# Problem solving In Mathematics

Derek Holton, Jim Neyland, and Julie Anderson
Abstract:

Not all teachers are comfortable with using the technique of problem solving, which is an attempt to find the answer to a problem when the method of solution is not known. This research shows how problem solving can be used in secondary school mathematics classes.

Journal issue:

# PROBLEM SOLVINGIn Mathematics

Derek Holton Department of Mathematics and Statistics, University of Otago

Jim Neyland Mathematics and Science Education Centre, Victoria University of Wellington

Julie Anderson Department of Mathematics and Statistics, University of Otago

## WHERE ARE WE?

In 1993, a new national mathematics curriculum which emphasised the importance of problem solving was introduced in New Zealand. Three years later there was evidence that a significant proportion of teachers believe problem solving to be important and many are making some effort to incorporate problem solving into their teaching. However, despite this evident commitment, problem solving does not seem to occupy a regular place in most classrooms.

By problem solving we mean the attempt to find the answer to a problem when the method of solution is not known. In problem-solving situations the solver has to use strategic skills to find appropriate mathematical techniques which will settle the question. Neyland (1995) discusses the similarities and differences between the problem-solving approach and seven other approaches to teaching mathematics.

It is worth noting that what is a problem to one person may not be a problem to another. For a five-year-old determining the number of legs three sheep have will almost certainly be a problem. However, it should not be a problem for most 15-year-olds. The five-year-old may need to draw, use equipment, or employ some other method to solve the problem. The 15-year-old will just say “3 times 4 equals 12”.

Word problems do not necessarily involve problem solving. If a word problem is introduced to practise a technique which has just been acquired, it should be reasonably obvious what technique has to be used. On the other hand, problem-solving situations may be presented as wordy problems, though this is not always the case.

According to the National Council of Teachers of Mathematics:

Problem Solving should be the central focus of the mathematics curriculum. As such it is a primary goal of all mathematics instruction and an integral part of all mathematical activity. Problem Solving is not a distinct topic but a process that should permeate the entire program and provide the context in which concepts and skills can be learned.

We would like to underline the last sentence of the above quote as we find it difficult to differentiate between problem solving on the one hand and mathematics, or its essential nature, on the other.

In Australia, these views are reflected in A National Statement on Mathematics for Australian Schools (1990), where problem solving is one of four sub-headings in the Mathematical Inquiry strand:

The mathematical processes described within this strand cannot be developed in isolation from the work of other strands. They should pervade the whole curriculum.

In New Zealand, the importance of problem solving has been underscored by the Ministry of Education publication Implementing Mathematical Processes in Mathematics in the New Zealand Curriculum (1995):

Problem solving is the first heading in the Mathematical Processes strand; unless the ability to solve problems is developed, there is little point in studying mathematics.

The Mathematical Processes strand in the Mathematics in the New Zealand Curriculum document provides the umbrella for learning in the other five strands of Number, Measurement, Geometry, Algebra, and Statistics. It incorporates three areas: problem solving, developing logic and reasoning, and communicating mathematical ideas. We have decided to focus our attention on problem solving because it can provide the medium for both developing mathematical processes and interlacing the learning of specific knowledge and skills. However, is problem solving valued in our classrooms?

## THE STUDIES

The work in this paper is based upon two pieces of research. The first was a questionnaire that was given to 11 secondary school mathematics departments in a provincial city of New Zealand. Some 53 teachers responded to the questionnaire (representing a 100 percent response rate from the schools) and the issues raised were discussed with seven experienced mathematics teachers.

The second study involved observations of problem-solving sessions in five secondary classrooms at the form 3 and 4 level [year 9 and 10 in New Zealand; year 8 and 9 in Australia].

The main questions in the questionnaire were:

In what ways are problem-solving activities being used at present and how often are problem-solving approaches being used?

What problem-solving strategies are being promoted in the classroom?

How can teachers integrate skill and content learning with a problemsolving approach to teaching and learning?

How do teachers want to use problem solving in their classrooms in the future?

What help do teachers need to implement problem-solving approaches to their teaching of mathematics?

### How is problem solving being used and how often?

There are a variety of approaches to problem solving in the classroom. The questionnaire avoided questions about specific types of activities (open-ended tasks, investigations, short challenging tasks, projects, or mathematical modelling) and concentrated on broader areas of usage that had been observed in classrooms. The six categories considered were homework tasks, lesson starters not directly related to the content of the lesson, lesson starters related to the content of the lesson, throughout lesson, at the end of a unit of work, and catering for the needs of more able students. See table 1 for results.

From table 1, we see that problem-solving activities are being used as homework tasks more regularly than any of the other five categories surveyed. However, of those teachers using homework tasks at least weekly, under half (48 percent) used problem solving through all of a lesson at least weekly. This suggests that homework tasks are often independent activities which are not supported by problem solving in the classroom.

