Algebra Teaching TodayAuthor(s): David HaleSource: Mathematics in School, Vol. 9, No. 1 (Jan., 1980), pp. 11-12Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211923 .

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Al Teaching Today

by David Hale

The teaching of geometry has been the subject of debate for well over a 100 years. Arithmetic teaching has received much attention during this century and is again under particularly close scrutiny following the "Great Debate" and technological advances.

Algebra in contrast has received little attention apart from some rather unspecific complaints from several quarters about the lack of emphasis on manipulative skills. Perhaps the most profound change in the teaching of algebra is one of scale, in that the majority of pupils now learn some algebra whereas its study was once restricted to the abler end of the ability range. Classroom observation suggests that pupils' response to algebra teaching is often disappointing. Much time is devoted to routine exercises which appear to have little meaning for many pupils and levels of interest and understanding are sometimes unacceptably low.

The purpose of this article is to stimulate a discussion about the teaching of algebra. It seeks to raise questions which are more fundamental than those so far generated by the manipulative skills debate. The matter appears to have some urgency because teacher response to criticisms about pupils' ability to manipulate symbols could result in an even narrower approach to algebra teaching.

Why Teach Algebra to All Pupils? This is a question which few teachers appear to have considered. Discussions on the purposes of algebra teaching often go little further than the statement: "It's in the CSE syllabus". Reasons which might be advanced in support of the teaching of algebra include:

(i) to help the pupil gain experience of generalisation as a mathematical process; (ii) to teach techniques which will be needed later in a post-16 course; (iii) to provide a range of techniques to strengthen the pupil's problem solving ability; (iv) to enable the pupil to attempt questions which appear on 'O'-level and CSE examination papers; (v) to provide for an algebraic approach to geometry. (The reader may care to add to the list.) Not all the above are equally relevant across the ability range, but it could be claimed that there is a sufficient basis to justify some algebraic content in all 11-16 mathematics courses. It is a

separate question whether we have the teaching force capable of carrying out the task of providing algebra for all.

How is Algebra Introduced in the Classroom? It would be a bold person who stated dogmatically when and in what sense a child starts thinking algebraically. (One com- mentator has pointed out that a 4-year-old who says "You are the bestest daddy", is already demonstrating algebraic awareness.)

However this section is not concerned with the complexities of children's psychological development. It looks instead at the more mundane matter of approaches to algebra in the early secondary school; in other words, what teachers do when they say they are starting algebra.

The evidence suggests that pupils today are introduced to algebra in a way that has changed little over the years. They solve simple linear equations (often by applying rules and not always checking to see if a proposed solution is correct); they collect like terms (sometimes to the accompaniment of grossly

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misleading statements from the teacher about the incompatibility of apples and bananas); they substitute numbers (often into expressions which have no obvious concrete meaning); they insert and remove brackets (the fact that calculators have a variety of hierarchies is seldom mentioned).

An alternative start, rarely seen, might be called the function- sequence method (exemplified by material in the DIME' and the South Notts Projects2). As an approach it has some obvious advantages: (i) it may be based on the use of cheap concrete materials - matches, cubes, grid papers, etc; (ii) symbolisation may be introduced more gently then in some of the more conventional activities; (iii) generalisation is a self-evident goal to most pupils; (iv) it grows naturally from number pattern work undertaken in many primary schools; (v) it appears to offer realistic challenges to pupils across a wide ability range; (vi) problems of symbolisation arise in a context; (vii) simple graphs can be drawn to depict relationships. There is one major caveat; it is an approach which appears to be significantly more demanding of the teacher than some of the others mentioned. There is no easy resort to the textbook exercise; the teacher needs to be able to handle a class or group discussion skilfully, to judge when to intervene in a pupil's work and when to introduce a particular symbol convention. In other words, the professional responsibility is firmly shifted from the textbook writer to the classroom teacher.

What is the Pay-off for the Pupil? For some pupils there is a return commensurate with the effort they have invested. They go on to use algebraic techniques in post-16 courses of various types. For other pupils, perhaps the majority, a return on investment is achieved mainly in their ability to attempt algebra questions in public examinations. For the remainder there is often little or no return: it is as if they have attempted to learn the symbolism and convention of knitting patterns without ever being able to produce a single knitted garment. For this last group the return on investment is negligible. They practise routines as time-filling activities and the symbols they manipulate have neither meaning nor interest.

To give algebra meaning and purpose across the whole ability range is no simple task but there are some strategies which are perhaps not used in the classroom as much as they might be.

(i) Not all mathematics teachers have at their disposal a comprehensive collection of simple formulae taken from other areas of the curriculum. (ii) The value of everyday applications of mathematics with a potential algebraic content is often not appreciated. (The table of stopping distances in the Highway Code for instance.) (iii) There is little use of simple algebraic techniques to solve recreational puzzles. (iv) Algebraic justifications for certain techniques are rarely discussed with pupils. (For example, divisibility tests.) (v) The use of differences to investigate sequences and to classify them as linear, quadratic, etc., is not often taught systematically. (vi) Investigation of sequences which have a more complex generating function than is suggested by the first few terms could be given more prominence. ("Regions of a Circle" is one of the classic examples of this type.)

Other Algebras How much does an experience of different algebras help the pupil to understand traditional algebra? Followers of Dienes might say that working with several algebras puts the pupil in a better position to extract the generalised notion of an algebra. But this view, as with that concerning the introduction of different number bases to facilitate the understanding of place value, seems to be based on a hunch rather than on reliable research evidence.

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Should we continue to assert the pre-eminence of traditional algebra, that is, generalised arithmetic? Yes, in the opinion of the writer, because of its arithmetical associations, its usefulness in a wide range of applications and its importance in post 'O'-level mathematics courses. But in addition we might also encourage more use of activities which lead to the study of some algebras with simple rules (the "Mark and Rub" algebra from Starting Points3 is one example).

Concluding Comments Perhaps more than any other area of mathematics, algebra needs skilled and imaginative treatment by the teacher if it is to have relevance and interest for the pupil. Evidence from the classroom suggests that, for pupils of average ability or below, algebra is often little more than the manipulation of symbols according to stated rules. Algebraic ideas are rarely developed from previous arithmetical work and there are few opportunities for pupils to take an active part in the learning of its techniques and conventions.

Two other matters which clearly need further investigation and discussion are the position of algebra in individualised schemes of work (is it particularly at risk?) and the teaching of conventional notation and certain algebraic techniques in the calculator age.

An unthinking response to the plea for more attention to be given to manipulative skills could easily make the present situ- ation worse. What is urgently needed is a professional discussion of the place and purpose of algebra in 11-16 mathematics courses with particular emphasis on the needs of the pupil who leaves full-time education at 16+.

References 1. Development of Ideas in Mathematical Education, University of Stirling. 2. Bell, Rooke and Wigley, Journey into Maths, Blackie. 3. Banwell, Saunders and Tahta, Starting Points, OUP.

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