European Journal of Psychology of Education1988, Vol. Ill, n? 3, 271-286© 1988, I.S.P.A.

The Computer as a Mediating Influencein the Developmentof Pupils' Understanding of Variable

Richard NassCelia HoylesInstitute of Education, London, UK

Within a Logo environment, pupils routinely use the notion of varia­ble. However, the process by which they come to develop understand­ings is more problematic. In this paper we describe a series ofsituationsin which we investigate how the computer and the intervention of aresearcher may facilitate the development ofpupils' conceptions of varia­ble. We report case studies of two thirteen-year-old pupils Nicola andNoel - the observations have been woven into two 'stories' derivedfrom data collected at different times and in two different settings: themathematics classroom and the university computing laboratory. Ourmain conclusions concern the multiple relationships among the resear­cher, the computer and the pupil - specifically the ways in which thecomputer can (but does not always) act as a mediating influence in deve­loping pupil understandings of variable.

Introduction

Our early work using Logo in the mathematics curriculum, was characterised by anemphasis on the openness of this environment; on it's appropriateness as a medium for explo­ring mathematical ideas, largely without intervention. Our stress lay on the mathematicalprocesses involved; investigating relationships, making conjectures and generalisations. Wewere able to identify how pupils spontaneously interacted with and used mathematical ideaswhile programming in Logo. While our attention was focussed in this way on Logo program­ming as a conceptual framework for the learning of mathematics, we identified a numberof the environment's characteristics including early learning strategies (e. g. 'degoaling' andself-imposed restrictions on the environment); categories of problem-solving activity (e. g.'making sense of' ideas, and global and local negotiation towards specific goals); individualdifferences in programming styles (which might be gender related). Work on attitudes alsoindicated that pupils were generally task-focussed in their Logo work, were highly motivatedand willing to take risks in an investigative way. We recognised that mathematical contentwas embedded within the Logo work, but at this point were less concerned with looking

This paper is a modified version of that appearing in the Proceedings of the Conference on Logo and Mathema­tics Education, Montreal, 1987.

272 R. NOSS & C. HOYLES

at the ways in which implicit use of this content might spill over into understanding, eitherspontaneously or after intervention. We now know that such understanding does not neces­sarily occur even within a Logo environment and that when it does, it tends to be con­text-specific - that is, children may not easily synthesise mathematical ideas acrosscontexts.

It was evident from our classroom observations that the computer could be a catalystfor challenging pupil dependence on the teacher. However, the atmosphere in the mathema­tics classroom - and in particular, the role of the teacher in allowing pupils the space todevelop autonomous behaviour and encouraging collaboration and investigation - emergedas crucial. With regard to pupils' learning, we found that teacher intervention was importantin a number of ways - to provoke reflection and prediction; devise structured tasks to achievespecific learning outcomes; identify conceptual obstacles and then structure the environmentso that the pupil would bump into them; and encourage pupils to use the powerful ideasof Logo. These findings were based on the data from our participant-observation work,together with results from structured tasks and exploratory studies designed to probe pupils'mathematical understandings (Hoyles & Sutherland, in press; Noss, 1985). In considering thecognitive effects of computer use, we also recognised the need to take into account the lear­ning culture as a whole, the educational guidance given, and the kinds of intuitions and fee­lings which the pupils bring to their learning.

We have argued elsewhere that Logo derives its power as a mathematical learning envi­ronment by allowing functional, experimental activity to take place as well as formalisation(Hoyles & Noss, 1987a). Viewed from a Piagetian perspective, the functionality can be seenas a means by which 'a new concept is at first a tool to... work out operational schemes',and the formalisation - like symbolising - 'a way of cutting invariants out of their con­text' (Vergnaud 1987, p. 51). We wish to emphasise that it is a unique attribute of the Logoenvironment that functional activity and formalisation take place simultaneously. This is nosmall claim given that the gap between pupils' activity and its syrnbolisation has been anoutstanding problem facing mathematics education. Expecting pupils to extract mathemati­cal meaning from algebraic formalism, is, as Freudenthal notes, «an appeal which falls ondeaf ears... at the same time asking for formalist acting and content-directed understandingis too much» (Freudenthal 1986, p. 484). It is our contention that this is no longer necessa­rily the case if appropriate computer-based environments are available.