### Lesson starters

One of the teaching approaches identified by Sigurdson et al. (1994) is the problem-process approach, which involves using simple problems related to the mathematical content of the lesson, solving these problems through student-teacher interaction in a whole class setting and focusing on the processes being used. While using problems as lesson starters seems a good starting-off point for the implementation of a problemsolving approach, over half of the teachers surveyed never used problemsolving starters or used them only sometimes. Less than four percent use lesson starters related to content daily. But, more surprisingly, stand-alone problems are used as lesson starters at least once a week by only 21 percent of respondents. This suggests that teachers should be encouraged to use short challenging tasks to initiate problem solving in their classroom. This might be a useful point to begin teacher professional development in this area.

### Throughout lesson

Over one-third of teachers (40 percent) stated that they were using a problem-solving approach throughout all of their teaching of a lesson, at least weekly. This suggests that a regular integrated approach to problem solving is occurring in a number of classrooms. However, evidence from later written responses suggested there may have been an overstatement of the actual situation.

The reason for the relatively small percentage of teachers using a problem-solving approach regularly through all lessons may be found in Burkhardt (1988). He believes that standard teaching methods are largely single track and depend on the method being explained by the teacher. On the other hand, problem solving is multi-track and is led by the methods proposed by individual students. Burkhardt feels that problem solving is difficult for the teacher. He identifies three areas giving rise to this difficulty—mathematical, pedagogical, and personal.

Mathematically the teacher must scan the different approaches that students are using and assess how useful each of these may be. The teacher then has to decide how best to complete the mathematical task from each of these starting points.

Pedagogically the teacher must decide when to help and when not to help, and what support and questioning to provide for each student or each group of students.

Personally there is the problem of teacher confidence. Because teachers may be confronted with approaches that they have not considered before, they need to believe that they can determine, possibly with student help, which approaches lead to solutions and which don’t. This is a potentially non-trivial barrier to tackling problem solving in class.

### End of unit

In the past it has been the practice to include word problems at the end of a section in order for students to practise newly learned skills in potentially novel situations. Some 27 percent of the respondents are using problem solving in this way at least monthly. It is worth noting though that all but one of the teachers who used problem solving at the end of a unit of work was also using problem-solving lesson starters or was also using problem solving throughout all of their teaching weekly. Applying skills to problems at the end of a unit of work seems to be only one component of a teacher’s problem-solving repertoire.

### Extension students

Many schools encourage their capable students to experience independent problem solving in a variety of ways. These include mathematics or computer clubs, small groups working apart from the normal class, through mathematics competitions, or via individual extension material. In a survey of talented mathematics students, Curran, Holton, Daniel, and Pek (1992) found that such students enjoyed the challenge and the open approach of problemsolving tasks. Burkhardt (1988) highlights a common belief in the need for problem solving for the gifted child but is reserved about how best to balance this need in the gifted students’ mathematical diet.

The survey shows that, at least monthly, individual students are being given specific extension opportunities in the area of problem solving by 43 percent of teachers. Other able students will benefit from being exposed to regular classroom problem-solving activities. However, of the 57 percent of classrooms which are not providing specific problem-solving experiences regularly for their capable students, some 41 percent do not provide opportunities for problem solving in the regular classroom at least monthly. Hence, in over 23 percent of classrooms, bright students are only occasionally being exposed to any form of problem solving.

### What strategies are being promoted?

Pólya (1973) stresses the importance of strategies for solving problems. Strategies are just means to discover a solution. They are almost always not the method of solution nor a justification of the answer. They are, however, a mechanism by which the answer may be found and then justified, if necessary, by some other means.

The suggested learning experiences in the problem-solving section of Mathematics in the New Zealand Curriculum include the “devising, using, and modifying of problem solving strategies”. The questionnaire in our study referred to 12 strategies from Mathematics in the New Zealand Curriculum. These were the strategies that we thought would be commonly used by teachers. The results are summarised in table 2.

The predominance of Find a pattern, etc is perhaps to be expected. It is one of the easiest techniques to promote because of the natural occurrence of patterning in students’ environments. In addition, children come to secondary school with a history of patterning experience from primary school for teachers to draw on. Further, the emphasis on patterning is encouraged by Mathematics in the New Zealand Curriculum. Exploring patterns and relationships is an achievement objective for all levels.

We believe that Make a list and Guess and check are also popular because they are easy to teach and can be applied to a range of problems. Equally it may be that these strategies are well suited to the types of problems that are frequently used in the early years of high school.

Surprisingly though, Draw a diagram was only rated as a commonly-used strategy by just over half of the teachers. It is not clear whether this is because teachers feel that this is not “proper” mathematics at the secondary level or whether its value as a tool is not appreciated. In the research project we have seen diagrams being of great value, especially to weaker students. It provides a closer representation of the problem than, for instance, algebra does, and hence may make some problems more accessible to students.

