Implications and Challenges of the Development of IT for the Mathematics Curriculum inSecondary SchoolsAuthor(s): John HiggoSource: Mathematics in School, Vol. 23, No. 1, Primary School Focus (Jan., 1994), pp. 49-51Published by: The Mathematical AssociationStable URL: http://www.jstor.org/stable/30215080 .

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Implications and

Challenges of the Development of lT

for the Mathematics Curriculum in

Secondary Schools by John Higgo

Readers will have been sad to learn of the untimely death of John Higgo in August 1993. At the time of his death he was in the process of writing an article for "Mathematics in School" in the series being produced by the sub-committee on "The Mathematics Curriculum for the year 2000". The article would have expressed his present views about how the school mathematics curriculum should change in response to the arrival of computers in the classroom. Although his article was not completed, fortunately it was based on a paper which he wrote in January 1993 for a DFE Conference. We are therefore reprinting this paper here, both because of its intrinsic interest, and as a tribute to John Higgo's work for the Mathematical Association.

Introduction 1. As a result of IT, mathematics education is probably at its biggest watershed of all time. It will be taken for granted throughout this paper that computers both have profound implications and offer great opportunities for the mathematics curriculum'. Some significant progress has been made in translating these into classroom realities over the last decade, despite a number of very real impediments. In truth, however, the effects seen in the typical British classroom in 1993 remain disappointingly modest. 2. The rapidly changing nature of the technology both in and outside the school environment exacerbates the difficulties. Continuously changing technology implies a continuously changing curriculum. Sadly, as Kenneth Clarke conceded privately in 1991, opportunities for such necessary evolution are not built into the National Curriculum. Furthermore, changes in examination sylla- buses arguably offer the most effective agent for change in secondary schools; but traditional written assessment is particularly unsuited to testing the most valuable work that can be done with computers in the classroom. The picture at A-level may be less gloomy. Progress will

depend crucially on how rigidly cores and criteria are applied. The more freedom of movement the best teachers (and innovative projects such as Nuffield Secondary Mathematics) can be given the more readily worthwhile curriculum development can take place.

Classroom Opportunities 3. By using IT, pupils can learn existing curricular content, skills and concepts more readily, more enjoyably, with better understanding and with enhanced ability to apply the mathematics. Opportunities for such improve- ments in learning abound. They are now extensively documented in educational journals and publications2 4. There is perhaps space to mention just a few examples here:

0 Through initially using numerical "trial and improve- ment methods" and/or "zooming in" with a graph plotter or graphic calculator, pupils can understand and solve more complex equations at a younger age.

0 Similar dynamic, numerical and graphical approaches can afford ready insight into the principles of differen- tial or integral calculus.

0 The program REACT3 has a dynamic, interactive visual display and itself carries out the arithmetic needed for basic statistical processing. Thus, pupils at Key Stage 3 using REACT to capture their own reaction times, are able quickly to gain an appreciation both of techniques of data processing and display, and of relatively sophisticated statistical measures.

Curricular Implications 5. Mathematics as practised in industry, commerce and universities has itself changed in response to the availability of computers. It is therefore to be expected that mathemat- ics courses in schools and colleges should reflect, at least in some measure, the relative importance of different aspects of mathematics in the world outside the classroom. As well as a change in emphasis, this change will involve the topics which are taught, the order in which they are introduced and the methods which are used to approach them. 6. At both primary and secondary levels, experience has repeatedly shown that concepts hitherto regarded as difficult, become accessible to children much earlier when computers are used appropriately4. In some cases, because computer approaches are available, the optimum time for

Mathematics in School, January 1994 49

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a child to be introduced to a topic may be very much earlier than laid down in the National Curriculum. Designing a curriculum which gets this timing right could play an important part in fulfilling children's entitlement to be given opportunities to develop their full potential. 7. Some of the shifts in emphasis implied by IT have already begun to take place independently, but there needs to be much more. Computers favour numerical and geometrical approaches. Perhaps more important, through taking away the need for repeated drudgery and allowing much more extensive computation or processing, they allow a greater concentration on understanding than on practising techniques. In statistics,

the emphasis can move away from the techniques of processing data and focus instead on interpreting the information which this processing provides.

