In this section we discuss Why Teach Problem Solving under two headings: benefits of problem solving, and difficulties of teaching problem solving.
Benefits of Problem Solving
Problem solving is the process part of mathematics that has often been overlooked in the past in favour of other skills (see What is Problem Solving?). But there are other reasons for it to be included in your mathematics programme:
- It bases students’ mathematical development on their current knowledge
- It is an interesting and enjoyable way to learn mathematics
- It is a way to learn new mathematics with greater understanding
- It produces positive attitudes towards mathematics
- It makes the student a junior research mathematician
- It teaches thinking, flexibility and creativity
- It teaches general problem solving skills
- It encourages cooperative skills
- It is a useful way to practice mathematical skills learned by other means
- It is similar in approach to the way that other subjects are taught in primary school.
Based on current knowledge. A constructivist view of learning proposes that we construct our knowledge through our experience rather than absorbing what we are told. The constructivist views the student as an active learner. The more traditional approach to teaching mathematics sees the student as an empty vessel that has to be filled. Correspondingly we have two contrasting models of "the guide on the side", the coach trying to encourage the learner along, and "the sage on the stage", the lecturer imparting knowledge. In actual fact, the best teaching approach is probably some combination of the two.
Most of the problems used in problem solving have more than one solution. So each of them can be approached in a variety of ways, some of which are sophisticated and some of which are less sophisticated. Hopefully, every student in your class can find one approach that they can use to solve the problems that you present. Over time, and from seeing what other students have done, you should be able to develop and extend the range of strategies that students have at their disposal.
Interesting and enjoyable. Problem solving seems to be implicitly interesting to students. This is partly because it does not involve a sequence of very similar questions that are designed to practice the same skill. The novelty of the problems seems to add to their interest.
Problem solving can be made even more engaging if you personalise problems to include characters that the students in the class know. This also makes them more interesting and relevant to the students.
Then again the questions can be very interesting in themselves. This is partly because they involve some detective work, which most students enjoy. It’s also partly because we all enjoy getting the answer after having struggled with a problem. And it’s partly because students enjoy having "ownership" of the problem. The ownership issue is an important one. By working on a problem, students can get quite deeply involved with both the mathematics that is required to solve it, and the mathematics that may be required to solve it.
Greater understanding. In the process of struggling with a problem, students can obtain a deep understanding of the mathematics surrounding the problem. This understanding is often enhanced, when, in a whole class setting, teachers draw together the various threads from all of the students in the class. (We say more about this in the 'Reporting Back' section of Organising the Teaching of Problem Solving.)
Positive attitudes. Because students enjoy the problems it helps them to gain a positive attitude towards mathematics more generally. Some of them have even expressed the opinion that what they were doing was not mathematics and have asked to do more problem solving instead of mathematics! But we want them to see that problem solving is mathematics and that it is an enjoyable subject.
Junior research mathematician. The way that students tackle problems can be virtually the same as the way a research mathematician tackles research problems. Hence, through problem solving, students get a much better feel for what mathematics is actually about than they get in the more traditional type of teaching. Hopefully they begin to see that the subject is a live one, get some feeling for the way it is created, and see why certain things are done in certain ways. This then increases their insight into the subject as a whole and gives them a better feeling for what the subject is and what it is trying to do.
Flexibility and creativity. Problem solving provides an opportunity for students to explore ideas and gives them the chance to extend their creativity. Students are continually coming up with ways of tackling problems that we hadn’t thought of before. The interesting thing is that the students who are producing these ideas are not always the ones who we generally think of as being good at mathematics. Sometimes though, it can take a little work on the your part before the consequences of some ideas are seen.
General problem solving. It is important to point out at this stage, that though we are concentrating here on mathematical problem solving, many of the strategies and techniques that are used in mathematics are used in any type of problem. The four stages of problem solving described by Pólya (in What is Problem Solving?) are quite general steps that can be applied to any problem whether mathematical or not.
Cooperative skills. Traditionally, mathematics has been taught to individuals working by themselves. Very little encouragement has been given to cooperation in the traditional didactic approach to teaching mathematics. There has not been the emphasis on students working together that there has been in other curriculum areas. But working in cooperative groups does have advantages. Discussing mathematics out loud helps learning and understanding, and it also helps many students produce original ideas. Emphasising group work in problem solving appears to increase not only enjoyment, but also learning and social skills such as communication.
