Tony Gardiner on Problem Solving

This is our final excerpt from Teaching Mathematics at Secondary Level Tony Gardiner’s 2016 commentary and guide to the English Mathematics Curriculum. (The first two excerpts are here and here.) It is a long and beautifully clear discussion of the nature of problem-solving, and its proper place in a mathematics curriculum (pp 63-73). (For Australia’s demonstration of improper placement, see here, here and here.)

Gardiner frames his discussion around four bullet points from the English Key Stage 3 (early secondary) Program of Study:

Solve problems

• develop their mathematical knowledge, in part through solving problems and evaluating the outcomes, including multi-step problems
• develop their use of formal mathematical knowledge to interpret and solve problems, including in financial mathematics
• begin to model situations mathematically and express the results using a range of formal mathematical representations
• select appropriate concepts, methods and techniques to apply to unfamiliar and non-routine problems.

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These four bullet points are clearly meant to encourage pupils and teachers to see school mathematics as more than endless practise with dry-as-dust formal technique. But beyond this admirable aspiration, it is far from clear what exactly is being advocated. We base our commentary on three questions.

• What is meant by a “problem”, rather than (say) an “exercise”?
• What does it mean to “solve problems”?
• And why are “multi-step” problems important?

2.3.1

We begin by clarifying the distinction between “exercises” and “problems”.

An exercise is a task, or a collection of tasks that provide routine practice in some technique or combination of techniques. The techniques being exercised will have been explicitly taught, so the meaning of each task should be clear. Each sequence of exercises is designed to cultivate fluency in using the relevant techniques, and all that is required of pupils is that they implement the procedures more-or-less as they were taught in order to produce an answer. The overall goal of such a sequence of exercises is merely to establish mastery of the relevant technique in a suitably robust form. In particular, a well-designed set of exercises should help to avoid, or to eliminate, standard misconceptions and errors.

Exercises are not meant to be particularly exciting, or especially stimulating. But they can give pupils a quiet sense of satisfaction. Without a regular diet of suitable exercises, ranging from the simple to the suitably complex (including standard variations), pupils are likely to lack the repertoire of basic techniques they need in order to make sense of mildly more challenging tasks …. In other words,

exercises are the bread-and-potatoes of the mathematics curriculum.

Pupils in England clearly need more (carefully prepared) “bread-and-potatoes” exercises than they currently get. However, bread and potatoes alone do not constitute a healthy diet. Pupils also need more challenging activities both to whet their mathematical appetites, and to cultivate an inner willingness to tackle, and to persist with, simple but unfamiliar (or “non-routine”) problems. A problem is any task which we do not immediately recognise as being of a familiar type, and for which we therefore know no standard solution method. Hence, when faced with a problem, we may at first have no clear idea how to begin.

The first point to recognise is that a task does not have to be all that unfamiliar before it becomes a problem rather than an exercise! In the absence of an explicit problem solving culture, an exercise may appear to the pupil to be a problem simply because its solution method has not been mentioned for a week or so, or because it is worded in a way which fails to announce its connection with recent work. The second point is that the distinction between a problem and an exercise is not quite as clear-cut as we have made it look, and is to some extent time- and pupil-dependent. For example, an “I’m thinking of a number” problem from Year 5 or Year 6 should by Year 8 be seen to be a mere exercise in setting up and solving a simple equation.

Most useful techniques involve a chain of simple steps, and the technique as a whole is only an effective tool if the complete chain can be carried out entirely reliably—a requirement which may only be achieved after extensive practice. Examples include: any of the standard written algorithms; the process of turning a fraction into a decimal; the sequence of steps required to add or subtract two fractions, or to solve an equation or inequality, or to multiply out and simplify an algebraic expression. Hence each set of exercises should include tasks that force pupils to think a little more flexibly, and that require them to string simple steps together in a reliable way. Too many sets of exercises get stuck at the level of “one piece jigsaws”—with one-step routines being practised in isolation, ignoring key variations. Pupils need to learn from their everyday experience that the whole purpose of achieving fluency in routine bread-and-potatoes exercises is for them to learn to marshal these techniques to solve more demanding multi-step exercises, and more interesting, if mildly unsettling, problems.

