let's talk maths!

Let's Talk Maths!Author(s): Richard BainSource: Mathematics in School, Vol. 17, No. 2 (Mar., 1988), pp. 36-39Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30214457 .

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Let's Tats

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by Richard Bain, Head of English, Stirchley Upper School (Shropshire

Talk Project)

When I have a personal problem it often helps to talk it through with someone else. It's not that I expect them to solve the problem for me, but rather that they may help me to look at the problem in a different way, and that the mere fact of explaining the problem will force me to clarify it in my own mind.

When pupils have problems they may talk to their parents, they may talk to a trusted adult, perhaps a teacher, but most likely they will talk to other pupils who are experiencing similar problems and who share their perspec- tive. This is true for many kinds of problems.

Even Maths problems?

Why not? Pupils with problems need to talk, and their most natural audience is each other. As teachers we too easily deny or fail to exploit the opportunity for pupils to find counsel in each other within lessons: often we assume that we are the only reliable source of help available; sometimes we are afraid of the noise of pupils talking, afraid of losing control; often we confuse learning with assessment, and discourage pupils from talking because at the end we will not easily be able to assess exactly who did what.

However the benefits of using pupil talk are considerable:

Othe very act of explaining what they do and do not understand will help pupils to consolidate their understanding.

Opupils at talk will explore more freely and hypothesise more willingly than pupils working individually or under the eye of the teacher.

Othe teacher cannot be everywhere at once: pupils talking to each other will free the teacher to give specialist help where it is really needed.

Osometimes pupils are able to explain difficulties to each other more effectively than the teacher could.

Opupils talking together will exchange ideas and push each other further.

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O pupils talking in groups are likely to be more adventurous and more independent of the teacher.

But talk in Maths? Pupils and teachers too easily suppose that pupil-talk is the province of English or Humanities lessons; it is not. There are many opportunities for pupils to use talk as part of their learning in Maths: indeed, it is often more easy and more natural for pupils to talk about their Maths than about their English.

In order to use pupil talk:

You don't need to buy lots of new books and materials - though good materials are a great help.

You don't need to be a charismatic performer - let them do the performing instead!

You don't need to change to a completely new syllabus: talk is an effective means of learning whatever the syllabus content, and the new GCSE syllabuses place a strong emphasis on it.

You don't need special "talk" lessons: like reading, writing, and listening, talk is best if it is absorbed into the natural pattern of everyday activities.

A few minor changes can create much more pupil talk in your classroom:

Use pairs or small groups rather than whole-class discussion wherever possible. In teacher-led discussion:

Oindividual pupils have little opportunity to talk. Opupil answers tend to be short and undeveloped.

Opupils have difficulty concentrating because they are not fully involved.

Oshy pupils will fear exposure. Othe teacher is wholly occupied. Othe progress of the entire class can be interrupted by the

misbehaviour or red-herring of a single pupil.

In small-group discussion:

Oindividual pupils have far more opportunity to speak.

Mathematics in School, March 1988



pupils are more likely to develop their answers. pupils are more involved, and therefore less likely to have problems with their concentration. shy pupils can speak with less fear of exposure. the teacher is freed to monitor/intervene/assess/talk to individuals. the teacher can deal with distractions and misbehaviour without stopping the work of the entire class.

Encourage pupils to work and talk together unless you have a specific reason for wanting them to work separately. There are many good reasons for wanting pupils to work entirely on their own:

Oto force pupils to think on their own as a preparation for pair or group work.

Oto allow the teacher to monitor what each individual understands.

Oto prevent pupils from relying too heavily on others. Oto allow periods of intense concentration.

All these ends can be achieved by using brief occasional periods of silent individual work where they are specifically needed. It is not essential to move your furniture, but pupils sitting in groups of four facing each other are more likely to talk usefully to each other than pupils sitting in rows facing the front.

Use what pupils already know. Pupils will come to any new topic with a great deal of

knowledge - albeit disorganised and imprecise. They will understand new ideas more quickly if they are able to relate the new knowledge to the old.

At the beginning of a topic, pupils might:

(a) work individually to note down everything they know or think they know about the topic.

(b) look at each others ideas in small groups and try to formulate (i) a series of propositions (ii) a list of questions and doubts.

(c) exchange lists of propositions and questions with another group. Are the propositions acceptable? Can they answer any of the questions?

(d) Report to the class on their conclusions.

The new work could then be seen as a development of their existing knowledge, and (hopefully) as a means of answering questions which they themselves have asked.

