Comput. Educ. Vol. 8, No. 1, pp. 85-92, 1984 0360-1315/8453.00+0.00 Printed in Great Britain Pergamon Press Ltd

I N V E S T I G A T I N G C A L ?

JIM RIDGWAY 1'*, DAVID BENZIE 2, HUGH BURKHARDT l, JON COUPLAND 2, GRAHAM FIELD 2, ROSEMARY FRASER 1'2 and RICHARD PHILLIPS l

The ITMA Collaboration

tShell Centre for Mathematical Education, University of Nottingham, University Park, Nottingham NG7 2RD and 2College of St Mark and St John, Plymouth PL6 8BH, Devon, England

Abstract--Can CAL facilitate discussions and problem solving? Surveys of classroom activities during mathematics lessons in British secondary schools present a rather gloomy picture. Disproportionately large amounts of time are devoted to teacher explanations, and to pupil exercises. Activities such as discussions, practical work, problem solving and investigations happen so rarely that they are collectively referred to as "missing activities". Can CAL help? Tutorial CAL, of course, reinforces the exposition- imitation mode of teaching. Other kinds of CAL exist, though.

An observational study watched 170 mathematics lessons in which the microcomputer was used as a teaching aid. Teachers chose freely from a collection of over 90 programs. The most popular programs, in terms of how often they were chosen, and how their usefulness was rated by the observers, were investigative in style. This paper describes the lessons which were based on the four most popular programs used (JANE, SUBGAME, EUREKA and VECTOR). Each one facilitated some of the "missing activities". The role of the microcomputer as an agent of change in teaching style towards more open methods is described and discussed.

I N T R O D U C T I O N

The recent Cockcrof t Repor t [1] into mathematical educat ion ,"Mathemat ics Counts" asserted

Para 243 "Mathemat ics teaching at all levels should include opportunit ies for exposition by the teacher; discussion between teacher and pupils and between pupils themselves; appropr ia te practical work; consol idat ion and practice o f fundamental skills and routines; problem solving, including the application o f mathematics to everyday situations; investigational work" .

Of these, discussions, practical work, problem solving and investigational work take place far too infrequently in British classrooms. The change in emphasis in favour o f these activities should be at the expense o f teacher exposit ion to the whole class and at the expense o f drill and exercise which are both over represented. Can C A L help? Or is most C A L material obviously on the side o f the conservatives, focusing on "tell 'em and test ' em"? A good deal o f C A L material is available which facilitates exposit ion either by the teacher or by the p rogram ("teaching machine" programs; "film mode" programs; "electronic b lackboard" programs, etc.) and which focuses on drill and practice (rote learning programs). However , there is also a good deal o f material developed by A T M members[2], the A U C B E group[3], and the I T M A collaboration[4] which falls outside these two broad classes of programs, and which is consistent with Cockcrof t ' s recommendat ions. These materials have been designed to facilitate classroom discussion, and to facilitate learning via investigation and problem solving. We will call this group "investigative programs" . Wha t happens when teachers are offered a large collection o f programs which include programs of all types?

In the 1981/1982 academic session we watched 15 mathematics teachers using C A L material to aid their teaching o f second and third year pupils. Teachers were provided with about 90 programs, together with suggestions for lessons, and a weekend in-service training course. In return, they agreed to use C A L at least once per week, to comment on each program, and to allow an observer to watch.

*On leave from Department of Psychology, University of Lancaster, Lancaster LA1 4YF, England.

85

86 JIM RIDGWAY et al.

Which programs did teachers choose?

Teachers are likely to choose programs which obviously relate to the curriculum. This was the case here. They were provided with an index which related specific curriculum-relevant topics to programs. The correlation between the number of entries in the index, and program use was highly significant (r = 0.48, d.f. = 89, P < 0.01). Of the programs that were run on five or more occasions, investigative programs were used more often than non-investigative ones. (Mann Whitney U-test, U = 8, n~ = 9, n+ = 7, P < 0.02). Of all the programs used, the six most often used programs were investigative in style (These were JANE, SUBGAME, EUREKA, VECTOR (unpublished), PIRATES and ERGO.)

Which program style did teachers like best?

After using each program, teachers were asked to comment on the program, and to indicate their willingness to use the program again. Comments and ratings were put into envelopes, then sealed. All ratings were confidential, and teachers were told that the envelopes would not be opened until the trial was over. There was no significant difference between the investigative and non- investigative programs in terms of teacher willingness to use them again (Mann-Whitney U-test, U = 3 4 , n~=9 , n2=7 , P > 0 . 0 5 ) .

