Spatial Ability, Visual Imagery, and Mathematical Performance
Author(s): Glen Lean and M. A. (Ken) Clements
Source: Educational Studies in Mathematics, Vol. 12, No. 3 (Aug., 1981), pp. 267-299
Published by: Springer
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GLEN LEAN AND M.A.(KEN) CLEMENTS
SPATIAL
ABILITY,
VISUAL
MATHEMATICAL
IMAGERY,
AND
PERFORMANCE
ABSTRACT. 116 Foundation Year Engineering Students, at the University of Technology,
Lae, Papua New Guinea, were given a battery of mathematical and spatial tests; in addition,
their preferred modes of processing mathematical information were determined by means
of an instrument recently developed in Australia by Suwarsono.
Correlational analysis revealed that students who preferred to process mathematical
information by verbal-logical means tended to outperform more visual students on mathematical tests. Multiple regression and factor analyses pointed to the existence of a distinct
cognitive trait associated with the processing of mathematical information. Also, spatiil
ability and knowledge of spatial conventions had less influence on mathematical performance than could have been expected from recent relevant literature.
1. INTRODUCTION
In a letter to JacquesHadamard,Albert Einsteinstated that he alwaysthought
about anything in termsof mental picturesand that he used wordsin a secondary capacity only (see Einstein's letter in Hadamard,1954). In the field of
mathematics, some mathematicianshave claimed that all mathematicaltasks
require spatial thinking (see Fennema, 1979). Indeed, as early as 1935 H. R.
Hamley, an Australianmathematician and psychologist, wrote that mathematical ability is a compound of generalintelligence, visualimagery,and ability
to perceive numberand space configurationsand to retainsuch configurations
as mental pictures (McGee, 1979). Given statements such as these, it is not
surprisingthat there is a substantialliteraturein which relationshipsbetween
spatial ability, mental imagery, and mathematical performance have been
investigated(Bishop, 1973, 1979; Fennema, 1974, 1979; Guay and McDaniel,
1977; Lin, 1979; Sherman,1979; Smith, 1964). The presentpaperis a contribution to that literature.
It will be useful to begin by commenting on how we shall use the terms
'spatial ability', 'mental imagery', and 'mathematics'(it being recognizedthat
no agreementon the definitions of each of these terms is evidentin the literature). By 'spatialability' we shall mean the ability to formulatementalimages
and to manipulatethese images in the mind (see McGee 1979, for a reviewof
definitions of spatial factors; see also Guay, McDanieland Angelo, 1978). By
'imagery' we shall mean, following Hebb (1972), 'the occurrence of mental
activity correspondingto the perception of an object, but when the object is
Educational Studies in Mathematics 12 (1981) 267-299. 0013-1954/81/0123-0267$03.30
Copyright ? 1981 by D. Reidel Publishing Co., Dordrecht, Holland and Boston, U.S.A.
268
GLEN
LEAN
AND M.A. (KEN) CLEMENTS
not presentedto the sense organ'and by 'visualimagery'we shall meanimagery
which occurs as a picture in 'the mind's eye'. (See Pylyshyn (1973), Kosslyn
(1979), and Evans (1980), for discussions of difficulties associated with the
notion of visual imagery.) By 'mathematics'we shall mean the course content,
teaching and learningassociated with the subject 'mathematics',as it is studied
in schools and tertiaryinstitutions throughout the world.
In addition to this introduction the present paper contains six sections. In
the first a list of mathematicaltopics in which spatial ability and visualimagery
are needed is provided.The next section reviewsthe literatureconcernedwith
the relationshipbetween mathematicalperformance, spatial ability, and visual
imagery. Then follows a description of an investigationwhich we carriedout
with first-year engineering students at the University of Technology, Lae,
Papua New Guinea, and an analysis of the data which were obtained. The
implications of the analysis for mathematical education, and, in particular,for
mathematicseducation in Papua New Guinea, are then discussed,and, finally,
a summaryof the main points arisingfrom the investigationis given.
2. SOME ILLUSTRATIVE
EXAMPLES
In view of the fact that there is very little direct discussion in the existing
literature on why spatial ability, mental imagery, and mathematicalperformance might be expected to be related, the following examplesare offered.
Considera student who is asked to find all values of x for which sin 3x > 4
and 0 x < 27r.The student's first reaction might be to think: 'The graph of
y = sin 3x is the graphof y = sin x "squashed"together, so that it has a period
of 27r/3 not 2rr;to find x so that sin 3x > I've got to superimposethe line
y = i on the graph of sin 3x and then find the values of x correspondingto
those parts of the graph above y = 4. I'll probably need to solve equation
sin 3x = 4'. Following this line of thought the student might then sketch the
graphsof y = sin 3x and y = 1 on the same axes, and proceed with his planned
solution. Note that although the question made no mention of graphs the
student thought in terms of them, and made considerableuse of visualimagery
and spatial ability in planninghis solution procedureand in deducingthe graph
of y = sin 3x from that of y = sin x.
While it is true that some mathematicaltopics requiregreateruse of spatial
ability than others, the following list of topics, compiled from seniorsecondary
and lower tertiary coursesin PapuaNew Guinea and Australia,drawsattention
to the difficulties which students with poorly developed spatialabilitiesmight
experience in their mathematicsprograms.
1. Sketch graphs.
SPATIAL
E.g.
ABILITY,
IMAGERY
y = -x3
+x
y = 1-2
sin 3 x +
AND MATHEMATICS
269
y = sin-'x (reflected y = sin x in the line y = x).
2. Conic sections. E.g. focus-directrix definitions, normalsto curves.
3. Interpreting or drawing two-dimensional representations of threedimensional situations. E.g. the angle between two planes, geometry of the
earth, engineeringdrawing.
4. Linearprogramming.
5. Geometricaltransformations(translations,reflections,rotations,dilations,
expansions).
6. The Calculus.E.g. concept of a limit, areas under curves, solids of revolution.
7. Probability.E.g. the normal curve (Find Pr (z <- 1) given tables which
list values for Pr (0 < z < a), a > 0).