This same process can be used as a means of encouraging pupils to reflect back on what they have learned at the end of a topic.

Encourage pupils to be critical of textbooks and worksheets. They might ask:

Owhy has each of these examples been set? What knowledge or skill is being exercised/tested?

Owhat better examples might be given to exercise/test the knowledge or skill in question?

Show might these ideas be presented more effectively?

Groups might:

Owrite questions for other groups

Oexplain concepts or problems to other groups Orewrite sections of worksheets or textbooks

Give pupils the opportunity to think and talk during "question and answer". During "question and answer" pupils are often given little time to think; as a result they may not be fully involved in the process of argument. If after asking a question the teacher gave everyone a minute or so to think and talk about


it, it would ensure that every pupil had considered and talked about the question posed, and that every pupil was therefore involved.

Ask pupils working in pairs to answer alternate questions and to think aloud to their partner as they do so.

Ask pupils to read and to mark each others work.

Ask pairs of pupils to explain to other pairs how they have solved a problem.

Models for small-group discussion One problem with using pupil talk in lessons is that it can seem too easy. (" The trouble," teachers will tell you, only half joking, "is not in getting them to talk, but in getting them to stop.") A consequence of this is that pupils are given talking tasks which are vague and unstructured, which will prob- ably lead to irrelevant chat, and at best fail to exploit the full potential of talk activities. However with careful planning it is possible to use a variety of small-group activities which are disciplined and purposeful. These are some examples:

Simulation Set up a situation in which members of the group take a role and respond to a changing situation in role.

e.g. The group are a builders' merchant. They have an opening statement of how much stock they have, how much cash they have, and a list of prices. The group also has a pile of cards. Each time they turn over a new card it will give them new information: if it is an order, they must calculate the cost, adjust the record of stock, order fresh stock if appropriate, and adjust their record of cash. The simulation can be complicated with rates bills, tax changes, and other delights. If you are feeling really ingenious you could arrange that each group is interacting with the other groups in the class.

Role Play Like a simulation, but much more contained. Pupils respond in role to an unchanging situation.

e.g. In pairs. One of you is a shopkeeper the other is a Martian who has never seen or used money before: explain to him what money is, how it is used, and what the differences between the coins are.

or Different groups look at employment statistics from a government point of view, a trade union p.o.v., an investor's p.o.v., an unemployed person's p.o.v., etc. and then try to persuade each other of their different interpretations.

Drama Act out situations in which Maths is used in everyday life.

e.g. Prepare a short play in which you use halves, thirds, and quarters as often as you possibly can. Each time you use a fraction you must work it out correctly and give the answer. Try to use ADDING, SUB- TRACTING, MULTIPLYING, and DIVID- ING. Different groups might be given different titles, e.g. "In school", "On holiday", "At the disco", "Sport", "A t home".

Interviewing There is a lot for pupils to learn from answering questions, from hearing each others answers, and from preparing questions to ask. Interviewing can be used to explore the applications of mathematics:

e.g. Look back over your work on percentages. Prepare 10 questions to ask another group about percentages.

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or 1 pupil is a Great Inventor, who has just invented percentages. He needs to go to the Minister for Great New Inventions to obtain the World Copyright and make his fortune. Conduct the interview where the Minister questions the Inventor, and the Inventor explains what percentages are, how they might be used, how they are different from fractions and decimals.

or (for homework) Interview a worker about all the different ways mathematics is used in their job.

Sequencing This is an effective way of teaching pupils to understand a written passage: you chop it up, and they need to understand the logic of it in order to reconstruct it. This might well be applied to a mathematical argument: chop it up line by line, and ask pairs or small groups to reconstruct. A flow diagram might be reconstructed in the same way.

Ranking Pupils take a series of statements and place them in order of importance, explaining to each other why as they do so.

e.g. give pupils details of several similar articles from a shop: which is the best value? How do you determine this?

Sorting Pupils are given a series of statements or problems, preferably on small cards, and they work in small groups to sort them according to different given criteria.

e.g. sort a series of numbers according to size or type; sort problems according to type.

Problem solving Many problem-solving activities already exist for Maths. These can be used quite individually, or by individuals talking but working separately: there is a great deal to be gained by using these as group activities.

Radio play Ask group to write or record a radio play (perhaps a detective story?) where the outcome depends on the solution of a series of mathematical problems. Or present groups with snippets of "evidence", e.g. shopping bills, etc. Ask group to construct a play which uses and relates these snippets.

