Rational Number Project Home Page

Reiss, M., Behr, M., Lesh, R., & Post, T. (1988, July) The assessment of cognitive structures in proportional reasoning. In J. Bergeron, et al. (Eds.), Proceedings of the Eleventh International Conference, Psychology of Mathematics Education PMR - XI Volume II (pp. 310-316). Montreal, Canada: PME.


Matthias Reiss (PH Karlsruhe)
Merlyn Behr (Northern Illinois University)
Richard Lesh (WICAT Systems)
Thomas Post (University of Minnesota)*

It is the goal to assess the conceptual background for proportional problems. In terms of cognitive science we have to distinguish between declarative and procedural knowledge. Although the conceptual background is part of declarative knowledge it is usually assessed by procedural tasks. We present a more direct way to cognitive structures in proportional reasoning using NOWAK's concept maps. The students have to describe the relations between concepts verbally and write them down on a poster. The students' implicit theories can be described pictorially. The individual student is viewed as a theoretician rather than a problem solver.

An increasing number of studies have concentrated on issues related to rational numbers and proportions (Behr & al., 1984; Noelting, 1980; Siegler & Vago, 1978, Hasemann, 1987). Traditionally, difficulties in this domain are assessed by interviews. Students solve rational number tasks and the authors describe the most common errors. These errors are the basis for analyzing strategies involved. Karplus & al. (1974) and Noelting (1980) suggested that faulty qualitative reasoning was the basis for many incorrect solutions. Hart's studies show that students tend to use additive operations where multiplicative operations would have been appropriate. The same author (1985) reported in her study about proportion tasks that students prefer whole numbers where the use of fractions would have been more adequate.

Hasemann (1987) emphasizes that already existing concepts and schemata are of central importance for mathematical learning. Often they exist before they become an issue in mathematics education. Therefore it can be productive to see the student not only in his or her role as a problem solver, but also as a theoretician. Often students have an implicit theory about mathematical concepts and they see relations between concepts not intended by teachers and textbooks. Students' errors may be seen in a different perspective taking this implicit theory into consideration.


In cognitive psychology a distinction is made between a procedural knowledge about rules and a declarative knowledge about facts. Anderson (1983) incorporates these two forms of knowledge into his psychological model of memory. He distinguishes between procedural rules formulated in condition-action pairs and factual knowledge in the form of strings, spatial images, and abstract propositions. The procedural rules make up a production system and are stored in the production memory whereas the factual knowledge can be retrieved from the declarative memory.

Anderson not only distinguishes between declarative and procedural memory but adds a third component, the working memory. The declarative memory contains rather general elements of knowledge and can be described as long term memory. In contrast, the working memory consists of rather volatile elements of knowledge that can be characterized as short term memory. Finally, the procedural memory includes productions, i.e. a set of rules which expresses the contingency between elements of knowledge in the form of condition-action pairs. In our case productions could be rules for handling fractions whereas the declarative knowledge would be involved while talking about the conceptual background of fractions.

(Anderson, 1983, p. 19)

In Anderson's model information originates in the environment and comes into the cognitive system via perception; it is encoded and stored in working memory. In our case the student perceives the fraction bar which is encoded in working memory. It does not have any meaning so far. The information from perception is transmitted to the declarative memory and the fraction bar becomes a signal for fraction tasks. Because the working memory only has a limited storage capacity, storage of perception is temporary and retrieval is fast. Perceptions are finally stored in the declarative memory for a long period of time. They are related to other objects and events, which is the basis for complex information retrieval from the declarative memory. More general aspects of the object or events are constructed so that information retrieval becomes more efficient.

It is not only the object that is stored, but this object as part of a whole class of objects. Fractions e.g. can have the attribute to be reduceable. This information is transmitted to the working memory and from there to the production memory. If the conditions for a reduceable fraction match with the numbers of a given fraction, the production memory initiates the action of reducing the fraction in the working memory. This is a cognitive activity which is finally performed by actually writing the reduced fraction on a piece of paper. If the conditions do not match with a given fraction, more information about fractions in general and about the specific task is retrieved from the declarative memory and transmitted to the production memory via the working memory. This process of information retrieval and matching condition-action pairs is continued until the solution is reached.

Problem solving in this model is conceptualized in terms of declarative and procedural components. As mentioned earlier research related to rational number concepts usually starts with the calculation procedure applied by the student. Errors in the procedure are interpreted as deficiencies in conceptual understanding. Using Anderson's terms, mathematics education usually starts with procedural knowledge and infers the structure of declarative knowledge on this basis.