8. Circularfunction. E.g. find sin 4 using a unit circle.
9. Complex numbers.E.g. write 1 - i in complex polar form.
10. Mechanics.E.g. drawingforce diagrams.
There are many other topics in senior secondary mathematicsand in primary
and junior secondarymathematicswhich, depending on the individual,might
involve the use of spatialabilities.
Equally important, but less obvious, is the fact that many children use
visual imagery when thinking about topics which do not appear to require
visual thinking (see Krutetskii,1976, pp. 158-159). Thus, for example, when
confronted with the problem of findingthe value of 3 -7, a junior secondary
pupil might, as a result of instruction, imagine someone walkingforwardsand
backwardsalong a numberline (see Figure 1).
walk
forward
turn
walk
around and
backwards
Fig. 1.
You begin at the origin 0; the '3' instructs you to face towards the right and
walk three units; the subtraction operator tells you to turn aroundand face
to the left; the '7' tells you to walk backwardsseven units. By this kind of
thinking, and not necessarilywith the aid of a diagram, a junior secondary
GLEN LEAN AND M.A. (KEN) CLEMENTS
270
pupil may determine that 3 -7 =+10. Similarly, many childrenconfronted
with the problemof finding the time three hours before 2.15 pJn. will attempt
to work out the answerfrom an imaginedcircularclockface; and some children
asked to find the value of i --, for example, think in terms of pictorial
representationsof fractions(see Figure2).
w@_
5-
4-
1e=
?
Fig. 2.
Of course, not all secondary pupils, or even most secondarypupils,would use
visual imagery when attempting tasks like 3 -7 and i -i. For the '3 --7'
task a very common method is to apply the rule 'when subtractingdirected
numbers, add the opposite number'. Thus, 3 -7 = 3 ++7 = 10. for the
'I - i' task, many pupils would use an algorithminvolvingequivalentfractions.
The fact that different children respond to the same written stimulus in different ways raises a numberof questions which are of interest to the classroom
teacher and the educationalpsychologist.
For a given task, is one form of response preferableto another?Whichform
of response is most widely used? To what extent is a person'spreferredmode
of response attributableto the form of instruction he has received?Do some
people consistently prefer to use a visual solution mode over a range of problems and others a verbal-logicalmode for the same problems?Whichis the best
form of instruction for a person who prefers a visual mode of response (or,
similarly,a verbal-logicalmode)?
While there is a considerablebody of researchpertainingto most, perhaps
all, of the above questions (see Gagn6 and White, 1978), the present paper
describesan investigationinto three issues which have not been the subject of
much research:
1. Can the construct, 'preferred mode of processing mathematicalinformation' be operationalised to the extent that reliable measures of the construct can be obtained for individuals?
2. Are 'preferredmode of processingmathematicalinformation'and spatial
ability related to mathematicalperformance?
3. Do persons with high spatial ability tend to prefer visual modes of processing mathematical information and persons with average, or low spatial
ability, verbal-logicalmodes of processingmathematicalinformation?
SPATIAL ABILITY, IMAGERY AND MATHEMATICS
271
3. SPATIAL ABILITY AND THE USE OF IMAGERY IN THE
PROCESSING OF MATHEMATICAL INFORMATION-A
LITERATURE REVIEW
The three questions listed at the end of the previoussection have,in fact, been
the subject of two pioneeringstudies by Moses (1977, 1980). Her first study
involved 145 fifth-gradestudents from one elementary school in Newburgh,
Indiana, who were given a battery of six tests. Five of the tests were spatial
tests (Punched Holes, Card Rotations, Form Board, Figure Rotations, Cube
Comparisons)and the other was a problem-solvinginventory consistingof ten
non-routine mathematical problems. For each individual three scores were
computed: a spatial ability score based on the z-scores from the five spatial
tests, a problem-solvinginventory, and a 'degree of visuality' score based on
'the number of visual solution processes (e.g. pictures, graphs, lists, tables)
present in the written solutions' to the problem-solvinginventory. Moses
found that although the correlations of spatial ability with problem-solving
performance and 'degree of visuality' were significantly different from zero,
the correlationbetween problem-solvingperformanceand 'degreeof visuality'
was not significantlydifferent from zero. She concluded that spatialability is a
good predicatorof problem-solvingperformance,and that althoughindividuals
with high spatial ability usually do well on pencil-and-paperproblem-solving
exercises, their written solutions do not always give a properindicationof the
extent to which visualsolution processeshave been used.
In the secondstudy, Moses(1980) investigatedsex and age-relateddifferences
on spatial visualization,reasoningand mathematicalproblem-solvingtasks, and
the effects that a sequenceof visualthinking exerciseshad on these differences.
An experimentaland a control group, each containingmiddle-classstudentsat
the fifth-grade,ninth-gradeand universitylevels were defined, and both groups
were given seven pencil-and-papertests as a pre-testand post-test battery. Four
of the tests were spatial tests (Mental Rotation, Punched Holes, Form Board,
and Hidden Figures), two were reasoning tests (Nonsense Syllogisms and
Reasoning), and the other was a problem-solvingtest containing ten nonroutine mathematicalproblems. It should be noted that in this second study
Moses employed a slightly different set of spatial tasks and a slightly different
problem-solvinginventory. Her results tended to confirm those of her earlier
study. The correlationsbetween scores on the problem-solvinginventory and
the measures of spatial ability, reasoning, and 'degree of visuality' were all
significantlydifferent from zero. Once again, 'degreeof visuality'was measured
by analysing students' written responsesto the problem-solvingtasks. In this
second study Moses found that instruction in visual thinking affected spatial
ability and reasoningability, but not problem-solvingperformanceor 'degree
of visuality'.
