

COGNITIVE PROCESSES AND PRODUCTS IN PROPORTIONAL REASONING M. REISS (Northern Illinois University) M. BEHR (Northern Illinois University) R. LESH (WICAT Systems) T. POST (University of Minnesota) 

Numerous studies have
shown that children and many adults have a great deal of difficulty with
basic concepts of fraction, ratio, and proportion, and especially with problems
involving these concepts. These studies have discovered correct and incorrect
thinking strategies children employ in problems involving, these concepts.
Little research seems to have been addressed to the question of how these
concepts develop in children. Some studies (Noelting, 1980; Karplus et al.,
1983) indicate that many children use faulty qualitative reasoning, both
incorrect and inappropriate. Some children use additive comparisons where
multiplicative comparisons are required (Hart, 1981). Apparently many children
do not see cause and effect relationships between components of equations
such as a/b = c/d or a/b = k. observation of important cause and effect
relationships are possible through qualitative reasoning alone.
In a multisite project about rational numbers and proportional reasoning, children are being studied while learning these concepts (clinical Interviews), individual differences between students are analyzed (by large scale testing) and an instructional program is being developed (computer implementation of a partwholeworld). The schema construct will be the center of interest. We will distinguish between two kinds of schemata, the PartWholeSchema and the EqualizedWholesSchema, to describe the cognitive processes involved in proportional reasoning tasks. We hypothesize two kinds of schemata to be involved in proportional reasoning that deal with two kinds of ratios. The first concerns the ratio of two extensive values given with respect to a common unit of measure, whereas the second concerns two dissimilar extensive quantities A and B, that is, two extensive quantities whose measures cannot be given in a common unit (distance and time for example). Involved in the first type of proportional reasoning schema is the cognition of an A and a B as conceptual entities, each of which are part of a common whole. If A and B represent two "mixable" quantities whose measures, extensive values, are respectively a and b, where a and b are given with respect to the same unit of measure, then one can conceive of a partwhole system with components A, B, and A B. Measures associated with these parts are then a, b, and a + b. Of interest in this system is how the values of a, b, and a + b interact with and affect the ratios a : b and the fraction a/(a + b), an intensive quantity which is a measure of the concentration of A in A union B. Of particular initial importance, at this point, is the realization that any proportion problem of this type can be conceptualized as a partwhole situation. To describe the cognitive processes involved we talk of a partwhole schema. The second kind of schema refers to the ratio of two nonmixable quantities A and B; that is, two extensive quantities whose measures cannot be given in a common unit. Involved in the cognition of this type of ratio are three important cognitive structures: (a) the conceptualization of subparts of A into a conceptual entity  A, (b) the conceptualization of subparts of B into a conceptual entity  B, (c) A conceptual equalization of the two wholes A and B. Because of this last characteristic we are referring to it as the EqualizedWholesSchema. From these initial cognitive structures for this ratio situation, together with conservation of proportionality under transformation within and on the two conceptual entities A and B, arise the important notions: (a) That ratio means the rate of change in A with respect to change in B, (b) The concept of unit ratio, i.e., the extent of A in one unit of B. Many situations in which two extensive quantities are expressed in different measure spaces the multiplicative relationships between them can be expressed in terms of EqualizedWholesSchema (EWS). An EWS is useful to represent the ratio (i.e., rate) between the two extensive quantities which are not mixable into a common whole because of the measure spaces from which they are derived. For situations where an EWS is applicable the relationship of interest is the intensive multiplicative relationship between the two quantities. So what are the most relevant characteristics to distinguish between the PartWholeSchema and the EqualizedWholesSchema? 1) PartWholeSchema Important in children's ability to judge the equality of inequality of two fractions, and in doing proportional reasoning is (a) to know when additive comparisons are appropriate of inappropriate, and the same for multiplicative comparisons, and (b) to know the cause and effect relationship between and within additive and multiplicative comparisons due to changes in components of a partwhole structure. Moreover, it is important from our theory base that a child be able to reason qualitatively about the cause and effect relationship which exists between and within additive and multiplicative comparisons. The comparison of the multiplicative relationship between the P_{1} : W ratio in two partwhole structures could be considered to be the initial and final states of a transformation of a partwhole structure (PWS). Under certain partwhole transformations the value of a parttowhole ratio (as well as others) is invariant, but not under all transformations. Under certain partwhole transformation, whether or not the value of the parttowhole ratio is invariant or whether the change is an increase or decrease, can be determined qualitatively; under other transformations this will be ambiguous and will require quantitative (i.e., computation based) reasoning. Underlying these questions are PWS transformation principles. The extension of a child's partwhole schema to include at least implicit knowledge of these principles we hypothesize to be essential for children's ability to meaningfully perform on proportional reasoning tasks. 