P. L. G A L B R A I T H

A S P E C T S O F P R O V I N G :

A C L I N I C A L I N V E S T I G A T I O N O F P R O C E S S

ABSTRACT. A clinical methodology was used to investigate the perceptions which pupils of secondary school age have concerning modes of mathematical argument which have an agreed status within the world of mathematics. The analysis of data obtained from three extended contexts led to the identification of dusters of characteristic response types. Differences were found to exist between the agreed meaning of some mathematical terms and procedures and the meaning ascribed to them by students. By considering levels of performance it was possible to identify particular components, the presence or absence of which consistently determined the capacity to structure or follow proofs and explanations.

1. B A C K G R O U N D

Increased interest in the nature o f mathematical p roof and its associated skills

seems to be one product of an era which has seen the not ion o f formal proof

exorcised from many curricula.

Lester (1975), has identified several problem regions within the literature

and has drawn at tent ion to conflicting evidence concerning the development

o f logical reasoning ability. Thus on the one hand studies by Lovell (1961) and

LoveU, Mitchell and Everett (1968) following Piaget (1928) were cited as

support ing the belief that while children in the age range 11-13 years are able

to handle certain formal operations they carmot synthesize a fully exhaustive

proof. On the other hand reference to studies by Suppes (1968), following

Hazlitt (1930) acknowledged an alternative viewpoint that there is no relation-

ship between age and logical reasoning beyond that determined by experience.

Bell (1976, 1979) described proof as an "essentially public activity which

followed the reaching o f conviction, though it may be conducted internally,

against an imaginary doubte r . " He saw the meaning o f p roof as carrying three

senses,

(a) verification or just if icat ion; i.e., concerned with the t ru th o f a pro-

posit ion.

(b) i l lumination; i.e., conveying insight into why a proposi t ion is true (or

false),

(c) systematisat ion; the organization o f results into a deductive system of

axioms, major concepts and theorems.

Bell claimed that an understanding o f p roof grows out o f the internal

testing and acceptance or rejection which accompanies the development o f a

EducationalStudies in Mathematics 12 (1981) 1-28. 0013-1954/81/0121-0001 $02.80 Copyright �9 1981 by D. Reidel Publishing Co., Dordrecht, Holland, and Boston, U.S.A.

2 P.L. G A L B R A I T H

generalisation. In the early stages a: re-assertion of a statement may be deemed

adequate by the pupil. At later stages there is appeal to evidence, the awareness of a need to write down the arguments so as to better display the structure. Only at advanced levels is the need for cogent reasoning from explicit starting

assumptions recognized. Consequently, according to Bell, pupils will not use formal proof with appreciation of its purpose until they are aware of the public status of knowledge and the value of public verification. This position was

accompanied by an analysis of responses from a group of 14-15 year old pupils

to a series of items requiring them to provide explanations and justifications. The responses were classified into two main categories viz. Empirical and

Deductive within which differential levels of functioning were identified. For empirical explanation (six levels) the quality of the responses varied from a

failure to fred any correct examples or adhere to the given conditions (the lowest level), to a fully systematic approach which culminated in a complete check of a full finite set of cases. The deductive approach (seven levels) was typified at the lowest level by those who, having managed to correctly con-

sider one or more special cases were unable to link data to conclusion. The highest level (rarely achieved) involved giving a connected argument deriving the conclusion from the data using generally agreed facts and principles.

Van Dormolen (1977) also invoked the notion of levels of functioning. He introduced a psychological-emotive aspect in which knowledge and skill are subservient to the acceptance of the need for a logical argument. He suggested alternative ways in which a given task might be approached by a pupil, these alternatives representing successively more sophisticated thinking. Thus to

prove that the diagonals of an isosceles trapesium are of equal length, pupils

might proceed in alternative ways:

(a) The pupil measures the diagonals with a ruler. Such a pupil sees only a particular trapezium and given another trapezium would begin all over again.

(b) The pupil mentally cuts out the trapezium, turns it over and replaces it

back into its hole. Now the trapezium is seen as representative of a class of similar objects rather than as a particular case and the pupil is examining pro-

perties of the species rather than the object. (c) The trapezium is defined as a quadrangle with an axis of symmetry

(not through a vertex) and general properties of a line reflection are used to deduce that the diagonals are equal. This solution is limited neither to special objects, or sets of similar objects - rather appeal is made to rules that apply in any logical argument independent of the subject matter. Van Dormolen relates his examples to levels of thinking as suggested by

van Hiele (1973), viz.

ASPECTS OF P R O V I N G 3

Ground Level: Student is limited to thinking about the particular example:

organization is local.

Level 1: This is achieved when, for example, the trapezium is no longer

seen as a form but by virtue o f investigation is recognized as a concept.

The organization is much less local, for example, certain properties o f this

trapezium go for all trapezia. However organization is still limited to the domain of discourse, and proving properties o f trapezia is viewed as a different activity from proving about squares etc.

Level 2: Experience in level 1 type thinking brings the realisation that

arguments in dissimilar areas have elements in common. These arguments

provide stepping stones to an understanding about local organization - o f reasoning about reasoning.

Van Dormolen claims that these levels form a strict hierarchy in that it is impossible for a learner to operate on a higher level if a lower one has not

yet been reached. Doubt is cast on the wisdom of curricular thinking which denigrates practice, for to gain insight into new subject matter it is necessary to do many exercises in order to achieve the various levels of thinking. If this is not done, claims van Dormolen, the risk of missing a level with its sub- sequent penalty is greatly enhanced.

2. C O N C E P T U A L F R A M E W O R K

The world o f Mathematics Education contains (as far as the communication process is concerned) three significant components, viz. Mathematics, Teachers,

and Pupils. Success requires a smooth mediation on the part o f teachers

between the pupils and the world of mathematics. A necessary (if not suf- ficient) condition for this to occur is that there be agreement between the

content and process o f mathematics, and the teachers view of this content and process, and also agreement between teachers and pupils as to the meaning of objects in the mathematical world. As Thorn (1973) expressed it:

The real problem which confronts mathematics teaching is not that of rigour, but the problem of the development of meaning, of the existence of mathematical objects.?

