MODELS FOR TEACHING MATHEMATICS

1 The job of the teacher is to make it easy for students to learn. Or is it?Alan Wigleyinvites us to take a closer look at the curriculum we offerto learners of mathematicsMODELS FOR TEACHINGMATHEMATICSThe current sceneOne potential advantage of a National Curriculum isthat, with the content at least partly specified andordered, we can move our energies from considera-tion of thewhatto consideration of is achallenge for teachers to work together on effectiveways of approaching chosen topics. How this is doneis likely to have a more lasting effect on pupils' learning and their attitudes to the subject than theparticular content selected. Perhaps, since of its verynature a national curriculum cannot be idiosyncraticand must compromise, it will seem rather conven-tional to forward-looking teachers.

2 This does notmake me pessimistic. With some teachers at least, Isense an emerging re-orientation in which a chosenpublished scheme, rather than defining the course tobe followed, is being used more selectively to meetnational curriculum requirements. This opens thepossibility of teachers taking greater control overwhat they offer their students, potentially to thegreat gain of the a previous article (MT132) I suggested threemajor issues on which we in ATM ought to beworking during this decade:(i) resolving the content/process dichotomy;(ii) developing ways in which pupils can behelped to reflect on their learning ;(iii) removing the unfortunate polarisation of theteacher's role into that of either instructor issues are germane to the wholeeducational debate and are not confined tomathematics.

3 Hence the need to discuss them withfellow professionals and others with a publicinterest. More particularly, and despite changesover the years, I believe that we are far fromachieving a consensus about approaches to teachingwithin the MATHEMATICS education community , for example, do many authors of texts take onthe impossible task of trying to create the whole4context for the learner on the written page when, asI believe, MATHEMATICS must necessarily be created'in the air' of the classroom? I long to see morestraightforward treatments of mathematical topics,enriched by descriptions of the historical andcultural context of the subject and with appropriatechallenges for the reader.

4 Lets have moredictionaries and reference books! Even the old-fashioned mixed bag of exercises is a good resource,particularly when pupils have to classify examplesby type, sort out which they can solve and whatmethods are appropriate!If we are to get our own house into better order,it is as important to tease out significant differencesof interpretation as it is to emphasise similarities. Inthis article I shall first explore a model of teachingand learning which still seems to me to be tooprevalent in MATHEMATICS classrooms. I shallconsider some of the reasons why this modelremains socially acceptable. I shall then describean alternative model and indicate some of the waysin which it might be developed in the path-smoothing modelFirst, the main features of the model, the essentialmethodology of which is to smooth the path for thelearner:1 The teacher or text states the kind of problem onwhich the class will be teacher or text attempts to classify thesubject matter into a limited number of categoriesand to present them one at a time.

5 There is animplicit assumption that, from the exposition,pupils will recognise and identify with the natureof the problem being Pupils are led through a method for tackling key principle is to establish securepathways for the pupils. Thus it is important topresent ways of solving problems in a series of stepsMT141 DECEMBER 1992 ATM 2008 No reproduction except for legitimate academic purposes for permissionswhich is as short as possible, and often only oneapproach is considered seriously. Teachers questionpupils, but usually in order to lead them in aparticular direction and to check that they Pupils work on exercises to practise the methodsgiven aimed at involving learners more are usually classified by the teacher ortext writer and are graded for difficulty.

6 Pupilsrepeat the taught processes until they can do so withthe minimum of RevisionLonger term failure is dealt with by returning tothe same or similar subject matter throughout this model emphasisesrepetitiveratherthaninsightful activities, almost all teachers who useit as their basic approach will also consciously offersome insightful experiences. They will, for example,attempt plausible explanations, or encourage pupilsto gather data about particular cases before offeringa generalisation. However, there is usually apressure of time felt by teachers, and consequentlyby their pupils, to move on to the 'work', which isperceived as doing exercises. The teacher may findthe time to offer explanations but not to provoke thedebate needed to clarify meanings.

7 Inevitably,pupils' perceptions remain unexamined if theypassively agree to the arguments in order thatwork can proceed. So attempts to justify andexplain, although genuine in intent, can fail toconvey understanding to the is important to note that the model isperpetuated in most textbook schemes. Individua-lised schemes almost inevitably follow the model,because they are dependent on the pupil being ableto take small manageable steps, without constantlyreferring to others. So does any approach whichbasically uses a sequence of pre-structured ques-tions and does not give pupils the space to exploretheir own responses to situations or to participate inmaking significant choices for themselves.

8 We evenhave structured investigations, which attempt toreduce exploratory work to a series of algorithms orpattern-spotting exercises!Teachers who hold to this model of TEACHING andlearning certainly exercise professional care for theirpupils and help them to achieve an importantmeasure of success in public examinations. Thiscare is shown in a variety of ways: by providing astructured framework with a clear work pattern, bymarking the pupil's work on a regular basis andexplaining where the pupil has gone wrong, bybeing available to sort out difficulties as they to public examinations, these tend to fall into aset pattern over the years and are therefore oftenamenable to a path-smoothing approach.

9 Themodel is also one which parents and the public canrecognise - a popular, if only partial, view of howMT141 DECEMBER 1992learning takes place. These, perhaps, are the pre-eminent reasons as to why the model is so persistentin the face of an increasing body of knowledge andunderstanding about learning which tells a morecomplex limitations of the model may emerge invarious ways. Sometimes learners flounder whenpresented with an unfamiliar problem, perhapsbecause they lack the strategies to explore theproblem or the insight to recognise how it relatesto problems which they have met before. Fromadults we hear comments such as 'I was no good atmaths' or 'I could do the maths but never reallyunderstood what it was about'.

10 On the one hand, wehave a story of repeated failure and on the other, alack of insight into mathematical relationships andtheir application to different contexts which hasbeen only too prevalent in the adult being 'no good at maths' has itselfbecome socially acceptable in some quarters, thusfurther perpetuating the problems in TEACHING andlearning mathematicsBefore setting up an alternative model, I wantbriefly to consider some general is a tendency in debate to polariseteaching and learning styles into one of two camps:explorationinstructioninvented methodsgiven methodscreativeimitativereasonedroteinfo rmalformalprogressivetraditionalopenclos edprocesscontenttalking (pupil)talking (teacher)listening (teacher)listening (pupil)The standard response is to declare oneself tofavour a mixture of methods, neither entirelydidactic nor entirely exploratory.