Organising the teaching of problem solving

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Our purpose in this section is to suggest ideas for lesson structure and organisation. We break this section up under three headings: lesson structure, scaffolding good questions to ask, and the year plan.

On this site, you will find lesson plans for problems sorted by strand of the curriculum. We have focussed mainly on providing single lessons with one-off problems for you to incorporate into your units of mathematics although it is of course possible to link these lessons or extend them to create problem solving units.  There are some lessons, however, that are more of an investigative nature and these will involve more than one teaching period. 

Lesson structure

One-off problems  
There are a number of ways of organising one-off problem solving lessons. One way that we have found to be quite successful is the following three-stage format.

Three-stage lesson format

  1. Introduction
  2. Group Work
  3. Reporting Back.

1. Introduction.   
This is a whole class phase where the problem is presented. The introductory stage of the lesson may last from a few seconds to as much as 15 minutes, depending on what  you are aiming to get from it.

In the initial stages of problem solving, or when a new strategy is being introduced, the introduction to the lesson may take some time. During this period, with the help of the students, you may model the problem solving process or a particular strategy, before sending the students off to practice the point for themselves.

The simplest introduction, for older students who have had some problem solving experience, might be to just say "Can you do the problem on the board?" But this will not always suit your purpose. As in any introduction it is important to interest the students so that they are motivated to participate.

2. Group work.    
After the introduction stage, the students go off in their groups to tackle the problem. While they are working, you move around and provide suitable help. (We’ll say more about this help under Scaffolding below.)

There are a number of things that can be said about group work, but the main two questions to consider are (i) how big should groups be and (ii) how should you assign students to groups?

We have watched quite a lot of groups of different sizes at work and we think that for problem solving, groups of two seems to work best. This appears as much as anything to be a problem of the actual physical arrangement rather than a social problem. Larger groups certainly work well for activities involving equipment. Problem solving groups bigger than four, seem to break up into sub-groups unless the students are using an Act it Out strategy. Other strategies don’t seem to be able to hold everyone’s attention and co-operation.

One thing that defeats groups of three and four is the geometry of their workspace. If the group is trying to work around a table, there are usually two students on either side. When they are working with pencil and paper, pairs on each side of the table tend to split off and work together. It is just too inconvenient for them to look at work on the other side of the table that for them is upside down. Groups of three often isolate one student. This is generally not for social reasons but because it is difficult for all three to share a piece of paper and work together.

So for problem solving purposes, two appears to be the optimum number of students to engage in co-operation.

On the matter of co-operation, New Entrants students especially need some assistance before they can really co-operate with one of their peers. It is not something that comes automatically. One way to facilitate this is to have the pair work together on a single sheet of paper.

Then there is the question of how to group the students. We know that most students prefer to work with a friend. They may be more likely to share ideas with someone they know reasonably well. You can always try this method of pairing. If friendship groups are working well, then continue with them. Some teachers prefer to put students together more or less randomly. This can be done, for instance, by picking counters out of a box. Students with the same colour counter form a group.

Whatever way groups are chosen it is probably a good idea to change them from time to time. In that way students get to work with a range of others and groups that are not working well don’t have to stay together for a long time.

It may sometimes be beneficial for able mathematical students to be grouped with other able students. Certainly such pairings will often be beneficial for both the students concerned. Generally though, it is recommended to use mixed ability groups, though it is important that these be monitored. The more able student might end up doing all the work and not provide learning experiences for the other member of the pair. If the more able student begins to feel put upon, that student may stop co-operating completely and the group may well break up as a result.

However the groups are constituted, you will need to monitor their progress. This can be partially done by having both members of the group report back to the class in the final stage of the lesson and partially by looking in on each group in the second stage of the lesson.

One of the problems with groups of size two is that it may take you some time to get to around to all of them. You should realise that you will be unable to help all the students in your class all the time but you should be able to monitor everybody and make sure that they are all on task. You don’t necessarily have to spend a great deal of time with every group in every lesson. Rather, you should ensure that each group is given more than cursory attention at least once every other lesson. Later, when the students have learnt to work well in pairs, you might consider putting them into larger groups. With larger groups it is easier for you to visit each group at least once every lesson and respond to individual students' needs.

3. Reporting back 

Generally we expect some students, from some of the groups, to report back to the whole class at the end of each problem. We deliberately said that the reporting back was to the class and not the teacher. Although you may use this part of the lesson to assess students’ progress in a number of areas, reporting back can be a very positive learning experience for all the class, not just those doing the reporting.

The main purposes of reporting back  

  • To expose students to a range of thinking. In problem solving, there may be more than one way to solve the problem. Students will learn to use new strategies that they see that other students have used.
  • To foster mathematical communication skills. This in itself is an important process. However, it does have the valuable by-product of increasing understanding. In verbalising their own solutions, students will understand them better themselves.
  • To increase students’ confidence. In a friendly classroom atmosphere we have seen many students’ confidence increase, both in themselves and in their mathematical ability, by reporting back to the class. Some students will require more help than others, of course.
  • To clarify common misunderstandings. Verbalising helps many students see their errors more quickly than writing their answers out on paper. By discussing points of difference, valuable learning can be achieved.
  • To provide a foundation for the extension of a problem. Once the students have made sure that their solutions are correct and that they understand the solution, you can move them forward to generalise and extend the problem This is an important aspect of mathematics.
  • To highlight the mathematics inherent in the problem. You should take the opportunity to point out any new strategies that a group has found and to show how this fits in to the other strategies that have been used. The ideas produced can also be related to the problems that have been attempted and to other aspects of the curriculum. Any generalisations and extensions can be developed to show that mathematics is about big ideas that cover many problems.