Since the advent of the New Mathematics movement considerable effort has been expended to improve the congruence between the world of mathe-

matics and the teachers' view of the same. Many courses have been conducted to this end and while quite obviously much remains to be done for present purposes this aspect is being set aside. The emphasis in this paper will centre on communication with pupils concerning understanding about and use of objects in the mathematical world. In particular, attention will be focussed

4 P . L . G A L B R A I T H

upon discrepancies between pupil thinking and accepted reasoning in relation

to the same mathematical objects. In seeking to increase the level of mathematical knowledge of pupils certain

assumptions (pedagogical axioms?) are made in regard to the communication process. One common assumption seems to be that while pupils may be ignorant of a particular concept or result (which is the current object of a teaching segment) they share a common meaning with the teacher regarding important other objects from the mathematical world which are used in the development of the concept or result. These objects are the mathematical vocabulary used and the meanings attached to particular techniques, e.g., of explanation and proof. For example if a teacher provides a counter-example to a proposition it is assumed that teacher and pupils share a common appre- ciation of what has been achieved. Similarly if a pupil agrees separately with each component of a linked chain of statements it is assumed that an inference deduced from the complete chain by the teacher will be convincing to the

pupil. (This assumption underlies guided explanations and step by step solution of exercises.)

In terms of the van Hiele description the foregoing means that while ground level and/or level 1 examples may be provided to introduce and consolidate certain aspects of a new topic the verbal communication and mathematical reasoning used to texture the development frequently requires level 2 appre-

ciation, i.e., the objects of mathematical reasoning (such as proof skills) are assumed to have a universal and objective meaning which is shared by teacher

and pupil. The investigation reported in this paper was particularly concerned to seek

insight into the understanding which pupils have of particular mathematical objects (concerned with explaining and proving), and the meanings that pupils may attach to forms of argument which have an agreed status in the mathe- matical world. Such understandings will be critical in determining what pupils regard as relevant for the purposes of synthesizing an explanation or proof or following a reasoned argument. The detail of the work sought to research pupil perceptions of particular techniques and concepts identifiable as contributing to an ability required in constructing and evaluating proofs and explanations.

A list of these is given below:

A. The importance attached to variety and thoroughness in checking finite cases.

B. Factors involved in embedding an explanation in knowledge external to the particular context, e.g., using an accepted external principle to achieve

justification.

ASPECTS OF PROVING 5

C. Factors involved in the capacity to link a sequence of inferences in order

to reach conviction about the conclusion. D. Awareness of the set over which a particular generalisation is valid. This

involves a systematic construction of classes for examination, awareness of the

need for representative examples, and of the significance of counter examples. E. The importance attached to the literal interpretation of statements and

conditions. F. Factors involved in appreciating and using the distinction between

implication and equivalence.

G. Awareness of the arbitrary and general properties of definition in

mathematics.

H. Proof as a linked network of steps. The capacity to undertake proof

analysis as a means of exposing the details of an argument.

3. METHODOLOGY

The method of investigation was by clinical interview. Judgments about

student thinking based upon written responses inevitably to some extent must assess process by product. In the absence of articulate explanation it is often

impossible to distinguish between the quality of some responses and in such

cases it can become a situation of all incorrect answers are equal. The clinical situation with its opportunity for probing questioning (pushing

pupils for a particular judgment where necessary) is essentially dynamic and consequently more appropriate for investigating process variables which are

themselves dynamic. It became evident that answers with the same surface response were often representative of quite different levels of understanding

when the deeper structures of the underlying processes were probed. The three items used in the study are printed below. They were chosen from

a pool of six items which were developed by the author, trialled in a Notting-

ham Comprehensive school in November 1978, and subsequently used in a

pilot study involving 8 pupils in December 1978. The initial impetus was pro- vided by the currently running SSRC research project on Teaching Methods

and the Acquisition of Strategies in the Learning of Mathematics, directed by

Dr. A. W. Bell and centred at the Shell Centre for Mathematical Education at the University of Nottingham.

The intention was to interview each pupil on all items with total interview time typically taking 30 to 40 minutes. The pupils were encouraged to use pencil and paper to carry out calculations, or otherwise clarify their thoughts.

Kantowski (1977) has drawn attention to difficulties in maintaining consis- tency and reliability in clinical research.

6 P . L . GALBRAITH

To provide a structured framework to assist the interviewer maintain con-

sistency between subjects, interview protocols were devised for each of the items. This was felt to be an important part of the preparation in view of the

fact that a series of helps and hints was to be supplied as necessary to probe the

depth of understanding of the subjects. The protocols have the form of im-

provised flow charts. A pupil who is fully successful (S) would be taken by the interviewer down the left hand column of the chart. Failure or incompleteness

(F) at some stage is followed by a prompt from the interviewer along the lines

of the content of the boxes in the right hand column of the chart. Success

following a prompt causes control to pass back to the next stage in the left hand column. If the prompt does not gain immediate success a stronger or additional prompt is given which includes a more detailed structure in the hint than previously. If success is still elusive the interviewer continues to inject

information - this stage is called TEACH. Here evidence is supplied which

points the pupil directly to the key variables at hand and brings them to the point where a direct inference can be made. The pupil must still however

recognize the situation for what it is and be able to correctly make the appro- priate inference. Not all pupils are able to do this. Some are totally confused

and unsuccessful while others may have identified relevant aspects but are not

able to assemble them coherently. At this stage the interview is concluded as

far as the particular part of the item is concerned - otherwise it is concluded

by success.

This technique of free response followed by standard prompts enabled the

interviewer to probe depths of pupil reasoning at various levels and at various

points of the items. Tape recordings of the interviews were analysed retro-

spectively. Comparison of the interview data was then undertaken to identify

clusters of common responses which provided descriptions and examples of pupil reasoning across the categories and items. Originally the complete bank

of 6 items was given to a group of 8 pupils by the author. The pupils ranged in

age from 12 to 15 years and were chosen to represent a range of ability as measured by achievement in conventional coursework. An account of this

pilot program is given in Galbraith (1979). Consistency across subjects was maintained as carefully as possible using the interview schedules. Even within such a small sample different process characteristics appeared and clusters of particular (and different) types of responses began to emerge. Subsequently a follow up investigation was conducted in Brisbane schools during September 1979. For this study a group of post graduate students in Education were given instruction in the methodology. They then proceeded to interview pupils in the

school in which they were practice teaching. This was regarded as part of their observation schedule. The pupils were taken from the junior school, represented

ASPECTS OF PROVING 7

all ability levels and ranged in age from 12 to 17 years with the majority

being in the 13 to 15 age group. After the removal of interviews which

for technical reasons were deemed unsatisfactory each item was represented

by a minimum of 170 individual cases across the variety of age and ability

ranges. The question of reliability in such circumstances is particularly sensitive

and needs discussion. No attempt was made to assign numerical scores to the

responses (i.e., to compare pupil with pupil) but rather the goal was to identify

clusters of mathematical reasoning characteristics within categories and items.