The assessment of declarative knowledge about proportions

It appears to be worthwhile to try to assess declarative knowledge directly. In studies of artificial intelligence declarative knowledge is often described in the form of semantic nets. The concepts are nodes, the relations between the concepts are links. These relations are well defined: a concept can be a superconcept of another one (ISA link), it can be characterized by certain properties (HASPROP link), and it can have certain parts (HASPART link).

This form of describing declarative knowledge did not seem to be appropriate for the 7th graders we interviewed. The links in semantic nets are too well defined and do not allow space for ambiguities in cognitive structures. That is why we decided to make use of the experiences Novak & al. (1983) had in physics education. Novak developed an assessment procedure that he called concept mapping. This involved concepts written on cardboard that had to be ordered on the table: similar concepts near each other, dissimilar concepts away from each other. The students had to explain what they saw on the table and to describe relations between concepts orally. Finally these descriptions were written down in the form of arrows between concepts and sentences defining their relation to each other. The result was a map on the table that described the students' implicit theory about the domain on a conceptual basis.

In order to demonstrate the assessment method, a concept of one of our students is included. The concept map should be read in the following way: one has to start where the arrow originates, read that concept, read the link, and finally read the concept where the arrow points to. In the case of double arrows it can be read in both directions.


The concept map on the preceding page is the product of an intensive interview (45 minutes) with Jon. He is a good student, he usually was one of the first students able to answer a difficult question. In spite of his remarkable qualities in solving numerical problems, his concept map reveals some misconceptions in the domain of proportions. He is able to describe what a fraction is, but he cannot relate the concept proportion to other concepts in a consistent manner. He seems to mix up proportion and portion. As long as he describes concepts within applications of proportionality concepts, he finds a more or less meaningful way to use them (distance, miles, time, hours, speed; water, orange concentrate, mixture). But rate, ratio, proportion, part and number do not really fit into the system and are used incoherently. Although for Jon proportion is the same as ratio and as part, he does not see a direct connection between ratio and part. Following this logic ratio had to be the same as part. The fact that he does not mention this connection reveals ambiguities in the use of these concepts.

Using this technique in the domain of proportions. It was possible to discover conceptual misunderstandings even in good students. Usually they did not connect all concepts that were related to each other. The results were islands of concepts which often were consistent in themselves, but proved to be relatively isolated from each other.

Concept maps reveal important aspects of declarative memory but certainly are not identical with this form of knowledge. They may be useful for diagnostic purposes as well as for monitoring the students' conceptual background. In contrast to Dörfler (1987) the underlying model assumes that it is possible to separate operative aspects and the conceptual background (or in Anderson's terms declarative and procedural knowledge). Dörfler argues that these are two sides of the same coin. Anderson's model views them as two components of a dynamic system interacting with each other. We used Anderson's model because we wanted to focus on the students' conceptual background and therefore had to separate it from operative aspects.


Anderson. J. R. (1983). The architecture of cognition. Cambridge, London: Harvard University Press.

Behr, M., Wachsmuth. I., Post. T. R. & Lesh. R. (1984). Order and equivalence of rational numbers: A clinical teaching experiment. Journal of Research in Mathematics Education, 15, 323-334.

Dörfler, W. (1987). Empirical investigations of the construction of cognitive schemata from actions. In A. Bergeron & al.: Proceedings of the Eleventh International Conference for the Psychology of Mathematics Education. Montréal: Université de Montréal.

Hart, K. (1985). Untersuchungen über Schlülerfehler. Der Mathematikunterricht, 31. 26-37.

Hasemann, K. (1987). Alternative Begriffe der Schüler und die Rolle begrifflicher Konflikte im mathematischen Lernprozeß. Der Mathematikunterrich, 33, 21-31.

Karplus, E. F., Karplus, R. & Wollman, W. (1974). Intellectual development beyond elementary school IV: ratio, the influence of cognitive style. School Science and Mathematics, 76, 476-482.

Noelting, G. (1980). The development of proportional reasoning and the ratio concept: Part 1, Differentiation of stages. Educational Studies in Mathematics, 11, 217-253.

Novak, J. D, Gowin, D. B. & Johansen, G. T. (1983). The use of concept mapping and knowledge vee mapping with junior high school students. Science Education, 67, 625-645.

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

* This research was supported by contract DPE-8470077 from the National Science Foundation. The first author lives in West Germany, the others in the United States. We would also like to thank Ann Hamilton, Guershon Harrel, and Shari Larson for their cooperation.