272
GLEN LEAN AND M.A.(KEN) CLEMENTS
In our view Moses' interpretationof her results which relate to her 'degree
of visuality' construct are of doubtful validity. Thereare at least two criticisms
which can be made of the studies, one concerning the method she used to
obtain 'degree of visuality' scores, and the other the questions she used in her
problem-solvinginventories.Withrespect to the first criticism,Mosesmeasured
a student's 'degree of visuality' by analysingwritten responsesto the problems
on the problem-solvinginventories and by noting the number of occasions on
which certain skills which she called 'spatial skills', such as making pictures,
diagrams,graphs, lists, tables and constructions, were used. The trouble with
this procedureis that some students might not have expressedin their written
solutions the visual imagery they used when solving problems.Mosesadmitted
that this point caused her difficulty, and interviewswhich she conducted with
students confirmed that many written solutions gave no hint of the large
amount of visual imagery used. Another difficulty with the procedurearosein
the first study because one of the questions on the problem-solvinginventory
actually asked respondents to draw diagrams. Given the manner in which
Moses measured 'degree of visuality' it is not surprisingthat she found that
more students used a visual processing mode with this question than any of
the other nine questions; the problem-solvinginventory for the second study,
however, contained no question which specifically asked for diagramsto be
drawn.
The second major criticism we would make of Moses' studies is that the
problem-solvinginventorieswere too difficult for almost all the students, and
this probably meant that many written solution attempts represented not
much more than guesswork. In the first study there was only one question
out of ten which more than one-third of the fifth-gradeobtained the correct
answer;for three questions less than one-tenth of the students gave the correct
answer. In the second study the mean scores, with a maximum possible score
of ten, were 1.22, 2.25 and 3.33 for fifth-grade, ninth-gradeand university
students respectively. Given the difficulty of the tests it is almost certainthat
Moses was forced to attach 'degree of visuality' measuresto solution attempts
by children who probably had little idea how to solve the problems.Thus, the
validity of her procedurefor measuring'degreeof visuality'is open to question.
In fairness to Moses we would wish to point out that her studies involved
much more than the definition and measurementof the 'degree of visuality'
construct, and that other aspects of the studies, and especiallythe attempts to
improve spatial performanceby means of spatialtrainingprograms,are worthy
of careful attention. Furthermore,the criticisms we have offered of Moses's
attempt to operationalizethe 'degree of visuality' construct drawattention to
several issues which must be considered by any person intending to use this,
or a similarconstruct,in future research.In particular:
SPATIAL ABILITY, IMAGERY AND MATHEMATICS
273
1. It needs to be recognizedthat persons who use visual imageryin solving
mathematicalproblems do not always give any indication of this when setting
out written solutions;
2. Questions whose formats involve diagrams,and questions which indicate
that diagramsshould be drawn, should be avoided in problem-solvinginventories constructedfor the purposeof 'degreeof visuality'research;
3. The matter of how incorrect solution attempts should be scored for
'degreeof visuality'measuresneeds to be considered.
VisualImageryand MathematicalPerformance
There have been a number of studies of the importance of visual imageryfor
solving questions which appear on spatial tests. In an early correlationstudy
Carey (1915) investigated the use of visual imagery by 7-14 year-old British
children on two spatial tests and concluded that ability to use visual imagery
does not influence performanceon spatial tests. Barratt(1953), after noting
that Kelley (1928), El Koussy (1935), and Thurstone(1938), had all suggested
an interpretation of the spatial group factors in terms of the mental manipulation of visual (and perhapskinaesthetic) imagery, pointed out that none of
these writers had specifically attacked this hypothesis with an experiment
designed to confirm or infirm it (Barratt, 1953, p. 155). Barrattindividually
questioned undergraduatestudents after they had taken each test of a battery
of twelve tests which included a number of spatial tests (Thurstone'sP.M.A.
Space test, Flags,SpatialEquations,CubeSurfaces,Raven'sProgressiveMatrices,
MinnesotaPaper Form Board);he asked the students to indicate the extent to
which they had used visualimagerywhen attemptingthe questions on the test,
how vivid their imageryhad been, and whether they had difficulty in manipulating visual images whenever such manipulationshad been needed. Barratt
found that on all twelve tests subjectswho had used visualimageryextensively
in their solution attempts tended to do better than those who had made little
use of imagery. Further,those who had used visualimageryextensivelytended
to do especiallywell on those tests with high loadingson a spatialmanipulation
factor, but their performanceson tests with high loadings on a reasoningfactor, but not a spatial factor, were no better than those by students who had
not made much use of imagery.Thus Barratt'sconclusions were not in agreement with those of Carey.
Smith (1972), in reviewingthe literatureconcerningthe relationshipbetween
spatial ability and visual imagery, commented that although many psychologists who have worked with spatialtests have been convincedthat personswho
are endowed with good visualimageryhave a considerableadvantagein doing
274
GLEN LEAN AND M.A.(KEN) CLEMENTS
these tests, work by Haber and Haber (1964) on children possessingeidetic
imagery suggeststhat such high imagery children are, typically, no more intelligent than other children, and do not perform better than others on spatial
tests. Siipola and Hayden (1965) reported a high incidence of eidetic imagery
children among a group of retarded children. The view that extensive use of
visual imagery might be a disadvantageto someone attemptinga mathematical
problem would surprisethose mathematics educators who hold, as an article
of pedagogical faith, that children'sconceptual understandingis enhanced by
their use of visual imagery (Lin, 1979). On this point, Twyman (1972) distinguished between the ability to form 'memory' images and the ability to
form 'abstract' images, and commented that if both abilities exist then flexibility in moving from one to another, and not being bound by the level of
imagery being used, could be an important factor in humanabilities.Twyman
added that if one is trying to correlate imagery with ability in some task situation one may have to introduce another variable,namely the use of imagery,
and that it is possible that a good reasonerwith poor imagerymay do better
than a bad reasonerwith good imagery. He also posed the question whether
there is any actual barrierto being a good reasoner with good imagery. On
this question of whether strongvisualimagerycan interferewith mathematical
problem solving, Twyman commented that the creation of an image can introduce difficulties associated with decoding the image. For example, the image
might possess irrelevant details which distract the problem solver from the
main elements in the original problem stimulus, and make it more difficult
for him to formulatenecessaryabstractions(see also McKellar,1968; Krutetskii,
1976).
