Versatile Learning of Mathematics

It was many years ago now,that as a young undergraduate in mathematics at University, I was first introduced to Richard Skemp’s ideas by my lecturer . I read the paperback The Psychology of Learning Mathematics with great interest. It was not until I was a graduate student of mathematics  that I re-read some of Richard’s work, and yet, as a teacher of mathematics in schools it had remained with me. I read Intelligence, Learning and Action and was struck by the power of the ideas expressed in it and the simplicity of the language in which they were presented, especially in comparison with other texts I read. I learned the valuable lesson that powerful ideas can be communicated in simple terms and do not require a facade of convoluted definitions and expressions in their presentation.

Another educationist, Ausubel (1968, p. iv), well-known for his concept of meaningful learning, said, “If I had to reduce all of educational psychology to just one principle I would say this: The most important single factor influencing learning is what the learner already knows. Ascertain this and teach him accordingly.” Similarly, today, many constructivist mathematics educators would maintain as a central tenet that the mathematics children know should be the basis on which to teach mathematics (e.g. Steffe, 1991). Skemp, agreed, remarking that “our conceptual structures are a major factor of our  progress” (1979c, p. 113). Since our existing schemas serve either to promote or restrict the association of new concepts, then the quality of what an individual already knows is a primary factor affecting their ability to understand. The existence of a wide level of agreement on this point indicates very strongly that it is something which mathematics educators should take to heart. However, one does not have to have been a teacher for very long before being faced with a dilemma. Many of us will have experienced the reaction of students when, after we have spent some considerable time trying to develop the ideas and concepts of differentiation, we introduce the rule for antidifferentiation  . They may say “well why didn’t you just tell us that was how to do it?” Too often it seems, the students’ focus is on how ‘to do’ mathematics rather than on what mathematics is about, what its objects and concepts are. This procedural view of mathematics is, sadly, often reinforced by teachers who succumb to the pressure and only tell students how to do the mathematics.

I am particularly interested in the way in which our conceptual structures enable us to relate the procedural/process aspects of mathematics with the conceptual ideas, such as why the formula is correct. The essentially sequential nature of algorithms often contrast with the more global or holistic nature of conceptual thinking, and, the ways in which we, as individuals, construct schemas which enable us to relate the two in a versatile way is of great importance. The full meaning of the term ‘versatile’, as used here will emerge during this discussion but, suffice it to say at this stage, that it will be used in a way which will have the essence of the usual English meaning, but, will take on a more precise technical sense, which will be explained during a discussion of the nature of our conceptual structures. Having acknowledged the importance of our mental schemas in building mathematical understanding, some questions worth considering

in any attempt to encourage versatile learning of mathematics include:

1. How are schemas constructed?

2. How can we identify and describe the quality of our constructions?

3. How do our schemas influence our perception of the objects and procedures in mathematics, and their relationship?

4. What experiences will help us improve the quality of our constructions so that we can build a versatile view of mathematics?

Skemp’s theory has much to offer towards building answers to these sorts of questions show how valuable it is. We should consider his model of intelligence and its implications for improving the quality of learning and understanding.