An Example of the Passage from the Concrete to the Manipulation of Formal Systems

Everybody knows that mathematics is an abstract subject. It follows that most people who have studied the problem of learning such an abstract subject, would agree that some passage from the concrete to the abstract must be mapped. By concrete, we mean usually, our immediate contact with the real world. We come into contact with objects and events and we re-act to them. This is the concrete level at which all organisms behave until they are able to organize their re-actions to events into re-actions to sets of events. This is the first stage towards abstraction — when the organism begins to classify. It is of interest to try and investigate the details of the abstraction process, not only from concrete experiences to classification, but to the learning of extremely complex abstract systems such as exist in mathematics and which more and more people, including children, are called upon to learn. The difficulties, in the way of such studies, are great, one of the main difficulties being that mathematicians on the whole are not interested in learning, and psychologists on the whole do not know enough mathematics to be able to formulate the problem in a way in which a possible solution might be sought. At the Sherbrooke Psychomathematics Center, for the past few years, we have been studying the abstraction process as it proceeds from the concrete to the final stage of wielding a mathematical formal system. We have rather few laboratory results of this aspect of our work as yet but we have enough classroom evidence that we can postulate certain regularities that seem to occur and certain pre-requisites that appear to have to be satisfied before certain stages of learning can successfully be undertaken.