Microcomputers in Mathematics Teaching

Derek Ball

Derek Ball trains mathematics teachers at the University of’ Leicester School of Educution, is the author oj‘Microcomputers in Maths Teaching’ which is published by Hutchinson, and is the editor of‘ ‘Micromath ’. a nationa1,journal ubout mathematics teaching and computers which is published by A T M und Basil Blackwell.

Abstract The use of computers is of considerable value for those learning mathematics. Their effective use depends upon adequate teacher education and the provision of sufficient hardware in classrooms where mathematics is taught. The purpose of using computers in mathematics classrooms is to engage learners’ interest and involvement in the mathematics they are learning. To achieve this learners need to be in control of what they are doing. At present, turtle geometry and the use of relatively limited software packages appear to be most effective for this, but in the near future it should be possible to develop software tools which are both powerful and easy to use. These are likely to include information-retrieval packages; spreadsheets; and also ‘microworlds’ written in a language such as Logo or Prolog.

Introduction As the life of the Microelectronics Education Programme (MEP) was coming to an end the Department of Education and Science commissioned a series of critical reviews of the ways in which computers were being used in school classrooms in connection with the teaching of a number of different subjects. In March 1986 a conference was held in Stanhope, County Durham, to produce a critical review of the use of microcomputers in mathematics education. This review was published in the spring of 1987 under the title Will Mathematics Count? It contained chapters on the place of computers in mathematics education, on classroom learning styles, on the mathematics curriculum, on teacher education, and on equipping classrooms for mathematics. The introduction to the review considered the relevance of computers to mathematics classrooms. It also reiterated the firm belief of all involved in producing the review that computers are of particular significance for the teaching of mathematics, and have profound implica- tions, both for the styles in which mathematics can be learned, and also for the content of the mathematics curriculum. At the beginning of the 1980s there was apparently a policy of giving priority to developing the use of computers in other areas of the curriculum. This was a consequence of a desire to try to dispel the myth that mathematics and computing were inevitably linked. The policy was remarkably successful in achieving this goal, but perhaps the price paid was the failure to make the most of a new teaching and learning aid of considerable relevance to mathematics. In its later stages, MEP sponsored the production of a series of primary in-service packs, several of which were concerned to a greater or lesser extent with mathematics, and with the Secondary Mathematics with Micros In-service Pack. These packs not only provided materials for classrooms, but also raised issues concerning effective ways of using computers in connection with the learning of mathematics.

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The provision of hardware Will Mathematics Count? makes clear recommendations concerning the provision of computers in classrooms where mathematics is taught. One significant reason why computers are not used more frequently to aid the teaching of mathematics is because they are not readily available. Frequently, the only computers that are available are housed in a computing room, use of which has to be booked in advance. This encourages teachers and pupils to think of the use of computers as a special ‘one-off event, relatively unrelated to the normal work of the classroom. Where computers are available for classroom use they frequently have to be shared among several departments, and transporting them may be as much of a problem as booking them. Will Mathematics Count? recommends that, as a minimum, a computer should be provided for every mathematics department in a secondary school and for every three or four classes in a primary school. It further recommends that we should move speedily towards the situation where a computer is available in every classroom in which mathematics is taught.

Educating teachers Will Mathematics Count.? also has a chapter on in-service education. It suggests that in- service work is most likely to be successful if those providing it acknowledge the fact that teachers are in the best position to judge what they themselves need. An in-service course currently based at Leicester School of Education is bringing together teachers who teach English, humanities and mathematics in secondary schools. They spent three weeks at the School of Education, discussing, sharing ideas about what approaches might be fruitful, and preparing themselves to make substantial use of computers in their classrooms. They are now spending more than a term working at some of their ideas in their own schools. At the end of the course they will spend a week evaluating what they have done and writing a report. At the time of writing, halfway through their term back in school, teachers have just spent a day sharing what they have done so far and discussing difficulties. Very few of these difficulties concerned the use of software, the shortage of ideas or the attitudes of pupils. They were almost exclusively to do with school management and policy. One teacher could not gain access to the computer she had been promised, and all the other school computers were urgently required at the times when she planned to use them. Another teacher had great difficulty gaining access to the computer room, where all available computers were housed. The head of computing was reluctant to put her software on the computer network, and expressed surprise that an English teacher, and especially a woman, would have any interest in computers at all. Eventually her software was loaded on to the network and she spent several hours writing datafiles for her pupils to use. All her careful work was subsequently wiped off the network ‘in error’. Yet another teacher had great difficulty persuading his head of department to part with even a modest amount of money for the purchase of software. The persistence of several of these teachers has brought its rewards: some schools have found money for the purchase of an additional machine, for example. But the stories demonstrate amply that more than vague goodwill is required if children are going to be able to use computers regularly to aid their learning of subjects in the school curriculum.

