The following is an article by Tony Gardiner, originally published in 1996 in the Mathematical Gazette. It is reproduced here with the kind permission of Tony, the Chief Editor of the Mathematical Gazette and the Mathematical Association. The original article is available on JSTOR, here (via an educational library), and we’ve also separately posted the problems in Tony’s article here.
‘Problem-solving’? Or Problem solving?
Problems: the lifeblood of mathematics and education
David Hilbert (1862-1943) was one of the most outstanding mathematicians of the modern era. At the International Congress of Mathematicians in Paris in 1900 he presented twenty-three major research problems which he felt would be important for the development of mathematics in the twentieth century. These problems all seemed very hard, but in bringing them to the attention of other mathematicians Hilbert felt the need to stress that this should not be used as an excuse to put off trying to solve them.
‘However unapproachable these problems may seem to us and however helpless we stand before them, we have, nevertheless, the firm conviction that their solution must follow by a finite number of purely logical processes. This conviction of the solvability of every mathematical problem is a powerful incentive to the worker. We hear within us the perpetual call: There is the problem. Seek its solution. You can find it by pure reason. In Mathematics there is no ignorabimus.’ 
Hilbert’s judgement that these problems would play a central role in the mathematics of the present century was remarkably accurate. But the most interesting thing for those of us who teach mathematics is his rallying call: ‘However unapproachable these problems may seem at first sight, and however helpless we stand before them, we have the firm conviction that their solution must be possible by purely logical processes. There is the problem. Seek its solution. You can find it by pure reason.’
This is the message which all mathematics teachers should seek to convey to their students, no matter what their age or ‘level’, and no matter how hard they may find the subject.
Every problem can be solved by combining simple, familiar steps. If only one step is needed, there is no challenge – the task is an exercise rather than a problem. Exercises are important – for it is absolutely vital that one-step routines be mastered completely and fluently. But mathematical problems begin when two simple steps have to be selected and combined to achieve the goal.
This activity of solving problems is something all our pupils are capable of on an appropriate level, if we provide the two key ingredients. First, the necessary one-step routines have to become completely automatic, so that the pupil’s mind is free for the higher tasks of selecting and combining the steps in the required way to solve the problem. Second, even when we are developing pupils’ fluency in some one-step routine, we should still include a smattering of two-step problems and make sure that pupils realise that the one-step routine is a means to an end, not an end in itself. (The books [2, 3] contain hundreds of attractive, elementary problems which require the solver to select and coordinate two or more simple steps.)
To find the area of a square given the edge length is an exercise. If the diagonal is given, one must know something about isosceles right-angled triangles, or use Pythagoras to find the edge length, or appreciate that the diagonal cuts the square into two right-angled triangles whose areas can be calculated easily from the given information. Yet when this question was given recently (in multiple choice form, with options 2, 4. 4√2, 8, 16) to one hundred Year 11 and 12 pupils in an excellent English comprehensive, not one chose the correct answer. This may be an extreme example but it illustrates the nature of our dilemma.
If one only reads the question, one should wonder how three quarter-circles and one three-quarter circle could possibly fit together to make any shape at all. One glance at the diagram shows that the three quarter-circles are curved inwards, so that one should clearly not simply add the areas of three ‘quarter-discs’ and one ‘three-quarter disc’. Yet this meaningless addition is now routinely presented as a ‘solution’ by apparently well-qualified students (including almost half of the 1995 first-year Single Honours Mathematics intake at a good English university). This suggests strongly that the questions such students have grown up on have trained them to believe that they will never have to do more than carry out one-step procedures. This same misconception undermines even those students who avoid the above crass mistake. For they appear to have no notion that the solver has to actively create stepping-stones between what is given and what is sought (in this case, marking in crucial points and lines, such as the centres of the four circles and the lines joining them).
When faced with an unfamiliar and apparently very difficult problem, there is always the temptation to imagine that it is too hard, that progress towards a solution requires some trick or technique that we have not yet learned, and that the solution is therefore beyond our powers. This defeatist view is all the more plausible because it must sometimes be true! But to be blunt, it is nearly always a cop-out. Let me try to explain why.
During the nineteenth century it became clear that the more scientists discovered about nature, the more they realised how little they knew, and that one could never hope to discover the whole truth. This realisation was summed up by one philosopher, Emil du Bois-Reymond, in the catch-phrase ‘ignoramus et ignorabimus‘ – ignorant we are and ignorant we shall remain. This catch-phrase certainly caught on! And as the new century dawned, David Hilbert felt it important to state as clearly as he could that mathematics is different. In mathematics, said Hilbert, we can tackle problems ‘with the firm conviction that their solution must follow by a finite number of purely logical processes’. And, as if to underline his assertion, one of his twenty-three outstanding problems was solved almost immediately.