To practice skills. Some teachers use problems to reinforce technical skills that may have been taught in other ways. Certain problems are chosen because they will use certain algorithmic skills. The divide by 2 or multiply by three and add 1 problem (in What is Problem Solving?) is an example of this. To get a feel for this problem you have to do quite a bit of arithmetic. If you want to practice the 5 times table, then do the extension where the 3 is changed to a 5. This problem then gives some point to multiplication. Hopefully, after doing lots of examples, the students will start to see some patterns. We have also mentioned the practising of skills under the strategy Think. By choosing problems of this type the students have an opportunity to work on basic skills in an interesting situation.
Similar approach to other subjects. Approaching mathematics through a problem solving perspective puts the subject much more on a par with other subjects, especially those in the primary school. The general philosophy of the teacher as a facilitator helping the student to learn and understand, is much more akin to the philosophy adopted in other areas of the curriculum than it is in the more traditional approach to mathematics. We believe that problem solving may provide a way to teach mathematics that is more in sympathy with primary teachers’ approach to teaching generally.
Difficulties of Teaching Problem Solving
There are generally thought to be a number of disadvantages to the teaching of problem solving in class. We list and discuss some of these below.
- It produces teacher discomfort
- It produces student insecurity
- It puts constraints on the curriculum and takes too long to teach
- It is not possible with students of low ability
- It takes a lot of preparation.
Teacher discomfort. The main cause of teacher discomfit regarding teaching problem solving is the worry that students will come up with ideas that they won’t understand. In a way this shouldn’t happen. In problem solving we expect students to be able to explain their methods. So you should be able to understand most of the ideas and solutions students produce because the students should be able to explain them. However, you can’t be expected to know everything about everything. So you shouldn’t feel embarrassed if you are not sure if the student's idea is a good one or not. There is nothing wrong with telling a student, class or group that you are not sure but will try to find out. Often things can be resolved by a quiet moment with a coffee, a paper and pencil or with a chat to colleagues in the staff room. However, as time goes by the answers to these unexpected ideas will mount, as will your strategies for dealing with them.
Student insecurity. This may occur because the students have never met open-ended problems before. Some teachers in mathematics have traditionally given students relatively closed tasks to work on. It is not surprising that in more open problem solving situations, some students will feel insecure. However, by careful handling and by introducing things gradually, students should be able to overcome their initial insecurity.
Curriculum constraints. Many teachers, especially initially, feel that problem solving takes a considerable amount of time. Hence they are concerned that parts of the mathematics curriculum at a given Level, will need to be omitted. Our experience is that teaching problem solving is time consuming initially. It does seem to take a while for both teachers and students to get the feel of how it works. But after this initial period, time can be actually gained. Many teachers find they are able to cover material more quickly than in previous years. They put this down partly to the fact that the students were looking for, and seeing, patterns and connections everywhere. This enables teachers to cover ideas more quickly.
There does seem to be another factor though. The time that students spend on problem solving seems to help them to come to grips with a topic - to own it. This produces greater understanding and provides a solid base for later learning.
Low ability students. There is some feeling that it is all right to undertake problem solving with bright students but it is of little value for lower ability students. Anecdotally however students of all abillities have been seen to make significant gains in mathematics after having problem solving lessons once a week for two terms. These gains are across the curriculum and not just confined to problem solving.
But there is an issue regarding both students who are not good readers and ESOL students. Clearly these students may not be able to read the problem. Because it is the mathematics that is important, lack of reading ability should not be a barrier to these students.
All students will need to read the problem more than once. If you are starting students off with a problem from a whole class setting, then it will almost certainly be read more than once. You can assess whether or not all students have understood the problem by asking them to restate it in their own words. As you go around from group to group, you can also check that every student is working on the problem that you actually posed. With New Entrants students, you will probably do most of your problem solving starting in a whole class situation. During this time you can make sure that the problem is understood.
Preparation time. There is no doubt that this is a problem for teachers undertaking problem solving for the first time. The main difficulty is finding the right problem to use to introduce a given strategy or to fit in to a given Strand or Level. One of the points of this website is to provide problems that are easily accessible in Strand and Level format. On the other hand, time can certainly be saved if informal teacher networks are established. Then ideas and problems can be shared.