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Different uses of computers The remainder of this article will be concerned with the subjects of the other two chapters of Will Mathematics Count?: classroom learning styles, and the mathematics curriculum.

There are many different ways of using computers to support teaching and learning. While this is true of all subjects, the range of uses to which computers can be put in mathematics classrooms is particularly wide. In looking at the uses of computers for mathematics it is useful to remember that not only the use of prewritten software but also programming are seen as having significant roles to play.

Using the software uuailahle There are various ways of classifying software written to aid the teaching and learning of mathematics. One way is to look at the degree to which the topic to be studied is prescribed by the computer program. Programs which do prescribe the topic are sometimes called content-specific while those which do not are called content-free. The distinction is not a tidy one; what needs to be considered is the extent to which a program specifies the mathematics to be considered. Very little will be said here about ‘drill programs’ or about ‘tutorial programs’. Such programs are intended to test learners or to teach them specific facts. Few teachers in this country make serious use of such programs and it will be assumed that they are of little value. But there are other types of content-specific program of which valuable use is made. ANGLE90 and ANGLE360, for example, are programs which help pupils to develop their under- standing of angles and their ability to estimate the size ofangles. PLOD is a program which sets a specific problem for learners to solve and provides them with a way of testing out ideas they may have for solving it. There are other programs which specify the problem to be solved but leave considerable choice concerning the way in which this is to happen. CARS: MATHS IN MOTION is such a program. Then there is a very important group of programs which deal with a fairly specific mathematical context, but which provide tools for learners and teachers to use in whatever way they choose. ATM’s COUNTER is such a program on the subject of sequences and functions. Function graph plotters (such as Richard Phillips’ program of that name) also fall into this category, as does CIRCLE PATTERNS, a program which will be described later. At the content-free end of the spectrum are utilities such as spreadsheets and information-retrieval packages. These are well known to educationa- lists and their usefulness is widely proclaimed, but the frequency of their use in mathematics classrooms to date does not appear to be great.

Writing progrums Getting pupils to write programs with the aim of helping them to learn mathematics is an idea about which much has been written. The best documented examples concern the use of LOGO. Some teachers have given children in primary schools, and in the early years of secondary schools, experience over several months or years of writing LOGO code for ‘turtle geometry’. Considerable claims have been made concerning what can be learned from such experiences. Children not only learn about lengths, angles and the use of large numbers: they also learn by discovery, and develop

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confidence from planning and executing their own projects, and from correcting for themselves the mistakes they inevitably make in the process. The thinking required to program in LOGO-to use procedures and recursion, for example-is also claimed by some to be of benefit to those learning mathematics, and such concepts as that of a variable may be met in contexts which make them easy to understand and use.

Children have also been encouraged to write short programs in BASIC. The advocates of this have suggested in the main that such work is most appropriate for older pupils in the middle and upper years in secondary schools. It has been suggested that one way of discovering whether you have understood a mathematical idea or process is to write a program for it. It is claimed that BASIC can help pupils understand what is meant by an algorithm, and can also help them to grasp particular algorithms of importance to those using mathematics. Many collections of ideas have been published, but it is not clear how widely they have yet been tried in classrooms. I t may be that most of the BASIC programming in mathematics classrooms has so far been done not by pupils, but by teachers who have wanted to use a program as a visual demonstration of some mathematical concept, or to provide a program for their pupils to use to help them explore a particular mathematical situation or idea.

There are other languages whose usefulness for the classroom have been discussed. One of these is PROLOG. Little use has been made of PROLOG so far in school mathematics classrooms. This is partly because the language is still relatively expensive, partly because the versions of PROLOG available on some school computers do not give users access to graphics, and partly because the language does not have a screen editor which is easy to use. PROLOG might also be considered difficult to use for another reason: there are very few commands available as part of the language itself. This weakness of PROLOG may also be its strength; with PROLOG, as with LOGO, it is easy for a programmer to provide ‘microworlds’ for users of the language. To create a microworld a programmer writes program code which provides users with commands in addition to those available within the language itself. It is possible to do this in such a way that the user is unaware of any difference between the commands provided within the language and commands added by the programmer. The programmer can add commands, all of which are relevant to a particular topic: for example, the drawing of graphs, transformation geometry, Euclidean geometry or Newtonian mechanics. Users of the microworld can then include these in their own programs. Tf the set of commands provided is chosen judiciously users can be provided with sufficient support, but retain considerable freedom to do what they want. To date little use has been made ofmicroworlds in mathematics classrooms, and little work has been done on ways of designing microworlds which are effective in supporting pupils and teachers in classrooms.

