diagnosing students’ difficulties in learning mathematics

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Diagnosing students’ difficulties inlearning mathematicsDavid Tall a & Mohamad Rashidi Razali ba Department of Science Education , University ofWarwick , Coventry, Englandb Department of Mathematics , Universiti TeknologiMalaysia , Locked Bag No. 791, 80990 Johor Bahru,MalaysiaPublished online: 09 Jul 2006.

To cite this article: David Tall & Mohamad Rashidi Razali (1993) Diagnosing students’difficulties in learning mathematics, International Journal of Mathematical Education inScience and Technology, 24:2, 209-222, DOI: 10.1080/0020739930240206

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INT. J. MATH. EDUC. SCI. TECHNOL., 1993, VOL. 24, NO. 2, 209-222

Diagnosing students' difficulties in learning mathematics

by DAVID TALL

Department of Science Education, University of Warwick, Coventry, England

MOHAMAD RASHIDI RAZALI

Department of Mathematics, Universiti Teknologi Malaysia, Locked Bag No. 791,80990 Johor Bahru, Malaysia

(Received 17 September 1991)

This study considers the results of a diagnostic test of student difficulty andcontrasts the difference in performance between the lower attaining quartile andthe higher quartile. It illustrates a difference in qualitative thinking betweenthose who succeed and those who fail in mathematics, illustrating a theory thatthose who fail are performing a more difficult type of mathematics (coordinatingprocedures) than those who succeed (manipulating concepts). Students who haveto coordinate or reverse processes in time will encounter far greater difficulty thanthose who can manipulate symbols in a flexible way. The consequences of such adichotomy and implications for remediation are then considered.

1. IntroductionThe aim of mathematical education is surely success for all pupils, yet it seems to

be a fact of life that whilst a few prosper in mathematics, a much greater number findmathematics difficult. Thus it is that, however successful a course may appear to be,there are students who begin to struggle and who will need appropriate help to beable to pursue mathematics further.

In this paper we will discuss a theory which indicates that there is a continualdivergence in performance between those who succeed and those who fail which isexacerbated by qualitative differences in their thinking processes. Gray and Tall [1]give evidence of younger children performing arithmetic which shows that whentaught arithmetic procedures, the less able seek security in carrying out the processwhilst the more able soon develop a more flexible kind of thinking in which notationis used to mean either a process to give a result or a concept to be manipulated at ahigher level.

Here we are concerned with the performance of students transferring to theUniversiti Teknologi Malaysia at 16 + who find difficulty with the mathematics thatis pre-requisite for courses in engineering and other sciences. The studentsattending this university are in the 50 th to 90 th percentile of the ability range for thewhole population and therefore represent a broader ability band than that normallyattending university in many countries. It therefore increases the perceived gap inability between the most and least able and, if there is a qualitative difference inthinking processes to be observed, then it is likely to be seen here.

A research group called the Diagnostic and Remedial Mathematics Group wasformed at the university in September 1987 with the primary task of identifying thedifficulties encountered by students and then to use this information to developremedial procedures to overcome these difficulties.

0020-739X/93 $10-00 © 1993 Taylor & Francis Ltd.

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210 D. Tall and M. R. Razali

At first the group considered various methods of diagnosing difficulties. Forinstance, they considered interview techniques, such as getting the students to thinkaloud [2,3], with students being asked to verbalize every thought that comes to theirmind in the course of solving a mathematics problem. A variant of this technique,developed originally by Jean Piaget, is the clinical interview [4, 5]. This differs from'thinking aloud' in that the teacher may follow the thoughts of his students by askingquestions here and there during the course of solving the particular problem.

Interviews with students, whilst giving subtle indications of the existence ofcognitive difficulties, need to be complemented by broader studies which indicatethe extent of such difficulties in the population under consideration. Therefore theDiagnostic and Remedial Group planned to develop a test paper consisting ofmultiple-choice questions whose design was based on suspected conceptualdifficulties with questions formulated to induce conceptual errors. They were awarethat the errors encountered might come from a number of sources, includingsensory, mental, emotional, motivational, cultural, social, reading or instructionaldeficiencies [6]. They were also aware of difficulties known at school level which haveyet to be properly diagnosed and remediated, as reported by the Malaysian Ministryof Education [7,8].