We have seen many valuable reporting back sessions where the students have been engrossed by what their peers have said and done. This can be because students sometimes understand what their peers say better than they understand what their teachers say. Maybe this is because a peer has had to struggle through a problem from scratch and shows the solution in smaller steps than a teacher might.  It is also possible that students will express the solution as more of an idea in progress than a finished, polished solution.  Sometimes it is easier to transmit ideas than the final product.

During this stage of the lesson you should take the time to monitor students' ability. This will give you the opportunity to assess the progress they are making and accordingly vary the experiences that you are providing them with. Particular aspects of this are listed below.

Points to monitor

  1. presentation of a reasoned argument
  2. making of conjectures (guesses)
  3. making deductions
  4. proving and reporting other students' statements
  5. demonstration of flexible thinking
  6. the thought processes of the students 
  7. the explanations students give.

It should be emphasised at this point, that a lesson may not necessarily proceed nicely through the sequence 1. Introduction, 2. Group Work, 3. Reporting Back, above, in that order. For instance, there may be good reasons for repeating the stages 1, 2, 1, 2, and sometimes there are advantages in leaving stage 3 until the following lesson. However your lesson goes, we feel that these three basic stages will  provide a useful framework for problem solving lessons

Scaffolding good questions to ask 

One of the important goals of problem solving in school is to prepare students to tackle problems in later life.  When dealing with these problems, they will not have a teacher by their side to prompt them towards a solution.  Part of the reason for scaffolding, good leading questions, is to help prepare students for later problem solving. 

Below we list some of the types of questions that you might like to ask during problem solving lessons. We break these scaffolding questions into three types. These types correspond roughly to both Polya’s four stages of problem solving (What is Problem Solving?) and to the three stages of the problem solving lesson we have just been talking about.

A. Getting Started

  1. Has anyone seen a problem like this before?
  2. What are the important ideas in this problem?
  3. Can you rephrase the problem in your own words?
  4. What is this problem asking you to find out?
  5. What information has been given?
  6. What conditions apply?
  7. Can you guess what the answer might be?
  8. What strategies might you use to get started?
  9. Which of these ideas are worth pursuing?

B. While Working on the Problem

  1. Tell me what you are doing?
  2. Why (How) did you think of that?
  3. Why are you doing this?
  4. What will you do with the result of that work when you’ve got it?
  5. Why is this idea better than that one?
  6. You’ve been trying that idea for 5 minutes. Is it time to try something else?
  7. Can you justify that step?

C. At the Finish

  1. Have you answered the question?
  2. Have you considered all possible cases?
  3. Have you checked your solution?
  4. Does the answer look reasonable?
  5. Is there another answer?
  6. Is there another solution?
  7. Can you explain your solution to the class?
  8. Is there another way to solve the problem?
  9. Can you generalise or extend the problem?

All of the above questions can be used with almost any problem. They are generic questions that fit almost any situation. The difficult scaffolding questions are the ones you have to use with particular students working on particular problems. It takes practice to be able to say the right thing at the right time. There are a few guiding principles, however.

Wait time. Research suggests that most of us are too anxious when it comes to receiving answers from students. We don’t leave them sufficient time to think about their answer and then reply. Try to give students the time they need to respond.

Another aspect of this relates to group questions. It is useful to stand and observe a group for a while before asking any questions at all. During this time you can gauge what they are doing and where they are with the problem, before you commit to asking questions.

Of course, it is not always necessary to ask a question.  You might simply say "very nice idea" and walk away. All of us respond very favourably to praise so we should not be afraid to use it with groups.

The guide by the side, not the sage on the stage. The teacher should be there to give assistance and to help the student over hurdles. It is difficult to develop good problem solving in a class by standing at the front and explaining. Problem solving in that sense is more like coaching. Remember that the person who does the thinking does the learning. 

Open rather than closed. The questions that you ask should be open rather than closed (as in the list above under Getting Started, While Working on the Problem and At the Finish). Questions should provoke thinking and encourage ideas. They should enable students to make their own progress. The idea is to try to ask questions that do not have a fixed answer. Good questions can be answered in a number of ways, each of which is effective.

Small grunts. Minimal encouragers are also valuable as students will then tell you their ideas. "Right", "OK", etc., provides support. Phrases such as "I like that idea" also give positive reinforcement. At the same time, none of these responses takes away the opportunity for students to find the key to the problem themselves.

In all of this, you are modelling behaviour that is expected of all students. One of the main objectives is to get students to internalise the questions (scaffolding) that you provide. When a student runs into difficulties and you are not around to provide assistance, they can go through their scaffolding armoury and ask the questions that you might have asked. In this way, they may be able to surmount the current difficulty without external assistance.

While ideally you should support students to find answers for themselves, there are times when you have to give up and provide the answers. This may be because you can think of no more open questions to ask. Again, it may be because the problem is beyond the student’s current ability to understand. No amount of encouraging and scaffolding will help here. For this reason the choice of problem is of the utmost importance. If the problem is too hard, then the best strategy may be to back off and come back to it later when the students have developed the required mathematical tools and understandings.

Finally in this section, it is worth while saying something about metacognition. This is essentially "thinking about your own thinking". It is a useful process that enables a person to monitor what they are doing. There are at least two reasons for teacher scaffolding. The first is to get students over the hurdle that they are currently facing. The second is to model practice that they will find useful in future situations. When they have internalised this scaffolding it becomes part of their metacognitive processing and should make tackling problems easier in the future.