Hilbert was talking about mathematics research. However his principle applies just as well to solving challenging problems at school level. When mathematicians begin to explore a problem, they take it for granted that it must be possible to make progress towards a solution using only the tools currently available. Of course, they are sometimes wrong; but that is not the point. Mathematicians know perfectly well that the assumption is, strictly speaking, irrational, in that it cannot be justified logically–and is, in general, clearly false. But it is an invaluable working hypothesis. Though strictly illogical, the assumption that every mathematical problem can be solved has justified itself so often in practice that it has become a powerful conviction – a conviction which is psychologically invaluable each time we experience that feeling of helplessness when trying to get to grips with a hard mathematical problem.
Faced with an unfamiliar problem, the mathematician–whether young or old–is like someone with a hopelessly small bunch of ‘keys’ who is trying to open up some fiendishly difficult Chinese puzzle box. At first glance the surface seems totally smooth, without a single visible crack. If one were not convinced that it is indeed a Chinese puzzle box, and that it can in fact be opened, one would soon give up. Knowing (or rather believing) that it can be opened makes one willing to persevere, to keep searching until one begins to discern the slightest hint of a crack here and there. One still has no idea how the pieces are meant to move, or which of the available ‘keys’ may help to open up the first layer of the puzzle. But, by trying the most appropriate-looking keys in the most promising cracks, one eventually stumbles on one that fits exactly, and the pieces begin to move.
In a good puzzle success is never the result of pure chance. Indeed, once one has discovered the way in, one often realises that it should have been ‘obvious’ where to begin. But things usually become obvious only in retrospect. If the puzzle is unfamiliar and well made, success may require persistence, faith, and much time. Exactly the same is true of any mathematics problem. So one should never give up too easily, and should always be prepared to look back after solving a problem to see wha should perhaps have done differently. That is how human beings learn.
‘Problem-solving’: the death of mathematics and of education
One of the many paradoxes we have to face up to is that the collapse in our best students’ ability to tackle simple mathematical problems (such as Problems A and B above) has coincided with the increased emphasis on teaching and assessing ‘problem-solving’ explicitly. We have to ask whether the two phenomena are linked. More generally, we must reconsider our failure to distinguish clearly between problem solving (which should be a fundamental requirement of all mathematics teaching to children) and the metacognitive theory of ‘problem-solving’ (which can be a fascinating diversion for consenting adults in private, but which is not obviously helpful – and which may even be harmful – for beginners). In exploring such issues, we have to go beyond the touching, but superficial, belief that, since ‘problem-solving’ was intended to help pupils solve problems, it cannot therefore be implicated in the observed decline in the ability to solve simple problems. Moreover, we must not shrink from accepting uncomfortable conclusions, nor cling to the status quo, simply to avoid giving comfort to some imagined ‘opposition’.
In recent years English education has witnessed a veritable epidemic of change. If some have objected, they have usually done so ineffectually. Too many of us have looked on – choosing not to challenge (and hence failing to clarify) both the ostensible ends and the declared means. All that is needed for evil to prevail is that good men should remain silent.
One reason we have made life so easy for the proponents of change is that they have based their claims on a rhetoric of plausible sounding ‘self-evident truths’. Which of us would choose publicly to oppose those who exhort us to ‘raise standards’? Or to provide ‘exciting lessons and good teaching for all pupils”? Or to improve the participation of minority groups? Or to ‘prepare our pupils for adult life”? Or to insist that teachers should ‘motivate’ what they teach? Or to embrace ‘modern technology”? Or those who suggest that ‘learning how to learn’ is more important than any individual fact, and that ‘problem-solving’ is more important than solving any individual problem’? We may know in our bones that such rhetoric is one-sided, that ill-considered change does untold damage, and that improvements in education cannot be so easily achieved as these slogans pretend; yet it is not easy to challenge this naive optimism without sounding like a remnant of some prehistoric age. Some may have felt obliged to learn (and even use) this educational ‘Newspeak’. A few may have succeeded in splicing these additional demands onto their traditional fare. But most struggle to pay lip-service to the new orthodoxy while remaining silent – hoping against hope that, despite their gut-feeling, some unexpected good may come of the whole business. The rhetoric may be half-baked, and the schemes based on it may be hastily cobbled together; yet at least their declared purpose seems well-intentioned. The road to hell is paved with precisely such pious hopes.