A similar accessibility can be provided through the strategies Make a model or Act out. These heuristics (or the means by which a solution is sought) give more control of the problem to the students and can provide motivation. Lovitt and Clarke (1988) advocate these approaches, referring to them as “kinesthetic”. It is now acknowledged that children learn in different ways and the tactile, physical, pictorial thinkers could well benefit if these strategies and Draw a diagram were used more often. It may be that some teachers avoid these strategies because they view them as being more appropriate for primary schools. They may also find the strategies too difficult to implement because of the resources needed and the time constraints on their lessons. Only a quarter of respondents regularly discuss Thinking creatively. It is difficult for a teacher to model with students that “flash of brilliance” which seems to come from nowhere. However, the fostering of such insights by discussion and brain storming is vital if we are to extend the horizons of students’ thinking and to enable them to strike out in new directions.

### Strategies need to be taught

Our research supports the conclusion that strategies need to be taught. They also need to be practised otherwise they become forgotten. This seems to be especially true for young students and students who are weak mathematically.

Generally a number of strategies can be used in one problem. For example:

Greedygrimes Charlie ate a total of 100 jellybeans in 5 days, each day eating 6 more than the previous day. How many jellybeans did he eat on the third day?

This problem was solved by students in form 4 [year 10 in New Zealand, year 9 in Australia] using:

The Australian National Statement on Mathematics says that the methods of good problem solvers are likely to be “idiosyncratic” and that students need to discuss a variety of strategies to increase their “awareness of the range of techniques available”. This is supported by Beagle (1979), Pólya (1973), and Schoenfeld (1992). The first of these authors states:

… problem solving strategies are both problem and student specific often enough to suggest that finding one (or few) strategies which should be taught to all, or most students, is far too simplistic.

It is not clear, though, how heuristics should be taught. Holton (1994) and Neyland (1994) share a concern that teaching should not lead to strategies being employed like another set of algorithms or rules. They also say that students should be encouraged to see how to translate strategies to other problems and situations. Strategy usage and teaching is not straightforward and much more research is required in these areas.

### How would you like to use problem solving in the future?

Just under 60 percent of the teachers surveyed want to use problem solving more often, suggesting that a receptive environment for future development does exist. They do identify a number of common difficulties though. Among these are:

the time necessary for resource collation and staff preparation,

a feeling that problem solving takes more time,

a feeling that assessing problem solving is difficult,

a feeling that lessons are difficult to control,

a perception that problem solving is difficult for weaker students.

Our experience is that problem solving does take more time, especially in the early stages. However, time can be saved later. One of the teachers in the research project commented that using a group investigative approach to construction tasks and locus problems enabled her students to develop a number of concepts in one lesson which usually took her two or three lessons of more structured teaching. This approach also freed her to help individuals who were struggling, resulting in all students achieving more than usual in the time.

There is a real concern for the needs of less-able students. This relates in particular to their reading ability. One of the classes in our research project is a low-ability class. They work effectively at problem solving. This may be because all the students are of comparable ability, because the teacher uses problems that can be completed in ten to fifteen minutes, because the reading level required by the problems is not too high, or because the teacher chooses problems in a single lesson that require the same strategies.

### What help do you need with problem solving?

The teachers in the survey overwhelmingly requested pre-prepared resources and problems that would fit in with content strands. Some 74 percent of the respondents asked for resources of this kind. Some typical comments regarding teacher needs were:

a scheme of work which fits in content/skills with a problem-solving approach in the limited time we have available with students;

ready to roll resources in connection with syllabus content/skill exercises;

more time to make up or find problems that are relevant, open-ended, multi-level, etc;

teachers need a lot of ideas and resources to build up a pile of relevant problems to fit in with the curriculum;

heaps of problem-solving activities catalogued into levels and topics.

As Neyland (1994) and Lovitt (1995) say, problem-solving activities by themselves are not enough to guarantee good problem solving. However, a start in this direction cannot be made without these activities. There is an urgent need to produce them in a form which makes them easily used.

Other comments by teachers included the provision of appropriate physical environments for problem solving. This meant appropriate equipment as well as space. Teachers also felt they needed help and training in scaffolding, heuristics, and metacognition (thinking about thinking—an important aspect of problem solving). Some assistance in extension work for bright students would also be appreciated.

Two valuable suggestions were made which could usefully be explored. These are:

watching other staff use problem-solving techniques;

the pooling of resources between schools.

With the change of the political environment, teachers may find the Ministry of Education in New Zealand will no longer provide the continuing services it has done in the past. Resources for teachers are less likely to be seen as a central responsibility and if not contracted out may be left to private enterprise to produce. Until these appear on the scene the next phase of teacher development will be largely left in the hands of the teachers themselves.