Especially at A-level, sophisticated modelling techniques are now accessible which would have been unthinkable a few years ago. Teachers who have used IT much in their classrooms will affirm that their whole conception of what many mathematical topics are about has undergone a radical change as a result of the way the technology can do it. 8. The approach which the computer requires is essen- tially one which encourages pupils to play an active part in their own learning; and this has implications for both pupils and teachers. Access to computers encourages the less traditional styles of learning and teaching mathematics advocated in the Cockcroft Report6, such as problem solving, investigation, discussion and extended work. 9. In such a learning environment, the task of the teacher changes from the giver of information to the mentor who can guide pupils towards methods and approaches which will deal with the problem under consideration and who, most importantly, can establish through discussion the links with related concepts and topics which are essential for sound mathematical development. This represents a significant departure from the more usual perception of a teacher as one who, from a position of authority, passes on knowledge to pupils and directs their work in consider- able detail.

Difficulties 10. A large number of obstacles stand in the way of implementing radical curricular change of the kind outlined above. Important among these are natural limitations on the possible pace of progress in education, large resource requirements (both human and financial), problems in defining subject boundaries (particularly between Mathematics and IT), and shortages of time, conviction and experience among teachers. Add to this the fact that, in such a new and developing situation, there cannot always be definitive ready-made solutions. Little can yet be said to be "proven" and some future technological developments cannot be foreseen. In such circumstances, full agreement between "experts" - let alone the ground troops - cannot be expected; room for debate and experiment are essential.

Assessment 11. On top of all these problems comes the old adage, which I believe to be largely truthful, that - at secondary level at least - what is not assessed does not get taught. Suppose, for example, that we wish to specify that learning mathematics through programming a computer should form a necessary ingredent of the curriculum for some pupils. Then this would be very difficult to assess in a

50

written timed test, even supposing that enough suitable machines were available. The fact that different pupils might choose, variously, to use a symbol manipulator, a graph plotter, a graphic calculator, BASIC, Logo, a spreadsheet or a modelling system such as Numerator as a medium for their "programming" would just compound the difficulties further. 12. Or take the National Curriculum. It includes several helpful references to the use of computers. But, because this is not a part of written testing (nor, indeed, much encouraged any longer in coursework), many secondary mathematics departments still manage to avoid using computers at all. They use the argument, for example, that database work is done within the Technology curriculum anyway.

Politicisation 13. In the decade up to about 1987, there developed a consensus, supported by research, as to desirable styles of teaching and learning mathematics, which were remarkably in harmony with what the advent of computers implied for the mathematics curriculum. Existing curricular struc- tures were well in line with what might best help in promoting rapid and widespread acceptance of IT-gener- ated changes in the classroom. 14. Since about that time, central government has exerted an unprecedented influence in education. It has to be said, unfortunately, that some of the directions taken and some of the current political wisdoms act counter to bringing about rapid evolution towards an "IT aware" mathematics curriculum. The big three problems areas are the rigidity of the National Curriculum for Mathematics, the down- grading of coursework and perceptions about "rigour". 15. In addition, the inhibiting effect of increased demands upon the time of teachers and - in respect of testing - upon pupils, should not be underestimated. 16. We may take pleasure in the fact that the National Curriculum for Mathematics is (perhaps nearly) as good a one as could have been devised- given the premise that it is necessary or desirable to prescribe what amounts to 100% of the content for nearly all children. But that premise is fundamentally flawed. Neither can there ever be such a thing as an agreed, ideal mathematics curriculum, nor could any static curriculum, allowing no choice of content, remain a good one for long. As stated earlier, at a time of growing technological development, there is more need than ever for the curriculum to be susceptible to organic change'. 17. A problem to address, then, is how to make necessary, progressive changes to the curriculum without resorting entirely to leaps in the darks. As was true of the internationally-admired English tradition of the immediate past, it may be argued that

Freedom to experiment at local level is a crucial precondition for informed national curricular choices based on actual classroom experience to be made in the future.

18. It would not be appropriate here to argue the general case for coursework in school mathematics, which has been established for a decade or more. However, it must be stated that, at all levels, allowing a substantial proportion of assessed coursework has proved to be an excellent way to give pupils scope and incentive to develop their skills in mathematics, using a computer, to the full. 19. In the first place, there are aspects of mathematics, such as investigations, extended endeavour, programming and other ways of using a computer, that are difficult or impossible to assess in other ways. A syllabus with a significant amount of coursework can enable pupils to

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demonstrate prowess in these aspects and so encourage these actually to receive due attention in the classroom. The very diversity of software and other technological aids available and of the ways in which they might sensibly be utilised in the curriculum also suggests that computer work can best be tested, and hence encouraged, via a flexible medium of assessment. 20. Finally, but perhaps most important of all, course- work could serve as a testbed for tomorrow's curriculum by providing a flexibility of content not otherwise available within the National Curriculum. I am by no means the only teacher who has encouraged pupils, as part of their coursework, to gain credit from experimenting with suitable topics or processes not in their GCSE syllabus.

