An important quality of mathematics is its applicability to a wide range of uses in science, technology, industry and commerce. This is why it is so useful practically, and so important in today's world of rapidly advancing science and its embodiments in technology. Many of these uses are routine, but we also encounter new requirements for which we do not have a ready-to-hand method. In other words, we have a problem to solve. This is the essence of a problem, that we want to do something but do not know how.
So we need two kinds of mental equipment: routine methods for routine situations, and problem-solving ability for situations we have not met before, or those we have met but have not yet found how to deal with. The latter may subsequently also become routine: so we need also to be able to learn new methods, and then routinize them without detracting from the possibility of future adaptation. This aspect has been discussed at length in Relational Understanding and Instrumental Understanding (Skemp, 1976, 1989a).
So far, most teachers have shown themselves much more successful at the first than at the second. Routine methods can be taught as something to be memorized, but in this form they lack adaptability. Since a new method has to be learnt for each new kind of task, the memory load in time becomes impossible. Moreover, without relational understanding students are often in doubt which method to use. Most of us are familiar with the query "Please, is this an add or a multiply?"
At the NCTM conference at Seattle in 1980 the committee resolved that it was time for a new approach to the teaching of mathematics, based on problem solving. I waited with interest to hear how this was to be done, but they never told us. Nor did the following few years provide any evidence of progress in this direction, to judge by the title of a paper published in April 1983 by the (American) National Commission on Excellence in Education.
This was entitled A Nation at Risk: the Imperative for Educational Reform, and the remedies proposed include taking more mathematics in high school, increasing the school day and year, and assigning more homework. In his telling criticism of this approach, Witney writes: "It should be very clear that we are missing something fundamental about the schooling process. But we do not seem to be interested in this; . . . we try to cure symptoms in place of finding the underlying disease, and we focus on the passing of tests instead of on meaningful goals." (Witney, 1985) A decade later, there is still little or no evidence of progress. In my view this was predictable, and will persist as long as efforts are based on what is promoted by the powerful commercial interests of the major publishing companies.
Can we do better ? Yes, if and only if we understand and accept the need for theory. Common sense alone is not enough, or it would have succeeded long before now.
To begin with a much simpler example, common sense knows that iron sinks. So how is it that we can make ships of iron which float? To understand this we need a simple but powerful theory, the principle of Archimedes. Some of us may remember this from our school days. If we don't, it doesn't matter because we aren't boat builders. It is this principle which enables us to predict whether a ship made of iron will float or sink. It takes us far beyond what we can do with common sense. Whoever would think of making a boat out of reinforced concrete? But these are made, and they do float. And powerful as it is, the principle of Archimedes doesn't provide enough theory to make a boat that will be stable. If we were going to set up as boat builders there is a lot more theory we would need to know.
Likewise, to understand how we can teach for problem-solving ability, common sense is not enough. We need the power which only theory can provide. And since we are teachers, it does matter whether or not we know it, because without it we aren't likely to succeed in helping our children any better in the future than in the past. And the amount of theory we need is more than will go into the present paper. Nevertheless, we can make a good start.
Common sense might suggest that the best way to teach students to learn problem solving in mathematics is to base teaching from the start on real-world problems which require mathematics for their solution. Even a little theory will be enough to indicate that this is not so.
A while ago I had a real-world problem, that of finding my way home from a school in an area quite new to me, which I had been visiting with a maths adviser. He gave me the following directions for getting back onto my familiar road home. "You need to go left, left, left, right, half right, and then left onto the A45."
You can imagine what happened. I took no notice of the left turn at the school gate and was soon lost. It took me a long time to get home. A plan of this kind is closely tied to action. And if one of the actions is wrong, it's liable to throw out all which follow. In this case, if one gets off the path, one doesn't know what to do to get back on. The cognitive element is low. Traditional teaching, based on habit learning, is of this kind. It has low adaptability. But would a problem-solving approach, in which I had to find my own way without even the help I was given, have been any better? I hardly think so.
In time for my next visit, my friend sent me a street map. Here is part of it. (The original also has street names.)
Now I had structured knowledge, which is a much more useful and powerful kind. For a start, I could see where I had gone wrong last time. And even if I did go wrong, I could use the street names to find where I was, and then work out a new plan for getting home from there. And in general, this map contains the potential for working out how to get from any one location to any other location on the same map. Suppose that I wanted to go home by a route which took in, say, a petrol filling station and a convenience store to pick up some milk. I did not have a ready-to-hand route, so I had a problem. Without a map it would have been a difficult one, but with a map it was easy to work out a way of doing it. To do this by learning all the possible separate routes would entail memorizing 3080 of these. This shows rather well the problem-solving power which structured knowledge makes available to us.
This principle is quite general, and the idea of a road map generalizes well into all kinds of knowledge structures (psychologists call them schemas) with the help of Tolman's useful metaphor of a cognitive map.
When I was studying for my psychology degree, I read about an experiment which I have remembered ever since (though unfortunately not the reference). There was a short wire mesh fence, with food on one side and an animal on the other, thus.
The food was clearly visible through the fence, and the hen made straight for it. Coming up against the fence, it could get no further, and continued indefinitely trying to get through the fence. The dog, on the other hand, immediately turned aside, ran round the end of the fence, and got the food. The dog could make a detour to achieve its goal, the hen could not.