## A TEACHING PERSPECTIVE

In the short period since the implementation of the new curriculum, it appears that most teachers are making at least some effort to incorporate problem-solving approaches into their mathematics classrooms. At present, problem solving is not a regular feature of most classrooms but it is clear that teachers are prepared to increase their use of problem solving. This increased use can be facilitated by the production of appropriate content-related problems. Indeed the teachers surveyed see this as the first priority to the further implementation of a problem-solving approach to mathematics.

The advantages proposed for problem solving in school are that it:

bases students’ mathematical development on their current knowledge;

is an interesting and enjoyable way to introduce and learn mathematics and engenders positive attitudes towards mathematics;

is a way to learn new mathematics with greater understanding;

is a useful way to practise mathematical skills learned by other means;

encourages students to learn together in co-operative small groups;

makes students into junior research mathematicians and helps them see more of the culture of mathematics.

Unfortunately simply giving teachers good reasons for teaching problem solving, and giving them good problems to pass on to their students, does not produce good problem solving. As Lovitt (1995) says:

All the early problem-solving efforts were mostly devoted to the creation of suitable problems in the belief that teachers could present these in classrooms and generate effective learning with the same maths they used for expository teaching. It has taken some time to recognise that this is not the case …

Neyland (1994) expresses the same view:

Some well designed [problem-solving activities] do not result in the active learning event intended. There appear to be two main reasons for this. Firstly, the way the activity was originally presented to the class diminished its potential. And secondly, the interactive component of the teaching–learning process was not adequately prepared and unanticipated problems arose. [emphasis in original]

Before starting to teach problem solving, teachers should be clear why, apart from the fact that it is in the curriculum, they intend teaching it. It is important too, to note that the problem-solving approach to teaching mathematics is different from what has become the traditional approach. However, much of the problem-solving approach is not new. Scaffolding, which is not limited to problem solving, has its origins in the Socratic method. Heuristics too have a long pedigree. So how is problem solving different from what has become the standard “chalk and talk” method for teaching mathematics?

The main difference is one of attitude or philosophy on the part of the teacher. The shift is from the so-called “sage on the stage” to the “guide by the side”. Philosophically, the teacher needs to change from a giving role to an encouraging role, from “here is how to solve a linear equation” to “how might we solve this linear equation? ”

Naturally the students will not be able to invent all the mathematics they need for themselves. There will still be things that they need to be told. They will still need to practise both skills and problemsolving processes. However, more time needs to be spent by the students exploring mathematics with the teacher as a guide. During this exploration, seeds will be planted and students will develop connections between various parts of their learning which will increase learning, understanding and retention later (see Hiebert and Carpenter, 1992).

So, instead of teachers taking students down well-trodden paths in a sequential unfolding of mathematical structures, there should be more emphasis on the students themselves structuring mathematical knowledge for specific problem contexts. Certainly the structures that are formed this way have to be justified but not necessarily in the one way shown in the textbook. Where possible, students should be given the opportunity to provide their own justifications. Many of these will be correct and may well be different from the text-book proof.

This is not to say that the development of mathematics during the year is anarchic. Teachers should know from the start what material is to be covered. However, there should be some flexibility in the manner and order in which the content arises. Of course, in highstakes years, when there are external examinations, the course may have to be slightly more structured. Nevertheless, it is important even on these occasions for teachers’ questioning to be open rather than closed in order to stimulate students’ thinking.

In the problem-solving approach then, there is less emphasis on students applying rules to problems which have been carefully chosen to fit those rules. There is greater emphasis on the creative construction of mathematical structures and solutions in non-routine situations and over a range of contexts.

### Incorporating process and model

In order to give the following five-part overview of the problem-solving process we combine the model of problem solving given in Holton and Neyland (1996) with the heuristics and metacognition discussed in that paper (see figure 1).

We recall that heuristics are means by which a solution is sought. Most of the heuristics of Begg (1994) are covered in this overview of the problem-solving process. We also show that certain heuristics tend naturally to appear at certain stages. In what follows, heuristics are emphasised in italics.

### I Getting started

This is the first part of problem solving and includes “problem” and “experiment” from figure 1. It also covers the first two phases of Póya’s four-phase model.

### 1. Understand the problem

The problem will either be posed by the students themselves, presented in writing, or given verbally. If the problem is not one posed by the students, a first and crucial step is for them to make the problem their own; to become familiar with its conditions, characteristics, and variables. If it is in writing the solver has to read the problem and read it carefully. Many a solver has begun by solving a problem they thought was there but was actually not.

As the solvers read the problem they should be asking the various questions posed by Pólya. They should also be looking for key words. These are words which, if changed, will lead to a quite different problem. They are words or phrases which are essential in the solution of the problem. For novice problem solvers, underlining key words is a useful strategy. It is also a good idea for the problem solver to restate the problem for themselves. This helps to emphasise the key aspects of the problem. Other strategies such as experimenting with special cases also help the student to understand the problem.

Consider the problem below.