A preliminary mathematics diagnostic investigation was given to 350 first yearstudents in August 1988, using questions requiring a written response which weredesigned to provoke possible misunderstandings and errors. The first multiple-choice diagnostic test was then designed using evidence from the preliminaryinvestigation to construct problems that may lead to observed errors. Thismathematics diagnostic test was implemented for first year students in October1989. It proved highly successful in discriminating between students of differentcapabilities and has since been partially modified for a test in July 1990 by replacing,or modifying, those questions which were less successful discriminators.

In the next section we will discuss the overall design of the test and follow thiswith a preliminary analysis of the success of the questions as discriminators. Then welook at erroneous responses to some of the individual questions which are indicativeof possible qualitative differences between the thinking of the more and less able. Wego on to postulate possible qualitative differences in thinking processes and considerthe consequences of this analysis for possible remediation. We come to theconclusion that it would be of little use to remediate the problems whilst the studentsare attempting to start a new course and suggest an intermediate foundation course inwhich the less able are given explicit teaching of the qualitative thinking skills whichthe more able develop implicitly.

2. Overall design of the diagnostic testFor the purposes of analysis, two groups of students were identified from those

taking the diploma, where difficulties were more likely to occur. These were those inthe highest 27% (group H) and those in the lowest 27% (group L). For each of the 40questions on the diagnostic test, the proportion of correct responses was calculatedfor groups H and L. The average of these correct responses gives a number (calledthe P index) which is a measure of the probability that a given student will obtain acorrect answer; the difference between the proportions for the groups H and L is anindication of the extent that the question discriminates between the high and lowperformers (called the D index). Questions with a P index of about 1 would be tooeasy, because nearly everyone would get them right, and questions with a P index of

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Diagnosing difficulties in learning mathematics 211

around 0 would be too hard because almost no-one would be able to do them. 'Good'questions would have a P index of around 0-5 and a suitably high D index (indicatinga high discrimination between those of high and low performance).

Overall the October 1989 diagnostic test proved to be an excellent instrument fordiagnosis. Only seven questions out of 40 had a D index ^0 -25. Of these, two had alow P index and were too difficult; the other five had a high P index and were too easy.Thirty-three questions out of 40 were good discriminators. Some of the effort ofredesigning the test was addressed to modifying those which were not such gooddiscriminators, however, the results of the second test were very similar to the firstand so the remainder of this article considers the 1989 version.

3. Analysis of error typesAn overall analysis of errors was performed by considering the wrong choices

made on the multi-choice questions. Two types of scenario were distinguished:

(1) The 'hard' questions which had a high proportion of errors in both the L andH groups;

(2) the 'discriminating' questions where the H group did well and the L groupobtained less than 50% correct responses.

Type (1) was indicative either of a peculiarly difficult question or of a problem whichapplied to almost all students. Type (2) indicated areas where the more able weregenerally successful but the less able were making significant errors. The formerrequires overall remediation and perhaps reconsideration of the earlier teachingmethods, the latter requires more specific remediation for individuals who fail tocope with topics which have been better understood by the more able.

An initial scan of the errors reveals a wide array of difficulties with specificknowledge. However, the nature of the process failures followed a much smallernumber of different patterns. We hypothesise that these patterns are indicative of aqualitative difference in the thinking between more and less able students. Thishypothesized difference is based on the theory developed by Gray and Tall [1],which gives empirical evidence that less able children cling to procedural methods inmathematics whilst the more able develop flexible thinking processes that manipu-late symbolism with equal ease as process or concept. It is essential in both diagnosisand the design of remediation to be aware of these differences.

4. Examples of errorsErrors arising on the diagnostic test suggested underlying differences in the

thinking processes of the students. Here we give a few selective examples from the 40questions on the test.