It would be foolish to claim that there have been no positive outcomes from the upheaval of the last fifteen years. What I would claim is that any benefits have been social, not mathematical: mathematics classrooms are generally friendlier places than they were. However the overall mathematical impact of recent changes could be summarised by saying that we have not only lost our way, but have thrown away the map, and are rapidly reaching the stage where relatively few teachers, the more experienced, can remember very clearly what the map used to show, or why it was important.
Knowledge, we are told, is growing so fast that there is no longer any point concentrating on ‘facts’ which may become obsolete before our pupils reach their forties. Instead (so the argument goes) we should focus on more permanent ‘skills’, such as
(a) ‘research skills’, which might equip pupils to find out for themselves;
(b) ‘creativity’, which will give pupils an insight into where literature or mathematics comes from, and help them think in non-standard ways;
(c) ‘problem-solving’, which could equip them better for, and make them more useful in, the real world.
I have been working with groups of pupils and teachers at many levels for over twenty years. In everything I do and write I seek to cultivate such approaches as part of a larger goal. Nevertheless I contend that the current emphasis is totally misplaced – and for three reasons.
First, such skills do not exist in their own right. There is no disembodied ‘problem-solving’ skill; there is only the practice of solving, and the consequent ability to solve problems in a particular context. Similarly, there is no generalised skill called ‘creativity’, and there is no such thing as ‘learning how to learn’. The words have meaning only in a context. The context must come first, and must be valued in its own right. Any meta-technique which one wishes to emphasise has to be firmly embedded within that context. To wrench it out of context creates something entirely artificial, which will quickly degenerate (as has the ‘Attainment Target’ Using and applying mathematics in the National Curriculum Mathematics Orders), and which will divert valuable time and energy from more important tasks.
Second, while middle-aged educationists may enjoy discussing ‘generalised skills’ in seminars, adolescents are far more interested in the power and pleasure they experience from getting their minds round specific techniques. The young cellist revels in playing spread chords; the young rugby player enjoys mauling, or fielding the ball under pressure and running with it; and the young mathematician delights in learning to juggle numbers in his or her head, or ‘angle-chasing’ in a geometry diagram. To make technical progress, to simplify one’s mental schemas, it is important from time to time to reflect on and to reorganise one’s experience. But having ‘reflected’, young minds want to make progress by moving on to further technical challenges. In particular, in mathematics pupils need a diet of short, multi-stage problems with exact solutions, not the long-winded, self-indulgent, ‘methodologically correct’ variety (each broken down into minute steps to guarantee success) which are now peddled by the exam boards and by those who construct SATs. Methodological self-indulgence is entirely alien to adolescents (as it should perhaps be to others). Deeper insight into a discipline comes slowly – more Freude durch Kraft than Kraft durch Freude.
Third, the proposed shift of emphasis (from ‘knowledge’ to ‘skills’) makes mathematics teaching far more difficult. Insofar as one is justified in being dissatisfied with the effectiveness of mathematics teaching in the past, it is hard to see how things can be improved by imposing a far more demanding role on precisely those teachers who gave rise to the original dissatisfaction. In short, the official curriculum should concentrate on carefully chosen, and clearly specified content, while encouraging teachers, textbook authors and examiners to approach that content in gently imaginative ways.
Many unsuspecting teachers welcomed the emphasis on ‘problem-solving’ when it was first introduced. They may not then have realised that the idea would be hijacked by curriculum theorists, and used for their own ends (to transform the traditional focus of education on content and knowledge to a new diet of imaginary ‘skills’); or that the resulting activities would concentrate on ‘problems’ – both ‘pure’ and ‘applied’ – that could not be ‘solved’ at all (at school level), but which could only be ‘explored’, or ‘investigated’; or that the commitment to assess everything that is taught (within a system increasingly driven by ‘market forces’) would guarantee that an essentially unpredictable activity (tackling problems) would degenerate into a totally predictable ritual having nothing to do with either mathematics or education. It is time to take stock.
1. D. Hilbert, Mathematical problems, Bulletin of the American Mathematical Society 8 (1902) pp. 437-479 (the German original appeared in the Göittinger Nachrichten (1900) pp. 253-297).
2. A. Gardiner, Mathematical challenge, Cambridge (1996).
3. A. Gardiner, More challenging mathematics, Cambridge (1996).