With the use of different interviewers the interview schedules became a crucial

control. A measure of confidence in the reliability of the procedure and results can

be claimed on two counts. Firstly there is the consistent appearance of particular types of response

given by pupils of different ages, schools, and interviewers. Pupils were ob-

served to give almost identical responses (the same wording)when confronted

by particular situations. This characteristic is widely emphasized in the Results

section of this paper. It should also be mentioned that the situations were not ones in which the interviewer guaranteed a particular kind of response by a form of questioning - in many cases they were initial responses to a free

situation. The second encouraging feature with regard to reliability is that the

classes of response showed good agreement with those first identified within the pilot program. A further safeguard exists in the record of the interviews themselves since the nature of any response is immediately relatable to

the degree of prompting provided. Because not all pupils required prompting, some required more prompting than others, and some interviews were not

complete in all parts the numbers shown in the RESULTS section as

responding to particular inputs varies from item to item and almost never

equals the total number of interviews recorded. What the analysis has

attempted to do is provide measures of the relative numbers of pupils dis- playing particular types of thinking when it has been ascertained that they

have responded from the same base level of either free response or a parti-

cular prompt. However it is the identification and interpretation of the response classes (rather than numbers) which is the most instructive part of the

work. Occasionally pupils were distressed at the prospect of being examined or re-

corded and in such cases no interview data was collected. In the vast majority

of cases the pupils quickly forgot about the electronics and settled down to their tasks often with obvious enjoyment.

8 P . L . G A L B R A I T H

4. ITEMS AND I N T E R V I E W S C H E D U L E S

Game o f 25

Game of 25

ry to prove that the player I who makes the total 18 can

l l win. Explain.

Try to find numbers less I than 18 from which you can [ F win. i.e. be first to 25. I If you think there are nonel explain why.

ensure he(she) will win? Explain why or why not.

S

,F

S

Play game through to ensure understanding I l

!

s<

I Prompt: What do you know about the numbers 4,11,18? Can that help you? How?

Prompt: Suppose you are 18 I say 21; you say ...

I say 19; you say ...

sI

n u m b e r f r o m w h i c h y o u c a n b e f i r s t t o 1 8 . ( T h e n y o u

c a n b e f i r s t t o 25)

I tF |Prompt: "iIf a wrong answer l is given). Play game through showing error. Do you want to think some more?

y I Prompt: What about ii? Why? Any others? Why?

F

' I I f give up or says none

,l

Rules

Player 1 picks a number from the list 1 ,2 , 3, 4, 5, 6. Player 2 picks a number from the same list and adds it on. The game continues like this with each player taking turns; the player who

makes the total 25 is the winner.

(1) Prove that the player who makes the total 18 can always win. (Give your

reasons in full.)

ASPECTS OF PROVING 9

(2) Find all other numbers (less than 18) from which a player can always win. (Show all your working and explain your reasons.)

(3) Can the player going first always win? Explain why or why not.

Sevens

Sevens

Introduction: Explain conditions to the pupil Is 49 one of the numbers in the list? 383? List L contains all such numbers less than 70. You are given the first few, What is the next one?

43 Good!! Now we are ready to start.

l Do you think Gary is right?~ ~ IAre you sure it will work from what Explain why you think so. ~--L--~you have don~

. j

I S iYou don t need to check any more?

What are the first few of~ IBrenda's numbers? | " I Answer question 2 fully.

IS

I Now decide whether Henry 1 is right. live your reasons

We can find one number | (59) which do~sn' t work

i for Brenda. Is that

l enough to make her wrong? I Do we need more to be I sure?

I Prompts: Is Brenda's rule the same as

]Does that make their statements the i

lwhamt?is a word Gary d .... 't use? I IDoes "every" make a difference? I ~ry to answer question 2 now. I

prompts: Which numbers are given by starting with 7 and adding 9's?

IAre these the only kinds of number

J whaL?abo ut 59?

Try question 2 again.

This question is about numbers for which the sum of the digits can be divided by 7.

10 P. L. G A L B R A I T H

34 is such a number since 3 + 4 = 7 185 is such a number since 1 + 8 + 5 = 14 and 14 can be divided by 7.

If we make a list L of all such numbers which are less than the 70 the start of it looks like this:

7, 16, 25, 34

Write down the next largest number in the list. []

Gary says " I f you start with 7 and keep adding nines you always get a number in the list L".

(1) Is Gary right? Yes/No Use the space below to explain why. (Include all working.)

Brenda says "Every number in the list L can be found by adding 9 to the previous number. You start with 7".

(2) Is Brenda right Yes/No Use the space below to explain why. (Include all working.)

(3) Henry says "Gary and Brenda have said the same thing".

Is this true? Yes/No Explain why.

Quadrilaterals

In this question the meaning of a quadrilateral is as given in the following

statement.

Definition

A quadrilateral is what you get if you take 4 points A, B, C, D in a plane and join them with the straight lines AB, BC, CD, DA.

Horace says "The angles in a quadrilateral at A, B, C, D always add up to 360 ~

B ~ C Here is my proof.

Every quadrilateral can be made in-

A D

to 2 triangles by joining a diagonal. Each triangle has 180 ~ and 2 • 180 ~ = 360o. ' '

ASPECTS OF P R O V I N G 11

Quadrilaterals

I Introduction : Cover diagrams and ask pupil I to draw a quadrilateral from the definition 1

study what Horace and Wa~i~_~Prompt: Why hasn't Warwick drawn' lhave said. What do you think~ la quadrilateral?

- - - - �9 ' ' iv iabout question i. Why? ~ [Look at the deflnltlOn.. " [ ~ IAre you still certain? Explain

[ ~ [~hy again.

~ s ~ ]

[Look at Horace's first statement.[ [Prompt: Does what Warwick says �9 . . F .... ~-~ [ (polnt it out agaln) ~matter. How?

[ I I - ]Are you still ..... f your [ Explain what you have decided [ ~answer to questlon z? [about question 2. I [

t [ Look at question 3. [ 4 [Prompt: How many steps are in

~Horaee' s proof? IWhat is right about H ..... 's ] [What are they? [ Proof? [ Is anything wrong with it? [ ~ [,Try question 3 again.

L Explain- L< I < J

i [Step i: (Every .............. ) [Warwick gave just ....... ple I S~-~ Step 2: (~triangl ....... ) [ to try to prove Horace wrong. ~ ~Is step 2 right? Step i? [I ........ ple which doesn't I F~----~Now what about Question 3? work (counter example) enough to prove a statement wrong? To prove a statement right?

Warwick says "Look at m y quadrilateral.

50 ~ ~ ~ ~ ~ My angles make up 240 ~ .

Horace is wrong."

12 P. L. G A L B R A I T H

(1) Has Warwick drawn a quadrilateral? Yes/No Explain why or why not.

(2) Is the first sentence that Horace says correct? Yes/No Explain why or why not.

(3) Explain what is right or wrong about Horace's proof.

Results

The results will be communicated in a form which integrates response clusters with a discussion of their characteristics.