Order of precedence of arithmetic operations (question 4) (P=0-5, Z) = 0-4)

When faced with the choice:A. 22

2 x 6 - 4 + 40 -=-2 =

B. 24

C. 28

D. 42

E. other

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a majority (61%) of group L responded by asserting that 2 x 6 — 4 + 40 -f- 2 = 24,which indicates reading left to right with no regard to the precedence ofmultiplication. This is the way that the calculation is carried out on some calculators.It indicates a treatment of the arithmetic symbols as a sequence of operations to becarried out in the order that they are read. Meanwhile a majority of group H scannedthe symbols and gave the answer 28 which indicates that they have understood theorder of precedence of operators. Thus one may hypothesise (in line with the theoryof Gray and Tall [1]) that the less able see the symbols as a process to be carried out inthe written order, whilst the more able can chunk the symbols together as sub-expressions to be carried out according to given conventions.

The meaning o/21og105 + log108-log102 (question 5) (P=0-62,

The problemA. Iog1016

B. logl031

= O51)

21og105 + Iog108—Iog102 = • C. 2

D. 100

E. otherpresents the students again with the problem of processing a group of symbols.Students who see the symbols as representing a process to carry out will have toconceive this as multiplying the logarithm (to base 10) of 5 by 2, then adding thelogarithm of 8 and substracting the logarithm of 2. This is a long sequence ofoperations which is complicated to coordinate, yet if the student can chunk it intosub-expressions and knows the properties of logarithms, then 21og105 may be seen asIog10(5

2) = log1025, and then the three sub-expressions combined as

logl025 + logl08 - l o g l 0 2 = logl0(25 x 8/2) = logl0(100) = 2

Manipulating algebraic equations (question 15) (P=0-66, Z) = 0-51)

Group L had some difficulty with the question

r + kp

If k= , then r in terms of p and k isr-p

A.

B.kp

k-\

C. kr-kp

D. kp\-k

E. other

Algebra is seen as a difficult subject for which the traditional cure when a student failsis lots of practice exercises. However, we would contend that such exercises may onlyserve to prolong the misinterpretation that algebra is a menagerie of disconnectedrules to deal with different contexts ('collect together like terms', 'turn upside downand multiply', 'do the same thing to both sides', 'change side, change sign', etc; etc.).

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Tall and Thomas [9] suggest that there are several cognitive obstacles in the earlylearning of algebra, for instance the fact that an expression is seen as a process to becarried out (2*+ 3 is 'multiply x by 2 and add 3') and not as an object which can bemanipulated. Since the less able are often not willing to accept such an expressionas meaningful (because they cannot work it out until they know the value of x, and, ifthey know the value of x they needn't do algebra, they could do arithmetic), they areuneasy about handling it and sink into the use of (ill-)remembered procedures.

Solve (x-2)(x + 3) = 6x (question 10) (P=0-7, D = 0-48)

An interesting, and not unexpected, difficulty occurs with the following:

A. - 6 , 1

If (x — 2)(a: + 3) = 6a:, then the values of* are-

B. 6, - 1

C. 2, - 3

D. 2, 3

E. other

A likely difficulty with the L group is that they may evoke (incorrectly) part of theprocess of solving equations in that the left-hand side is factorized. This acts as adistractor and may lead to the selection of one of the incorrect solutions. Once morethe less able have problems with the meaning of the symbolism, evoke part of theprocess of solving equations by factorization and, being unable to coordinate theprocedure correctly, fall into error. If students are unable to give appropriatemeaning, then procedures are likely to be carried out by rote and highly prone tofailure. Tall and Thomas [9] have shown how a blend of BASIC programming,computer software to give meaning to algebraic expressions and games to carry outanalogous procedures to those of a computer manipulating variable stores, can leadto more versatile understanding of algebraic notation.

x— 1 1The singular case of simplifying — (question 12) (P=0-62, D = 0-43)

Only 6% of group L could give the correct answer (B) to the following problem:

Simplify 2-x x-2'A. -ix

B.2-x

c x

U x-2

D. (2-*)(*-2)

E. other

They would have probably done better if the denominators had not been the same upto a minus sign. This is an example of a singular case. That is a case which seems not

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to follow the general pattern: find the lowest common demoninator, which thestudents might be able to do faced with, say, denominators x(x — 1), (x + 3)(x — 1).The 45% of group L giving the distractor (x2 — 4x)l((2—x)(x — 2)) shows the desire tofollow the 'general method'. Such singular cases will always cause problems.(Another is to solve ax = a for x, which has a singular case when a = 0). They requirespecific teaching to show how the singular cases fit in the general pattern. In this caseit requires revision of the process of handling rational expressions and, within thisprocess, to deal with singular cases which may not appear at first sight to fit thepattern.