With this background mind, we have devised and begun to refine a model for learningmathematics which involves the dynamically related components of using, discriminating,generalising and synthesising. The essence of the model is: first, to use mathematics to pro­duce an outcome (at this stage there might be little appreciation of what is being used); second,break down procedures that previously only operated in whole units - that is, discriminatethe function of the components; third, rewrite in explicit form what was previously implicit- that is try to form generalised relationships; and fourthly, attempt to apply the structuresin new domains - that is, synthesise across different contexts of application. A full descrip­tion of this UDGS model and its application is given in Hoyles (1986). This model providesour framework for designing computer-based microworlds - learning situations which canaid in the restructuring of the pupil's knowledge from its initial basis within theorems inaction' to more abstract cognitive structures. The model has also provided a means by whichwe have reanalysed and reinterpreted case-study data that we have collected in mathematicsclassrooms. Thus we now lay more explicit emphasis on mathematical content, but do notsee this as a move away from the study of process. Indeed, in trying to analyse the pupil­-computer interactions within a microworld, we inevitably have been forced back to an inves­tigation of process issues, that is to a study of how transitions come about between thecomponents of the UDGS model - how and why change occurs. Thus we maintain thatthe separation of process from content is a false dichotomy, and that a central task confron­ting mathematics educators is to focus on their dialectical relationship.

THE COMPUTER AS A MEDIATING INFLUENCE

Pupils' conceptions of variable

273

In this paper we seek to identify how this model applies to the development of pupils'conception of variable. In a Logo environment, pupils quickly use variables once they con­struct programs with inputs; we wish to focus on how pupils come to understand the mean­ing of these inputs, their effects, and the relationships between them. Hillel & Samurcay (1985)have illustrated how such understandings are far from spontaneously generated. However,Noss (1986) has indicated that pupils' Logo experiences can provide a conceptual frameworkfor subsequent algebraic intuitions, and Sutherland (1988) has shown that, provided appro­priate pedagogical input is offered, pupils can and do construct linkages between variablesin a Logo context and in a more traditional school mathematics setting.

The empirical core of this paper will describe a series of situations in which we investi­gate how the computer and the intervention of a researcher may together or separately faci­litate the development of pupils' conception of variable. There are two sources of data:

Case-study work in the classroom: We have been working with a group of seven chil­dren aged between 13 and 14 years with considerable Logo experience (around 120 hoursover four years). During the year 1985-6, we collected together a series of case studies oftheir Logo activities within their mathematics classroom with the aim of exploring the linksbetween Logo work and the mathematical curriculum of the school. Our interventions weredesigned to probe pupils' mathematical intuitions, uncover the basis of their strategic deci­sions and to plan and structure activities on the basis of our insights. We frequently cameto sessions with hidden agendas - particular mathematical ideas that we would encouragethe pupils to encounter and explore.

Laboratory work in the University: We designed some structured tasks to be undertakenduring two two-hour sessions in our computing laboratory. Data was obtained by referenceto the dribble files of the pupils work (which record all pupil-computer interactions automa­tically), the researchers' notes (both researchers were present during the activity but no inter­ventions were made by them), and the written work of the pupils.

In this paper, we trace the relationship between the material collected in the classroom,and the laboratory - that is between situations where the pupils were working on the com­puter towards their own goals with the researcher intervening, and situations where the taskswere tightly specified with no researcher intervention. We recognised that the «contracts»implicit in these two settings are very different from the pupil's points of view: we aim toexploit these differences in order to highlight the mediating role of the computer in pupils'learning, and the specific influence of researcher intervention. The analysis presented hererepresents our attempts to synthesise findings gained from both sources of data with respectto two stories - Nicola's and Noel's - which focus on the development of their understan­ding of variable.

Two case studies

Nicola's necklace project

We begin with a study of three sessions involving Nicola. For the first session, Nicolaworked with Julie; both children were above average in mathematical performance as rated bytheir teacher, Nicola and Julie were involved in building a procedure for a chessboard, andthey adopted a highly modular strategy. They built a state-transparent procedure SQUAREfor their empty squares, and a. procedure COLOUR for the filled-in ones, as follows:

TO SQUAREREPEAT 4 [FD 48 RT 90JEND

274 R. NOSS & C. HOYLES

TO COLOURREPEAT 12 [FD 48 RT 90 FD 2 RT 90 FD 48 LT 90 FD 2 LT 90]END

Their final procedure (ALLL) was:

TO ALLLREPEAT 3 [LINE M LINEI M]LINE M LINEIEND

Figure 1. L'echiquier - ALLL

a)~1A_ _ _ _ _ _ _ 1

LINE M

b)~

1A_ _ _ _ _ _ _ I

L1NEMLlNE1 M

1A 1

REPEAT 3 [LINE ML1NE1 MI

Figure 1. The chessboard - ALLL

d

ALLL

At this point, the researcher intervened to suggest that they make a general sized chess­board; our research agenda was to provoke the use of variable. After some time they hadaltered their procedure SQUARE giving it an input :S, and COLOUR giving it an input :C,to produce the following modification of ALLL:

TO ALLL :5 :CREPEAT 3 [LINE :5 :C M :S LINEl :5 :C M :S]LINE :3 :C M :3 LINEl :3 :CEND

By looking at the processes involved in producing this procedure, (see Hoyles 1987 fordetails) we can be fairly sure of the following points. Nicola and Julie, if left to themselves,always adopted an 'adding-on' strategy for their inputs in which inputs are simply appendedto procedures. The use of this strategy implies that the pupils do recognise the variants involved,but do not at this point pay any regard to the relationships which may exist between them.There is therefore redundancy in the use of inputs; (note that ALLL actually requires onlyone input as it will not produce a square board unless ALLL is called with :5 equal to :C).However, the researcher did intervene to help the girls to consciously conceptualise the rela­tionship between the number of repeats (:C/4) necessary to fill the square (in COLOUR)

THE COMPUTER AS A MEDIATING INFLUENCE 275

and the SiZI: of the square (:C) - the pupils' first reaction had been that 'we need anotherinput'. Presumably partly as a result of this intervention, they spontaneously operated onan input in their interface procedure M as indicated below; this was the first time that thishad occurred,

TO M:SBK :SLT 90FD :S * 8LT 90END

From a detailed analysis of the pupils' programming, it is apparent that Nicola and Julietended to see the inputs as related to the physical representations within their picture, andthis seemed to lead them to introduce a new input for every new variable encountered. Weshould beware, however, of concluding that they were completely unaware of the relationshipbetween the inputs; on the contrary, their subsequent composition of the procedures, andthe paper-and-pencil calculations which preceded calls of their procedures (to ensure the 'right'relationship between inputs) testified to some fragmented understandings. They were howe­ver, unable or unprepared to make relationships explicit in any formal sense.

We have argued (Hoyles & Noss 1987b) that discrimination of relationships generallyoccurs only after many opportunities to tryout, use and make sense of the underlying con­cepts. In particular, we would expect Julie and Nicola to need time, further diversity of Logoexperience, and pedagogical intervention before we could in any sense say that they 'unders­tand' the idea of variable. Fortunately, we have some data which throws further light on theprocess by which Nicola's partial understandings were synthesised into a more coherent whole.

A few weeks later, Nicola was working alone on constructing a necklace which had beeninspired by a photograph shown her by one of the researchers (see Figure 2).

Figure 2. Modele dont s'est inspire Nicola

Figure 2. The inspiration for Nicola's necklace

She began by writing a fixed procedure using the following REPEAT line:

REPEAT 4 [COLOUR 100 FD 50 RT 45 COLOUR 100 LT 45 PU BK 50 PDj.

Firstly, we should note that Nicola had changed her task, and made the tilted squaresthe same size as the 'upright' squares - either she had not realised that the tilted ones nee­ded to be smaller to make a 'straight' necklace, or she had chosen to ignore this constraint.Secondly, her procedure utilised the procedure COLOUR which she had designed for herearlier chessboard. When challenged to make the pieces variable size, she responded by saying,«I know what it is, dots C», and built the following procedure:

TO SQUARES :C :NPU rr 90FD 500 RT 90 PDREPEAT :N COLOUR :C FD :C/2 RT 45 COLOUR :C FD :C LT 45 PU[BK :C/2 PDjEND

276 R. NOSS & C. HOYlES

It is significant that in changing her REPEAT line from its original fixed form, Nicolauses a substitution strategy: that is, she substitutes .C for 100. In addition, because she imme­diately sees that 50 is 10012, she is prepared to substitute :C/2 for 50, rather than introducea new input. The next intervention aimed at challenging her to make the number of squaresvariable. In the light of her apparent adeptness with the idea of input, Nicola's responseis perhaps surprising:

N: «Can you use an input for a numberb

Nicola's response is a salutary reminder of the difficulty of synthesising understandingsacross contexts. It seems clear that she has achieved her generalisation (inserting the input:C) while maintaining a correspondence in her mind with the length it represents (value 100).Nicola has used variables for the number of repeats in other contexts; we suggest that thereason for her raising the question at this point provides further evidence for the way inwhich she is thinking of the input here as tied to its referent.

After receiving a positive response to her question, Nicola replaced REPEAT 4 withREPEAT :N. She was then challenged to start a new line of the necklace - that is, to trans­form the task into one of covering the screen with the necklace pattern. So, having produceda single line of variable length, the problem now was to return to the start of the line. Nicoladid this by a homing-in process -- gradually edging her way to the start in direct drive ­and arrived at the 'answer' of 960 for four pairs of squares:

N: «It would be FD :C * 8 if they were all straight».

Thus she sees (although not explicitly) the length as being 2 * :N * :C + (an extra bitderiving from the slantiness of the lines). Note that in fact, each of the four 'extra bits' arediagonals of the squares of length :C * n. (Her initial attempt gives a value for {2of 1.4).

At the instigation of the researcher, interest then focussed on the length of a single dia-gonal; (Nicola had already stated that for :C equal to 100, the diagonal would be 140):

Researcner: How far would it be if the length was 50?Nicola: Mm... 70R: How do you know?N: Half it.