According to Neisser (1967), an individual never uses only mental imagery
when performinga task, because mental images are rarelyvery clearand other
processingmodes are needed to complement them. Paivio(1971,1973, 1978)
maintains that non-verbal and verbal symbolic systems are involved in any
thinking task, but the proportion of one system to the other variesfrom task
to task and from individual to individual. He points to three variableswhich
influence the amount of visual imagery a person uses when performinga task.
First, there are stimulusattributes,that is to say, the characteristicsof the task;
usually a task which requires thinking about familiarphysical objects evokes
more visual imagerythan one which does not involve physicalobjects. Second,
there is the extent to which the type of thinking is specified in the definition
of the task; if the instructions for a task suggest an approachwhich does not
make much use of visual imagerythen persons doing the task might be expected to use less visual imagery than otherwise would have been the case. Third,
different processing modes are employed by different persons doing a task:
SPATIAL ABILITY, IMAGERY AND MATHEMATICS
275
Johnson-Laird(1972), and Wood, Shotter, and Godden (1974) point out, for
example, that a personwho is familiarwith a task tends to use linguisticprocessing more than visualimagerybecause the former processingmode requires
a minimum amount of information to be stored in the short-termmemory
while the task is being attempted. In a similar vein, Bishop (1978, 1979), has
conjectured that University students in Papua New Guinea, unlike University
students in Westerncountries,perform memory tasks with little or no verbal
mediation, that less acculturatedstudents have better visual memories than
students who are more acculturated, and that students coming from areas
where the local languagecontains no easy conditional mood will tend towards
a greater use of visual memory and ikonic processing. Swanson (1978) has
reported that childrenwho verballyencode visual stimuli outperformchildren
who do not use verbalcodes on visualmemory tasks involvingthe same stimuli.
By contrast, Clements and Lean (1980), in an investigation involving community school and internationalprimaryschool childrenin PapuaNew Guinea,
have reported that the use of verbal codes depresses performanceon visual
memory tasks.
Hadamard (1954), Menchinskaya (1969), Poincar6 (1963), Richardson
(1969, 1977) and Walter(1963) are among those who have contended that
individuals can be classified into three groups with respect to a visual-verbal
dimension. The first group, consisting of 'visualizers', contains individuals
who habitually employ visualimageryor pictorial notations when attempting
to solve problems; the second group, the 'verbalizers',contains those who
tend to use verbal codes rather than visual images or pictorial notations; the
third group, the 'mixers',consistsof individualswho do not have a tendency to
prefer either a verbal or visual processingmode. According to Walter(1963)
most people belong to the last group, but there appearsto be some difficulty
in obtaining an instrumentwhich will enable people to be classifiedreliably
into the groups. Indeed, researchershave not been able to agree on the processing modes individualsuse when attempting well-definedtasks. Forexample,
Lunzer (1965), Huttenlocher(1968), Huttenlocher and Higgins(1971), Clark
(1969a, 1969b, 1971), Johnson-Laird(1972), and Rosenthal (1977), who have
examined the processingmodes used by children attempting three-termseries
problems (e.g. "Sarais taller than Jane; Jane is shorter than Mary.Whois the
shortest?"), have not been able to agree on which processingmode, visual or
verbal,childrentend to preferwith such problems.
In the area of mathematicslearning,V. A. Krutetskii,the Russianpsychologist and mathematics educator, has also concluded that individualscan be
divided into three categoriesso far as the processingof mathematicalinformation is concerned (Krutetskii, 1979). First, there is the 'analytic'type who,
276
GLEN LEAN AND M.A.(KEN) CLEMENTS
accordingto Krutetskii,prefers verbal-logicalmodes to visual-pictorialmodes;
second, there is the 'geometric' type, who prefer visual-pictorialmodes; and
third, there is the 'harmonic'type, who uses both verbal-logicaland visualpictorial modes freely. Given the similarresearchfindingsof linguists,psychologists, and mathematicseducatorsit would appearto be importantthat mathematics educators conduct researchwhich clarifiesthe implicationsof information processing theories for mathematics teaching and learning.Needless to
say, care must be exercised in the design of such research.Mathematicseducators can learn from A. R. Jensen (1971) who demonstratedthat although,
for over a decade, many educationalpsychologistshad been conductingresearch
which was based on the assumption that 'auditory' and 'visual'learnerscould
be identified, there was no unambiguousevidence for the existence of these
kinds of learners.Subsequentresearchhas failed to providesuch evidence (see
DeBoth and Dominowski, 1978).
In a recent paper,Webb(1979) analyzed the problem-solvingprocessesand
performancesof forty high school students (from four schools), and found that
of thirteen component variablesconsidered,the three which accountedfor the
most variance in performanceson a problem-solvinginventory were Math
Achievement, Pictorial Representation, and VerbalReasoning. According to
Webb, Math Achievement and VerbalReasoning were conceptual knowledge
factors but Pictorial Representation, which was interpretedto representprocesses related to drawingor using pictures, was a process factor. Webb found
that students who drew and used pictures when attempting mathematical
problems tended to obtain higherscores on the problem-solvinginventory, and
concluded that the fact that the heuristic components, in particularPictorial
Representation, accounted for a sizeable proportion of the variancein scores
in addition to what was accounted for by the pretest components, suggests
that the use of such processesare importantin solvingproblems(Webb, 1979,
p. 92). Such a conclusionshould encourageteacherswho believe they influence
the thought processes their students use, and should provide incentive for
researchersinterested in investigating the extent to which process variables
influence problem-solvingperformance.
SpatialAbility and MathematicalPerformance
After analyzingthe spatialability literatureSmith (1964) concludedthat while
spatial ability is positivelyrelatedto high-levelmathematicalconceptualization
it may have little to do with the acquisition of low-levelmathematicalconcepts
and skills (such as those requiredfor simple calculations).Guay and McDaniel
(1977), however, have reported data which not only suggest that a positive
SPATIAL ABILITY,
IMAGERY AND MATHEMATICS
277
relationship exists between mathematicaland spatial thinking among elementary school children,but also that this relationshipholds for low-level as well
as high-levelspatial abilities (where low-level spatial abilities were defined as
requiring the visualizationof two-dimensional configurationsbut no mental
transformationsof these visual images, and high-levelspatialabilitiesas requiring the visualization of three-dimensional configurations, and the mental
manipulationsof these visual images). In a large longitudinalstudy involving
senior high school students, Sherman (1979), after careful analysisin which
the effect of spatial ability on mathematicalperformancewas considered,with
a number of other cognitive and affective variablescontrolled, reportedthat
the spatial ability factor was one of the main factors which significantly
affected mathematicalperformance.