Group product Groups can work together to produce a display dem- onstrating what they have learned; they can work together to produce a talk or a recording for the class; they can work together to apply what they have learned to a specific real situation; they can work together to design a game or a building.

Tv documentary This can be an effective means of exploring written text or statistics. Pupils are given the information, then have to work in groups to prepare a TV documentary in which various interested parties are interviewed about their responses and interpretations. (See role play above.)

Games There are many simple mathematical games which involve pupils actively in puzzling out problems together. These provide an entertaining opportunity both for Maths and for talk. A useful extension is to

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ask pairs or groups of pupils to devise their own games similar to the ones they have tried.

Group discussion/analysis Material such as videos can provide a lively stimulus for Maths work. Small group discussion/analysis of videos will allow the pupils to reinforce and develop their understanding of what they have seen, especially if the groups are given a clear but open brief. Groups might be asked to analyse a series of questions/problems to determine what skills and ideas are being tested in each.

Transactional Ask pupils to describe a game or a process to other pupils; giving directions; giving/receiving instruc- tions.

e.g. give 1 pupil a complete map, and the other the same map, but with a large area missing. Without seeing each others maps the pupils have to reconstruct the missing area.

Problems you may encounter Some pupils dominate the group

Rotate groups so they don't always work with friends. The design of the task (e.g. role play) can change how pupils respond.

Some pupils don't want to speak

Working through talk can be threatening, but some tasks are "safer" than others; working in pairs, with no reporting back, can ensure more participation. The weakest pupils are those who need the most practice!

If the teacher isn't in the group they'll make mistakes

As they do in most lessons and activities; we have to find ways of helping them to cope and to improve.

You can't tell what they're talking about

But you can observe and listen; you can control the amount of reporting back, or recording of results, that you want.

Pupils and parents think talk is a waste of time

Some do, but with time they will see the gains, especially if the talk is disciplined and purposeful.

Talking makes for discipline problems

It can do, but equally it can free the teacher to devote more attention to discipline. Decide what is manageable - what kinds of task, how often, for how long.

What about the attitudes of colleagues or management?

You can't be sure, unless you talk honestly about it. The advent of GCSE with its increased emphasis on the pro- cesses of learning, its specific references to the need for oral activities, and the requirement for an oral assessment in Maths by 1991, should help to change attitudes.




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There's no time to organise

There isn't; in education there never is. Start a bit at a time. Work and plan with colleagues: share ideas and resources. But remember: developing pupil talk is not an extra impo- sition, it is fully congruent with the developments and changes you are already having to make for GCSE, and indeed it will help you to make those changes more effectively.

Assessment The problems of oral assessment are not new: your colleagues in English have been struggling with them for some time. They, of course, are assessing the ability to speak rather than mathematical understanding, but their ex- perience is relevant.

English teachers have rejected the single "interview" assessment where you have 1 pupil and 1 examiner, on the grounds that it is hugely expensive in teacher time, and furthermore it is crude and inaccurate: it only assesses the pupil on one occasion and in one limited situation. They have moved instead to a process of continuous assessment, where each pupil is assessed on at least 5 different occasions during the 2 year course. These assessments take place within ordinary classwork rather than on special occasions.

Such a process could be used in Maths, provided that pupils are warned of the possibility of their being assessed on their talk, and provided that occasions are chosen where talk is a necessary part of the activity, and where the activity gives pupils a genuine opportunity to display their mathematical ability. Such activities might include:

Spreparing a display to show a mathematical idea.


Sexplaining a problem to the class, or another group. cpreparing questions to ask another group. cinterviewing, or being interviewed by other pupils. cexploring a problem together.

Any such process of assessment depends upon close cooperation between teachers within a department to ensure standardisation, and to provide mutual support.

So ............. cPupil talk matters. * It is as natural and as important for pupils to talk in their

Maths lessons, as it is for them to talk in their Science, Humanities or English lessons.

SUsing talk - like using writing - is a learning strategy: its usefulness is not limited to a particular syllabus. However: The use of talk is particularly well suited to the new GCSE syllabuses.

cUsing talk need not require a huge new investment of time or resources: simple strategies could be adopted in your lesson tomorrow.

STalk lessons do not require teachers who are either fanatics or brilliant talkers: they require teachers who have taken the trouble to think about what is going to happen and why.

*When you get over the initial guilt at not running every aspect of the lesson all the time, it is tremendously revealing relaxing and rewarding to stand back and listen to pupils thinking for themselves.

Good talking!

N.B. The Shropshire Talk Project team consists of Richard Bain, Paul Francis and Richard Mathews.

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