The modulus sign (question 2) (P=069, D-OAb)

The modulus sign is known to pose difficulties to students [10]. They have difficultyin thinking of \x\ as — x when x is negative, probably because of the difficulty ofconceiving of something starting with a minus sign as being positive. Then they havedifficulty in manipulating expressions involving the modulus sign. Difficulties areindicated by the question:

A. - 3

If x=\ and y = 4, then \x—y\ =

A significant minority in the L group say

\x-y}=-3.

B. iC. 3

D. yryI E. other

The meaning of the symbolism is plainly unclear to these students, and it may bemade worse by the difficulty coordinating two processes, here the action ofsubtraction and the taking of the modulus.

The inverse function (question 7) (P=049, £> = 0-58)

Functions give students great difficulty [11-13]. The inverse function causes evenmore problems, as witnessed by responses to the following:

x— 1A function/ is defined by/: x-* , x#2

x—2

The expression / " 1 in a similar form is

A.

B.

C.

D.

E.

f~1' X

other

1~*2

x-2x-l

2a;-1

1-x

2a;-1x-l

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This is a really difficult concept! Professional mathematicians often misunderstandthe notion of inverse, so we should not be surprised by the difficulties of students!This is part of the difficulty of undoing operations. The powerful idea here is that ofreversing a process. A function y=f(x) is a process: input x, carry out a procedure,output y. The reverse process, what x must be put in to get y out, is much harder. Ingeneral it can't even be done. The function may not be one-one: there may be severaly s which give the same x. Thus the 'inverse function' may not even be a function—unless one artificially restricts its domain. Here are a few typically awful cases. Iff(x) = \x\, what is/"*(*)? If/(*) = INT(x) (the integer part of x), what isf~1(x)?In the case of something \ikef(x) = x5 — 1 Ix7" + Ax2 — 1, even finding / ~ 1(0) is hard,let alone f~l{x) for general x\

The curriculum in school needs revising in the light of computer technology tostudy the general (powerful) idea of reversing a process, first by trial andimprovement. To find x such that/(x) = y for given y, put in various values of x andexperiment to find one such that/(x) is as close to y as is practicable. Then better'guess and test' methods can be introduced (e.g. solving an equation by bisection—currently a possible option in A-level, but now definitely possible for youngerchildren using computer technology). The idea that under certain, very restricted,circumstances one can reverse a symbolic process to find an inverse should be seen as avery special case.

Here the routine is ostensibly much easier by algebraic manipulation. Puty = (x — 1 )l(x — 2) and solve for y in terms of x. This gives yx — 2y = x — 1 so x(y — 1)= 2y— 1, so x = (2y—l)l(y — l). This is f~1(y). Hence (changing variable!),f-Hx) = (2x-\)l(x-\).

However, we have seen earlier that the algebraic manipulation is already hard forthe less able, and coordinating and reversing the procedures in this complicated wayis compounding the difficulty.

Given the circumference: to find the radius (question 11) (P = O55, Z) = 0-46)

Students find difficulties when asked to link visual and arithmetic processes in away which is not in the same order as normally encountered. For instance, inthis diagram, where information is given about the circumference, rather than

.A

2;tcmI i>{ i

lOJtcm

the radius. The two points A and B are on the circle centre O, the arc AB is 2JI cm,and the large arc from A to B is 1 OTI cm. Students are asked to say whether the radiusof the circle is:

A.

B.

C. 5 cm

D. 6 cm

E. other

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It is highly likely that the students know the formula c = 2nr. They may fail in anumber of possible ways, and the equal likelihood of three responses to the solutionindicates a variety of possible causes. First they may fail to properly relate differentrepresentations (graphical and algebraic). They may be unable to reverse the process(to find r in terms of c). Then they may not properly coordinate the processessuccessfully.