Next, again at the researcher's instigation, a table was drawn up (values on the left pro-posed by the researcher; values on the right by Nicola):

100 -> 14050 -> 70200 -> 28010 141 -> 1.4 (without hesitation):C -> :N (!)

One interpretation of this final response is that Nicola was once again adopting anadding on strategy for inputs, that is she introduced a new input :N without constructingany relationship with :C (here in a non-computer context). We conjecture that this is becausethe relationship is not obvious (in contrast to the earlier 'divide by 2' in the SQUARESprocedure).

We next attempted to provide some scaffolding for Nicola so that she could see the pos­sibility of operating on variables in a exploratory way when the relationship between themis not obvious. The researcher turned the page and wrote:

I 1.45 ....

THE COMPUTER AS A MEDIATING INFLUENCE 277

to which Nicola's response was to try and turn the page back! It seems clear that she wasreferring vertically to previous entries, rather than looking for a (horizontal) relationshipbetween the entries. Such a strategy is, of course, not uncommon. After some considerabletime, but without any further prompting (other than a refusal to turn the page back), Nicolaturned to the computer and typed:

PRINT 5 * 1.4

and on receiving the response 7, inserted it into the table. On then being asked to fiII in:

200

she again tried to turn the page back.After a. further period of time, Nicola decided that 'its times 1.4' and typed 200 * 1.4

which she then checked to see that it was the same as the answer on the previous page. Atthis point, Nicola's attitude changed from one of curiosity and frustration to certainty andresolution:

N: «Ail / do is put :C * /.4»

The third chapter of Nicola's story occurred some weeks later, when she participatedin an experiment in our computing laboratory. We set out to provide her and six other pupilswith a highly structured and progressive set of tasks, some of which were to be attemptedon the computer, and some off the computer. The task was designed to place the pupils ina situation in which they would inevitably bump up against the ideas of ratio and propor­tion. Specifically the task involved the construction of various N-shaped figures, and ourinterest was concentrated - among other things - on the ways in which the participantsconceived of the relationship between the diagonal of the 1'1, BC, and its vertical sides ABand CD (the angle - marked <p in Figure 3 - was varied). The significant factor here isthat this relationship is not obvious - the pupils did not know trigonometry - so the onlystrategy open was an experimental one. A full description of the task and the results is givenin Hoyles and Noss (in press).

Figure 3. La tache du 1'1

B D

Figure 3. The 1'1 task

A c

Nicola's initial response was to use the computer to help her predict the lengths on paper:she saw the diagonal length as the length of the vertical side plus a 'little bit', and usedher empirical findings to predict how much the 'little bit' should be: «For a 450 1'1, [450

was the angle between AB and BC, marked q, in the diagram) every 50 [along the verticalside] means an additional 25 [along the diagonal].» This is a similar approach to the oneshe used to calculate the length of her necklace in the earlier session.

In one of the tasks, each pupil was asked to design a pattern of Ns and then to con-

278 R. NOSS & C. HOYLES

struct a procedure for a general N-shaped figure (called SUPEREN), which could be usedto draw all the Ns in all the different patterns. Nicola came up with the following:

TO SUPEREN :FO :RI :01'FD :FORT :RIFD :01'LT :RIFD :FOEND

Here the input :FO refers to the length AB, :RI to the turn between AB and BC, and:01' to the length BC - (see Figure 3).

What can we conclude from the fact that Nicola did not try to relate the input :01'to the input :FO, but instead reverted to an adding-on strategy? Firstly, it is clear that ina structured task such as that in the laboratory, we need to be aware of the constraints onNicola arising from the situation -- the experimental contract between Nicola and the resear­chers which generated a pressure to produce a product. The adding-on procedure workedfor her, since she made quite complex paper-and-pencil calculations for the value of the input:01' to ensure that it did. We have traced the development of Nicola's work and are confi­dent that she is well able to operate on variables when she sees a relationship immediately.Here, for example she perceived the symmetry of the N and used her inputs :FO and :RItwice in a correct form rather than introduce new inputs for the second vertical line andthe second turn. When given some assistance by the teacher, she can also build up morecomplicated relationships between variables as described in the necklace episode. We nowconjecture that if we had designed the situation differently, she might have adopted an expe­rimental approach on the computer, and come up with a relationship between :OT and :po.

The introduction of our structure into the learning situation thus becomes problematic; ouraim was to provoke the pupils to make predictions away from the computer, and this hadunforeseen consequences. We now have some evidence that the computer can provide thesupport by which pupils can search for the relationships that seen to be just beyond theirgrasp and thus, to a certain extent, take on the role of the teacher (Hoyles and Noss, inpress). We therefore now see that by forcing the pupils away from the computer we almostinevitably left Nicola with her more 'primitive' strategy by denying her access to an experi­mental approach. We now turn to a description of some further episodes, in an attempt tothrow more light on these issues.