Which lessons went best?

Observers rated the contribution of CAL to each lesson, on a scale - 3 to + 3. The data suggest that investigative programs made a greater contribution to the lessons, but the pattern is not clear cut (Mann Whitney U-test, U = 14, n~ = 9, n~ = 7, P < 0.05, one-tailed).

These broad summary statistics give some feel for the body of the data, but at a certain cost. Programs are aggregated together; and program features other than +'investigative-or-not" are relevant; this group of teachers is probably not representative of mathematics teachers in general. In any event, if we are to gain an understanding of the mechanisms of problem solving activities in the classroom, we have to examine actual classroom events, and sketch out computer based activities which seem to help. The main thrust of this paper will be to describe the use of the four most popular programs in the trial, in different classrooms, to show the opportunities offered by CAL to foster Cockcroft 's "missing activities".

C L A S S R O O M USES OF P R O G R A M S

J A N E (Author: ITMA collaboration, published b+v Longmans )

The program provides an introduction to the idea of mathematical function. In common with most of the ITMA programs, JANE can be used in a wide variety of ways-~omprehens ib le yet challenging to pupils of a wide ability range. For example, a simple use of JANE begins by +'giving JANE a small number". On the command of the user, JANE transforms this number to another one. The pupils' task is to discover the rule which is being used. A more complex use of JANE is to provide a number which has been transformed by two operators. Pupils have to discover the two operators, and thus be able to deduce the starting value. A screen dump is shown in Fig. 1. Despite its apparently simple structure, JANE was used to teach a large number of mathematical topics, including:

extending children's intuitive ideas on mapping to inverse mapping and joint mappings introducing the notion of functions teaching the multiplication of directed numbers co-ordinate systems introducing algebraic notation.

JANE was the program used most frequently in the trial; 13 teachers used it, three of them using it twice. In all cases, the program was used for its mathematical content. It was never used as a game, or a bribe, or just to fit in with the observation study.

In general, teachers all used JANE in much the same way even though they differed greatly in their teaching style. Pupils guessed answers, successful answers lead to hypothesis generation. These

Investigating CAL? 87

,.] Cq..-IN H / F ; Y

Fig. 1. JANE in action.

hypotheses were then checked by the teacher and by using other examples on the computer. Classes differed a great deal in:

the amount of unfocused guessing the emphasis placed on tests to distinguish between rival hypotheses the way results were recorded discussions about the identifiability of different hypotheses heuristics for distinguishing between different operators emphasis placed on "proof".

Observers commonly refer to children as being "responsive", "interested" and "amused". JANE can be used to set problems for the class, and to be the sole arbiter of the correctness or otherwise of children's hypotheses--liberating the teacher and permitting other teaching roles to emerge. Emphasis on generating hypotheses and checking them with examples, naturally facilitates an "open" style of teaching, characterised by questions and answers, and teacher-pupil dialogue, rather than the relatively "closed" style which consists largely of teacher exposition and pupil imitation. Two observers reported that their teachers adopted a more open style when using JANE. For example

" . . . JANE provided the only [high level intellectual] activity in the whole lesson I and this completely changed the lesson, far more pupil hypothesising etc".

"JANE made the teacher more open in style than usual--but he still used a structural approach to JANE".

The range of other teacher roles included:

passive keyboard operator classroom manager--selecting answers and checking them with pupils fellow pupil--referring to the class as "we", and trying to discover JANE's functions along with everyone else facilitator--stimulating ideas and encouraging the pupils catalyst--helping children to develop useful strategies for discovery, and helping them to articulate them clearly, and to see how they work.

Despite the remark made by one teacher that "JANE is a work of genius!" and an observer comment "JANE is very robust and it is difficult to do a bad lesson with it", some teachers did succeed in producing unsatisfactory lessons.

One teacher produced a worksheet to support JANE activities that neither the observer nor the children could connect with JANE, and lost half the class.

One teacher used JANE, but did not ask for pupil hypotheses! The observer's rating of usefulness of CAL to the lesson (on a scale - 3 to +3) was -2 .

Summary of JANE's contributions

JANE offers an interesting and accessible entry to a wide range of related mathematical topics (mappings, functions, directed numbers, algebra, systems of co-ordinates). It rewards skills like the systematic collection and recording of results, and the formal identification and use of strategies. It can focus attention on process aspects of mathematics--i.e, discovery skills like generating and testing hypotheses, rather than traditional aspects like learning algorithms. It can facilitate role changes by the teacher, away from exposition, and towards co-discovery and fellow pupil.