Determining the angle at the circle given the length of arcs (question 12) (P=0-62,Z> = 0-43)

Using the same figure as question 11, students are asked to state whether the acuteangle AOB is

A. 30°

B. 36°

C. 60°

D. 72°

E. other

Again problems occur in relating representations and reversing/combiningprocesses.

Scale (question 14) (P=0-44, D = 0-47)

Ratio is a known difficulty, because it relates not to an explicit number, but torelationships between numbers. On the diagnostic test the following question wasgiven:

A map is prepared using a scale 1: 5 000 000. Using this scale, a river whoseactual length is 100 km will be represented by a length

A.

B.

C.

D.

E.

0-2 cm0-5 cm

2 cm

5 cm

other

The wide range with five almost equally favoured responses in group L suggestedserious misunderstandings. Here there are so many different things to coordinate:the use of appropriate mathematical symbolism, relating actual measurements to apractical graphical representation, coordinating the relationship between thelengths.

5. Possible qualitative differences between the thinking processes of themore able and the less able

Having looked at specific examples from the diagnostic test, we might take thesuperficial view that the questions are hard and the less able students are not goodenough to do them. But this is a shallow and unhelpful attitude. We must determinemore succinctly what the problems are. At least we should provide a hypothesis forthe differences in performance of the more able and less able.

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It has been hypothesized that the objects of mathematics learning are facts andskills. The child must learn the facts and will be shown procedures (addition,multiplication, manipulation of fractions, of algebra, methods of solving equations,proofs or geometric theorems, formal calculation of derivatives, formal integration,computation of probabilities, etc.). So often the learner is shown how to do aprocedure (preferably in a meaningful way) and then practises this until it isroutinized and can be carried out (almost) automatically. Often this routine becomescrystallized into a concept. For instance, the process of 'counting on' to calculate asum such as 2 + 3 gives the concept of the sum of two numbers, the process of formaldifferentiation gives the concept of derivative, and so on. It usually happens that thesymbolism may indicate either or both of these, for instance 2 + 3x represents boththe process 'add two to the product of three times x' and also the result of thecalculation, the expression '2 + 3*'. In this way, a dynamic process is crystallized intoa static concept.

It is now more than this. The flexible thinker can switch easily from the process tothe concept and back again in such a fluent way as to conceive hardly any distinctionbetween the two. Prof essional mathematicians become so fluent in this ability that theyoften fail to distinguish explicitly between the two ways of thinking [14,15].

One finally masters an activity so perfectly that the question of how and why(students-don't understand them) is not asked anymore, cannot be askedanymore and is not even understood any more as a meaningful and relevantquestion. (Freudenthal, 1983, p. 469)

The flexible, crystallized concept can be mentally manipulated and used for higherlevel thinking. For instance 2 + 3x can be part of a more complex expression such as(2 + 3x)2 —15X(2 + 3A:). The more able faced with the problem of factorization maychunk this sub-expression as a single entity and see the factorization

(2 + 3x -15*)(2 + 3*) = (2 -12x)(2 - 3*)

The less able may only follow rules... multiply out brackets, collect together liketerms, look for common factors, etc. In this way the more able are manipulatingmental objects whilst the less able are having to coordinate the processes of algebra.Thus the more able succeed because they are performing qualitatively easier tasks,manipulating symbols, whilst the less able are performing harder tasks or coordinat-ing processes.

A major hypothesis that we propose is:

The failure in conceptualization:The more able are better at crystallizing a process into a concept and so maymanipulate the concept with greater fluency than the less able who may stillbe operating at the process level. Processes occur in time and therefore aremore difficult to conceive simultaneously in the mind than concepts.

For instance, the expression

21ogi05 + Iog108 - Iog102

(which appeared in the diagnostic test) can be thought of either as a process or as aconcept. As a process it is an instruction to multiply the logarithm of 5 by 2, to addthe logarithm of 8 and to subtract the logarithm of 2. As a process to be carried out intime it is an arduous task whereby one may easily forget the beginning of the task as

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one approached the end of it. Thus someone working at the process level will find itdifficult to comprehend and hold in the mind for simplification purposes. However,seen as an object, it can be 'chunked' together as sub-objects, so that one mayconcentrate on the part 21og105, which equals Iog1025, and the whole expressionreduces to Iog10(25 x 8H-2) = log10(100) = 2.