Styles of learning The Cockcroft report appeared in 198 1 and, since then, the paragraph from this report which has most frequently been quoted is paragraph 243. This describes six styles of learning mathematics and suggests that mathematics teaching at all levels should provide opportunities for all six. It has been widely assumed that four of these styles of learning (discussion, practical work, problem-solving and investigational work) have

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been missing from many mathematics classrooms. The interest in turtle geometry in primary schools and the inclusion of coursework in GCSE mathematics can both be considered as attempts to remedy this situation.

What was the intention of those who drafted paragraph 243 of the Cockcroft report? Why should there be interest in encouraging a variety of styles of mathematical learning? It seems self-evident that learning is most likely to occur when the learner wants to learn, and what is presented to the learner engages the learner’s attention, interest and involvement. This, it is supposed, is more likely to be achieved if the learner is encouraged to use a variety of learning styles.

It is also claimed that computers can help engage the learner’s attention, interest and involvement. Is this true? And what uses of computers do we offer learners of mathematics to make it true? To help me begin to answer this question I considered what uses of computers in connection with mathematics have engaged by interest during the last few months.

My use of computers for mathematics I have selected three examples. The first of these involves my role as editor of Micromath. One of the articles appearing in Micromath 2: 3 concerned the decoding of secret messages. At the end of the article were some messages which readers were invited to decode. I thought I ought to try this for myself, partly to ensure that there were no misprints. Decoding is an algorithmic activity which computers are good at, and the obvious thing to do was to write a short program in BASIC to do the job. I did this and derived satisfaction from using it to help me decode the messages successfully.

The second example concerns a task I was given by a teacher. Pupils in some local schools had been investigating ‘growth patterns’ in connection with their GCSE coursework. In the simplest case you start with a square, and colour in the four squares which touch it. Then you colour in all squares which touch one side, but not more than one, of these squares. And so on. (See Figure 1 .)

In this simplest case the number of squares added at each stage produces a recognizable pattern:

4 4, 12 4, 12, 12, 36 4, 12, 12, 36, 12, 36, 36, 108 and so on.

The situation can be modified in various ways. You can start with more than one ‘seed’ square. You can grow triangles or hexagons instead of squares. The teacher who sought my help wanted to know about growing triangles, What was the number of triangles added at each stage: did it form a regular pattern? Could I get a computer to generate the first hundred terms, so that she could look for patterns? I took up the challenge. I had no piece of software available which would do the job. I had to write the software myself. BASIC did not seem a suitable language to use. The problem was not specifying the algorithm (‘keep doing the same thing over and over again’) but communicating the problem to the computer. PROLOG seemed a far more appropriate language for this purpose and, as I already had experience of program-

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Figure I.

ming in PROLOG, this is what I used. It did not prove too difficult to obtain what I wanted. However, for my own interest and satisfaction, and also to check that the program was doing what was intended, I wanted to get the computer to draw the pattern of triangles on its screen. PROLOG was useless for this purpose: it does not support graphics. Consequently I rewrote the program code in LOGO. This was not too dificult, because LOGO and PROLOG have many similarities. On the other hand, the code was considerably less elegant and less comprehensible in LOGO than in PROLOG.

The third example is the program CIRCLE PATTERNS. This program draws on the screen a circle whose circumference is marked with a number of equally spaced points. The number is chosen by the user. Each point is assigned a number in order, starting with 0 at the top. Pairs of points are then joined by straight lines in accordance with a rule, also specified by the user. If the number of points is 12, and the rule is N +N + 3 , the pattern produced is as shown in Figure 2.

If the number ofpoints is 36 and the rule is N+2N the pattern is as shown in Figure 3 . I had been using this program with pupils aged from ten upwards for some time

before I became interested in the situation myself. Then a teacher on an in-service course suggested that we look at patterns where the number of points was 12 and the rule was a linear function. (See Figure 4.)

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Figure 2. Figure 3.

n d 3 n n -3n + I

Figure 4 .

This proved to be a fascinating problem and one which I have still not finished exploring. This made me realize that CIRCLE PATTERNS was a function graph plotter. Since then 1 have looked at quadratic and cubic functions and discovered surprising regularities in the patterns. 1 have also begun counting the number of lines and trying to make sense of that data.

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Lessons for the classroom What conclusions can we draw from this? Firstly, and perhaps most importantly, computers really do help people to do mathematics. At any rate they help me.