Peter is 18 years older than his daughter Sue who is 7 years old. How old will Peter be when he is twice as old as Sue?

In this problem, “18”, “7”, and “twice as old” all appear to be key words or phrases. The question, and more importantly the answer, is altered if key words are changed. (In actual fact, “7” is an extraneous piece of information here.) On the other hand, “Peter” could be “Peta” and “Sue”, “Sam” without changing the answer or the method of solution.

In this particular problem it is obvious that both Peter and Sue age at the same rate, one year at a time. This is nowhere explicitly stated but it is an hidden assumption. Many problems have implicit conditions that turn out to be important in finding a solution. So it is vital to make sure that hidden assumptions have been noted.

One of the metacognitive aspects of problem solving is to know when to come back to the original problem and read it again. This may be necessary to see if you are on the right track of a solution or if you have inadvertently started to solve another problem. It may also help you to try another approach if you have made no progress with a particular method of attack or change the point of view. Once an answer has been obtained it is important to reread the question to ensure that the solution you have obtained does indeed solve the original problem.

### 2. Think

Apart from the metacognition (or “thinking about thinking”) at the start of a problem—which largely relates to monitoring and controlling the problem-solving processes—it is necessary for solvers to go through a mental list of heuristics in an effort to find a few which will get them started.

It is worth asking the following Pólya questions:

What is the unknown?

What are the data?

Is drawing a figure useful?

Is there a related problem that has been solved before?

Could it be used here?

In addition it is worth considering:

What area of mathematics can be used? (algebra, geometry, number, etc)

What approaches might work?

### 3. Experiment

This is often an early stage in the problem-solving process. It has two main functions. The first is to get a feel for the problem. The second is to start to produce some evidence for a conjecture. The experiments may be calculations or measurements or a variety of things depending on the problem in hand.

The experiments should be systematic. The aim is to get central, useful information that can be collated in some way, rather than an unconnected jumble of, say, numbers. And the results of experiments should be recorded in some logical fashion—in a list, table, or diagram. When appropriate, consider the case n = 1, then n = 2 and so on. It is worth keeping these experiments until the problem has been completely solved. They may well be useful to provide or inspire a counterexample later.

Another problem that is useful to illustrate some ideas here is the frog problem.

Four spotted frogs (S) and four green frogs (G) are sitting on lilypads as in figure 2. Frogs can move to the next lilypad if it is free or they can jump over another frog if there is an empty lilypad on the other side. What is the smallest number of moves which can interchange the green and the spotted frogs?

There is a certain amount of symmetry in this problem. Whatever happens next, certainly it doesn’t matter whether a spotted or a green frog starts first.

This is a good problem too, to illustrate solving a simpler problem first. It’s easier to do the problem above once you’ve tried moving one frog of each type, two frogs of each type, and so on.

### 4. Panic

This is a common feeling at the start of a new problem. Success and experience will give the solver confidence to continue. However, even expert problem solvers face panic on occasions. Be prepared to think “there is no way I will ever do this problem” and then get on and realise you can.

### II Conjecturing

The next process in problem solving involves conjecture.

### 1. Pattern

This can be the most enjoyable and interesting part of the problemsolving process. In more difficult problems it is necessary to find a pattern, rule, or relation in order to produce a conjecture.

Some problems give up the correct conjecture immediately, other problems take more time. In harder problems it may be necessary to discard a number of conjectures before obtaining the right one. Some people find some problems easier to make conjectures about than others. And conjectures require practice. So it is necessary to start with simple problems and work up.

### 2. Sense

Does the conjecture make sense? Is it consistent with previous knowledge and the data that has been assembled in the experimental phase? Does the conjecture imply anything that feels wrong? Intuition and common sense can and should be used here. A conjecture which implies that the height of a mountain is five centimetres must surely be wrong. Any conjecture should be justifiable. Even though it cannot be proved at this stage, there should be good reasons for choosing one conjecture over another.

### III Proof/Counter example

The more difficult problems will require the solver to trade off conjecture against proofs against counterexamples. The process is a dynamic one which is only completed when the final step in the solution is written down. Frequently, trying to prove a conjecture gives an idea for a counterexample and looking for a counterexample can give an idea for a proof. Whether starting down the proof or counterexample trail the solver may see that the conjecture needs adjusting and the process starts again.

It is difficult to see whether to first try to justify a conjecture or show it is wrong. The decision as to which to try first depends on the solver and the problem. If testing a few cases will cover all possible counterexamples, then the solver should go that route. This will either produce a counterexample or strongly confirm the conjecture. In the latter case, of course, a proof will still be required.

Some problems look like others and this may suggest a way to proceed to a proof. A slight change of a known proof might work. Or the problem may be one which suggests a well-known proof technique such as proof by contradiction or mathematical induction.

