workcard schemes

Workcard SchemesAuthor(s): Bertram BanksSource: Mathematics in School, Vol. 11, No. 2 (Mar., 1982), pp. 32-34Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30213713 .

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Workcard Schemes

by Bertram Banks, Kent Mathematics Project

I mean by "workcard schemes" those which use workcards instead of textbooks and, in view of several such schemes appearing in the market, a discussion about them might be timely. It is always expensive for a school to change its maths scheme, and teachers should be aware of the implications and pitfalls of a change in method and material. Unlike buying a new house, when one can see what one is buying, the real value of a workcard scheme cannot be seen from the material or the publisher's literature, unless one is knowledgeable about underlying aspects of design, development and philosophy.

A teacher must understand and accept that much of his/her work will be handed over to workcards, and typical of every workcard scheme ever tried is that pupils promptly get down to work with a surprisingly high degree of motivation. This initial display of industry has startled many teachers but it is understandable when one realises the implications to the pupil of the change of learning model. From a learning-from-teacher mode the pupil transfers to a learn-by-yourself mode in which there is no compulsion to learn at a rate considered to be most suitable to a group of children. What a relief it must be. If accustomed to waiting until others catch up, the pupil can now go ahead. If one of the unfortunates who needs more time to understand but is embarrassed to admit it, the pupil can now consolidate. If in difficulty, the pupil can consult the teacher without the rest of the group being concerned. Nobody should be surprised to see children settling down to work. The only query at this stage is how long the pupils will keep it up, and this is the real test of a scheme.

The following is a list of notes of nine important characteristics of workcard schemes which will affect long-term success in various ways.

1. Lock-step or Personal Courses I understand by "lock-step" the organisation of pupils into groups and each group treated as a unit and taken through a syllabus for a period, usually a year at a time. "First Year", "Second Year" and so on syllabuses are usually the basis of learning mathematics in secondary schools, and are at the root of most of the mathematical failure of children. I mean by "personal courses" the organisation of a personal course for each individual child, but not in intellectual and social isolation. Children learn from each other in a most remarkable way and if not isolated, valuable social education is developed.

Lock-step teaching has certainly failed with much of the large bulk of non-examination and lower CSE grade pupils. To keep pupils quiet, some teachers have turned to any measure likely to interest them, and I suspect that this might be the reason for some of the workcard schemes appearing in the market. But why only for reluctant learners? The inefficiency of the lock-step model is exactly the same for examination pupils - fast learners waste time, slow learners drop behind and there is never a match between every pupil and the teacher - and a workcard scheme of a lock-step nature is doomed for long-term learning. Its only outcome will be an initial novelty motivation.

So one aspect to look for is whether the workcards are in any way based on a lock-step system. Some modular schemes start the whole group on a particular topic and maintain work on it

until it seems to be exhausted, then another topic is started. The topics are not related, so the scheme is a series of lock- step, jump-start studies. Other schemes relate the mathematics to year groups, so are decidedly lock-step syllabuses. The idea of personalised courses in which pupils, whatever their chrono- logical ages, can forge ahead and always meet material which will challenge but be mastered, is totally ignored. For an efficient scheme, it must be possible to cater for children in the same year group being in some cases as far as years apart in concept development in different topics. With such a scheme, one could be confident of better work from all pupils over a very long period, and for examination candidates, improved results.

2. Teaching or Practice Cards The most sensitive part of teaching mathematics is the presentation of the concept. The follow-up consolidation in the form of practice is usually straightforward. So what happens if a teacher uses a scheme which is essentially one of practice cards with no presentation components? Obviously, the teacher will have to teach the presentation and if, as happens with any workcard scheme, pupils all work to different stages, the organisation to keep all the pupils busy all the time becomes impossible. Queues are inevitable, and the frustrations of waiting for attention generate behaviour problems. Practice cards used for only a short time can produce chaos.

3. Concept Hierarchy All mathematics can be analysed and fitted into a hierarchy of concept development. For example, a concept of column values is required before a concept of decimals can be established; concepts of area, square numbers and square roots are a necessary foundation for the concept of Pythagoras' theorem. The mathematical demands of a workcard should therefore not require a concept which is not already established. This means that workcards must be very carefully organised and different topics arranged so that invisible cross-connecting concept lines operate. If they do not, there will be unmanageable demand on teacher-time followed by collapses of learning and loss of interest.