2.3.2

This distinction between exercises and problems affects how we choose to introduce each new topic or technique. Should we concentrate on relatively simple examples that minimise pupil difficulties, and which seem likely to guarantee a quick pay-off? Or should we—when working with the whole class—move quickly on to examples that provide a significant challenge, and so require pupils from the outset to grapple with (carefully chosen) tasks of a more demanding nature?

How challenging one can safely be will depend on the pupils. But experience from those who observe lessons in other countries suggests that the English preference for concentrating the initial worked examples on easy cases increases the extent of subsequent failure. Easy initial examples lead to cheap apparent success; but this initial pupil success may be based on pupils’ own inferred methods that appear to work in easy cases, but which are flawed in some way; or on backward-looking methods, that seem (to the pupil) to work in simple instances, but which do not extend to the general case. So we need to consider the benefits of starting each new topic with a harder “class problem” that brings out the full complexity of the method that we want pupils to master, and then to follow this up with exercises that may start simply, but which oblige pupils to think flexibly from the outset, and to handle standard variations including inverse problems.

2.3.3

The last 30 years have witnessed a consistent concern about pupils’ ability to “use” the elementary mathematics they are supposed to know. Previous versions of the [English] mathematics National Curriculum displayed an admirable determination to incorporate “Using and applying” within teaching and assessment. But such determination is not enough. The experience of the last 25 years in England is more useful as a guide to what does not work than to what does work. Much effort has been expended in trying to do better—but with limited effect. In particular, ambitious attempts to coerce change—using extended investigations, coursework, and “modelling”—have mostly served to demonstrate what should not be officially required at this level.

Somewhere along the line we seem to have lost sight of simple word problems. Word problems typically consist of two or three short sentences, from which pupils are required

• to extract the intended meaning and any required information,
• to identify what needs to be done,

and then

• to carry it out, and interpret the answer in the context of the problem.

Everyday uses of elementary mathematics tend to come in some variation of this form. Yet the simplest exercises, which might be solved routinely if they were presented without words, become powerful discriminators when given this gentle packaging. The need for pupils to read and extract the relevant data from two or three English sentences may appear routine—but it is a skill that has to be learned the hard way, and that constitutes the initial stepping-stone en route to the ultimate solution of almost any problem. This simple format can be tweaked to cover the standard variations of the underlying task (e.g. so that it appears both in direct and in the various indirect forms).

During Key Stage 1 [the first two years of schooling] word problems are important because they reflect the fundamental links between

• the world of mathematical ideas and mathematical reasoning,

and

• the world of language.

Indeed, for young children, the logic of mathematics is inextricably bound up with the grammar of language.

At later stages word problems continue to serve as an invaluable way of linking the increasingly abstract world of mathematics and the world where its ideas can be applied. That is, they constitute the simplest exercises and problems in any programme that seeks to ensure that elementary mathematics can be used.

The suggestion that improving mathematical literacy depends on rediscovering the world of carefully structured word problems is both more ambitious and more modest than what has been attempted in recent English reforms.

• It is more ambitious in that the evidence from other countries shows just how much more we might achieve were we to incorporate a permanent thread of such focused material from the earliest years.
• It is more modest in that it explicitly encourages more focused (and hence more manageable) tasks—short problems with a clearly specified beginning and end, but with the path from one to the other left for the solver to devise. Such problems have “closed” beginnings and “closed” ends, but are open-middled. Almost any mental arithmetic problem, or word problem, might serve as an example. Suppose we ask:

“I pack peaches in 51 boxes with 16 peaches in each box.

How many boxes would I use if each box contained just 12 peaches?”