### 1. Extreme cases

In trying to find counterexamples it is often valuable to try extreme examples or extreme cases. These examples are somehow at the edge of the spectrum of values that are being used. For instance, what happens if a number is very small or what happens as “n” approaches infinity? What happens if the triangle is isosceles? Extreme cases are usually easier to handle than more general situations so it is worth testing a few before trying a proof. Even if they do not provide a counterexample they may well give the solver some useful information which can be used later. Here is a problem which can be quickly solved by looking at an extreme case: Do all one-litre milk containers have the same surface area? The answer, no, can be easily obtained by imagining a one-litre box-shaped milk container with a base the size of a tennis court and a height a fraction of a millimetre. Clearly this one-litre container has a surface area much larger than the one in the fridge.

### 2. Special cases

A conjecture can also be tested against special cases rather than extreme cases. Special cases are typical cases such as the ones during the experimental phase. Where possible, cases would be tested because they are straight forward or easy to test. But sometimes more complicated cases are forced on the solver.

### 3. Simpler problems

Sometimes problems are far too difficult to solve the first time. They may involve far too many variables to be able to understand easily. In that case reduce the problem to a simpler problem in some way. For instance, in the frog problem using fewer frogs is a good way to start. Similarly a gambling problem involving five dice might be reduced at first to one involving only two. Insight gained from the two dice case may well lead to the solution of the original problem. Here the aim is to keep as many of the essentials of the problem as possible while reducing the problem to a manageable size. Using simpler problems is also something that is useful at the experimental stage, as we saw earlier. Sometimes problems can be simplified in more than one way and students should learn to choose the best from the range of available simplified situations.

### 4. Exhaust all possibilities

One of the very simple methods of proof is to exhaust all possibilities. If there are sufficiently few cases to handle (under 50, say), then by looking at each one in turn, the solver can learn about the entire problem. This way any conjecture about the situation is immediately verified or disproved. Simple combinatorial problems such as the behaviour of two dice are often open to this approach. For instance, how many different ways are there for two dice to give a sum of four?

When a solution has been obtained by older children by the exhaustive approach, it is worth asking the Pólya phase four question “Can you derive the result differently?” They should be looking for more sophisticated approaches. There are times, however, when no other method is available. Certainly this will be the case with younger children. And finding a justification, any justification, is better than no justification at all.

### 5. Guess and check

One of the simplest ways of tackling a problem is guess and check. This is sometimes also called trial and error. In the problem of Peter and Sue, an answer can be found by guessing and checking. It is therefore a good strategy for simple problems. There are drawbacks, however. First, if the problem involved has a large number of cases, for instance the five dice gambling game we alluded to earlier, then it may not be easy to guess the correct answer in a reasonable time. Second, guessing and checking will give an answer. However, the method cannot tell you whether the answer is unique. It may well be that the problem has a number of answers. Only an exhaustive search will be able to determine all answers in such a problem. Finally, the guess and check method in complicated situations may only reveal the conjecture required; it may not justify that conjecture.

The efficiency of guess and check can be increased with metacognition by using guess and improve. Here subsequent guesses are improved using the data of past guesses. Again in simple problems where a unique answer is almost certain, this is a useful strategy. In Peter and Sue’s problem, we could first guess that Peter was 30, in which case Sue would be 12. The check 2 × 12≤ 30, shows that we have guessed incorrectly. We could then guess, say 40. Now 2 × 22 = 44 which is more than 30. The next guess should then be directed between 30 and 40. Guessing and thinking allows us to hone in on the correct answer.

To make sure that we don’t keep making the same guesses, the results could be recorded in a table such as that in figure 3. Another advantage of the table is that it enables us to move our guesses in the right direction; it enables us to see how best to use guess and improve.

We actually came across a nice solution of this problem where the student had made a model. On the edges of two pieces of paper he wrote the numbers 1 to 30. He then put the edges of the paper side by side and moved one of them until corresponding numbers were 18 higher on one edge (see figure 3). He then looked along the edges to try to find where one number was twice the other. (Actually he didn’t put down enough numbers, but what he had was enough to enable him to see the right answer.)

### 6. Work backwards

Work backwards is often useful in problems which involve a sequence of moves, events, or operations leading to a known end point. By reversing the process the starting configuration can be obtained; this is a bit like running a video in reverse. It is useful for investigating two-person games. Start from the last move and work back to the starting position. Work backwards is the basis of many commonly used methods for solving equations. This strategy is also useful when dealing with proofs of trigonometric identities.

### 6. All information

If the solver is unable to make any headway it is often worth re-reading the question in an attempt to ensure that all of the information given in the problem has been used. Sometimes there is an implicit detail that is not mentioned specifically but which is vital to the solution. Knowing general facts such as the sum of the dots on opposite sides of a dice is seven may well be required to solve a problem. (See hidden information in the experimental stage.)