A useful value of a hierarchy is to define it into stages, steps or levels, allowing pupils to see for themselves how they are climbing a ladder of learning. It is a powerful motivational component and is certainly long-term, especially in later examination work when the rungs of the ladder are modified by test results and used for assessment purposes.

4. Learning Strategies One clearly defined dichotomy is between discovery and didactic strategies and as I see it in simple terms, the choice is between whether you want the pupils to find out for themselves or whether you want to tell them. If we borrow terminology from the programmed learning (PL) world, and talk about mathematics being in terms of rules and examples (ruls and egs), the comparison is between eg-rul and rul-eg strategies. Although mathematics concept building is more complex than

32 Mathematics in School, March 1982



simple rules and examples because there are such ingredients as spatial thinking, problem-solving and psycho-motor skills, the use of this terminology makes the matter easier to discuss.

Using the trivial concept of complementary angles, one could present either:

"Two angles are complementary if together they form a right angle, or 900. For example, the complement of 400 is 500 (900 - 400 = 500). Exercise: Find the complements of 300, 350, 550, 68 .. ." or:

"We say that 40' is the complement of 50' 30' is the complement of 600 350 is the complement of 550 320 is the complement of 580

(a) What are complementary angles? (b) Find the complements of 100, 200, 250, 730 ..."

Clearly, the first approach is rul-eg. It states the rule, which is then consolidated with practice. The second approach presents examples, asks the pupil to find the rule, and then con- solidates with practice. This is really an eg-rul-eg approach and might be called guided discovery. Such a strategy is signi- ficantly different from the didactic approach of the first.

The question is, what effect will each strategy have on long- term learning? Rul-eg strategies can establish a concept in less time than eg-rul approaches, but the concepts will be poorer. An eg-rul presentation demands more inductive thinking so the concept will be richer in mathematical ideas rather than merely another piece of information to be memorised. Eg-rul pays the pupil a compliment by asking for something to be worked out; rul-eg is purely instructive. It is not surprising to find that, although eg-rul learning takes a longer time, it carries better retention and transfer.

Richer learning is not, however, the only benefit from eg- rul strategies. Most pupils find these strategies much more interesting than the instructive rul-eg approaches, so there is a powerful motivational quality that promotes persistence over a long period. Actually, the most effective strategy is a clever mix of the two approaches, and as well as straightforward examples, there are non-examples and incomplete examples to be considered and used. The important thing is that all these options have been considered and decisions made, appropriate to the concept and to the age and ability of the pupil. One can see whether this has been done by inspecting the material. If the strategy throughout a series of tasks is solely rul-eg followed by practice, one could wonder whether the author(s) knew about different strategies and their implications. Certainly, a repetitive rul-eg presentation will be boring to most pupils.

5. Supply of Answers When a teacher presents even a brilliant exposition of a mathematical topic, sets an exercise, marks pupils' work and returns it later, it is a very inefficient learning mode for the pupil. To read out answers at the end of the lesson is not much better. Pupils who have misunderstood a point, let their attention wander for a minute or two, or have even been disturbed by an interruption, can hand in work that is incorrect. More important, they do not find this out until later, whereas if they had known at the time that they were on the wrong lines, they could have modified their thinking as their thought processes were developing.

PL talked about "reinforced learning" when a pupil was supplied with each answer as the programmed lesson unfolded, and likened it to the way animals (and pigeons) learnt prescribed behaviours. Without appealing to behaviourist psychology, it can be argued as common sense that if children know whether they are working along right or wrong lines as they proceed, they will learn more efficiently.

What worries most teachers is that if pupils are supplied

with answers, they can look at them before working on the questions, and this is cheating. The Americans said it was not really cheating but "peeking", or "inadvertent cueing" but whatever, it only needs pointing out to a pupil that answers are there to be used intelligently.

In fact, intelligent use of answers can be a powerful aid to learning. To read some mathematics, do the exercise, check with answers and find oneself incorrect, can establish real understanding when one finds out why one was wrong. This is impossible to do if answers are not available, as is a pupil being able to check thought processes as a concept is developing. After the question "What is a complementary angle?" in the eg-rul sequence, the pupil should be able to check the answer.

