Robert Karplus, Elizabeth Karplus, Marina Formisano, and Albert-Christian Paulsen
The wide diversity of interests and abilities encountered among secondary school students is well known to teachers. The present study was intended to illuminate this range in regards to logical mathematical reasoning, or what Jean Piaget has called formal thought. We developed two tasks to assess proportional reasoning and control of variables. On these tasks, which could be administered to classroom groups in several countries, the students were asked to solve certain problems and then explain or justify their answers. The logical mathematical reasoning was exhibited by the form and content of the explanations.
Both areas of thought we selected for investigation are vitally important in science instruction. Proportions play a key role in quantitative relations in science. The idea of a controlled experiment, "keeping all other things constant" to isolate the effect of one variable, is essential for determining unambiguous cause-and-effect relationships in science or other subjects. In view of the general ferment in science education, with new course materials and new teaching methods being introduced into secondary schools on an increasing scale, it seemed worthwhile to investigate students' logical-mathematical reasoning.
Traditionally, reasoning has been investigated by means of clinical interviews in which a subject confronts a phenomenon or a problem and is asked to make a prediction, explain what he observes, or find a solution(1). The subject is then asked to justify his reasoning, and is pressed by the interviewer to provide additional reasons or alternate theories regardless of his first response. The resulting conversation is eventually analyzed for the types of relationships, "convincing" arguments, and resolution of ambiguities employed by the subject.
Written responses to a group test clearly do not provide the same depth of in. formation about a single individual. There are many reasons why an individual's first response--and that is what he is likely to write down--may not tap all his intellectual resources. Copying from other students and lack of interest also influence the results. Nevertheless, the written task does have an advantage also, in that it permits rapid surveys of large numbers of subjects interna-____________________