### 7. Check

Having obtained a complete solution the solver should go through the justification and check every step. Even if the proof passes the check test, sometimes the solver still has a nagging doubt. In all cases it is a good idea for the solver to get someone else to check their solutions. At this stage too, sometimes a solver may see a quicker, more efficient, way to solve the problem. The new method needs to be written down and checked too.

## IV GENERALISATION/EXTENSION

### 1. Generalisation

A generalisation is a problem which contains the original problem as a subset or special case. Generalisations can often be found by increasing the number of digits (in some number problems), stepping up the dimension (in a geometry problem), or by increasing the number of variables in some way. For instance, you may want to do the frog problem with an arbitrary number of frogs (n, say) on either side. This is a generalisation of the four frog case.

The reason for considering generalisations is that they give a result which is true for a much wider class of objects. One difficulty with generalisations is that they may require a quite different justification from the justification used in the original problem.

### 2. Extension

An extension of a problem is one which is related to the original problem but not by way of a generalisation. These can be found by changing one of the conditions of the original problem in some way. It may be that a problem can be extended by changing addition to multiplication. As with generalisations, the proof of an extension may not be linked in any way to the proof of the original problem.

### V GIVE UP

The final part of the problem-solving process may in fact be to give it up!

### 1. Why?

Some problems are too difficult to be solved right now. So, at times, students will have to abandon some problems. If they are forced to abandon all problems they tackle then the problems are too difficult.

### 2. When?

This will depend on the teacher and the student. It becomes a matter of priorities. However, any reasonable problem will take more than 10 minutes and less than two hours (unless it is an extended investigation). In general, we suspect that students can go further than we usually expect. This is especially true of the better students. And it should be remembered that expert problem solvers like to sleep on a problem if no solution is at first forthcoming.

### 3. Help

In cases where the problem is too hard for both the teacher and the student, outside help should be summoned. This may be from another teacher, a book, or a local university.

### Implementing problem solving

If you are a teacher who has been convinced philosophically of the importance of problem solving in mathematics, or if you see it in the curriculum and feel a responsibility to teach problem solving, the next question to be resolved is how? The first thing to point out is that there is no unique way to teach problem solving. Teachers and schools will need to develop their own styles. Each teacher will need to develop an individual style which will no doubt develop and be a function of the school, the students, and many other things. An easy way to start is by using a series of one-off problems that may not necessarily be related to the content of the remainder of the lesson. This allows the teacher to gain confidence and gather together a string of useful problems. However, the danger is that students will not realise the relevance and importance of problem solving that stands alone, not integrated with the rest of the curriculum.

Our research suggests that problem-solving lessons work well around a three-stage format. The first stage is a whole-class format, where the problem is discussed by teachers and students, and students may suggest possible heuristics.

In the second stage, students work in small groups. The aim of this stage is to enable students to become involved with the problem and attempt to solve it for themselves. During this period, the teacher is able to go from group to group to provide scaffolding. However, our research suggests that most scaffolding during group work is undertaken by the students themselves as they give peer tuition to their group members.

The third stage is a reporting back stage which gives the students an opportunity to say what they have done and how they did it. Here the spotlight is again on the students who are reporting on their work. Different methods of solution are to be encouraged during this stage. It is important that student thinking and learning takes place in this final stage. One reason for the reporting is to improve students’ ability to communicate but even more important is the chance for students to see alternative approaches to a problem. These alternative approaches should be useful on future occasions. In this stage too, there is an opportunity for students to see links between different parts of mathematics which is again another aid towards understanding.

We have observed these three stages in a variety of problem-solving classes. They seem to work equally well with weak and strong students and with young and old students. A general rule of thumb is that the three-stage cycle should be repeated with young students and weak students. For these children the three stages should take no more than 15 minutes. Very good students may only go around the cycle once in a one-hour lesson.

## CONCLUSION

Problem solving is not easy for many teachers as it involves a rethinking of their whole approach to teaching mathematics. The difficulties arise for many reasons. Some relate to potential loss of control in going from a closed to an open environment where it is never clear in which direction the students will head next. Some relate to the wider range of teaching skills required to handle the situation. Other difficulties are due to social phenomena, including student expectation. But it is our experience that most of these difficulties are surmountable by many teachers, even though at times they may be close to giving up. In our current research, one teacher was in despair after what she thought was a bad lesson. The next problem-solving lesson she taught was extremely successful, with the students seeing links that she herself had not previously seen. From that moment she has been sold on problem solving.

Another teacher had worked solidly doing a competent job for over two terms with a weak-streamed class. In a regular third-term exam (not based on problem solving) her students were 10 percent above similar students in another class. Her students had gained considerably on word problems. The practice they had had in problem-solving situations had given them confidence to successfully tackle the word problems in the exam.

