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Behr, M., Reiss, M., Harel, G., Post, T., & Lesh, R. (1986, July). Qualitative Proportional Reasoning: Description of Tasks and Development of Cognitive Structures. In Proceedings of the Tenth International Conference for the Psychology of Mathematics Education (PME-10) (pp. 235-240). London, England.

Description of Tasks and Development of Cognitive Structures

M. Behr, M. Reiss, G. Harel; Northern Illinois University;
T. Post: University of Minnesota;
R. Lesh: WICAT Institute


Qualitative reasoning is a significant variable in problem solving performance. Expert problem solvers are known to reason qualitatively about problem components and relationships among them before attempting to describe these components and relationships in quantitative terms (Chi & Glaser, 1982). It is not that expert problem solvers use qualitative reasoning and novice problem solvers do not, but that novice problem solvers direct qualitative reasoning to the surface feature of a problem, rather than structural features, and fail to anticipate relationships between problem components. For instance, when asked to explain the transformation of 2/5 to 6/x with respect to the equality relationship, the novice problem solver may reason the transformation to be additive, and that x = 9. Such an argument is inadequate because this particular additive transformation has a qualitative effect on the ratio relationship which is not taken into consideration.

The consensus of current research is that an expert's reasoning about a problem leads to a superior problem representation because it contains numerous qualitative considerations about problem components and their interactions. Such a problem representation enables the expert to know when qualitative reasoning is inadequate and quantitative reasoning is necessary.

Research about fractions, ratios, and proportions (e.g., Behr, Wachsmuth, Post & Lesh, 1984; Hart, 1981; Karplus, Karplus & Wollman, 1974; Noelting, 1980; Seigler & Vago, 1978) have described correct, incorrect, and inappropriate strategies and inadequate qualitative reasoning that adolescents use for such problems. It has been shown that adolescents frequently use additive comparisons when multiplicative comparisons are required. The effects of qualitative change in the magnitude of components of relationships such as a/b = c, are inadequately anticipated by adolescents and adults. Yet, important cause and effect relationships can be deduced through qualitative reasoning alone.

The focus of earlier studies on proportional reasoning has been on whether children have achieved quantitative proportional reasoning. Observed qualitative strategies have been characterized as inadequate without investigation of how they might interact with or serve as a basis for the development of quantitative strategies. One aspect of our research is to investigate whether children have, or can be taught, qualitative reasoning strategies to answer proportionality questions when appropriate, and to determine circumstances where quantitative reasoning is inadequate.

Work by deKleer and Brown (1984) responding to a need to explain the qualitative reasoning observed by expert problem solvers in scientific domains has resulted in a qualitative calculus based on the concept of confluence equation, or qualitative differential equation. In this calculus an equation of the form a/b = c is associated with the qualitative differential equation ?a - ?b - ?c (not to be confused with a standard differential equation). Interest focuses on values of +, -, and 0 assigned to ?a, ?b, or ?c, according to whether a, b, and c are increasing, decreasing, or unchanged (deKleer & Brown, 1984). The qualitative differential equation provides an algebraic description of the qualitative behavior among the three components of the equation a/b = c, as shown in the following table.


The Rational Number Project has developed tasks to investigate children's qualitative proportional reasoning. Success on these tasks requires ability to reason qualitatively about two ratio situations modeled by a/b = c and x/y = k. They require reasoning about how qualitative changes in a or b and x or y affect c and k, the intensive values of the respective ratios, and of how these qualitative changes affect the comparison of c and k.

Tesselation Tasks which we developed were based on the concept that any uniform pattern visually expressing a comparative relationship between two or more sets of congruent juxtaposed geometric figures, such as curves to squares, or squares to rectangles, can be used to tesselate the plane. Proportionality problems arise from questions about iterations, partial iterations, or partitions of the basic tesselating pattern. Comparison of rates in terms of equality or inequality relations are embedded in a comparison of two tesselating patterns. Missing value problems are embedded in situations when an iteration, partial iteration, or partition of a basic tesselation pattern is shown with one set of geometric patterns masked. Incorporation of perceptual distractors into a tesselation task serves as a test of the strength of a child's logic for proportional Judgments over perceptually based judgments.

The block task is strictly a non-metric proportionality task which involves two pairs of blocks ((A,B) and (C,D)). Corresponding blocks (A,C) and (B,D)) across each pair were constructed from the same kind of unit-blocks; blocks within a pair differed in the size of unit blocks used, A and C using larger units. The number of unit blocks in A and C differed, but remained constant across tasks. The three instantiations of each of B and D compared in number of unit blocks by one less, the same, or one more to A and C, respectively. Subjects were asked to judge the weight relationship between C and one of the instantiations of D based on one of three weight relationships given between A and an instantiation of B.