Most writers who have reported data pertainingto a relationshipbetween
spatial ability and mathematical performance (see Fennema, 1974, 1979;
McGee, 1979) havebased their discussionmainly on the patternsof correlation
coefficients which they calculated.Whilethis method is appropriatefor exploratory investigations(and, indeed, will be used in the present paper) the coefficients which are obtained are rarely easy to interpret. That a correlation
coefficient is significantly different from zero does not mean that the ability
associated with either one of the variableshas priority over the other in the
learning process, or that any causal relationshipcan be legitimatelyinferred.
While, for example, a Pearson product moment coefficient of 0.64 suggests
that about 40% of the varianceof either one of the variablescan be attributed
to variancein the other, there is always the additional question of why that
should be the case.
Webelieve that more clinicalinvestigations,which concentrateon the extent
to which spatial ability is used by personsattemptingdifferentkinds of mathematical problems, are necessary before relationshipsbetween spatial ability
and mathematicalperformancecan be clarified. Interestingly,Krutetskii,who
used clinical methods extensively in his study of mathematicalability, has concluded that gifted mathematiciansdo not always possess above-averagespatial
abilities and often prefer solution methods which make little use of spatial
ability (Krutetskii, 1976). Radatz (1979), in discussingmathematicalerrors
which can arise because of spatial weaknessesin pupils, has commented that
the ikonic representationof mathematicalsituations can involve great difficulties in information processing, and that perceptual analysis and synthesis
of mathematical information presented implicitly in a diagramoften make
greaterdemandson a pupil than any other aspect of a problem.
From the precedingreview of literatureit is clear that although there have
been many investigationsinto relationshipsbetween spatial ability, the use of
278
GLEN LEAN AND M.A.(KEN) CLEMENTS
visual imagery,and mathematicalperformance,very few, if any, definite statements can be made as a result of the investigations.Researchhas not thrown
much light, for example, on the question of whether personswho prefer to
use visual imagery, with little verbal coding, when processingmathematical
information are likely to do better on certainmathematicaltasksthan persons
who prefer a verbal-logicprocessingmode. In the investigationwhich will now
be described,a battery of spatial and mathematicaltests, and a mathematical
processing instrument, were administeredto a sample of tertiarystudents in
Papua New Guinea, and analyses were carried out which sought to clarify
relationshipsbetween spatial ability, preferencesfor certain modes of processing mathematicalinformation,and mathematicalperformance.
4. THE EXPERIMENTAL
STUDY
The subjects were 116 entrants into the Engineeringfoundation year at the
University of Technology, Lae, Papua New Guinea. (Hereafterthis University
will be refered to as 'Unitech'.) Of these, 111 were PapuaNew Guineansfrom
nineteen of the twenty provinces of the country; two were from Samoa and
three from the Solomon Islands. Entry into the Foundation Year occurs in a
number of ways. Students may be selected at the completion of Grade 12
from each of PapuaNew Guinea's four National High Schools; 57 of the subjects were in this category.Alternatively,students may be selected to enter the
University after completing Grade 10 at one of the ProvincialHigh Schools;
they must then complete a preliminaryyear at the Universitybefore entering
the Foundation Year;34 subjectswere in this category. The remainingsubjects
were 'overseas' students, or mature Papua New Guineanswho had had work
experience and had completed a certificate-levelcourse at a technical college.
The mean of the reported ages of the subjects was 19.6 years; the modal age,
however, was 18, the mean being affected by the higher ages of the mature
students. Of the 116 subjects, 114 were male and two female.
TheInstrumentsand theirAdministration
A battery of five spatial tests was administeredto the subjects during the
first two weeks of their course.The tests, in order of administrationwere:
1. Spatial Test EG by I. MacFarlaneSmith, published by the National
Foundation for EducationalResearchin Englandand Wales(N.F.E.R.).
2. Spatial Test II by A. F. Watts,D. A. Pidgeon and M. K. B. Richards(also
publishedby N.F.E.R.).
3. Gestalt CompletionTest by R. F. Street (1931), publishedby Teachers'
College, ColumbiaUniversity,New York.
SPATIAL
ABILITY,
IMAGERY
AND
MATHEMATICS
279
4. Standard ProgressiveMatrices,Set D, by J. C. Raven, publishedby the
AustralianCouncilfor EducationalResearch(Raven, 1938).
5. Three-DimensionalDrawingTest by M. C. Mitchelmore(1974).
The NFER Spatial Test EG, which deals with two-dimensionalmaterial,has
six sub-tests each precededby a practice test; the sub-tests are: fitting shapes,
form recognition, pattern recognition, shape recognition, comparisons,and
form reflections. The total working time is approximately one hour. The
NFER Spatial Test II, which deals with three-dimensionalmaterial,has five
sub-tests each precededby a practicetest; the sub-testsare: matchbox corers,
shapes and models, square completion, paper folding, and block building,The
total workingtime is approximately45 minutes.
Street's Gestalt CompletionTest comprisestwelve items, each of which is a
black and white picture, parts of which have been deleted. Each incomplete
picture was presented to a group of subjects as a slide-film projectedonto a
screen. Subjects were requiredto complete the pictures mentally, and to indicate in written responseswhat they thought the picturesrepresented.The first
two items presentedwere practice examples. The exposure time for each item
was 10 seconds and the total time for the test was approximately5 minutes.
Raven's StandardProgressiveMatricesSet D is a 12-item test. Each item
presents a figurative matrix constructed on some principle which may be
deduced from the design. For each item eight possible choices of the portion
of the design which is missingfrom the original matrix are given,and subjects
are requiredto select one. The total time for the test was 5 minutes.