Noel's Snakes and Ladders board

In the sessions we report here, Noel was initially working with Trevor although Noelwas very much in control of the situation; Trevor played very little part and subsequentlydropped out of the proceedings. Their initial goal was to make a snakes and ladders game,and their first task was to design the board. They started off directly in the editor and defi­ned a program for a 'saw tooth' pattern (see Figure 4):

TO SNLREPEAT 4 [FD 500 RT 90 FD 50 RT 90 FD 500 LT 90 FD 50 LT 90]END

Notice that their program did not have a variable input - entirely reasonable as theysimply wanted one large board. Also, notice that at this stage their board was not square.The researcher intervened to provoke the use of input(s):

R: 'Can you make it so that your board can be different sizes?'Noel: 'No problem!'

THE COMPUTER AS A MEDIATING INFLUENCE

Figure 4. Premiere etape dans la construction du cadre du jeu «Snakes & Ladders»

50

500

Figure 4. SNL's first saw-tooth

279

They edited SNL to SNL :8 :F and substituted :F for 500 and :5 for 50. They then addeda FD :F and a LT 90 to their program so that the turtle was on the top right hand cornerof the 'board' (see Figure 5). They immediately set about constructing the second saw-toothto produce the squared pattern by adding one more line to their SNL procedure:

Figure 5. Etape intermediaire

LT 90

FD:F

Figure 5. The interface in SNL

TO SNL :S :FREPEAT 4 [FD :F RT 90 FD :8 RT 90 FD :F LT 90 FD :8 LT 90]FD :FLT 90REPEAT 4 [FD :F LT 90 FD :8 LT 90 FD :F LT 90 FD :S RT 90]END

Firstly, the way the project had been built up resulted in the interface between the twosaw-teeth being such that it was not possible (as we believe Noel originally intended) to usethe same REPEAT line. However, there was an appreciation of symmetry - their second'saw tooth' was the same expression as the first with LT and RT interchanged. What is imme­diately apparent though, is that they seemed to think that :F was the long side of their saw­tooth pattern and :5 the short side. They have clearly not discriminated the components of theboard: that is, they do not appear to see that the way they have constructed their board must:

- imply a relationship between these two variables which depends on the number ofREPEATs they have used in line 1 of their program (in this case :F should be :S*8 for a'closer!' board)

280 R. NOSS & C. HOYlES

- imply that their board must be square because they have used REPEAT 4 again inline 5 of their procedure for SNL.

They tried SNL 50 250 and found that it did not fit (see Figure 6):

Figure 6. SNL 50 250

Figure 6. SNL 50 250

250

D

A

so,..-

-

50250 ch

C50

B

R: 'Can you work out what that length along the top must be to make it all fit?'N: 'Well lets suppose each of these is 100 then that is 8 times 100'.

So focussing on the top of the board Noel knew - by using his substitution strategy- that the distance was :S * 8. He then typed the following procedure:

TO SNL :S :FREPEAT 4 [FD :F RT 90 FD :S RT 90 FD :F LT 90 FD :S LT 90]FD :FLT 90FD :8 * 8LT 90REPEAT 4 [FD :5 LT 90 FD :8 * 8 RT 90 FD :5 RT 90 FD :8 * 8 LT 90]END

What is interesting to note is that he only changed :F to :S * 8 where the :F signifiedthe value of the length along the top of the board (emphasised in the text of the program).This indicates that he was simply using a substitution strategy, and not seeing a general rela­tionship between :F and :S. Thus, his program fits the second saw-tooth along the side CD(see Figure 6), but will only fit along the side DA if the values of SNL's inputs are adjustedcorrectly: that is the second input is 8 times the first. Noel was unaware of the implicationof the number of REPEATs in the second REPEAT line, which meant that :F must be repla­ced by 8 ... :S everywhere; this involves recognising a relationship between variables in a moregeneral way. To summarise, the problem was - given the procedure had two inputs - topredict that the total length 'downwards' was :S ... 8 because the number of REPEATs inthe second saw-tooth was 4, and simultaneously to relate this to the :F in the first saw-tooth.

For the time being at least, this proved too difficult. Noel typed SNL 100 500 (notestill the same relationship of I : 5). The researcher tried to help him to see that the secondinput must be 8 times the first but had a feeling that he was at a loss. This was confirmedwhen Noel said: 'Lets not have a square board. It will never fill the screen!' (The screenis, of course, not square).

THE COMPUTER AS A MEDIATING INFLUENCE 281

Noel then changed the number of REPEATs in SNL to 3, and typed SNL 150 500. Thushe seemed to have some idea that with a smaller number of REPEATs he must increase thesize of the first input in relation to the second, but he was not aware of the necessary rela­tionship explicitly at this stage. On running SNL, he saw that this did not work. His firstreaction was to debug the match along CD.