It is important that in our efforts to achieve our present goal we act more like the dog than the hen. And theory, even the small amount described so far, tells us that the best way to help students to become good at problem solving involves taking them via a detour, that of building up the necessary knowledge structures. Once they have these, they will be equipped to solve a number and variety of problems far greater than could be possible by any direct approach. This is particularly necessary in today's world of rapidly advancing science and technology. Some of the future problems which our present students will need mathematical knowledge to solve do not yet even exist.
Another aspect of theory which is of importance in the present context is the process of abstraction by which we form concepts of any kind. Since mathematical concepts are a particularly abstract kind of concept, we particularly need the help of theory for teaching these. My own theory, as presented in a number of publications (see especially Skemp 1971, 1979, 1989) predicts the following.
The kind of learning situation which best supports the formation of mathematical concepts, and the building of mathematical schemas (knowledge structures) is one which satisfies the following requirements. These predictions have been confirmed experimentally.
(i) It should provide embodiments of the concept having the least amount of irrelevant information which has to be ignored while forming the concept. (The technical term is 'low-noise.')
(ii) There should be a number of examples of the concept, close together in time.
(iii) There should be just one new concept to be learnt at a time, and
(iv) This should be one for which they already have an appropriate schema, so that they can connect the new concept to it and thereby learn with (relational) understanding.
The above requirements are difficult or impossible to achieve in 'real-world' problems, but are specifically catered for in SAIL through Mathematics. Can any reader suggest real-life problems from which children can learn the mathematical concept of "addition" which satisfy the foregoing requirements at all, leave alone better than Stepping Stones, Crossing, Slippery Slope, Explorers (to name but a few). And from what real-world problems can students gain understanding of the concepts which give validity to the method of long multiplication? These concepts, five in all, are among the main foundations for learning algebra in the future. One could go on and on with examples of this kind. Real-world problems are also unlikely to provide enough repetition to consolidate newly-formed concepts, and to establish new skills as well-established routines.
I have already suggested that the essence of a problem is that there is something we want to do for which we do not have a ready-to-hand method. Solving the problem means finding one or more suitable methods, and then applying them. For mathematical problems, we can distinguish the following main categories. A given problem may be in more than one category at a time, which naturally makes it harder. For beginners, it would seem desirable to ensure that this is not the case.
Category 1. A verbally stated problem for which a mathematical model has to be constructed (this is the non-routine part), using mathematical knowledge which we already have. When this has been done, there is a routine method for (e.g.) doing the calculation, solving the equation, or whatever. Finally, the result has to be interpreted in the context of the original problem. These are often called word problems; a more general term is problems in applied mathematics.
Category 2. A mathematical task which is itself of a non-routine nature, but which can be solved by using mathematical knowledge which we already have. These are problems in pure mathematics. For short, we may call them mathematical puzzles.
Category 3. A task in pure or applied mathematics for which we do not yet have the necessary mathematical knowledge. I think that problems of this kind are necessarily also in category 2, but not the other way about. A classical problem for the Greek mathematicians was to find the length of the diagonal of a unit square, when their available number systems did not go beyond rational numbers. Historically, this problem led to the invention of irrational numbers. These we may call problems outside our present domain. An important special case of these are problems in our frontier zone. Each new concept presented in SAIL through Mathematics is a problem of this kind, in a carefully devised and field-tested embodiment which (unlike real-world problems) satisfies the requirements listed in section 4.
We can distinguish three stages of abstraction relating to problem solving (there may be more).
(i) From real-life situations we abstract a conceptual model, often represented by words.
(ii) From this conceptual model we abstract the concepts which matter for our problem.
(iii) From these concepts we abstract a mathematical model.
We ourselves often do these in rapid succession, from long practice. But abstraction is often much harder than working with the mathematical model when we have it.
Next, at the abstract level, we manipulate the model in ways which correspond to the events we are interested in. So we need to be clear about this correspondence, too. Word problems start at stage (ii). Real-life problems start at stage (i). Finally, we work in the reverse direction.
Predicted result in the physical world or word problem we re-embody Result of these manipulations
Putting our plan into action we re-embody Making a mental plan
The three diagrams which follow offer an overview of the differences which have been discussed, between the direct approach, which suggests (reasonably enough at a common-sense level) that the best way for students to learn real life problem-solving with the help of mathematics is to start by giving them problems of this kind; and the theory-based approach, which suggests that students will get there more successfully by a detour. First, help them to build up the structured mathematical knowledge which is essential for solving for solving real-life problems. They will then have the mental equipment they need for successful real-life problem solving.
Skemp, R. R. (1976). "Relational and Instrumental Understanding in the Learning of Mathematics," Mathematics Teaching, no. 77. [Also the Prologue in Mathematics in the Primary School.] Skemp, R. R. (1971, 2nd edn 1986). The Psychology of Learning Mathematics. Harmondsworth: Penguin Books. Skemp, R. R. (1979). Intelligence, Learning, and Action: A Foundation for Theory and Practice in Education. Chichester: Wiley. Skemp, R. R. (1989). Mathematics in the Primary School. London: Routledge. The American Commission on Excellence in Education. (1983). A Nation at Risk: The Imperatives for Educational Reform. Witney, H. (1985). "Taking Responsibility in School Mathematics Education." Proceedings of the Ninth International Conference for the Psychology of Mathematics Education, vol. 2, The State University of Utrecht.