The Three-DimensionalDrawingTest developed by M. C. Mitchelmorecomprises four separate tasks which share a common feature in that subjectsare
required to representparallellines in space using the conventions appropriate
to representing three-dimensionalobjects two-dimensionally. In the first
exercise, subjects are given a diagramof a winding road with two light poles in
the foreground and are required to draw more poles alongside the road. The
time allowed was 3 minutes. In the second exercise, subjectsare shown an upright bottle half full of liquid and are then shown how to representthe liquid
on a diagramof the bottle. Diagramsof the bottle in variousorientationsare
then presentedto the subjects,who then drawthe liquid surfaces.Two minutes
were allowed for this exercise. In the third exercise the subjectswere shown a
cuboid made from small wooden cubes together with a diagramrepresenting
the cuboid. Subjectswere then given seven minutes to complete the drawings
of four other blocks (no models shown) to make them appearas if they were
constructed from small cubes. In the final exercise, subjects are shown a clear
plastic cube together with a diagramrepresentingthe cube which uses the convention that drawndotted lines representedges of the cube hidden from view.
280
GLEN LEAN AND M.A.(KEN) CLEMENTS
Subjects are then requiredto complete the diagramsof four prisms(no models
shown) by the addition of all the hidden edges. Six minutes were allowed for
this exercise.
Spatial Test EG and Spatial Test II were administeredto each groupof subjects during a two-hour session in the first week of their course. The three
remainingspatial tests were administeredduringa two-hour sessionthe following week. During a further two-hour session in the third week of their course
the subjects were given a mathematics test and an associated questionnaire
developed by Suwarsono. These will be described in detail shortly. Subsequently, ten students were interviewed in order to determine their preferred
methods of solving the problemsin the mathematicstest. The resultsobtained
by interview were then compared with those obtained by the questionnaire.
During their course the subjects sat for two further mathematicstests as part
of their course assessment.The first was a 'Pure' Mathematicstest with 24
items assessingroutine mathematicaltechniques. The second was an 'Applied'
Mathematics test with 27 items assessing the understandingof physical and
mechanicalconcepts. Data from both of these tests were used in the subsequent
statistical analysis. Two typical items from the 'Pure' Mathematicstest and
two from the 'Applied'Mathematicsare shown in Figure 3.
Suwarsono'sMathematicalProcessingInstrument
This instrument, which was developed in 1979 by S. Suwarsono,a doctoral
student at MonashUniversity,Melbourne,2has two parts: the first consists of
thirty mathematicalword problemswhich were chosen so that they would be
suitable for junior secondary pupils in Australian schools; the second part
contains written descriptions of different methods commonly used by pupils
attempting the word problemsin PartI. Usually three to five possible methods
are describedfor each problem.
Pupils are asked to attempt the problems in Part I and then to indicate
which (if any) of the methods describedin Part II they used. If a pupil believes
that his method for solving any problem was unlike any of those describedin
Part II then he is instructed to say so, and to describe his method in writing,
givingas many details as possible.
To illustrate the use of the instrument we give an example of one of the
questions in PartI, and the correspondingsection in Part II.
Question 13 (in Part1)
At each of the two ends of a straightpath a man planted a tree, and then every
5 m along the path (on one side only) he also planted another tree. The length
of the path is 25 m. How many trees were planted on the path altogether?
SPATIAL
Two typical
Question
l(b)
IMAGERY
ABILITY,
items
from the
An aeroplane
AND
mathematics
'pure'
speed and direction
Question
below.
given
i)
5(c)
(-5,
Question
at 120 km/h.
The rectangular
Find their
items
course
of 225?
by a wind
Find the resultant
of the aeroplane.
-5 JE)
Two typical
test
headed on a true bearing
at a speed of 850 km/h is being blown off
coming from the northwest
281
MATHEMATICS
coordinates
polar
ii)
of two points
are
coordinates.
(-3.46,
from the 'applied'
2)
mathematics
test
5
Both of the boxes A and B are in equilibrium.
weighs more?
If both weigh the same, mark C.
Answer:
A.
B.
Question
11.
Which box
C.
Which is
A-
the harder
way to carry
the hammer?
If both are equally
mark C.
difficult,
Answer:
Fig. 3.
A.
B.
C.
GLEN LEAN AND M.A.(KEN) CLEMENTS
282
The Section in PartII Correspondingto Question 13 (on PartI)
Solution 1
I solvedthe problemthis way:
Every 5 m along the path a tree was planted.This meansthat the path was
divided into I = 5 equal parts. Every path correspondedto one tree. But
at one of the two ends of the path the part correspondedto 2 trees.Therefore, the numberof trees was:
= (4
1) + (1 X 2)
=4+2
=6
Solution 2
I solved the problemby imaginingthe path and the trees,and then counting
the trees in my mind. I found there were 6 treeson the path.
Solution3
I solved the problem by drawinga diagramrepresentingthe path and the
trees,and then countingthe trees.
I found 6 trees.
I did not use any of the abovemethods.
I attemptedthe problemin this way:
In developing his instrument Suwarsono made use of results he obtained from
an extensive preliminary investigation in which he analysed not only the
written responses of junior secondary pupils in three schools to mathematical
word problems, but also verbal descriptions they gave, in individual interviews,
of the thought processes they had employed when attempting the problems.
When selecting the questions to be included in Part I of his final instrument
Suwarsono was guided by the following criteria.
1. The questions should range in difficulty from 'very easy' to 'moderately
difficult' for most junior secondary pupils. Very difficult questions were to be
avoided.
2. No diagram would be given, or requested, in any question.
3. For each question it could be expected that a variety of methods would
be used by junior secondary pupils. In particular, it could be expected that in a
large group of, say 200 pupils, some would use verbal-logical methods and
others visual methods.
SPATIAL ABILITY,
IMAGERY AND MATHEMATICS
from Suwarsono's
Four questions
Mathematical
Question
4
and half
a brick.
The scale
Question
ball
On one side
9
of a scale
On the other
Only four
side
there
is
there
is one full
a 1-kg mass
brick.
What is the mass of the brick?
football
teams took part
Each team played
competition.
teams once.