Noel: '1 know, we have to change that :S * 8 to :8 * 6'.

This he did but then typed: 8NL 75 600 (the relationship is 8 times, and would haveworked if he had not changed the number of repeats from 4 to 3 - what a pity!)

Noel: 'It's too small.'

He tried SNL 100 800 (note the relationship is still 8 times).Noel continued to experiment with different inputs in quite a systematic manner, purely

working on the symbolic code. For example, he tried SNL 100800, and then SNL 87.5 700.When asked why he was choosing these values, he replied «It's half way» (i.e. 87.5 is half­way between his first two tries of the first input - 75 and 100, and 700 is half-way between600 and 800 - his first two tries of the second input). In fact, his systematic strategy meantthat the relationship between the inputs remained the same (that is, one is eight times theother). He did not think to try to synthesise his symbolic manipulations with visual feed­back from the screen. Neither was he able to come up with the relationship that one inputneeded to be multiplied by 6 to give the other.

In the next session, Noel was working by himself. By this time, he had introduced aSTART procedure and reverted to his original program for SNL with 4 REPEATs (see thelast version of SNL above). He began the session by running SNL 87.5 700, which now works.Unfortunately, we do not know if this was merely chance and a left-over from the previoussession, or whether Noel saw the necessary relationship between the two inputs. He was invi­ted to try to make boards of other sizes to probe this question, and eventually made thefollowing remark:

Noel: The S has to be B times smaller than F.'R:'What inputs would it need if F was 200?'

Noel responded by trying SNL 16 200, which failed and then - without prompting ­typed PRINT 200/8. On receiving the answer 25, he typed SNL 25 200, which produced therequired result.

R: 'What if it was 300?'

Noel typed PRINT 300/8 receiving the response 37.5, and then typed SNL 37.5 300.At this point, the researcher suggested that it might be possible to write SNL in such a waythat the calculation 'could be done for you'. Without hesitation, he entered the editor, andbegan the process of replacing :8 by :F/8 throughout the program. When he reached FD:S* 8 he simply replaced it with FD :F.

R: 'Why did you put :F there?'N: 'Because that's the length... that's the way the grid's drawn... I knew I wanted asquare grid so each long bit should be however many squares along the grid longer thanthe size of one of the sides oj the squares:

Thus Noel was arguing geometrically, but he also understood his symbolic manipula-tions, since when questioned further about the equivalence of (:F/8)*8 and :F, he replied:

N: 'Because if you divide something by one number and then times it by exactly thesame number then you'll get what you started ojj with.'

Like Nicola, this story is continued through our work in the computer laboratory someweeks later. In this case, we set up some tasks to probe the pupils' understanding of a paral­lelogram (Hoyles & Noss, in press).

282 R. NOSS & C. HOYLES

Figure 7. La tache du parallelograrnme

Figure 7. The parallelogram task

In trying to draw Figure 7, Noel went straight into the editor and typed:

TO HaLO :R :YREPEAT 2 [FD 200 RT :R FD 400 RT :YjEND

This indicated that Noel had recognised the structure of a general procedure which couldbe used repeatedly to draw the given pattern. He then typed:

HaLO 9090RT 30HaLO 60120RT 30HaLO 30150

His first attempt at a procedure was however:

TO TIlHaLO 9090RT 20HaLO 60120RT 20HaLO 30150END

The fact that he had apparently inadvertently changed the interface turn from 30 to20 was significant. In this work in direct mode, Noel had not apparently been aware of thesignificance of the interface between the HaLOs, that is the relationship between the inputto RT (let us call it :A), and the consecutive first inputs to HOW (:Rj and :Ri + 1) , that mustbe satisfied in order that all the HaLOs remained on the same base; namely that :R j •

i+1 ;:A (see Figure 8). On trying his procedure TIl, he found that the bases of his paralle­lograms were moving. He changed the values several times. What interesting here was thathe eventually was able to work out what was wrong by reflection on the visual effect of thedifferent inputs. Finally he used the following sequence of inputs to HaLO which worked:9090, 70 110, 50130, 30 150, 10 170 commenting to himself:

"It was not working last time 'cos I was not coordinating the angles properly. Every­time I took 20 off the angle I have to add it on to the other."

Thus the necessary relationship was made explicit by developing a synthesis between themeanings in the symbolic code and the visual output. This is analagous to the progressiondescribed in the Snakes and Ladders task, although in this case there was no researcher inter­vention.