Thus the difficulties facing the less able (coordinating mental processes) aregreater than those facing the more able (manipulating mental concepts). A corollaryof the hypothesis of failure in conceptualization is:

The burden of the less able:the more able are faced with conceptually easier mathematics (manipulatingconcepts) and so are progressively more likely to improve, whilst the less ableare faced with a more difficult task (coordinating processes) and so areprogressively more likely to fail.

The difference is made even greater if the more able are manipulating symbols whoserelationships are in some sense meaningful to them. Meanwhile the less able.arelikely to be constricted within a context wherein the symbols have a moreinappropriate process meaning. For instance, (2 + 3)x and 2x + 2x are totallydifferent as processes. The first adds two and three and then multiplies the result by x,the second multiplies two by x, then multiplies three by x and then adds the twotogether. Thus the student with only a process interpretation will find it difficult toconceive that these two symbols represent the same thing. Meanwhile the studentwho sees these expressions as also representing the product of the two calculationswill be able to focus on the fact that the results are the same and thus see theexpressions as being equivalent ways of expressing the same end-result.

The more able are likely to benefit from seeing linkages and making connectionsbetween the crystallized concepts, because this new information reinforces andsimplifies what they already know by placing it within a more secure knowledgestructure. The more able need to remember less because they can reconstruct more. Theless able see more information as a burden, as even more disjoint pieces ofinformation to remember—an increased burden on a weaker back, leading to thegreater probability of inevitable collapse.

The less able have a fundamentally different viewpoint. In addition to having lessfacility with mathematics, their difficulties are likely to manifest themselves invarious ways, including:

(a) being less likely to crystallize processes into manipulable concepts, thusimposing the greater burden of process coordination rather than conceptmanipulation;

{b) having fewer manipulable concepts, preferring to rely more on the security ofcarrying out familiar routinized processes;

(c) through relying on routinized processes, being less likely to relate ideas in ameaningful way.

6. Implications of non-diagnosis of mathematical difficultiesIt is clear from this discussion that the less able students are likely to need special

treatment. If this is not done, the students' mind will be more confused and in thelong run these students will not survive in post-secondary mathematicsprogrammes.

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According to Robert Gagne's theory of learning [16], the four phases of learningare the apprehending phase, the acquisition phase, the storage phase and the retrievalphase. The more able can reach these four phases but for the less able, the last phasemay be a problem. The mind of the less able is like a flawed computer diskette.Sometimes, it will respond well to some mathematics problems (usually the easierones) and it will 'blank out' to more difficult ones.

According to Steen [17] in his survey lecture at the 1988 Sixth InternationalCongress on Mathematical Education in Budapest:

Teachers normally act as if each student's mind is a blank slate—or an emptycomputer disk—on which effective teachers can record whatever informationthey like. Research in cognitive science suggests otherwise: each student'smind is more like a computer program than a computer disk. Each studentbrings to the mathematics classroom a rich set of prior mathematicalexperiences that provide a unique mental framework in which the studentcreates new patterns derived from new experiences. Learning occurs not in theact of remembering, but in the gradual development of mental frameworksunique to each individual. In other words, students learn by modifying theirmind's program, not by storing new data in their mind's memory.

We believe that these analogies are helpful, yet incomplete. We believe that thedifference between the thought processes of the more able and less able lead to agrowing divergence in performance in which the more able can only get compara-tively better and the less able worse as the subject progresses. The more able flexiblymanipulate symbols either as process or concept, whilst the less able tend to be moreconcerned with process and get locked into the strategy of accumulating proceduresto 'give the answer'. Gray and Tall [1] refer to the use of a common symbol to duallyrepresent a process or a concept as a procept. Thus a procept is the amalgam ofprocess and concept represented by the same symbol. It is the ability of the more ableto switch from seeing a symbol such as 2 + 3x or

limsBn-»oo

as either a process of the product of that process which gives them the flexibility ofthought to make them effective mathematicians. This proceptual means of thought,we claim, plays a fundamental role in causing the difference in thought between themore and less able. Whilst the more able develop a flexible proceptual method ofoperation, the less able are locked into an inadequate procedural method of thoughtwhich relies on coordinating an ever increasing array of disparate procedures.