88 JIM RIDGWAY et al.

SCORES You Me draws

i 0 0 Use l h i s

I ~ma' s board Computer ' s hoard

SS 9 3 0 9 0 3

YOU WIN - WELL DONE!! Do you wan~ to ~ry aga in (Y o r N)?

Fig. 2. SUBGAME in progress.

Particularly important, it usually served to arouse and maintain children's interest in whatever mathematical topic JANE was focused on to.

SUBGAME (Author: Brian h, es, published by A TM and Longmans)

The program generates five random digits, one at a time, which have to be put sequentially into the blank cells of a subtraction sum. The aim is to make the answer as large as possible. The program also plays the game, in competit ion with the class. A screen dump is shown in Fig. 2.

S U B G A M E was used by 13 teachers, each for one lesson. On 9 of these occasions, it provided a focus for the lesson; on 4 occasions it was used at the end of the lesson as a game, to fulfil the teacher's obligation to the trial. Pupils keyed in responses in 6 of the 13 lessons. In this discussion, attention will be focussed on the 9 "serious" lessons. In two of these lessons, no at tempt was made to discuss strategies. In five lessons, the idea of methematical strategy was introduced, and in two more, it was discussed extensively.

One class which we observed were a lower ability group who were rather poorly motivated towards mathematics. The observer remarked that " S U B G A M E made the lesson really terrific for them". Computer use involving the whole class was kept to a minimum. Instead, initial class discussion (which introduced children to the idea of " random number" and so on), was followed by individual group work on the computer while the rest of the class worked on unrelated tasks.

In almost all of the other lessons observed, the entire class joined in, suggesting where to put the numbers. Class votes and competit ion between each pupil and the computer were also common.

Observers commonly used words like "interest", "mot iva t ion" and "enjoyment'". No one reported that children were disinterested or bored. Playing to beat the computer seems to be a great motivat ion--witness the success of arcade games like SPACE INVADERS. This competitive aspect of the program seems to be a considerable virtue, shifting the focus away from the usual competit ion between pupils to a competit ion between the computer and the class.

The program facilitated role playing by both the teacher and the pupils. This might be because of the dual roles played by the computer i t s e l ~ i t is both a random number generator (RNG) and it tries to beat the class. This duality sometimes leads to pupils accusing the program of being ~'a copy cat" or "a cheat". A common misconception was that the computer "knows" which numbers will come up next and plans accordingly. A second misconception is that the computer actually copies the pupils' ideas. Both these misconceptions stimulated some teachers into unusual (but illuminating) classroom activities. For example, in response to the pupils" complaint that you can only win if you know what numbers are coming next, one teacher took on the role of the computer, as RNG, and pupils played against each other.

To show that strategies are useful, even when presented with random numbers, one teacher asked a pupil to act as R N G , then competed against a second pupil on the board (with neither looking at the other's placings).

Investigating CAL? 89

One teacher arranged the children into groups of three. One child played the role of RNG, and the other two competed. The winner became "the computer".

Teachers found it easy to break out of their omniscient role. They do not know what numbers will be generated. This has two immediate effects. First, they are not forced into the role of task setter and tester. Second, they can actively help the children as much as they can--with no guarantee of success. In One lesson, the teacher simply adopted the role of "compere"--al lowing the computer and the pupils to take charge of the lesson. In the most successful lessons (from the observer's viewpoint), the teachers acted as catalysts for the children's ideas--facilitating the articulation of successful strategies, not just providing algorithms for the problem.

Of course, SUBGAME is not an educational panacea. Although there were many successful lessons, the observers commented of two lessons:

"S UB GAME was crowbarred into a very successful lesson on ratio. A good lesson was wasted, and SUBGAME was abused". "Traditional boys against girls competition. This meant that the teacher demanded disinterest from 50~o of the class. Not a successful way to use the program".

A large proportion of observers mentioned problems of screen visibility when SUBGAME was used. (This did not seem to lessen pupils' interest, though.)

Are there dangers in computer programs that are thought to cheat, copy or be schizophrenic?

Summary of SUBGAME's contributions

The program invites discussions on the topic of randomness and probability. It can offer an excellent vehicle for getting pupils to verbalise mathematical strategies. It capitalises on the great motivating challenge of a "machine to beat", and can facilitate pupil discussions amongst themselves and with their teacher. Teacher roles became quite fluid, because the dominant role of expositor and all knowing task setter were removed. Many teachers responded to their new freedoms quite creatively, in terms of the patterns of classroom interaction that were established and in terms of the classroom activities which they initiated. In two lessons, teachers focused on crystalising children's own conceptions (with great success) rather than simply telling pupils how to proceed. Overall, SUBGAME aroused and maintained the interest of almost all the pupils, while it was on.