Game o]'25

Success in this item revolves around the ability to identify and apply the principle of complementation with respect to 7. Part (1) can also be decided by a complete check of all possible numbers. Part (2) requires that the principle embedded in (1) be first recognized and then used t o identify 11 and 4 as further winning numbers. Part (3) requires that the rules of the game and the inferences from (1) and (2) be combined to deduce that the first player can certainly win by choosing 4 as the starting number. Before commencing the item the interviewer played one or more games through with each pupil to ascertain that its structure was completely understood.

Analysis

(i) Of the pupils interviewed 90 achieved a proof for (1) by using a complete

check of all cases from 18. (ii) Twenty six (26) gave a justification based on some form of partial

check. The quality of these varied from those at one extreme who were con-

vinced that a principle shown to work for a few cases could be assumed for the rest, to those (fewer in number) who consciously associated a restricted check with a necessary explanation. Thus Kelly (13 years) checked using only the extremes 1 and 6 but added "I t works for the smallest and biggest so it will be O.K."

(iii) Forty seven (47) pupils used the idea of complementation in order to prove (1). Typically such proofs included statements like that of Jane (13 years) "You need two numbers to add to 7." Several attempts contained an idea of complementation but fell short of a satisfactory explanation.

ASPECTS OF PROVING 13

(iv) Several pupils (between 5% and 10% of the total) made no independent progress at all. These had no expectation that the situation was rule governed in any way. Millie (12 years) first began to subtract instead of add and then proceeded to exceed the limit of 25. She was unable to establish (1) even by checking.

Part (2) of this question exposed a considerable range of reasoning types. (v) Jenny (13 years) after proving (1) using a complete check saw that 18

was the goal for the next lowest winning number, and was thus aware of the existence of a principle. However she was unable to identify any principle and when prompted to consider 11 and 4 she explained their status as winning numbers by conducting a painstaking check of cases. She then tried to argue on behalf of 5.

(vi) Noel (14 years) perceived an underlying pattern to be present but wrongly identified it as leading to the sequence 18, 12, 6 by apparently ab- stracting the six times table from the data. Similarly Joanna (15 years) typified

another group who identified a wrong pattern. She chose 13 as a winning number on the ground that 13 = 25 -- 2 x 6

(vii) Several pupils had no rational expectation of a pattern at work as illustrated below.

! "Now can you think of a number that leads to 18? In the same way that 18 led to 25?"

P "4"

/ (Hiding excitement at such insight) "Why 4?"

P "I just chose it because it goes into the 18 times table."

In a similar situation John (15 years) responded

'Nine, since it divided into 18 and is a pretty good number! '

(viii) Herbie (12 years) typified those who identified the correct principle but were unable to apply it. Herbie noted the 'build to 7' property had been used but could not apply this knowledge to find 11 or 4.

(ix) Perhaps the most interesting responses were from pupils who correctly identified the principle and then proceeded to attempt to apply it. Of the 55 pupils in this major cluster who proceeded without prompting 41 identified both 11 and 4 as winning numbers. The other 14 pupils would make no claim beyond 11.

For those pupils who were unable to proceed from 18 unaided a prompt to consider the equivalent of a game of 18 was provided. This generally assisted these pupils to perceive the principle.

14 P.L. G A L B R A I T H

Sixty five (65) pupils in this cluster then proceeded to identify both 11 and

4 while 29 would make no claim beyond 11. Reasons given for this limited application were typified by Terry (15 years) who after unhesitatingly selecting

11 was most reluctant to commit himself to 4 because "4 to 25 is too far through." This concern was supported by similar statements from other pupils. George (14 years) "Numbers below 11 are too small and anything could happen." Michelle (16 years) "There is too much space from 25."

Some few pupils combined empirical experiment with the principle to provide the assurance they felt was necessary, e.g., they verified 11 by indi- vidual checking from 18 and then used 'seven difference' to complete the chain 25-18-11-4 .

(x) In general those who used a form of complementation argument for (1) were more successful in applying the principle than those who answered (1) by means of a complete check. This might be expected in that the existence of a principle is implied by the complementation argument. Of the 90 pupils who employed a full check for (1) only 36 (40%) found 11 without prompting whereas 28 of the 47 (59%) who used a complementation approach initially

were correspondingly successful. (xi) In part (3) it became evident that identification of the chain

25-18-11-4 by no means guaranteed for many pupils that the first player

could manipulate to win. (It was established that the first player knew as much as they (the pupils) did about the game.) In fact 36 pupils who identified the chain 25-18-11-4 with confidence were not prepared to admit that this gave the first player a certain road to victory. The reason for this reluctance emerged as a difficulty associated with accepting the implications of the principle when applied to give numbers a long way from the starting point. Taken with the responses arising in part (2) of the question and quoted in (viii) the following responses provide corroborative evidence for the stated

interpretation. These latter responses were tendered within answers to (3) and arose when pupils had to consider a backward chain of inferences: the responses quoted previously arose while students were in a process of chaining

forwards from a starting point. Jenny (14 years) "Four is so far away from 25." Joan (15 years) "Can't win from 4 because there are too many numbers." Heather (14 years) "Different ways of reaching 18 might affect the out-

come." Sue (15 years) "Maybe because it's a long way from 4 to 25." (xii) Various other responses were given which when taken together help

to convey the wide variety of perceptions which pupils bring to their thinking

about this type of processing.

ASPECTS OF PROVING 15

(a) Pupils like Robyn (13 years) found the chain 25-18-11-4 by using the principle but remained uncertain until they divided the game into sections, viz., 4-11, 11-18, 18-25. After checking these in turn they became confident sometimes after making a few intermediate checks. Such pupils had effectively changed the structure from formal to concrete operational. Collis (1975) has discussed this type of functioning in relation to algebraic operations.

(b) Pupils like Mary Ann (16 years) who think that a principle can be thwarted. " I 'm not sure the first can win if the other is aware of the sequence."

(c) Pupils like Laurie (14 years) who think that the principle helps rather than assures "First player could only win most of the time."

(d) Pupils like Annette (15 years) who after working through the principle (the academic part) revert to trial and error when playing the game (the practical part).

Since interviews vary in detailed structure (see interview schedule) the quoted numbers in the major clusters do not generally sum to the total of

pupil interviews. The stated numbers should be regarded as containing a tolerance of about 5% to allow for 'grey areas' in responses and interpreta- tions. The relative frequencies within the clusters do however represent a good indication of the distribution of major response types.