Some workcard schemes do provide answers, but they are stored in booklets. Depending on how many answers in each booklet, how many booklets available and how the scheme organises their use, both learning efficiency and classroom organisation will be affected. There is no doubt that the best system is that when answers are put on the workcards with instructions to check them at vital points.

If there are no answers supplied, the teacher will collect piles of marking and the pupils will receive delayed knowledge about their trains of thought. If answers are supplied, the organisation of their use needs careful inspection from a logistic point of view. Queues and inadequate use of answers will be the outcome of a badly organised system.

6. Tests and Records Tests are necessary for two reasons, (a) the pupil needs to know how successful the work has been and (b) the teacher requires information about a pupil's achievement in order to select the next group of workcards in terms of the pupil's strengths, weaknesses and personal requirements. Records of all work and test results are valuable to both pupil and teacher, to be consulted and used in the establishment of mutual in- volvement. A scheme without tests and a system of record- keeping is ignoring two very useful ways of maintaining interest, apart from the fact that both are usually accepted as being necessary in any mathematics course. If absent from the scheme, teachers will have to devise their own tests and record- keeping system.

7. Try-out Procedures Somewhere in the pamphlets or teacher's guide should be information about try-out. A statement about qualifications and experience of the author or authors does not describe the effectiveness of the scheme as does a claim that material and system have been tried out in n schools for N years on pupils of specified age and ability, and modified from results. Of course, the greater the values of n and N the better the quality of the try-out, and if further information about criteria is published, the statements take on the character of a guarantee of long-term effectiveness.

One of the unfortunate aspects of workcard material is that everybody thinks it is easy to design. In fact, it requires a high degree of knowledge, skills and experience, only obtained by study and try-out. PL introduced an interesting reversal to the usual attitude to learning in that if material was not successful it was the fault of its design or a mis-match of pupil and objectives, and not because the pupil was stupid or lazy. Defining objectives and criteria and then trying out on appropriate pupils is the only way to test effectiveness and if there has been no proper try-out before publication, it amounts to offering experimental workcards which are bound to have something wrong with them with no facilities for modification. Buying such material could be a calamitous waste of money.

8. Authorship An interesting consideration about authorship is that even with

Mathematics in School, March 1982 33



our best teachers there is always a certain number of his/her pupils who fail, but given another teacher, some of these pupils will succeed. Is this a matter of matching pupil with teacher, perhaps a rapport between personalities? Or is it that a particular teacher's concepts match or mis-match a pupil's concepts? Whatever the reason, it seems right that multi-authorship is preferable to mono-authorship because it increases the prob- ability of pupils meeting an approach which is suitable.

Multi-authorship also carries another advantage because over a wide field it will bring greater experience to the material. Long experience in actually teaching a particular topic is the most valuable asset in creating successful material, and no one person can compare favourably with many over the experience of teaching a wide field of concepts and range of abilities,

9. Use of Colour We all know that colour is more attractive than black and white, but is it really necessary? I have already listed eight characteristics related to quality material which will motivate pupils for the whole of their learning careers and the addition of colour as a motivational component is trivial in comparison. It is, however, more than a trivial component of the cost and if used, one could be suspicious about the range of options known to the designers of the scheme.

Summary 1. Lock-step or personal courses. Any scheme of a lock- step nature is an inefficient learning model and defeats the idea of personalised courses. 2. Teaching or practice cards. If no teaching component in the material, the scheme becomes impossible to administer. 3. Concept hierarchy. There must be an organisation of concepts into a hierarchy with cross-connecting sub-concept lines. 4. Learning strategies. There should be a variety of approaches clearly identifiable. Repetitive use of rul-eg strategies is boring. 5. Supply of answers. Answers should be on the workcards. 6. Tests and records. If not built into the scheme, teachers will have to design tests and a record-keeping system. 7. Try-out procedures. There should be information in the publisher's literature about try-out. The larger and wider it has been, the better the scheme. 8. Authorship. Multi-authorship means more experience behind the material. 9. Use of colour. Is expensive and not necessary if scheme intentionally uses other motivating techniques.

