Why Don't You Write a Few Worksheets?Author(s): David McLarenSource: Mathematics in School, Vol. 4, No. 3 (May, 1975), pp. 8-9Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30211371 .

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Why don't you write

a fewwor eets?

by David McLaren, Mathematics Adviser, Glasgow

What's in a name? To judge by the recent rash of "buzz-phrase generators" or "jargon generators", very little. To make a jargon generator, all you need is a column of "subjects", one of "verbs" and one of "objects". Select an item from each column and a jargon sentence is born. This bit of fun can be given greater mathematician-appeal by numbering the entries in each column 1 through 9. Thus 351 might represent: "This structured operational facility will initialize optimal learning sequences. " To introduce a competitive element, replace each entry with a single letter and award points if the letter-triple generated by the three digit number selected represents a mathematics writing group. The remarkable response of teachers of mathematics to recent changes in syllabus and content has guaranteed an adequate supply of scoring selections. The introduction of a fourth dimension would be sure to provoke a response from physicists thus promoting much appreciated multi-disciplinary activity.

This version of the "numbers game" was inspired by a discussion with a friend of mine. His headmaster, for whom streaming was a "self-fulfilling prophecy", had announced that next session's intake of pupils were to be organized in mixed-ability classes--"even for mathe- matics and modern languages". Not surprisingly, we discussed the difficulties involved and a number of possible solutions including the headmaster's contribu- tion, "Why don't you write a few worksheets?".

Mathematics is highly structured and this imposes a measure of linearity on its development. There is little point in discussing the sum of the angles of a triangle with pupils who have no understanding of the concepts symbolized by "angle", "sum" and "triangle". For able pupils who will absorb ideas and acquire the necessary skills fairly quickly, this linearity is essential if they are to appreciate the vital differences between mathematics on the one hand and physics and engineering drawing on the other. The mathematical education of able children must not only encourage insight and creativity, it must also ensure that these insights be structured

into a cohesive whole. Although mathematics should not be unfolded as "a game played according to simple rules with meaningless marks on paper", it is important that our brightest pupils witness the power of the deductive method and experience the pleasure of applying it. To say that this approach is not ideally suited to the majority of children and that it would prove disastrous for the least able, is to state the obvious--a class lesson for the top set and the corresponding lesson for the bottom set have little in common. The differences in emphasis, style, content and pace are so great that such a pair of lessons could not be interchanged. It is hardly surprising that the predominance of class teaching and the technique of streaming are so closely linked.

How then is a teacher to cope with a class containing not only "top set" pupils and "bottom set" pupils but also pupils from all the other sets in between? The search for a suitable teaching method to meet this new situation should begin with the one we know best--the class lesson. This technique has served the profession so well for such a long time that one should not abandon it lightly. However, the class lesson is not at all suitable for unstreamed classes. It is simply not flexible enough to respond to the varied demands of the many levels of ability contained in a mixed ability class. For example, at which section of the class should the lesson be directed? If, in the interests of the majority, the lesson is aimed at the "middle" of the class, then it is likely to prove soporific and patronizing for the able while hustling and confusing the least able. Some of the method's supporters suggest a com- promise--a short lesson followed the issue of different sets of exercises, assignments,..., to ensure that each pupil is challenged but not beyond his capacity. However, this emaciated creature bears little resemblance to the class lesson so beloved of us all. Moreover, it does not cater for the special needs either of the least able or of the most able. This approach is profligate in its use of the teacher's energies without providing any obvious compensations. Class teaching

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and a policy of unstreaming classes would seem to be incompatible and once the decision to abandon streaming is taken, the most pressing need is for an alternative to class teaching.

The present suspicion of class teaching, however, is not to be attributed solely to the increasing popularity of unstreamed classes--the logical consequence of the comprehensive principle, no longer the controversial issue it was in the 1950s. The continuing improvement and increasing availability of items of "educational technology" alone would force a re-assessment of the methods used to realize educational goals. Yet again this does not explain the pressures for curricular reform. It may well be, as Hilda Taba claims, that "The strongest pressure for re-examination of the curriculum comes from the drastic changes in technology and culture, ranging from automation to atomic power, and, in the words of the Rockefeller Report, 'the constant pressure of an ever more complex society against the total creative capacity of its people'. " (Curriculum Development-Theory and Practice: Harcourt, Brace and World 1962) In addition there is a growing conviction among teachers that the learning process should be characterized by what the Nuffield Foundation describes as: "a new emphasis on the active part that the pupil should play in the learning process." To accept this view is to conclude that the class lesson is a technique to be used sparingly, if at all--during a class lesson it is the TEACHER who is active, the pupil is passive.