Secondly, I was only interested in these situations because I had considerable freedom over what I was doing. When I use a piece of software which tells me exactly what to do and how to do it I very quickly become bored. I hope I never think that such software would be more suitable for children than for me.

Thirdly, note the nature of the tasks undertaken. Two of them concerned programming: only one made use of available software. On the other hand, I find it hard to know how I would transfer the two programming activities into the mathematics classroom. The first is a relatively trivial task, and yet demands considerable knowledge of BASIC. Is it really worth encouraging children to learn so much about BASIC for uses such as this? The second activity is a more interesting programming task mathematically, and for these reasons is more appropriate for the mathematics classroom. But Prolog is a difficult language to use for the reasons stated earlier and few pupils have sufficient expertise with Logo programming to be able to tackle this sort of problem without considerable help which, if offered, might invalidate the experience. The third activity, which made use of software already available, can be undertaken in the classroom without any major problems, provided, of course, that appropriate hardware is available.

The present and the future It was suggested earlier that an important reason for using computers is to engage the learner’s attention and interest by encouraging the learner to adopt the four ‘missing’ styles advocated by the Cockcroft report. The HMI document, Mathematics from 5 to 16, in the ‘Curriculum Matters’ series, proposes a number of objectives for the mathematics curriculum. I t may well be that computers can help fulfil several of those objectives given under the headings ‘General Strategies’ and ‘Personal Qualities’. The only objective in Mathematics from 5 to 16 which mentions computers specifically is Objective 9. This concerns using computers as a tool for pupils when doing mathematical tasks. To meet Objective 9, the uses made of computers must be under pupils’ direction and control.

What uses are these? In spite of disparaging attitudes adopted by some proponents of languages such as LOGO and PROLOG, pre-written software has a great deal to offer. Even content-specific software can sometimes be used by pupils as a tool. Programs such as PLOD and SNOOK, which suggest specific problems to be tackled, can be used to introduce these problems to pupils, or to provide a means for pupils to check the results they have obtained, and the initiative for the work being undertaken can remain with the pupils. At present the most usable computer tools for the classroom are probably provided by programs such as FUNCTION GRAPH PLOTTER, CIRCLE PATTERNS, COUNTER and TILEKIT, which are only moderately subject-specific, together with LOGO used for turtle geometry.

There are many classrooms in which these tools are not yet available. This may be because the hardware is lacking, or because teachers have not yet gained the confidence

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and familiarity to offer them to their pupils. For some time yet work will need to be done to enable more teachers to offer tools such as these to their pupils.

But we need also to look to the future. With the advent of 16-bit microcomputers it will be possible to offer software to pupils and to teachers which is both more powerful and easier to use. Spreadsheets and information-retrieval packages have an enormous potential, unfulfilled at present principally because suitable software is lacking. Languages such as LOGO and PROLOG, when implemented on 16-bit machines, offer powerful facilities for the creation of microworlds. These microworlds will only be of use if they are well-designed; this means they must be designed by classroom teachers and others who are in regular contact with classrooms. Philip Leith, in an article in The Computer Journal in 1986, described what he saw as the misuse of PROLOG to create a logical legal database. His objection to this was that such a database encouraged misunderstanding of the nature of law. There is no comparable philosophical objection to using PROLOG in connection with mathematics. For that PROLOG will be useful if it helps those doing mathematics to solve the problems they want to solve.

Information-retrieval packages, spreadsheets and microworlds are all potentially powerful computer tools. Considerable work is required if tools such as these are going to be designed in ways which make them suitable for mathematics classrooms. But if that work is done, and if the hardware is made available, pupils in schools in the next decade should have considerable support for helping them to pose and to solve their own mathematical problems.

Note Readers may be interested in the sources of the programs mentioned in this article.

ANGLE90 is from the SMILE'S The First Thirty and ANGLE360 from SMILES The NcJut Sezwztecw. Both are obtainable from the SMILE Centre. CARS: MATHS IN MOTION is produced by Cambridge Software House, Town Hall, St Ives. PLOD is from Teaching with a Micro; M a t h I , available from the Shell Centre, University of Nottingham. SNOOK is from the Shell-JMB pack Problems with Patterns and Numbers, obtainable from the Joint Matriculation Board. FUNCTION GRAPH PLOTTER is from Some Lessons in Mathematics bvith a Microcon?putev, and COUNTER and TILEKIT are from Some More Lessons In Mathematics "ith a Microcomputer. Both these are obtainable from the ATM Office in Derby. CIRCLE PATTERNS is on the second Micromath Disk. The first Micromath Nimbus Disk contains example of microworlds written for Nimbus Logo. Both these are obtainable from the ATM Office in Derby.