What is given and what is required is “closed”—i.e. specified uniquely. But the mode of solution is left entirely open:

• some pupils might calculate the total number of peaches and then divide by 12;
• one would prefer to see a more structural version of this representing
  the total number of peaches as “” without evaluating, and the
  required number of boxes as  before cancelling

     

• others might notice that , and look for the number satisfying ““;
• while some might remove 4 peaches from each of the 51 boxes and
  group the 4s in groups of to get 17 additional boxes.
  

2.3.4

Pupils need a regular diet of problems and activities designed to strengthen the link between elementary mathematics on the one hand and its application to simple problems from the wider world on the other. Word problems are only a beginning.

Some have advocated using “real-world” problems. But though these may have a superficial appeal, their educational utility is limited. Problems which support the move towards using and applying beyond the limited world of word problems need to be very carefully constructed, so that the real context truly reflects the mathematical processes pupils are expected to use as part of their solution. (Problems which have to be carefully designed in this way are sometimes called “realistic”.)

The related claim that technology allows pupils to work with “real-world problems” and with “real (or ‘dirty’) data” becomes important once the underlying ideas have been grasped. However, for relative beginners the claim too often ignores the distracting effect of the noise which is created by “real” contexts, by “real” data, and by the non-mathematical interface that so easily prevents pupils from grasping the underlying mathematical message.

2.3.5

The official programme of study makes repeated reference to the need to solve multi-step problems. A multi-step problem is like a challenge to cross a stream that is too wide to straddle with a single jump, so that the prospective solver is obliged to look for stepping-stones—intermediate points which reduce the otherwise inaccessible challenge of crossing from one bank (what is given) to the other (the completed solution) to a chain of individually manageable steps. In elementary mathematics, this art has to be learned the hard way. It should not be seen as optional, or as a matter of taste. It is central to what elementary mathematics is about, and to how it is used.

One might think that—given the original emphasis on Using and applying—this goal has been an integral part of the [English] National Curriculum since its inception. But that is not quite true—for we have too often confused

• “solving problems”, and tackling “multi-step” problems with
• real-world problems, and extended tasks.

The limitations of “real-world” problems were outlined in the previous Section 2.3.4. An extended task allows pupils considerable freedom, and can be beneficial precisely because the outcomes lie to some extent outside the teacher’s control. However, this lack of predictability and control means that extended tasks are not an effective way for most pupils to learn the art of solving multi-step problems. For most teachers, this art is much more effectively addressed through short, easily stated problems in a specific domain (such as number, or counting, or algebra, or Euclidean geometry), where

• what is given and what is required are both clear,
• but the route from one to the other requires pupils to identify one or more intermediate stepping-stones (that is, they are “open-middled”)—as with
  • solving a simple number puzzle, or
  • interpreting and solving word problems, or
  • proving a slightly surprising algebraic identity, or
  • angle-chasing (where a more-or-less complicated figure is described and has to be drawn, with some angles given and some sides declared to be equal, and certain other angles are to be found—using the basic repertoire of angles on a straight line, vertically opposite angles, angles in a triangle, and base angles of an isosceles triangle), or
• proving two line segments or two angles are equal, or that two triangles are congruent (where the method of proof is not immediately apparent).

The steps in the solution to a multi-step problem are like the separate links in a chain. And the difficulty of such problems arises from the need to select and to link up the constituent steps into a single logical chain. Suppose pupils are faced with:

Question: “I’m thinking of a two-digit number N <100, which is divisible by three times the sum of its digits? How many such numbers are there?”

In Year 7 pupils may see no alternative to guessing, or to testing each “two digit number” in turn. But by Year 9 one would like some to respond to the trigger in the question

“three times the sum of its digits”

by gradually noticing some of the hidden stepping stones.

Steps toward a solution

1. The number has to be a multiple of 3 (“divisible by three times the sum
   of its digits”).
2. Hence the sum of its digits must be a multiple of 3 (standard divisibility test).
3. But then the number is divisible by 9 (“divisible by three times a multiple of 3”).
   