THE COMPUTER AS A MEDIATING INFLUENCE

Figure 8. Solution proposee par Noel a la tache du parallelogramme

283

20

.R, - :Ri+l =:A where

.R, =90, 70, ..,:Ri+1 = 70, 50, ...:A=20

Figure 8. Noel's HOW solution

Discussion

We are aware of the dangers of overgeneralising from a few cases, and in particular werecognise that it would he incautious to attribute any particular behaviour in the laboratoryuniquely to interventions made at some time previously. Nevertheless we suggest on the basisof many years of classroom observations of pupils working with Logo that the two case stu­dies described above could be considered as generic examples of how children make theirway around the unos model while working in a Logo environment. In particular these casestudies illustrate firstly how pupils use variables as inputs to procedures simply to 'flag' whatis varying, and secondly the situations in which they may come to make explicit any implicitrelationships between the inputs. They serve to illustrate the complementary roles of the resear­cher and of the programming environment in driving the progression towards an acceptanceof the need for a more decontextualised relationship. We have argued elsewhere (Hoyles &Noss, in press) that there are identifiable stages of discrimination in which the following arepresent:

i) discrimination of the features of the figure without regard to its available symbolicrepresentations;

ii) discrimination within the symbolic representation; that is perceiving its structure andpattern without regard to the visual outcome.

Thus there is a progression from implicit use of the relationships between inputs to amore expliciit appreciation of the task-structure, by synthesis of the symbolic descriptions interms of programs (or fragments of programs) with the visual image on the screen. This isillustrated in the case study of Noel who came to see both the simple relationship betweena) the two inputs to his SNL procedure and b) the turns between his parallelograms andthe inputs to his parallelogram procedure by reflection on the geometry of the situation ashe changed the values of the parameters.

Now let us turn to the construction of general relationships between inputs which arenot so obvious or straightforward - the focus of the case study of Nicola. Here, after aninitial researcher intervention, the visual output assisted Nicola in seeing how to operate oninputs in a simple way (for example :C/4 or :8*8) by adopting a substitution strategy. Howe­ver this method could not provide her with an answer for the length of her necklace, and

284 R. NOSS & C. HaYLES

the researcher played a major role in her solution. Nicola already knew, or at least had beenacquainted with, multiplicative relationships and with simple mapping diagrams. The roleof the researcher was to help her mobilise her existing knowledge to serve in her problemsolution - and the computer allowed her to think at a symbolic level while providing visualfeedback.

This leads us to consider how to include the role of the researchers' interventions andthe structuring of the situation into our understanding of the learning process. We wouldlike to refer here to a major thesis, of Vygotsky (1978) which suggests that every specific stageof a child's development is characterised by an actual developmental level and a level of poten­tial development. The pupil is not able to exploit the possibilities at the latter level on herown, but the provision of scaffolding - help in the form of interpersonal communication- makes possible the solution of tasks which would not be possible unaided.

In Noel's case, the researcher provided scaffolding by focussing attention on the rela­tionship between the visual and symbolic modes. In one sense, the computer could do therest in the laboratory (during the parallelogram task) because of the way in which it provi­ded dynamic interaction between the two modes.

The case of Nicola is rather different. In the classroom, the researcher provided scaffol­ding to assist her in finding a non-obvious relationship between inputs as mentioned above.Why did she not mobilise this experience in the laboratory? We now conjecture that the struc­ture of the situation in the tasks, in which she was forced to work away from the computer,might have been a key influence. We have argued elsewhere that the symbolic representationof a computer program can itself act as a form of scaffolding which allows the learner tosketch out the whole problem as she sees it, and then attend to the elements of the concepton which work still needs to be done. For example, we have described ways in which thecomputer has been used to test out some ideas in symbolic form, use them (knowing theywould not work) and then edit them as a result of the visual feedback generated. The symbolicform thus captures the pattern perceived although not as yet fully comprehended, and actsas a bridge towards generalisation. It was just this sort of activity which was not initiallyavailable to Nicola. Given that she still needed some 'assistance' she had therefore to revertto her more primitive approach.

We are thus led back to the main theme of this paper; that is the need to identify thecircumstances in which the chances of development in understanding a specific mathemati­cal idea - variable - are optimised. We have considered the role of pupil/computer inter­action, and how this can complement pupil/researcher interaction and vice-versa. Our mainfinding concerns the crucial importance in this environment of the synthesis between the visualand symbolic. In addition however, we also have shown that imposing a priori structures canfacilitate some progressions in understanding but can also inhibit others. This is probablyinevitable, and resurrects old questions of achieving the 'right' mix between guidance anddiscovery. However, the new factor in the equation is the computer - which imposes its ownconstraints but, at the same time, can provide scaffolding for pupil learning.

References

Freudenthal, H. (1986). Didactical Phenomenology of Mathematical Structures. Dordrecht: D. Reidel.

Hillel, J., & Samurcay, R. (1985). Analysis of a Logo environment for learning the concept ofprocedure with variable.Project Report, Concordia University, Montreal.

Hoyles, C. (1986). Scaling a Mountain - a study of the use, discrimination and generalisation of some mathematicalconcepts in a Logo environment. European Journal of Psychology of Education, I, 11l-126.