Sevens

A complete answer to (1) requires a full check of numbers generated by the rule as far as 70. Alternatively an argument can be given that adding 9 con- serves the sum of the digits for numbers in the required range. (It should be noted that 70 is not an arbitrary limit, for the next number in the sequence viz 79 does not have the required property). To answer (2) adequately careful

attention must be paid to the wording of Brenda's statement. The student must recognize that there are two classes of number in the list L ; those whose digits sum to 7 (given by Gary's rule) and those whose digits sum to 14;viz. 59 and 68. Recognition of this property is the key to providing a counter-example to refute Brenda. Part (3) requires the student to evaluate a comparison between statements of Gary and Brenda. In the light of (2) they are seen to be non-equivalent.

Before the item commenced the comprehension of the pupil was tested with regard to the nature of L. Each pupil was asked whether 49 and 383 were numbers of the type defined in the opening sentence. Pupils were required to answer correctly for both numbers (or others if more explanation was re- quired) before being permitted to proceed.

16 P.L. G A L B R A I T H

Analysis

(i) The majority of pupils managed to directly achieve success with (1). Ninety

seven (97) managed a complete check successfully with several commenting

upon the significance of the number 70 as the limit. (ii) Approximately half as many again (47) were satisfied with a partial

check. Typical of this group was Jenny (13 years) who said that Gary was right because of "7, 16, 25, 34 and so on." She saw no need for a complete check

because "If it works for three it should work." Lesley (14 years) responded similarly "If a rule goes for one it will go for

another." (iii) Some efforts were frustrated by an inability to adhere to conditions.

Thus Millie (12 years) first claimed that 49 belonged to L and then proceeded to add 6 rather than 9. With prompting she named 43 and 52 correctly and decided that Gary was thereby proved correct without further checking.

(iv) In (2) only 5 pupils obtained full success without prompting. In parti- cular a careful literal analysis of Brenda's statement was seldom achieved with the significance of her claim to have 'every number' being consequently over- looked. Similarly, despite the careful introduction only 11 pupils identified the two classes of number which comprise L, with only the successful 5 being able to relate this knowledge to the structure of the question. As a result of this lack of analysis the vast majority (170 pupils) initially stated that Gary and Brenda had said the same thing.

(v) Careful prompting was successful in promoting further progress. A prompt to consider the significance of every did not produce significantly greater success: only 6 of the 50 pupils given this prompt proceeded then to complete the question successfully. Many pupils were given a stronger initial prompt viz to consider the number 59 and/or 68 in relation to the statements made. (Some received this as a second prompt following lack of progress with 'every'). Ninety four (94) out of 130 pupils already receiving such a prompt were able then to proceed successfully. (The proportion unable to perceive the significance of the given counter example thus remained substantial at approxi-

mately 28%.) (vi) A number of pupils were prepared to engage in wishful thinking to alter

the context to conform to their own ideas. Thus one pupil asserted that "Gary meant to say every". One major mistake was to interchange the positions of defined knowledge and conjecture. In the context of the question the defi- nitive statement is the one prescribing L against which the conjectures of Gary and Brenda must be tested. Some students however made permissible numbers subject to the rules given by Gary and Brenda so defeating any attempt to produce convincing counter examples.

ASPECTS OF PROVING 17

Allan (15 years) "59 doesn't suit Gary or Brenda." Graeme (13 years) "59 isn't accepted because it doesn't fit the pattern." Craig (14 years) "59 isn't in the list because it doesn't fit Gary." (vii) There was reluctance on the part of quite a few pupils to ultimately re-

ject Brenda's statement even when the significance of 59 and 68 was reafised. Rather than judge the complete statement against the discipline of its own claim they preferred to argue that it must be partly right since some of the right numbers were generated. This lack of objectivity apparently caused by a centra- tion upon a part of the whole is typified by responses such as the following:

Helen (13 years) "Brenda is a little bit right." Jeremy (14 years) "Brenda is half right." Wayne (14 years) "Brenda is not completely wrong as it gives some num-

bers." Jane (14 years) "Brenda is right to a certain extent." (viii) With respect to the process of refuting by means of a counterexample

considerable variety in appreciating the technique was shown. Many pupils did not see the significance of a counter-example and this appeared to be attribut- able to two main causes.

(a) The mechanism o f refuting was not understood, viz. the example must satisfy the conditions but violate the conclusion. Misunderstanding here is typified by the responses given above in (vii), e.g., in spite of 59 "Brenda is half right." Of the students reaching this stage of the question (either directly or by receiving a prompt to consider 59) 98 recognized the meaning of the counter-example while 55 did not perceive its significance in refuting Brenda.

(b) The philosophy of disproof by counter-example was not appreciated, viz. a single counter-example is sufficient. Seventy three (73) answers were identified as clear responses to the question as to whether a single counter- example was sufficient to destroy a proposition. Of these answers 13 (approxi- mately 18%) indicated that the pupil felt that one such example was in- sufficient. (In the original pilot testing with 8 pupils this proportion was much higher. See Galbraith (1979).) Typical pupil responses included the following:

Bill (14 years) "One example is not enough to disprove it." Michelle (15 years) "One example is enough but the more you get the more

you're disproving." Gary (15 years) "59 and 68 are not enought to disprove Brenda." Ann (14 years) "Could be a freak accident - one in a tnillion chance." Jill (13 years) "Need about 11." Interviewer "Why 11 ." Jill "Oh, should it be 9?"

18 P . L . G A L B R A I T H

Quadrilaterals

Each pupil was initially given this question with only the verbal definition

visible (diagrams covered) and asked to draw a quadrilateral from the definition.

(1) Since Warwick's figure adheres to the conditions given in the definition Warwick has drawn a quadrilateral which serves as a counter example to the statement of Horace in (2). An appropriate answer to (3) will vary depending

upon what has gone previously. Following from correct answers to (1) and (2) exception can be taken to the use of the word every in the first part of Horace's proof. This proof is correct for the class of quadrilaterals represented

by his diagram but not for every quadrilateral. If (1) and (2) have been answered incorrectly viz Warwick's diagram is not accepted as a quadrilateral then an alternative approach to (3) is possible. This involves recognizing that Horace's diagram is not representative of all quadrilaterals (only convex ones) and that he needs to show the use of every (in a restricted sense) is justified by considering concave ones as well.

Analysis

(i) Twenty students attacked the definition insisting that it was incorrect. (ii) Of the students who accepted the definition 67 agreed that Warwick

had drawn a valid quadrilateral but 96 would not accept his figure as valid. These students showed through their responses that they used information apart from the definition to decide whether a figure constituted a quadri- lateral. Mere adherence to a definition was not seen as sufficient evidence. Typical responses were:

Carmel (14 years) "No, because a quadrilateral is an out of shape square." John (14 years) "Warwick is wrong since his doesn't add to 360 ~ Karen (15 years) "Warwick hasn't drawn one because I know what one is."