7. Proposals for remediationIn the writing of this paper we have used the physician's metaphor of diagnosis

(of weakness) and of prescription (of a cure). It is not always the most apt ofmetaphors. We do not believe that it is simply a case of giving the weak students themedicine which will cure them and let them re-enter the race. Instead we would seethe way ahead through using our analysis of difficulty on the one hand, and themethod of success of the more able on the other, to help the less able develop suitableexplicit strategies that will help them be more successful.

First the less able will clearly lack confidence. They will only gain confidencethrough success, so they must be helped to attain success. Simply giving them a

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practice at the procedures which are faulty will, according to our theory, not be asolution to their problems. For their difficulty is precisely that they do not crystallizethese procedures into manipulable mental objects that allows them to coordinatethem to solve more complex problems.

Given the hypothesis of failure of conceptualization, less able students are morelikely to operate by coordinating processes than by manipulating concepts. Theyalready suffer from having too many separate pieces of knowledge which overbur-dens their memory and leads to failure. Remediation through correction of specificerrors is likely to give them even more isolated pieces of information to remember,causing even greater cognitive strain and increasing the probability of failure.Furthermore, if the remedial students do not develop the more versatile form ofthinking available to the more able students, then they will be ill-equipped to copewith the mathematical courses which follow and continue to suffer cognitive strain.

This suggests that such students need to focus not just on the many and variedspecific errors which they have made, but on the overall powerful strategies used bythe more able students to succeed. Given that such remedial students have alreadysuffered loss of esteem from their failure compared with the success of their moreable peers, they need a positive approach which emphasizes a small number ofpowerful general strategies to give more versatile thinking and greater success.

This causes us to focus on the problem of such students and whether it is possiblethat they can cope with the changes that the more versatile kind of thinking requires.It raises a serious and fundamental problem which requires empirical research:

(Q) Are remedial students capable of benefiting from teaching which focuses onhigher level learning strategies in addition to specific mathematical content?

We would suggest that the answer to this question may be in the affirmative, but onlyif it is done in such a way that the cognitive strain is not increased by trying to teachsuch concepts and higher level strategies simultaneously. One possible solution is toteach problem-solving strategies in a separate course from those courses teachingmathematical content and procedures.

We hypothesize that several ingredients are necessary to help remedial studentsconsciously come to terms with the nature of higher level mathematical theory,including:

(1) Revision of procedures to give success and a sound basis for crystallization ofthese procedures as manipulable mental concepts.

(2) Reflection on these procedures to give them a foundational meaning thatgives the students greater confidence in handling them and begin toencapsulate them as concepts.

(3) Explicit extensions of the procedures to see how they should be coordinated,reversed, and mentally manipulated as 'chunks' of mathematical code in amore proceptual manner.

(3) Explicit experience and teaching of mathematical thinking processes whichgives them confidence in clarifying, formulating and solving mathematicalproblems.

It should be clear that such remediation cannot be done at the same time as startingthe main course, burdening an already overburdened mind with yet more to do. Onepossibility is to adopt remedial strategies in a constructive confidence-buildingmanner in a foundational course prior to taking further study. Such a course must

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contain some revision (item (1)) but it must use the experience and encouragematuration of the thinking processes through items (2), (3) and (4). In particular, themathematical thinking (4) can be a course which concentrates on mathematicalprocesses to give experience, success and enjoyment of mathematics, to teach thestudents to reflect on how they do mathematics without the expectation of teaching anynew content.

It is envisaged that items (1), (2) and (3) may be treated in sequence within anumber of different courses, beginning with revision of elementary procedures,wherever possible giving the student the experience which may hope to give newmeaning to the concepts being manipulated. For example, algebraic notation and theconventions of arithmetic may be given a more appropriate notation by using a littleprogramming using letters as variables [9]. In this way an initial meaning can begiven to the symbolism which enables it to be manipulated in a coherent way.