Hoyles, C. (1987). Tools for learning - insights for the mathematics educator from a Logo programming environment.For the Learning of Mathematics, 7, 2, 32-37.

.Hoyles, c., & Noss, R. (1987 [al), Synthesising mathematical conceptions and their formalisation through the cons­truction of a Logo-based school mathematics curriculum. International Journal of Mathematics Education inScience and Technology, 18, 581-595.

THE COMPUTER AS A MEDIATING INFLUENCE 285

Hoyles, C, & Noss, R. (1987 Ibn. Children working in a structured Logo environment: From doing to understanding.Recherches en Didactique des Mathematiques, 8, 12, 131-174.

Hoyles, C., & Noss, R. (in press). The computer as a catalyst in children's proportion strategies. Journal ofMathema­tical Behavior, 5.

Hoyles, C, & Sutherland, S. (in press). Logo Maths in the Classroom. London: Routledge.

Noss, R. (1985). Constructing a Mathematical Environment through Programming: a Case Study of Young ChildrenLearning Logo (Doctoral Thesis, Chelsea College). University of London, Institute of Education.

Noss, R. (198~). Creating a Conceptual Framework for Elementary Algebra through Logo Programming. EducationalStudies in Mathematics, 17, 335-357.

Sutherland, R. (1988). A longitudinal study of the development of pupils' algebraic thinking in a Logo environment.Unpublished Doctoral Thesis, Institute of Education, University of London.

Vergnaud, G. (1987). About Constructivism. In J. Bergeron, N. Herscovics & C Kieran (Eds.), Proceedings of the Ele­venth International Conference on the Psychology of Mathematics Education: Montreal.

Vygotsky, L. (1978). Mind in Society, the Development of Higher Psychological Processes. Cambridge, Mass.: Har­vard University Press.

Influence mediatrice de I'ordinateur dans Ie developpementde la notion de variable chez I'enfant

Dans un environnement Logo, les eleves emploient couramment lanotion de variable. Cependant Ie processus par lequel la comprehen­sion de cette notion se developpe souleve plus d'un probleme. Dans cetarticle, nous decrivons une serie de situations dans lesquelles nous avonspu examiner comment Ie recours it l'ordinateur et l'interventiond'un chercheur peuvent faciliter le deveioppement de la notion devariable.

Nous presentons deux etudes de cas reconstituees iJ partir des don­nes recueillies aupres de deux eleves de treize ans, observes iJ differentsmoments et dans deux sites distincts: la salle de cours de mathemati­ques et Ie laboratoire d'informatique de l'Universite. Nos principalesconclusions concernent les relations multiples s'etablissant entre Ie cher­cheur; l'ordinateur et I'eleve; en particulier Ie role de mediateur cogni­tif que peut jouer (mais pas toujours) l'ordinateur pendant l'acquisitionde la notion de variable.

Key words: Logo; Algebraic variable; Pupil conceptions; Computer.

Received: November 1987Revision received: March 1988

Richard Noss, Dept. of Maths, Stats and Computing, Institute of Education, University of London, 20, BedfordWay, London, WCIH OAL.

Current theme of research:

Children's mathematical thinking in computer-based learning environments. Influencing teacher attitudes towards mathe­matics with the computer.

Most relevant publications in the field of Educational Psychology:

Noss, R. (1986). Constructing a Conceptual Framework for Elementary Algebra through Logo Programming. Educa­tional Studies in Mathematics, 17, 335-357.

Noss, R. (1987). Children's Learning of Geometrical Concepts through Logo. Journal for Research in MathematicsEducation, 18, 343-362.

Noss, R. (1988). The Computer as a Cultural Influence in Mathematical Learning. Educational Studies in Mathema­tics, 19, no. 2, 251-268.

Hoyles, C, & Noss, R. (1987). Children working in a structured Logo environment: from doing to understanding. Recher­ches en Didactiques de Mathematiques, 8, 12, 131-174.

286 R. NOSS & C. HOYLES

Celia Hoyles. Dept. of Maths, Stats and Computing, Institute of Education, University of London, 20, Bedford Way,London, WCIH OAL.

Most relevant publications in the field of Educational Psychology:

Hoyles, C. (1986). Scaling a Mountain - a study of the use, discrimination and generalisation of some mathematicalconcepts in a Logo environment. European Journal of Psychology of Education, I, no. 2, 11l-126.

Hoyles, c., & Noss, R. (in press). The Computer as a Catalyst in Children's Proportion Strategies. Journal of Mathe­matical Behavior.

Hoyles, C. (1987). Tools for Learning - insights for the mathematics educator from a Logo programming environ­ment. For the Learning of Mathematics, 7, 32-37.

Hoyles, C., & Sutherland, R. (1987). Ways of Learning in a Computer Based Environment: Some Findings of the LogoMaths Project. Journal of Computer Assisted learning, Vol. 3.