Instrument
Processing
is balanced.
283
How many football
each of the other
against
matches
in a foot-
were there
in the
competition.
Question
11
The difference
A mother is
between
seven
their
times
is
ages
as old as her daughter.
24 years.
How old are
they.
Question
14
A balloon
moved 100 m to the east,
50 m to the east,
first
200 m from the ground,
then dropped 100 m.
and finally
How far was the balloon
rose
It then travelled
dropped straight
from its
starting
then
to the ground.
place?
Fig. 4
In the instrument he developed for his doctoral study Suwarsono included
thirty word problems, but in the present study we have used only fifteen of
these problems, together with the corresponding Part II sections. Four of the
fifteen questions are shown in Figure 4. Suwarsono scored pupils' responses
to Part II of his instrument in the following manner. For each method indicated a score was given according the the following criteria:
+ 2 if the correct answer was obtained and reasoning was based on a diagram (drawn by the pupil) or on some ikonic visual image (constructed by the
pupil);
+ 1 if an incorrect answer was obtained and reasoning was based on a diagram or on some ikonic visual image;
284
GLEN LEAN AND M.A.(KEN) CLEMENTS
0 if no answerwas given to a question or the pupil could not decide which
method he used;
- 1 if an incorrect answer was obtained and reasoning was based on a
verbal-logicalmethod which did not involve a diagramor the construction of
an ikonic visualimage;
- 2 if a correct answerwas obtained and reasoningwas based on a verballogical method which did not involve a diagramor the construction of an
ikonic visualimage.
Thus, for Suwarsono'sinstrument containing thirty problems an individual
could obtain an 'analyticality-visuality'score between - 60 and + 60. For the
present study, which used a modified form of the originalinstrumentcontaining fifteen problems only, an 'analyticality-visuality'score between - 30 and
+ 30 was possible.
Suwarsono's decision to allocate ? 1 for incorrect responses was made
because it was thought that often personswho give incorrectresponsesare not
confident that the methods they have used are appropriate.By contrast persons giving correct responsesare more likely to be confident that the methods
they have used are appropriate.
In December 1979 SuwarsonoadministeredParts I and II of his instrument
to 200 grade 7 pupils in a Victorian High School. He scored each Part II
responsemade by the pupils and then applied Andrich'smultiplicativebinomial
extension of the Rasch model (Andrich, 1975) to the set of scores. This
enabled him to place his word problemson an assumed'analyticality-visuality'
dimension and, further,to measurethe extent to which a pupil preferredvisual,
or verbal-logicalmethods on the same 'analyticality-visuality'dimension.
It is not appropriate,here, to give further technical details of Suwarsono's
validation of his instrument. (These will appear in Suwarsono's doctoral
thesis). However,because we believe that the instrument does enable the construct 'preferredmode of processing mathematical information'to be operationalized to the extent that valid and reliable measuresof the construct can
be obtained for individuals,we wish to report three results obtained from the
application of the modified form of the instrument, containing fifteen of
the originalthirty problems,to the 116 Foundation Year Engineeringstudents
at Unitech.
Suwarsono'smethod of scoring and analysis was applied to the Engineering
student's responses and an analyticality-visuality scaling (hereafter referred
to as an 'ANA-VIS'scaling) of the fifteen problems thereby obtained. It was
found that there was a Spearmanrank-ordercorrelation of 0.90 between the
rankings of the fifteen problems obtained from Unitech students and grade 7
pupils in Victoria. Considering the different educational levels and cultural
SPATIAL ABILITY, IMAGERY AND MATHEMATICS
285
backgroundsof the two groups this is an impressiveresultwhich suggeststhat
the ANA-VIS scaling procedure is largely independent of the sample being
used. (In one sense this is not surprising,for it will be recalledthat Suwarsono
used the Rasch model in establishinghis instrument, and this model should
providesample-freecalibrationsof items-see Andrich, 1975).
Furtherevidencefor the validity of the ANA-VISscalingwas obtainedwhen
one of the presentwriters(Clements) interviewed ten of the Unitech students
who had done the fifteen problems. During the interviews,which were conducted on a one-one basis and without the interviewerbeing aware of the
responses which the ten students had originally given to the problems, the
students explained how they did each of the fifteen questions. Each student
was then classified by the intervieweras an 'analytic' or an 'harmonic'or a
'visual' thinker, according to the amount of ikonic visual imageryhe seemed
to use, or the number of pictorial representationshe made when explaining
his solutions. Five of the ten students were classified as 'analytic', four as
'harmonic', and one as 'visual'. The original written responsesgiven by these
ten students were then analyzed, and the ANA-VIS scaling procedureused to
rankthe students on an analyticality-visualitydimension.The following results
were obtained:
The five 'analytic'studentswere ranked 1, 2, 3, 4, and 7;
The four 'harmonic'studentswere ranked5, 6, 8 and 9;
The one 'visual'student was ranked 10.
These data also providedimpressivesupport for the Suwarsonoinstrument.
Finally, we providesome evidence for the reliability of the ANA-VISscaling procedure. Six Unitech Engineering students who had completed the
fifteen questions were selected for further consideration;accordingto ANAVIS results, two of the students strongly preferredto use non-visualmethods
when processingmathematicalproblems, two strongly preferredto use visual
methods, and two showed no definite preference. The six studentswere asked
to attempt the following three problems,(which, althoughnot includedamong
the fifteen problems given to the Unitech students were among the thirty
problemsused by Suwarsono).
PROBLEM1: Tau has more money than Dilli and Mike has less money than
Dilli. Who has the most money?
PROBLEM2: In an athletics race Johnny is 10 m ahead of Peter;Tom is 4 m
ahead of Jim, and Jim is 3 m ahead of Peter. How many metres is Johnny
ahead of Tom?
GLEN LEAN AND M.A.(KEN) CLEMENTS
286
(1)
(2)
Tau 3x
Dilli
2x
Mike
x
. .
Tau has
John
10 m to Peter
Tom
4m
Jim
3m
John
to
Jo
10m
to
Peter
3
Jo - P
10 - 0 = 10
P
0
Jo - Jim
0
to
Tom
: m
Tom ?
