these interpretations are sorted out and understood. This is a difficult task for many. Major changes must be made in ways children encounter these concepts individually, and in the amount of attention paid to their collective interrelationships. The latter is almost totally absent in the school curricula. Important questions about childrens’ conceptual development in the rational number domain are being addressed by the research community. We do know much about the way in which children are able to understand these concepts and the accompanying difficulties in teaching them. It is time for such results to seriously impact school curricula. The ubiquitous nature of rational number concepts in all of mathematics surely make them one of the most important conceptual domains to be studied by children. Decimals, fractions, ratios, and other multiplicatively based relationships will literally dominate the junior high school mathematics curricula. These rational number ideas will eventually play a major role in the development of proportional reasoning abilities. These abilities will, in turn, become the intellectual and mathematical cornerstone of much of what is to come in the secondary years. References Cramer, K. et al. (1989, January). Cognitive restructuring ability, teacher guidance and perceptual distractor tasks. Journal for Research in Mathematics Education. Behr, M. et al. (1983). Rational number concepts in R. Lesh & M. Landau (Eds.), The Acquisition of Mathematics Concepts and Processes (pp 91125). New York: Academic Press. Kieren, T. (1976). On the mathematical, cognitive, and instructional foundations of rational numbers. In R. Lesh (Ed), Number and Measurement (pp 101144). ERIC/SMEAC. Lesh, R. et al. (1987). Rational number relations and proportions. In C. Janvier (Ed.), Problems of Representation in the Teaching and Learning of Mathematics (pp. 4158). Hillside, NJ: Lawrence Erlbaum. Post, T., & Cramer, K. (1987, October). Children’s strategies when ordering rational numbers. Arithmetic Teacher, 3335. 