Shaun (13 years) "The letters should be clockwise." Included among the 96 pupils were some (approximately 15%) who ac-

cepted Warwick's quadrilateral but 'only for his definition'. Joan (14 years) "It 's a quadrilateral by his definition but it isn't one really." George (13 years) "Yes it fits the definition but it isn't a quadrilateral." (iii) The question of Warwick's refutation of Horace's statement showed a

similar pattern of responses. Sixty one (61) pupils agreed that Warwick had achieved a refutation while 54 insisted that Horace's claim remained unscathed.

As with the definition many of the incorrect responses contained subjective and sometimes emotional aspects.

Jeff (15 years) "Warwick is wrong and his definition is stupid." Other responses in general fell into the following categories.

ASPECTS OF PROVING 19

(a) Responses which attempted to "turn the tables" on Warwick, e.g.,

Steve (15 years) "Horace's proof shows Warwick is wrong." James (15 years) "Warwick is wrong since the angles don't add to 360 ~ - so

it doesn't matter what the proof says."

(b) Responses which appealed to an authority outside mathematics, e.g., Amanda (14 years) "Mr Brown says angles in a quadrilateral add up to

360 ~ Robyn (13 years) "Horace is fight because of what I learned."

(c) Responses indicating ambivalence, e.g., Vicki (13 years) "Nothing wrong with saying both are correct." Cathy (13 years) "I can't see that Warwick disproves Horace - it's just his

word against Horace's." Greg (12 years) "They can both be right." (iv) The proof analysis required to answer (3) was rarely achieved and

success almost always required prompting. Of 83 pupils who reached this stage

of the question 38 managed to expose the structure of Horace's proof (into two separate statements) while 45 either failed to do so or failed to see the need to. Of those few who achieved success without prompting Toni (14 years)

gave the following analysis. "Each triangle has 180 ~ is right." "Every triangle is wrong because of Warwick." Some pupils tried to evade the issue entirely. Fay (15 years) "Warwick's example seems more at risk than Horace's

proof." There was almost no evidence of attempts to draw different shaped (e.g.,

concave) quadrilaterals to check out Horace's claim about every quadrilateral. Warwick having been rejected the given diagram sufficed to satisfy such pupils that it represented a generality.

(v) The concept of a counter-example was pursued also in this item. The proportion of pupils accepting one as sufficient was virtually the same as for Sevens, but there was inconsistency between the responses of several pupils within the two questions. Glenda (15 years) typified this uncertainty, "Some- times one is enough and sometimes it isn't."

Consequently the number of pupils having problems accepting the signifi- cance of counter examples almost certainly exceeds the number identified from a particular context. As with the previous item those pupils with doubts about the status of a counter-example expressed themselves clearly.

Peter (14 years) "Need maybe 5 to be definite - with 3 you might be

lucky." Denise (15 years) "One counter-example makes it neither right nor wrong -

you need more."

20 P .L . GALBRAITH

Jenny (14 years) "Even though Warwick has drawn a real quadrilateral one

example is definitely not enough- Horace is more right (90%) than wrong."

Robin (13 years) "If you had maybe 10 examples against Horace?"

One final comment should be made in regard to Quadrilaterals. In both of

the earlier items the actual mathematical objects being manipulated were familiar numbers. This choice was made to minimize the likelihood that con-

cern about content would interfere with the pupils capacity to use processes

they perceived as relevant. Since all pupils were familiar with the numbers

chosen they all began from a comparable standing point. With regard to the

present item many pupils had had experience (to varying degrees) with quadri-

laterals in course work. This could be raised as an objection to the item. Such a position could reasonably be sustained if the intention was to rank order the

pupils on the results from the items. Since the purpose was in fact quite dis- tinct from this, viz. to map the different process characteristics displayed by pupils in various situations Quadrilaterals provided a particularly useful con-

text. This context enabled the effect on pupils of an assimilation/accommoda- tion decision to be observed, Skemp (1971).

As the responses showed, those with entrenched schemas in relation to quadrilaterals often reacted vigorously (even emotionally) against the perceived threat of Warwick's example. On the other hand some perfect responses were

obtained from 12 year old pupils who had not received firmly entrenched prior

ideas. Of special interest was a perfect response from a 14 year old pupil

classed as very poor at mathematics by his school.

5. DISCUSSION

The patterns of responses will now be related to the categories of processing

activity previously listed. The following table shows how these categories A to H were distributed among the items.

Category Clinical item

A B C D E F G H

Game of 25, Sevens Game of 25 Game of 25, Quadrilaterals Sevens, Quadrilaterals Sevens, Quadrilaterals Sevens Quadrilaterals Quadrilaterals

A S P E C T S O F P R O V I N G 21

A. Variety and Completeness in Checking

Two main clusters of response types were identifiable. A majority of pupils did conduct a complete check of cases in part (1) of Game of 25 and Sevens. How- ever a significant minority thought otherwise and were satisfied with a limited check. This observation is consistent with the findings of Bell (1976) for pupils who adopt empirical approaches.

B. Identification and Use of Principle

Three essential components of successful functioning were identifiable from the clusters of response patterns.

(a) Recognition that a principle exists. (b) Identification of the principle.

(c) Application of the principle. Pupils failing to achieve (a) were typified on the one hand by Millie (see

earlier discussion, p. 13) who indulged in wild guessing. On the other hand Jane (14 years) achieved a proof that 18 was a winning number by using a full

check and then proceeded to use the same technique when prompted to con- sider 11. She did not think any winning numbers existed below 11.

Pupils who achieved (a) but failed at (b) are exemplified by Herbie, Noel and Joanna (see earlier discussion, p. 13). A feature of the latter two (where a wrong principle was identified) was the high level of fixedness shown. In those cases very strong prompting was required to prise the pupils away from their

own deductions to consider numbers (such as 11) which refuted their claims. Some such as Noel continued to revert to their perceived principle in the face of contradictory data. As already observed a substantial number of pupils achieved both (a) and (b) but only partial success with (c). This group typically applied the principle to obtain 11 but did not feel secure in pursuing further. An explanation for this reticence is suggested in the discussion of the next category.

C. Chaining o f Inferences

Factors emerged from the Game of 25 which seemed to affect the ability of pupils to follow (or create) a chain of inference to a logical conclusion. Typically the problem occurred when the principle was being applied to fred 4, or when having found the chain 4 to 11, 11 to 18, 18 to 25 uncertainty existed about its implications for success. The following extract from an interview by the author exemplifies the problem.

22 P. L. G A L B R A I T H

Interviewer: Would you like then to go over the reason again, why you've chosen 11?

Terry (14 years): Well, you know 18 is going to win. So from 18 to 25 is 7, from that is 11. Then if he says say 5, on to 11, that equals 16. You just add

any number to make 18, which is 2. Then you've won, the same as that one.