1. And so the sum of its digits must be a multiple of 9 (standard divisibility test).
   
2. So the number is divisible by 27 (“divisible by three times a multiple of 9”).
   

6. So we only have to check 27, 54, and 81. QED

The sequencing of the steps, and the connections between the steps, are part of the solution. In short, basic routines become useful only insofar as sufficient time is devoted to making sure they can be linked together to solve more interesting (multi-step) problems.

2.3.6

Expecting pupils to select and to coordinate simple routines to create a chain of steps in order to solve simple multi-step problems should be part of mathematics teaching for all pupils. In contrast, recent efforts to improve the effectiveness of mathematics instruction in England have concentrated on:

• the teacher, textbook author, or examiner breaking up each complex procedure into easy steps, and then concentrating on teaching and assessing the easy steps, or atomic outcomes (one-piece jigsaws),
• monitoring centrally whether these atomic outcomes can be performed in isolation, and
• ignoring the fact that we have neglected the most demanding skill of all—namely that of integrating the separate steps into an effective multi-step procedure.

The evidence from international studies confirms what should have been obvious: this reductionist process of de-constructing elementary mathematics into atomic parts, combined with central monitoring that rewards partial success, has distorted the way pupils and teachers perceive elementary mathematics in a most unfortunate way. Improved problem solving and more effective mathematics teaching depend on enhancing the skill of the teacher. In contrast, the policy of focusing on targets and testing, and our misplaced dependence on crude measures of “pupils’ progress”, have tended to undermine the authority, the professional judgement, and the perceived long-term responsibility of the teacher.

Solving problems is hard. Any system that uses targets and testing to exert pressure on schools soon discovers the awkward facts that assessment items that require pupils to link two or more steps

• have a high failure rate, and
• generate pupil responses whose profile is at odds with the contractual demands placed on those who design centrally administered tests.
  

Such problems are therefore deemed unsuitable, and the tests tend to concentrate on more manageable one-step routines (or break down longer questions into a pre-ordained sequence of one-step “subroutines”). As long as teachers are judged on test outcomes, and as long as unfamiliar, multi-step problems are largely excluded from the official tests, teachers will continue to conclude that “in the (short-term) interests of their pupils” they dare not waste time developing the only thing that matters in the long run—namely:

to provide their pupils with the skills and attitudes they need for the next phase.

In short, England has adopted an “improvement strategy” that guarantees neglect of the delicate art of solving multi-step problems, and that is therefore self-defeating. Central prescription, and political pressure to demonstrate relentless year-on-year improvement, have resulted in a national didactical blind spot, with curriculum objectives and assessment—and hence teaching—becoming atomised, so that pupils are only expected to handle “one piece jigsaws”. Exams have routinely broken down each problem into a succession of easy steps—in order to minimise the risk of failure, and to ease “follow through marking” for the examiner. Teachers have then concluded that the delicate art of interlinking simple steps can be safely ignored. And we have all pretended that

• candidates who can implement (most of) the constituent steps separately
• have thereby achieved mastery of the integrated technique.

This is a delusion. The individual steps may be a starting point; but the power and challenge of elementary mathematics lies in learning how simple ideas can be combined to solve problems that would otherwise be beyond our powers. That is, the essence of the discipline lies not so much in the techniques themselves as in the connections between its ideas and methods. Hence the curriculum (and, where possible, its assessment) need to cultivate the ability to tackle multi-step problems without them being artificially broken down into steps.

A curriculum or syllabus can specify the individual techniques, or steps; but this is futile if one then forgets that it is the linking of the material which determines whether it can be effectively used to solve problems. This interlinking is an elusive property, which depends entirely on the way the material is taught: that is, it depends on the teacher. So we need a system in which teachers are free (nay, in which teachers feel professionally obliged) to value this activity in their classrooms, even though its value will only become apparent at subsequent stages—after their pupils have moved on to other classes.