Strategic Priorities 21. We need to define and implement a longer-term strategy for the mathematics curriculum based on up-to- date perceptions of what the subject is about. One strategic priority should be to ensure that teachers and children do use appropriate available technology in the classroom. As a higher priority, we must grasp the curricular and assessment implications of IT, for example of the arrival in large numbers of graphic calculators or symbol manipu- lators, before we end up simply having to react to events! This will require considerable thought, debate, experiment and research. Solutions to curricular problems

must be found by drawing on the experience of those who work with pupils in the classroom and on the results of carefully planned research.

22. Although some of the changes which will be required can already be foreseen, the need for others will become apparent only as the result of experience in the classroom; for example, change at one level will have a knock-on effect at other levels. It is therefore essential that arrange- ments should be made to ensure that the situation is kept under continuing review. 23. Particular questions which will ... need to be addressed include:

0 to what extent should pupils still be required to carry out statistical calculations manually rather than make use of the powerful and accessible statistics packages which are increasingly becoming available?

. to what extent have graphic packages supplanted the need for accurate geometrical construction with pencil, ruler and compasses?

. to what extent has the availability of graph plotters removed the need for pupils to develop the skills of plotting and drawing graphs on paper?

. to what extent should the availability of symbolic manipulation packages lead to less emphasis on the practice of algebraic techniques such as factorisation and the solution of equations?

24. A great deal of the development of the last decade has been "technology led"; it is to be hoped that in the coming years more attention will be paid to development which is clearly focussed on important curricular and pedagogical issues.

The development of graphic calculators has shown that it is possible for mathematics education to influence manufacturers.

Some Possible Tactics 25. Some of the more obvious tactical suggestions for achieving the aims discussed, are:

0 Continue to encourage NCET, the subject associations and other groups developing classroom ideas, methods and materials.

0 Allow choices - reduce the statutory content of the National Curriculum to no more than 70% of what will be taught.

It is ... essential that those who have the task of revising the National Curriculum should seek to do so in ways which show awareness of the use of computers and take account of the possibilities which they offer for improv- ing the teaching and learning of mathematics. This could be accomplished by providing a greater degree of flexibility within the existing framework so that the stages within each attainment target are not too small and the order of their introduction is not defined too precisely. In this way teachers would be enabled to develop their thinking and their practice so as to find the best routes by which to approach the targets which the National Curriculum specifies.

0 Make appropriate use of a computer a compulsory aspect of Mal. (Use of computers should be considered just as important as, say, practical work - or even as open-ended tasks.)

0 Make desirable new topics or approaches in the curriculum examinable, initially allowing choices as appropriate.

0 Encourage the development of at least one "IT aware" GCSE (KS4) mathematics syllabus. (At A-level, encourage projects such as Nuffield Secondary math- ematics.) Free such projects from some of the shackles.

0 Continue to raise the profile of IT in initial and further training of teachers.

Challenges 26. To conclude, here are some of the major challenges which the development of IT poses for the mathematics curriculum in secondary schools. First, can educationalists and other interested parties agree on a set of principles and goals? Second, have we the will and vision to develop an "IT aware mathematics curriculum"? Third, can we loosen curriculum and assessment structures sufficiently to implement such a curriculum and allow it to be updated regularly enough? Fourth, can we influence technological development along lines appropriate to classroom needs? And, if nothing else, can we at least find ways of accelerating uptake of IT-based methods in mathematics classrooms? Finally, all this implies a heavy burden for teachers; can we help them more?

References 1. The case for this is well documented, most recently and comprehen-

sively in the Mathematical Association's report Computers in the Mathematics Curriculum, published, in November 1992, by the Mathematical Association, 259 London Road, Leicester LE2 3BE.

2. Most notable of these are perhaps the ATM journal Micromath, the DES-sponsored Stanhope report Will Mathematics Count, 1987 (avail- able from the MA or ATM), various NCET publications and the MA report Computers in the Mathematics Curriculum, 1992 (op. cit.).

3. See ibid. paragraph 145. The program is by Richard Phillips, Shell Centre, Nottingham.

4. Three examples of this are to be found in paragraph 4 above. 5. All indented pieces of text in this paper, such as the one here, are

taken verbatim from the Mathematical Association's report Computers in the Mathematics Curriculum (op. cit.).

6. Mathematics Counts, 1982 (HMSO). 7. See Not the National Curriculum, 1992, a booklet available from the

Mathematical Association, 259 London Road, Leicester LE2 3BE. 8. It may be noted in passing that we have recently witnessed a highly

successful curricular leap in the dark - namely, the "Chance" compo- nent at the lower levels of Ma5 in the National Curriculum.

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