I. Do you think there is another number as well as 11, between 1 and 25,

from which you can always win?

Now what have you decided?

T. Probably 7 from 11 would be the same.

I. Probably?

T. Probably, yes.

I. Just probably? T. Yes. I 'm not sure, I don ' t really know.

I. Are you sure about 18 to 25? You're sure i f y o u get to 18 first, you can

be first to 25. Sure of that?

T. Yes. I. And you're sure about 11 to 187

T. Yes. I. But you think only 'probably' from 4 to 11?

T. Yes. I. What interests me is why have you said 'probably ' when you were talking

about 4 to 11 and you said 'certainly' when you were talking about 18 to

25 and 11 to 18. T. It 's further to go through, to 25.

On further reflection Terry decided that the length of the chain shouldn't

make any difference. However his uncertainty was manifested again in his

answer to (3) in which he didn't see that the player going first could win. Other pupils were not prepared to concede even 'probably 4' without prompt-

ing. The difficulty appears to be in accepting that given an agreed logical basis (in this case the principle) a sequence of steps will lead to a certain conclusion

without the necessity of achieving separate closure of each intermediate step. The level of successful thinking is thus identifiable as formal operational. As previously indicated a number of pupils introduced their own concrete, re-

ferents in the form of intermediate numbers and with this assurance achieved success. It should be noted that the content of the question made this a poss-

ible strategy. Deductive chains are normally composed of logical statements with no obvious concrete referents so that uncertainties will not generally be

resolvable in that way. One group of pupils appeared at first sight to have complete mastery of the

problem yet further probing revealed that this was not so. Typically such

ASPECTS OF PROVING 23

pupils answered (1) by using the compensation argument and then identified

11 and 4 as winning numbers. However this was followed by uncertainty about

the status of the chain 4-11-18-25 (it was apparently non-commutative).

Successful application of the generalisation was thwarted because while its

form was observed its meaning was not understood. A link seems worthwhile

with the notion of levels of understanding as discussed for example in Skemp

(1976). A first reaction of the interviewer was to assume that pupils who derived

11 and 4 without help had a relational understanding of the structure. An

initial classroom reaction would be to leave them and move on to help else- where. Evidently such action would not be justified, which serves as a caution that verbalised generalisations may be instrumentally perceived while disguised beneath a veneer of relational clothing. Full understanding in the present con-

text involved both a recognition of form and a conviction (achieved for ex- ample by checking) that numbers 7 apart can be irrevocably and reversibly

linked. A teaching question which is raised by the present discussion is the status that pupils attach to a given chain of inferences. This is a teaching mode

frequently employed in both group and individual instruction and its success

requires that pupil and teacher share a common view of its logical power.

D. Domain o f Validity o f a Generalisation

In working from deffmitions or exploring the boundaries of generalisations it is

essential to construct the full spectrum of situations implied by the context.

The more sophisticated the context the greater the demand on the individual to unpack all the individual situations.

In Sevens while every pupil responded correctly to numbers like 49 and

383 (as defining the list) less than 10% recognized that the given list L con-

tained two classes of number and set out to relate this information to Brenda's claim. There was thus an almost uniform failure to define the domain of the

conjecture. In quadrilaterals no exploration was conducted to fred the variety of shapes consistent with the definition for the purpose of evaluating Horace's

statement, e.g., those who rejected Warwick might have been expected to con-

sider other configurations consistent with the definition (e.g., re-entrant

quadrilaterals) besides the one represented by the given diagram. This did not

occur and the diagram itself was seen as defining a generality rather than as a representative of a particular class. Perhaps teaching approaches which produce a particular diagram from a general statement without any mention of other possibilities contribute to this problem.

Also included within the compass of this category is the method of disproof

24 P. L. GALBRAITH

by counter-example. Two aspects in the use of counter-examples emerged as

important from the studies.

(a) Significance of a counter-example. (b) Mechanism of refuting by counter-example.

It has been previously shown that a consistent minority of pupils did not accept the meaning which the world of mathematics ascribes to the presence

of a counter-example. Further it was indicated that while some of these pupils exhibited stability in their views for others the significance of counter- examples was contextual. There are implications here for explanations and

proofs which proceed on the assumption of congruence between the views of

teacher and pupil in regard to the significance of counter-examples. The mechanism of refutation, viz. find an example which satisfies the

conditions of a proposition but violates the conclusion was unclear to many. In

Sevens confusion reigned among those who sought to interchange the roles of

L and Brenda's list. In quadrilaterals the manifestation was different - here

there was a refusal to admit that the conditions for a counter-example test

had been met by Warwick's figure. A further illustration of difficulties in this

area was obtained from the responses to one of the items (DIVISORS) used in

the original program Galbraith (1979).

E. Literal lnterpretation of Statements and Conditions

The responses contained information on this attribute at two levels.

(a) Local: Many pupils were not careful to adhere to given data nor to give proper attention the meanings of words. For example in Sevens virtually

all pupils failed to compare the two given statements' carefully enough so that the significance of 'every' was missed. Similarly in Quadrilaterals the given definition was often referred to as an after-thought, and statements about the need for Warwick to include 6 angles proceeded from a lack of attention to

the precise detail given. (b) Global: This major malfunction involved an alteration of whole

contexts. This was featured on a number of occasions by pupils who made no progress at all with an item. An example was afforded b) Millie (12 years) who in Game of 25 both subtracted numbers and went beyond the stated limit of 25. In Sevens after responding correctly to initial examples she applied her own rule and later changed the conditions in Brenda's statement. In Quadrilaterals her first attempt consisted of 4 separate dots. Millie's consistent context altering is obviously a debilitating fault - other pupils indulged in it as well

although not so obviously to the same extent.

ASPECTS OF PROVING 25

F. Distinction between Implieation and Equivalence

Problems associated with this category were specifically pursued in Sevens although it could be argued that Quadrilaterals also provided an implicit context. The major fault in Sevens has already been considered, viz. the incapacity of pupils to fully explore the ramifications of Brenda's claim. Further probing unveiled psycho-emotional viewpoints which interfered with processing, e.g., indignant resistance to the suggestion that Brenda was inadequate.

It was not possible to tell whether the ascribing of statements to named people created a human context which invited emotional involvement but this could have been an additional factor. As discussed in the previous section of this paper there were two discernible malfunctions which regularly appeared among the responses.

(a) The tendency to exhibit centration upon only part of the statement followed by an evaluation of the complete statement from this restricted basis.

(b) The inability to perceive that conditional statements must be judged according to the discipline of their own claim, and not from the viewpoint of a sympathetic and subjective observer.