Short Notices Playday National Girobank, 20 pp. free

Playday is a visually attractive colour magazine designed to provide information and create interest in the area of money manage- ment for the average and above average pupils in the 14-16 age range. The magazine can be read independently by the pupils, however the new Teachers' Guide offers sugges- tions as to how the magazine might be used in the teaching situation. A very useful additional free resource for teachers wishing to develop this area of work in a modern context. Copies should be ordered in multiples of 50 from, Lark Hill, Parsons Road, Bradford, West Yorkshire BD9 4DW.

Calculator Puzzles by Andrew Rothery Harrap, 48 pp. paper, x1.50 A set of calculator puzzles for the relative beginner, presented through amusingly illus- trated fantasy situations. Some of the puzzles lead to interesting mathematical patterns but the emphasis on having fun with the calculator seems to lead to some confusion in others.

Calculators in the Classroom by David Moursund John Wiley, 202 pp. paper, x5.35

A detailed introduction to calculators; how they work, their capabilities and their limita- tions, how they might be used in the classroom in the middle years and an insight into their potential impact on the curriculum. The exer- cises are designed to help the teacher develop skills and understanding necessary to ap- preciate the possible use of the calculator in the classroom.

Radio Challenge by I. R. Porteous and A. Veevers University of Liverpool, 44 pp. paper, x0.50 A booklet giving the scripts of the eight broadcasts which formed the final rounds of the "Radio Challenge" competition, the development of which is described elsewhere in this magazine (pp. 12-1 5).

34

The detail of the questions and the format of the contest will be of considerable help to any teacher contemplating organising a mathematical competition, whether in, or outside of school. The authors generously offer their questions for use in whole, or in part, or in a modified form, to anyone wishing to initiate a similar contest. Well worth the price for interest value alone.

Money with orders, please, to the authors at PO Box 147, Liverpool L69 3BX.

Electronic Learning Aids: Enquiry One by Brenda Briggs and Maurice Meredith University of Southampton, 100 pp. x2.50 post paid It is likely that, in the near future, a wide range of hand-held electronic learning aids will be marketed. Many will find their way into the home as educational toys, but what have such devices to offer in the context of the school?

This booklet attempts to provide evidence through a variety of perspectives about the

effectiveness of children's use of a particular electronic learning aid - DATAMAN. The work described in the report was conducted by a group of teachers from Hampshire. Their observations, and those of their pupils, are both interesting and informative.

Orders, with cash, to Department of Education, University of Southampton, High- field, Southampton S09 5NH.

The Problem Solver by David Wells Rains Publications, 16 pp, 25p post free

A new inexpensively duplicated magazine, containing approximately 50 problems and puzzles, designed for use in the secondary school. The range of problems covers a wide range of pupil ability and interest, but no attempt has been made to classify them. Furthermore solutions are not provided because some of the problems do not have simple answers which can be described in a few words. This will not deter the eager problem solver but may the busy teacher. Sample packs are obtainable from 194 Gold- hurst Terrace, London NW6.

Book Reviews Conquer That Cube by Czes Kosniowski CUP, 32 pp, paper, x1.35

Notes on Rubik's Magic Cube by David Singmaster Penguin, 68 pp, paper, x1.95

Mastering Rubik's Cube by Don Taylor Penguin, 96 pp, paper, x0.95 A pupil in school needs a solution which is not difficult to learn and which can be done at speed. There are a number of books on the solution to Rubik's Cube that have been rushed on to the market without much thought having been put into the solution.

Conquer That Cube does not fall into this category. It is a text which is simple to follow, with splendid coloured pictures of the cube at all stages of the solution, which should help the reader. The solution is written

in such a way with the use of brackets, that a pupil will find it easy to memorise. The book also provides a solution to the Octagonal Prism puzzle, and to Cubes with Pictures; plus a Pretty Patterns section. Very good value for money.

Notes on Rubik's Magic Cube by David Singmaster, one of the world's leading pioneers on the Cube, is your bible on the Cube. It has a four page solution in the centre pages which is concise and can be easily learnt by an able pupil. It also contains a wealth of additional material about the Cube.

Mastering Rubik's Cube has a solution which can be learnt, but it has a very small format and nearly half the pages are either blank or contain just a few words. MAL DAVIES

Basic Statistics A Step-by-Step Guide by F. Devine Harrap, 282 pp, paper, x3.95 The aim of the book is given as being to pro- I

Mathematics in School, March 1982



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