## NOTES

DEREK HOLTON is Professor in the Department of Mathematics and Statistics, University of Otago. E-mail: dholton@math.otago.ac.nz

JIM NEYLAND is Lecturer of Mathematics Education in the Mathematics and Science Education Centre, Victoria University of Wellington. E-mail: jim.neyland@vuw.ac.nz

JULIE ANDERSON is in the Department of Mathematics and Statistics, University of Otago, and St Hilda’s College, Dunedin.

We would like to thank the Ministry of Education, Research Section, for the financial support which allowed us to engage in the research on which this paper is based. Our thanks also go to the many teachers and students who allowed us into their classrooms.

For more details of this research, see:

Holton D., Spicer, T., & Thomas, G. (1995). Is problem solving too hard? Proceedings of Mathematics Education Research Group of Australasia, 345–351.

Holton D., Spicer T., Thomas G., & Young, S. (1996). The benefits of problem solving in the learning of mathematics (Report to the Ministry of Education). Dunedin: Otago University, Department of Mathematics and Statistics.

The New Zealand curriculum documents are:

Ministry of Education. (1992). Mathematics in the New Zealand curriculum. Wellington: Learning Media Ltd.

Ministry of Education. (1995). Implementing mathematical processes in the New Zealand curriculum. Wellington: Learning Media Ltd.

The Australian curriculum documents are:

Australian Education Council. (1990). A national statement on mathematics for Australian schools. Carlton, Vic: Curriculum Corporation.

That problem solving should be the central focus of the mathematics curriculum is from page 23 of:

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematic. Reston, VA: Author.

The problem-process approach is described by:

Sigurdson, S. E., Olson, A. T., & Mason, R. (1994). Problem solving and mathematics learning. Journal of Mathematical Behaviour 13, 361–388.

That talented mathematics students enjoyed the challenge and the open approach of problem-solving tasks is noted by:

Curran, J., Holton, D., Daniel, C., & Pek, W. H. (1992). A survey of talented secondary mathematics students. Dunedin: Department of Mathematics and Statistics, University of Otago.

A common belief in the need for problem solving for the gifted child is highlighted in:

Burkhardt, H. (1988). Teaching problem solving. In H. Burkhardt, S. Groves, A. Schoenfeld & K. Stacey (Eds.), Problem solving: A world view (pp. 17–42). Nottingham, UK: Shell Centre.

The importance of strategies for solving problems is noted in:

Pólya, G. (1973). How to solve it (3rd ed.). Princeton, NJ: Princeton University Press.

The kinesthetic approaches are advocated by:

Lovitt, C., & Clarke, D. J. (1988). Mathematics curriculum and teaching program, activity bank (Vols. 1 and 2). Canberra, ACT: Curriculum Development Centre.

That the methods of good problem solvers are likely to be “idiosyncratic” is supported by:

Beagle (1979) pp. 145–146, see above.

Pólya (1973), see above.

Schoenfeld, A. H. (1992). Learning to think mathematically: Problem solving, meta-cognition and sense making in mathematics. In Douglas A. Grouws (Ed.), Handbook of research on mathematics teaching and learning (pp. 334–370). New York: Macmillan.

That students should be encouraged to see how to translate strategies to other problems and situations is stated by:

Holton, D. (1994). Problem solving. In J. Neyland (Ed.), Mathematics education: A handbook for teachers (Vol. 1, pp. 18–30). Wellington: Wellington College of Education.

Neyland, J. (1994). Designing rich mathematical activities. In J. Neyland (Ed.), Mathematics education: A handbook for teachers (Vol. 1, pp.106–122). Wellington: Wellington College of Education.

That problem-solving activities by themselves are not enough to guarantee good problem solving is stated by:

Lovitt, C. (1995). Personal communication.

That time needs to be spent by the students exploring mathematics with the teacher as a guide is noted by:

Hiebert, J. & Carpenter, J. P. (1996). The benefits of problem solving to the learning of mathematics (Report to the Ministry of Education, Wellington). Dunedin: University of Otago, Department of Mathematics and Statistics.

For more details of the heuristics used in the problemsolving process see:

Begg, A. (1994). Mathematics: Content and process. In J. Neyland (Ed.), Mathematics education: A handbook for teacher (Vol. 1, pp. 183–192). Wellington: Wellington College of Education.

Ministry of Education. (1995). Implementing mathematical process in mathematics in the New Zealand curriculum. Wellington: Learning Media, Ltd.

National Council of Teachers of Mathematics. (1980). An agenda for action. Reston, VA: Author.

National Council of Teachers of Mathematics. (1989). Curriculum and evaluation standards for school mathematics. Reston, VA: Author.

Neyland, J. (1995). Eight approaches to teaching mathematics. In J. Neyland (Ed.), Mathematical education: A handbook for teachers (Vol. 2, pp. 34–48). Wellington: Wellington College of Education.

For a good example of problem-solving teachers include:

Stacey, K. & Groves, S. (1985). Strategies for problem solving. Camberwell, Vic: Lattude Publications.