J

G. Meaning o f Definition

To succeed with Quadrilaterals it was necessary to accept the meaning of definition as it applies to mathematics (viz. arbitrariness) and to perceive the ramifications of the particular definition in context. The responses showed the effect of prior conditioning on the acceptance of a definition, which for

many pupils had connotations of infallibility rather than arbitrariness. Thus pupils such as Glenda (15 years) and Jeremy (14 years) ignored the given definition until completion of their (convex) figures and then consulted it only for the purpose of labelling. These able pupils both rejected Warwick's

diagram and strenuously resisted attempts to ~get them to review their judg- ment. Clearly for such pupils a quadrilateral was a closed figure of a certain

form and this property was sacrosanct. Their reaction is an illustration of

'monster barring' as described by Lakatos (1976). They have a theorem (in fact Horace's) which must be defended by outlawing potential threats as

monstrous distortions of the concept of quadrilateral. Here also a psycho- emotional factor is involved in that success usually required the restructuring of a firmly entrenched schema. (It should be noted that perfect answers to Quadrilaterals were given by young pupils who had had no thorough exposure to the concept.) From a teaching viewpoint it seems important to realise that

26 P. L. G A L B R A I T H

in using the word definition we are (for many pupils) hallowing a concept with a once and for all universal meaning which has no arbitrariness associated with it whatsoever.

H. Proof Structure

The capacity to analyse and evaluate proofs was only lightly tested. In Quadri- laterals it was necessary for pupils to identify the separate component state- ments in Horace's 'proof' and then to evaluate the statements. The first was achieved reasonably successfully by those pupils who retained an open mind. However the evaluation of the statements was not successful on the part of

most of the pupils who supported Horace. Most took the given diagram as fully representative, without any investigation of Horace's claim that every quadrilateral could be divided into triangles. A very few stated that a diagram

had to be representative but almost none applied this knowledge to investigate the class of re-entrant quadrilaterals not represented by Horace's figure. This linked consistently with the similar failure to explore the details of Brenda's claim in Sevens. There are obvious inter-relations between this category and D.

6. SUMMARY

Essential components of successful processing were identified from clusters of response patterns and discussed in the previous section. They are summarized below for convenience.

A. Variety~Completeness in Checking (a) Variety in choice of special cases. (b) Thoroughness of checks. (c) Avoidance of conjectures on insufficient evidence.

B. Proof/Explanation related to an External Principle (a) Recognition that a Principle is present. (b) Identification of the Principle. (c) Application of the Principle.

C. Linking of Inferences (a) Identification of chains. (b) Acceptance of Lack of Closure within chains.

D. Domain of Validity o f Generalisations (a) Need for system in generating/examining special cases implied by

definitions and statements.

ASPECTS OF PROVING 27

(b) Significance of a counter-example.

(c) Mechanism of refutation by counter-example.

E. Literal Interpretation of Data (a) Local interpretation of statements. (b) Global conservation of contexts.

F. Evaluating Statements/Distinguishing Implication and Equivalence (a) Avoidance of Centration. (b) Separation of conditions and conclusions. (c) Awareness of distinction between conjecture and defined knowledge.

G. Meaning of Definitions (a) Properties of definitions. (b) Awareness of need to restructure basic schemas.

H. Proof Structure (a) Analysis of a proof into components. (b) Evaluation of the components.

The central purpose of this paper has been to report on the perceptions pupils have with regard to, and the uses they make of process skills which have a publicly accepted meaning within the world of mathematics. As postulated within the Conceptual Framework successful learning in mathematics depends upon agreement between teacher and pupil with respect to the power and purpose of these process skills. It has been claimed that the demand made upon pupils with regard to the acceptance and use of process skills in different situ- ations requires level two appreciation in the van Hiele sense. The response types identified in the present work suggest that in general pupils do not have this objective and detached view, but rather exhibit restricted and on occasions psycho-emotional approaches to the use of process skills.

Rarely is it that objects of mathematical processing receive the treatment suggested as necessary for the development of high level understanding. Such a treatment would require many contexts across which key attributes of pro- cesses could be identified and stabilized for pupils - a process oriented curricu- lum. Finally against the background of the foregoing discussion the occurrence of conflicting views in the literature e.g. as described by Lester is unsurprising. All investigations using particular examples or contexts provide by their nature at most level 1 experiences. The likelihood of consistent generalisations from such studies must be regarded as small.

University of Queensland

28 P. L. G A L B R A I T H

R E F E R E N C E S

Bell, A.W.: 1976. 'A study of pupils proof-explanations in mathematical situations, ~ Educational Studies in Mathematics 7, 23-40.

Bell, A. W.: 1979, q'he learning of process aspects of mathematics ~, Educational Studies in Mathematics 10, 361-387.

Collis, K. F.: 1975, A Study of Concrete and Formal Operations in SchooIMathematics, Australian Council for Educational Research, Melbourne.

Galbraith, P. L.: 1979, Pupils Proving." Test Items and Case Studies, Shell Centre for Mathematical Education, University of Nottingham.

Hazlitt, V.: 1930, 'Childrens thinking', British Journal o f Psychology, 20, 354-361. Hill, S. A.: 1968, A Study of Logical Abilities of Children, Unpublished doctoral dis-

sertation Stanford University. Kantowski, M. G. : 1977, 'Processes involved in mathematical problem solving', Journal for

Research in Mathematics Education 8, 163-180. Lakatos, I.: 1976, Proofs and Refutations, Cambridge University Press, Cambridge. Lester, F. K.: 1975. 'Developmental aspects of childrens ability to understand mathe-

matical proof', Journal for Research in Mathematics Education 6, 15-25. Lovell, K., Mitchell, B. and Everett, I.: 1968. 'An experimental study of the growth of

some logical structures', in Siegel and Hooper (eds.), Logical Thinking in Children, Holt, Rinehart and Winston, New York.

Lovell, K.: 1961, 'A follow-up of study of Inhelder and Piaget, The Growth of Logical Thinking', British Journal of Psychology 52, 143-153.

Piaget, J.: 1928, Judgement and Reasoning in the Child, Routledge & Kegan Paul, London. Skemp, R. R.: 1971, The Psychology of Learning Mathematics, Pelican, London. Skemp, R. R.: 1976, 'Relational understanding and instrumental understanding', Mathe-

matics Teaching 77, 20-26. Suppes, P.: 1966, 'Mathematical concept formation in children', American Psychologist

21,139-150. Thorn, R.: 1973, 'Modern mathematics, does it exist', in Howson, A. G. (ed.), Develop-

ments in Mathematical Education, Cambridge University Press, Cambridge. Van Dormolen, J.: 1977, 'Learning to understand what giving a proof really means',

Educational Studies in Mathematics 8, 27-34. Van Hiele, 1973, 'Begrip en inzicht', Purmerend, Netherlands.