EUREKA (Author: ITMA collaboration, Published by Longmans)

E U R E K A is designed to teach elementary graph interpretation by considering how the level of water in a bath changes over time when someone has a bath. Bath time operations (like turning taps on and off and getting in and out) are shown pictorially in the upper half of the screen, while in the lower half, a graph is plotted of water level against time. EU REK A was supplied with a lesson plan and worksheet. A screen dump of EUREKA is shown in Fig. 3.

The program was used by 10 teachers, four of them using it twice. On all occasions, it was used as an aid to understanding graphs, rather than merely being used to satisfy the demands of the project. Once it was used as a bribe ("work hard or you won't see that man in the bath"), but even here, this was in the context of a lesson on graphs.

Teachers often followed the lesson plan provided exactly. This involves a sequence in which bath activities are associated with their time graph, followed by a straightforward graph which pupils are asked to interpret (focusing on different levels of explanation like "the water level had dropped suddenly"; "the man got out"; "he has gone to answer the phone"). Interpretations can be checked by playing the bath sequence (and by freezing the display at appropriate times). The computer is switched off. Pupils write their own sequence in words, then sketch a graph on a separate sheet. Pupils swap graphs, and write a description of their neighbour's graph; then compare notes. The computer can be used to arbitrate!

Six examples are stored in the program, and the worksheet shows the graphs associated with each. The interpretation exercise was sometimes carried out in class and sometimes set as homework. In general, E U R E K A seems to encourage written work. There was no written work in only three lessons--and each of these was followed by a lesson on which EUREKA based work took place.

c a e 8/! o

90 JIM RIDGWAY el al.

-i j ..7

> > Man out. > > Man in. > > T a p s o n . >>

Minutes

Fig. 3. EUREKA at work.

Observers often commented that children seem to enjoy the opportunity to embroider a story in a mathematics lesson. In general, pupils were described as being "attentive", "interested" and "enthusiastic" and "noisy" as well.

E U R E K A does not guarantee a successful lesson, though.

"The children were rowdy and unsettled and the teacher was not himself (he had been sick). E U R E K A was used as a bribe three times. A shallow use of the program".

The decision to show a sequence, then replay it repeatedly leads to very slow lessons. Class interest waned rapidly on the occasion when the teacher adopted this strategy.

The program was not always used optimally. Sometimes it was underextended (e.g. by omitting a period in which children translate graphs into words, and vice versa), and sometimes it was overextended (e.g. being on throughout the lesson, generating example after example).

Summary of EUREKA's contributions

E U R E K A has a role to play either as an introduction to graphs, or to extended pupils' knowledge, or to revise the topic. It focuses on translation skills between pictures, words and graphs-- th is is a far cry from emphasising the technical skills of plotting points, at the expense of understanding (technical skills are important too, of course). Pupils often worked in pairs on E U R E K A style p rob lems- -a rare event in British classrooms. Because the computer could always be used to arbitrate upon disputes between pupils, teachers could experience the management of groups and pairs, without the worry of being completely overloaded by pupil demands for attention.

VECTOR (Author: ITMA collaboration, Unpublished)

This is a treasure hunting game. Its main aim is to teach about directed numbers, and to emphasise the difference between coordinates and vectors. Instructions appear when the program is loaded--negat ive numbers are not mentioned. To play the game successfully, pupils have to discover their use either by experimenting or by teacher promptings. Clues are given about distance from the treasure. Information about the treasure's distance from the current location helps pupils locate the treasure. Overall performance is improved if children adopt a strategy of using all the information about the treasure's distance from each location that they have tried. An example of an ongoing game with VECTOR is shown in Fig. 4.

Ten teachers used V E C T O R ~ of them used it twice. On two occasions, the program was presented and used just as a game, with little or no emphasis being placed on its mathematical content. In all the other lessons observed, it was used for a serious purpose. Teachers used it both

Investigating CAL? 91

Places from the buried treasure 2

Time taken : 1 0 4

~-.~:::*.~:~ii::::~:~.~.~::~:~.~ ~ :::::::::::::::::::::::::::::::::::::::::::::::::::::

0 I 2 3 4 5 6 7

Present position

( I , I )

Moving vec to r [ ? ]

Fig. 4. VECTOR in progress.

in lessons devoted to directed number, and in lessons devoted to vectors. Its game format made VECTORS a good vehicle for revision too--pupils were not bored by recapping familiar material. The topics which VECTOR explores appear on all curricula; it was relatively easy to weave it into the lesson. VECTOR stimulated a good deal of written work; the accompanying worksheets were used by the majority of teachers.