Initial experiences might therefore concentrate on:

(a) interpreting information and translating it into an appropriate symbolism;(b) understanding the conventions of mathematics and the choice of symbolism

in different contexts;(c) being able both to give initial meaning to symbols and also eventually to

manipulate them in a routinized and efficient manner.

Once procedures have become routinized, attention can be turned to the morecomplex actions that cause remedial students difficulties:

(d) coordinating two or more processes;(e) reversing or undoing a process;(/) coping with singular cases (exceptional ones which seem not to follow the

general pattern and so require special treatment).

Attention needs to be focused on the important idea that so many symbols, e.g. 2 + 3,3x + 4y, —3, sine = opposite/hypotenuse, f(x), limn_0Osn, etc., represent both aprocess of calculation and the result of that calculation:

ig) focusing on the fact that same symbolism represents both the process to get aresult and the result itself;

(h) crystallizing processes into concepts (by the simple process of meaningfullyusing the same notation for both).

Knowing that the less able are more likely to want to operate procedurally, the task offocusing on coordinating procedures and manipulating them as concepts needs to bedone in contexts where the more powerful methods of operation clearly lead to easierways of doing the mathematics. Students need to also be aware of:

(i) using different representations of the same concept (e.g. graphical, numer-ical, symbolic);

(/) transferring between different representations, to use whichever is easier fora given task.

It cannot be expected that the less able will, by some magical process, be brought to aposition where they operate indistinguishably from the more able, desirable thoughthis may be. But cognisant of the different qualities of thinking processes betweenthose who succeed and those who fail, we hypothesize that a plausible way in whichstudents may become more successful is to become consciously aware of more

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222 Diagnosing difficulties in learning mathematics

successful thinking strategies and that this must be done in a context designed toimpose less cognitive strain. The implementation of such a remediation strategy isnow the subject of future research.

AcknowledgmentsWe would like to acknowledge the cooperation and the hard work done by our

colleagues of the Diagnostic and Remedial Mathematics Group, UniversitiTeknologi Malaysia for implementing and devising the diagnostic tests in this paper.

References[1] GRAY, E., and TALL, D. O., 1991, Proc. PME14, Assissi, 2, 72-79.[2] NEWELL, A., and SIMON, H. A., 1972, Human Problem Solving (New Jersey: Prentice

Hall).[3] BARTLETT, F. C., 19S4, Thinking (London: George Allen & Unwin).[4] EKENSTAM, A. A., and GREGER, K., 1983, Educ. Studies Math., 14, 369.[5] GREER, B., 1987, J. Res. Math. Educ, 18, 37.[6] BELL, F. H., 1978, Teaching and Learning Mathematics (in Secondary Schools)

(Malaysia: Wm. C. Brown).[7] Laporan Jawatankuasa Kabinet Mengkaji Perlaksanaan Dasar Pelajaran, 1979, pp. 18-

20 (in Malay).[8] Laporan Jawatankuasa Mengkaji Taraf Pelajaran di Sekolah-Sekolah, 1982, p. 18 (in

Malay).[9] TALL, D. O., and THOMAS, M. O. J., 1991, Educ. Studies Math., 22, 125-147.

[10] VINNER, S. (to appear), in Advanced Mathematical Thinking, edited by D. Tall(Dordrecht: Kluwer), 65-81.

[11] MARKOVITS, Z., EYLON, B., and BRUCKHEIMER, M., 1988, The Ideas of Algebra, K-12,N.C.T.M. 1988 Yearbook, 43.

[12] VINNER, S., 1983, Int. J. Math. Educ. Sci. Technol., 14, 293.[13] TALL, D. O., and BAKAR, M. N., 1992, Int. J. Math. Educ. Sci. Technol., 23, 39-50.[14] FREUDENTHAL, H., 1983, The Didactical Phenomenology of Mathematics Structures

(Dordrecht, Holland: Reidel).[15] THURSTON, W. P., 1990, Notices Am. Math. Soc, 37, 844.[16] GAGNÉ, R. M., 1970, The Conditions of Learning, 2nd edition (New York: Holt Rinehart

& Winston).[17] STEEN, L. A., 1989, The Australian Math. Teacher, 45, 19.

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diagnosing students’ difficulties in learning mathematics

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