There is a well-known anecdote about a professor of mathematics who, when addressing a learned audience, wrote a mathematical statement on the board and said ‘This, of course, is obvious’. Looking at it again, he mumbled ‘Well, I think it’s obvious’. After further pause, he excused himself from the room, taking pencil and paper with him. Returning twenty minutes later he declared in confident tone, ‘Yes, ladies and gentlemen, it is obvious’.

This anecdote, recounted by Richard Skemp, sets the scene for chapter 4, which discusses the relationship between intuitive and reflective intelligence. Although the book is replete with anecdotes and common place analogies, its ideas on educational psychology are centred wholly on the learning and understanding of mathematics.

For example, if a student were to write \(x + 3 = 7 = 7 - 3 = 4\), it could lead to discussion as to the nature of that student's understanding; and a central theme in this book is to examine in close detail exactly what is meant by ‘understanding’. Skemp gives the following three versions:

- Instrumental understanding is the ability to apply particular rules to the solution of a problem without knowing why it works (as in rote learning).
- Relational understanding is the ability to deduce rules or procedures from other mathematical relationships (as in an investigational approach to teaching and learning).
- Formal (logical) understanding is the ability to connect mathematical symbolism and notation into chains of logical reasoning (as in mathematical proof).

So, in the example \(x + 3 = 7\), it seems that the student has some understanding of the connection between addition and subtraction (relational understanding). But he/she may have blindly applied a rule for solving such an equation (instrumental understanding). Either way, the sequence \(x + 3 = 7 = 7 - 3 = 4\) is logically flawed and shows deficiency with respect to formal understanding.

Of course, the hierarchies of mathematical knowledge must begin somewhere, and they do so at the very basic level of ‘conceptual understanding’. For number systems, a primary concept is that of cardinal number. Higher order concepts include place value and the classification of natural numbers (odd, even, multiples, primes etc.). The ever growing body of knowledge that accumulates around the natural number system is called a ‘schema’, and any one concept within it is linked it to many others (e.g. subtraction is the inverse to addition).

Hence, relational understanding of an idea means being able to assimilate it into an existing schema, which is a conceptual framework that is continually adapted to new circumstances. So when the schema for number systems is extended to include fractions, the learner’s view of addition and multiplication have to be adapted by what Piaget referred to as the process of accommodation.

All organized learning is goal directed. One goal might be the wish to avoid punishment associated with rule-based learning (crosses in red ink and low grades). Relational understanding, which can connect seemingly different ideas, is its own reward due to the satisfaction of problem solving or the aesthetic pleasure arising from mathematical results. It is far less likely to evoke the question ‘Why are we doing this?’.

What is intelligence? Several years after being classed as an unintelligent schoolboy myself, I bought a book called ‘Test Your Own IQ’, and I was amazed to find that, the more tests I did, the higher my IQ became. So when I was compulsorily enlisted in the British army, I passed all IQ tests with ease, and was deemed to be above average intelligence. Such tests were redolent of Skinnerian principles, based upon non-adaptable stimulus-response habit learning.

In contrast, intelligent learning develops flexible plans of action that are modified and improved following each application (investigative learning). Intelligence is therefore a cluster of mental adaptable abilities whose application demands that thinking precedes action and that reflection follows it. The first chapter of Part B of this book (A New Model of Intelligence) discusses this in great detail, and there is subsequent discussion of the transition from behaviourism to constructivism.

Further dimensions to this ‘psychology of learning mathematics’ include interpersonal and emotional factors, use of imagery, symbolic understanding and the psychology of communicating mathematics.

There will be many very good teachers of mathematics with little or no formal knowledge of educational psychology, but they will appreciate Skemp’s analysis of what has made them good teachers. I read the first version of this book over 40 years ago, and I still believe that it is one of the best theoretical models for the learning and teaching of mathematics.

Peter Ruane taught in primary and secondary schools, and was then involved in the training of mathematics teachers from primary to high school level.