The criteria involved in the selection of an alternative are now emerging. The alternative strategy should make provision for:

(i) Pupil involvement. This would seem to imply that the system be, in some measure, self-instructional. (ii) Flexibility in content and approach. (iii) Mixed ability classes. (iv) Individualized/small group learning. The headmaster clearly meant his suggestion--"Why don't you write a few worksheets?"-to be taken seriously.

Like many of us, my friend had experimented with this kind of material and had even written some himself. His experiences had shown him that pupils respond well to such an approach and that, relieved of the burden of formal class teaching, he was able to establish much closer contact with his pupils as individuals. He was, however, somewhat daunted by the prospect of such a large scale operation and our conversation turned to the "mechanics" of producing a suitable system. The process may be illustrated as follows:

Planning/Design

Production <

Experiment (

Evaluation/Revision

Implementation

Planning/Design It would be difficult to overemphasize the importance of planning. The aim(s) must be clearly stated before anything is attempted. Otherwise the entire exercise is likely to be muddled and inefficient. The greatest of care must also be taken in designing the system. If the planning and design are skillfully and conscientiously undertaken then the remainder of the exercise may be completed fairly uneventfully.

Production The appropriate equipment and apparatus should be purchased or constructed and the instructional material written and duplicated. If proper attention has been paid to the planning/design stage, this task will be reduced to manageable proportions. Experiment It is usual to limit the experiment to a small number of pupils. This not only reduces the cost and minimizes the risk, it also makes it easier to collect the information necessary for a proper evaluation of the project. Evaluation/Revision At this stage the evidence collected during the experimental period should be analysed carefully to discover whether or not the aims of the project have been realized. The comparison of the intended learning with the actual learning, as indicated by assessment, will provide the information needed to decide whether or not a given part of the course should be modified or rewritten or even completely re- designed, whether or not the testing/assessment pro- gramme should be overhauled, whether or not the statement of aims and objectives should be altered. ... This cycle of refinement, planning-production- experiment-evaluation, should be continued until the actual learning is consistent with the aims and objectives.

The first stage in this process is worthy of further discussion. The classical approach begins by stating clearly the specific educational objectives to be achieved by the learner. These objectives, if they are to be useful, should be stated in terms which describe what the learner will be DOING when he is showing that he has achieved the objective i.e. in behavioural, or performance, terms. The four reasons commonly advanced for this procedure are: (i) It gives the mathematics curriculum a definite direction. (ii) It facilitates the selection and organization of content. (iii) It facilitates learning in that the learner knows what is expected of him. (iv) It makes possible the evaluation of outcomes. This approach is not without its opponents. Not all outcomes can be specified in behavioural terms in advance. In many areas of the arts the object of learning is an original idea, design, invention,..., and the outcome should "surprise" both teacher and pupil. Clearly the method has its limitations and equally clearly there are learning outcomes in mathematics which are not amenable to this kind of description. Again, some outcomes are not "quantifiable" and so are not available for evaluation by testing. The implications are not that behavioural objectives are harmful; the harm lies in asserting that they are universally applicable or that evaluation is to be equated with testing. It's a question of "choosing horses for courses". assessments to provide: (a) Feedback. The learner must be made aware of his achievement, or otherwise, of each specific goal. The system should contain a number of short "self-tests". (b) Diagnosis. The teacher needs this information if he is to provide remedial instruction.

(c) Prognosis. Before embarking on an enterprise as difficult as this, it would be prudent to investigate the ways in which other groups with access to the necessary finance and research facilities have responded to the challenge. IMU (a Swedish individualized mathematics instruc- tional system now published in Britain by Caffrey, Smith) is an interesting example of the "classical" approach and will be the subject of a later article.

In the meantime, why don't you write a few worksheets? I

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