The three within pair weight relationships crossed with the three within pair number of unit blocks relationships -resulted in 27 possible problem situations. Nine carefully selected problems were presented to subjects. The blocks were constructed so that proportional reasoning was required for problem solution, carefully avoiding the possibility of solution by transitive reasoning.

The triangle similarity tasks assume that similarity is a primitive cognitive structure to proportionality. Each task involved right triangles as the pre- and post-image of a uniform or nonuniform right-triangle preserving positive or negative dilation. Three transformations were represented such that from pre- to postimage, the leg of the right triangle in a counter-clockwise orientation to the right angle was stretched or shrunk by an equal, or less, or a greater multiple than the leg in the clockwise orientation to the right angle. Thus, six types of tasks were possible. For missing-value proportionality problems in this context, children were given values of two sides of the transformation pre-image and one corresponding side of the post-image along with a picture to characterize the transformation as one of the six possible, and asked to give the value for the length of a corresponding side of the post-image. For ratio comparison problems, children were given numerical data for two pairs of corresponding sides of the pre- and post-images and then asked to choose from a set of pictures the one of six possibilities which would result if the pre-image were superimposed upon the postimage.

The concept map task is concerned with the conceptual representation of a topic, in this case, proportional reasoning; not with numerical relationships or strategies. The children were given a number of concept names on cards and were asked to arrange them so that similar concepts were near each other. They then drew and described links between the concepts which exposed their understanding of the relationship between the concepts. The resulting concept map gives an impression about whether children see relationships between concepts and whether or not they group them hierarchically or as separated entities. The concepts given were general (proportion, fraction) or specific (numerator, denominator). Some were mathematical (rate, ratio) or from applications (speed, time, distance). So there were ample opportunities to find interconnections. The results from this task will be used to aid in constructing semantic nets of the student's cognitive representations of concepts associated with fraction, ratio, and proportion.

These tasks have been given in one-on-one interviews given at intervals in a 15-18 week teaching experiment. Further analysis should help to characterize the development of qualitative and quantitative proportional reasoning strategies, and the interaction between them.

A written test was developed to study the effects of problem context from the two perspectives of qualitative and quantitative reasoning. Quantitative tasks were of missing value (MV) and numerical comparison (NC) types. Qualitative problems contained no numerical values they required a decision about the directional change of a rate when its numerator or denominator is(are) changed, or to determine the order relationship of two rates according to the variables under consideration.

Four contexts were selected - speed, mixing, scaling, and density. For each context, separate tests were developed for each of two settings - more familiar and less familiar - resulting in eight tests. The contexts and settings are listed:

  Setting (more familiar) Setting (less familiar)
Speed Running Laps Driving Cars
Mixing Mixing Lemonade Mixing Paint
Scaling Drawing Classroom Maps Reading Road Maps
Density Standing in Movie Line Hammering Nails

Each test had eight quantitative problems (4 MV and 4 NC) with integer ratios both within and between rate pairs. Two types of qualitative problems (4 rate change and 4 comparison) were included in each test. A qualitative rate change task requires determination of the direction of change in a single rate: "If Nick drove less miles in more time than he did yesterday, his driving speed would be a) Faster b) Slower c) Exactly the same d) There is not enough information to tell." In a qualitative comparison task two rates are given and an order comparison is required: "Bill drove more miles than Greg. Bill drove for less time than Greg. Who was the faster driver?"

Test results from about 950 grade-7 and -8 children were analyzed in a context X setting X problem type X grade level design. Types of student solution strategies were examined for the quantitative problems.


Behr, M., 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, pp. 323-334.

Chi, M. T. H. & Glaser, R. (1982). Final Report: Knowledge and Skill Differences in Novices and Experts (Contract No. N00014-7 -C-0375). Washington, D.C.: Office of Naval Research.

deKleer, J. & Brown, J. S. (1984). A qualitative physics based on confluences. Unpublished manuscript, XEROX PARC, Intelligent Systems Laboratory, Palo Alto, CA.

Hart, K. (1981). Children's Understanding of Mathematics: 11-16. London: Murray.

Karplus, E. F., Karplus, R. & Wollman, W. (1974). Intellectual development beyond elementary school IV: Ratio, the influence of cognitive style. School Science and Mathematics, 74(6), pp. 476-482.

Noelting, G. (1980). The development of proportional reasoning and the rate concept. Part I - Differentiation of stages. Educational Studies in Mathematics, 11, pp. 217-253.

Seigler, R. S. & Vago, S. (1978). The development of a proportionality concept: Judging relative fullness. Journal of Experimental Child Psychology, 25, pp. 371-395.