A common feature of lessons using VECTOR was that strategies for playing were discussed by pupils and teachers--an activity which occurs but rarely in British classrooms. The level of involved, active problem solving by the pupils was often maintained at a high level throughout lessons. Concepts like negative numbers were "invented" by pupils in some lessons--for example, one teacher keyed in "LEFT 4, DOWN 3" and other invalid forms of input, then asked the pupils to help find something that the computer might accept.

The program was generally enjoyed by the pupils. One commented "I wish all Maths lessons were like that". We noted earlier that, despite its game-like nature, it was only used without a serious mathematical purpose on two occasions. Pupils tackled the hard work of understanding directed numbers and vectors with apparent joy and eagerness. This can only improve their view of mathematics, and of themselves as mathematicians.

VECTORS was not always a great success. One observer judged that the lesson had been inadequately prepared (e.g. the teacher mis-

understood how to use the worksheets). Another lesson achieved little apart from entertainment. Pupils were confused about the

difference between a co-ordinate and a vector, and were generally uninterested. One lesson treated VECTOR as a guessing game; strategies were not discussed at all.

One lesson was taken very slowly by the teacher (e.g. pupils were not allowed to make mistakes) and this detracted somewhat from pupils' enjoyment. Interest and enthusiasm did increase towards the end of the lesson though, as children's understanding and expertise increased. (One pupil commented "I like doing this instead of Maths".)

Summary of VECTOR' s contributions

VECTOR generally provided great motivation for pupils to learn about directed numbers and vectors. It was a good vehicle for classroom discussions on strategies for problem solving. It was sometimes used to facilitate "discovery learning" by the pupils.

92 JIM R1DGWAY et al.

R E V I E W

Well, Dr Cockcroft? These program vignettes do illustrate clearly that CAL can provide opportunities for changes in classroom activities of the kind that your committee recommends.

All four programs freed teachers from the role of expositor. Similarly, none of the programs focussed directly on the repeated practice of simple skills. So they have at least reduced the time spent on Cockroft 's over-represented activities.

Discussion was facilitated in many classes: SUBGAME and VECTOR often stimulated teacher pupil discussions on strategies for "winning"! JANE required a good deal of hypothesis generation by pupils. EUREKA facilitated pupil-pupil discussion, as they compared graphs and their interpretations.

Problem solving was central to each of the four programs. Pupils often developed good problem solving strategies and heuristics as lessons progressed.

Int,estigations were not stimulated by any of these programs, nor was Practical work. Programs which were written to facilitate investigations, like SNOOK, were rarely used (four times) probably because the task of predicting how many times a ball will bounce before going down a hole on a "mathematical billiard table" of a given size does not seem to fit easily into existing curricula!

Our impressions from the trial are that CAL can bring about a change in teacher style, in the direction of a more open approach [5]. We hope that CAL will prove to be a powerful in-service training aid to teachers. These programs afford the chance for teachers to explore and practise alternative teaching styles, while being strongly supported by the computer. Hopefully these styles will generalise to non-CAL teaching, too.

As a post-script, we should note that all the programs had a strong effect on pupils' motivation. One observer commented "The affective response when the computer was switched on was unbelievable". This enthusiasm did not seem to wear off as the children became accustomed to the computer, either. Anything that increases children's enthusiasm for mathematics must be worth- while!

Acknowledgements--We gratefully acknowledge the financial support of the United Kingdom Science and Engineering Research Council. We would like to thank M. Anderson, D. Pimm, G. Field, M. Martin, S. Sandle, H. Brown, P. Freeman, J. Godwood, G. Habbishaw, R. Hatcher, J. Jones, B. Morrell, G. Morris, P. Rylatt, C. Smith, I. Smith, K. Smith, M. Smith, T. Tomaney, J. Trenery, B. Tunbridge, P. Turpitt and G. Winter.

R E F E R E N C E S

1. Cockcroft W. H. Mathematics Counts. HMSO, London (1982). 2. Association of Teachers of Mathematics. Some Lessons in Mathematics with a Microeomputer. ATM, Derby (1982). 3. Advisory Unit for Computer Based Education. Computer Adided Learning. Hertfordshire County Council, Hatfield

(1980). 4. ITMA Collaboration. Micros in the Mathematics Classroom. Longmans. London (1982). 5. Fraser R., Burkhardt H.. Coupland J., Phillips R., Pimm D. and Ridgway J. Learning activities and classroom roles

with and without the microcomputer. Submitted for publication.