2) EqualizedWholesSchema Important in children's ability to judge the equality or inequality of two rates, and in applying concepts of rates to problem solving situations is to (a) know the effect that an additive transformation f the components of an EWS has on the multiplicative transformations of the components of an EWS on the multiplicative relationship between these components. Moreover, it is important from our theory base, that a child be able to reason qualitatively about the cause and effect relationship between the additive and multiplicative transformation and the ratio between the components of the EWS. Proportional thinking is frequently called for in problem situation in which three or four data are given and the problem solver is to find the missing one. The solution to such a problem essentially requires application of a restatement of the EWS transformations. The fact that two ratios form a proportion means that the multiplicative relationship between W_{1} and W_{2}, the two components of an EWS, remains equal under an EWS transformation. A missing data proportion problem then begins with two conditions: that (a) the multiplicative relationship (i.e., the ratio) is constant and (b) a given transformation on one of the two components of the EWS is given. The problem solver must then determine (a) the transformation to perform on the other component of the EWS and (b) the result of this transformation. That is, the problem solver must determine the required action under the given conditions. On that basis we have constructed a proportional reasoning test which takes into consideration (1) four different ratios: speed, mixture, scaling, and density; and (2) different setting within each type. The different settings for each ratio are (a) speed: running laps and driving cars, (b) mixture: orange juice, paint, (c) scaling: making a classroom map or city map, (d) density: standing in a movie line, hammering nails into a board, (3) the sequencing of quantitative and qualitative questions (missing value questions with numerical comparisons first vs. questions without any numerical comparisons first), (4) the sequencing of rational number and proportional reasoning questions (rational number first vs. proportions first). This test gives a 4 x 2 x 2 x 2 design so that the different variables can be analyzed in terms of their influence on the procedure. About 1100 students have been tested and these results will be presented soon (Post, 1985). As this approach concentrates on the outcome of the test and not on the process of the task solution, we are going to observe a limited number of students more intensively in the context of teaching experiments. For that purpose a partwholeworld has be developed for personal computers which serves as a microworld in order to explore the concept of proportion by means of visualizations and active modifications of partwholediagrams. Children learn to transform real live situations into partwholediagrams, to compare these situations by comparing the two diagrams and to construct new situations after transformations of the partwholediagram. In that sense the microcomputer can be used to develop a concept of part and wholes, of fractions, of ratios, and later of proportions. The reactions of the students while working on problems on the other hand can be used to determine where they have difficulties. This in turn will be used to modify the presentation of the problems. Abbreviations P_{1}: Part 1 P_{2}: Part 2 W: Whole W_{1}: Whole 1 W_{2}: Whole 2 PWS: PartWholeSchema EWS: EqualizedWholesSchema References Behr, M., Wachsmuth, I., Post, T., & Lesh, R. (1984). Order and equivalence of rational numbers  a clinical teaching experiment. Journal for Research in Mathematics Education, 15, 323341. Hart, K. (1981). Strategies and errors in secondary mathematics  the addition strategy in ration. In C. Comiti (Ed.), Proceedings of the Fifth International Conference for the Psychology of Mathematics Education. Grenoble. Hart, K. (1983). Tell me what you are doing  discussion with teachers and children. In R. Hershkowitz (Ed.), Proceedings of the Seventh International Conference for the Psychology of Mathematics Education. Rehovot, Israel: Weizmann Institute of Science. Karplus, R., Pulos, S., & Stage, E. (1983). Proportional reasoning of early adolescents. In R. Lesh & M. Landau (Eds.), Acquisition of mathematical concepts and processes (pp. 4590). New York : Academic Press. Noelting, G. (1980). The development of proportional reasoning and the ratio concept. Educational Studies in mathematics, 11, 217235 (part I), 331363 (part II). Vergnaud, G. (1983). Multiplicative structures. In R. Lesh & M. Landau (Eds.), Acquisition of mathematical concepts and processes (pp. 127174). NEW YORK: Academic Press.


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