P
Jim
m
: m
to Jim
to
4m to
=
10 =3=7
5 = 10
-
3 -
. John
13)
more money.
4 =
3
3 metres
ahead
of
Tom.
Kuni
+ x
5x
+ 2x
If
Kuni = 10
.
5x
-
:'.
10x
-
.
Fig. 5(a).
Jack
2x
3x
is
yr old.
=
3x
=
7
7 years
old.
Solutionsby a non-visualstudent (unedited).
PROBLEM3: Jack, Luke and Kuni all have birthdayson the 1st January,but
Jack is 1 year older than Luke and Jack is three years younger than Kuni. If
Kuni is 10 years old, how old is Jack?
When the six students had completed the three problems they were asked to
indicate, by ticking appropriateboxes on the instrument used by Suwarsono
in Victoria, the methods they had used.
When their responses were scored (using the ANA-VIS scaling procedure),
both non-visualstudents obtained scores of -6, one visual student obtained
a score of + 6, and the other + 5, and the other two students scores of- 1
and - 2. Figure5(a) and Figure5(b) show unedited solution to three problems
given by a non-visual student and a visual student respectively. While the
written solutions do not enable each student's methods to be identified fully,
it is clear that the respective students prefer non-visualand visual processing
modes.
IMAGERY AND MATHEMATICS
SPATIAL ABILITY,
M
D
T
(1)
'rau has the most
J'
(2)
287
money.
J3'
T
.4
to0
<
(
Pr.
p - 41i
1
,i
4
4gon
If
of
Jim
Peter,
is
John
(3)
is
_
3m ahead
and if
3m ahead
of
John
of
93.
Peter
is
Tom is
then
10m ahead
7m ahead
Peter
then
Tom.
I YEAR
\
of
I
10 yArtS
Jack
?
-
Fig. 5(b).
Jack
is
7 years
old.
Solutions by a visual student (unedited).
Hypotheses,Results and PreliminaryAnalyses
A multiple regressionanalysiswas plannedin orderto investigatethe influence
on mathematicalperformanceof the cognitive abilities and preferencesmeasured by N.F.E.R. Spatial Tests E.G. and II, the Mitchelmore3-D Drawing
test, Street's Gestalt CompletionTest, the twelve items from Set D of Raven's
ProgressiveMatrices,and an ElementaryMathematicsTest (the fifteen-problem
modified versionof Suwarsono'smathematicalprocessinginstrument).Although
288
GLEN LEAN AND M.A.(KEN) CLEMENTS
it was recognized that the sample of 116 students in the present study might
not be representativeof senior secondary or lower tertiary mathematicsstudents in other places, it was decided, neverthelessto employ inferentialstatistical procedures, it being understood that any suggested inferences should be
regardedas tentative hypotheses suitable for further investigationwith other
samples.
For the regressionanalysisscores on the two cumulativeEngineeringMathematics tests (the 'Pure' Mathematicstest requiringmanipulationof algebraic,
trigonometric, and vector expressions, and the 'Applied' MathematicsTest
containing problems in elementary mechanics) would constitute dependent
variables,and the other variablespossible predictorvariables.For the regression
analysis with 'Applied' Mathematicsas the dependent variable,'Pure'Mathematics would also be included as a possible predictor variable,but 'Applied'
Mathematicswould not be included as a possible predictorvariablefor 'Pure'
Mathematics.It was hypothesized that either or both of the dependent variables, 'Pure' and 'Applied' Mathematics,should be expressed as a linear combination of some or all of the possible predictorvariables.For each possible
predictor variable a standardizedregressioncoefficient (Beta value) would be
estimated and the probability calculated that an estimated standardizedcoefficient of at least this magnitude could be obtained by chance if, in fact, the
population coefficient were zero. The proportionsof variancein the dependent
variablesarising from each of the possible predictor variableswould also be
calculated. All calculationswould be performedby a computer, a non-causal
multiple regressionmodel in the computer subroutine REGRESSIONof the
Statistical Package for the Social Sciences (Nie et al., 1975, pp. 333-350)
being used. The variablenames,variablelabels, and respectiverangesof possible
scoreswere as shown in Table 1.
5. THE REGRESSION AND FACTOR ANALYSES
Before beginning the proposed regression analysis the chi-square test for
normality (Book, 1977, pp. 325-355) was appliedto the set of scores defining
each predictor variableto check whether the distributionof scoreswas sufficiently close to normal distrubution to justify its use in the regression.This
proved to be the case for each of the seven possible predictor variables.Also,
Pearson product-moment correlation coefficients between the possible predictor variableswere calculated in order that any difficulty due to multicollinearity, which can arisewith highly correlatedpredictor variables,might be
avoided (see Nie et al., 1975, p. 340). It was decided, a priori, that no two predictor variables should have a correlation of 0.60 or more. The productmoment coefficients obtainedare shown in Table 2.
SPATIAL
ABILITY,
IMAGERY
AND
MATHEMATICS
289
TABLE I
Variable Names, Labels, and Ranges of Possible Scores
Name
Label
RLangeof Possible
SIcores
1. Unitech 'Pure'
Mathematics Test
PM
0 to 100
2. Unitech 'Applied'
Mathematics Test
AM
0 to 30
3. N.F.E.R. E.G. Test
NFER (EG)
0 to 100
4. N.F.E.R. II Test
NFER (II)
0 to 100
5. Mitchelmore's 3D
Drawing Test
3D
0 to 40
6. Street's Gestalt
Completion Test
Gest.
0 to 10
7. Raven's Progressive
Matrices (12 items)
RPM
0 to 12
8. Suwarsono's elementary
Maths. problems
(15 problems)
Elmath
0 to 15
9. Suwarsono's mathematical
processing instrument
ANA-VIS
- 30 to + 30
TABLE 2
Pearson Product-moment Correlation Coefficients between Possible Predictor Variables
NFER
(EG)
NFER
(I)
3D
Gest.
RPM
Elmath
ANA-VIS
NFER
(EG)
NFER
(II)
3D
Gest.
1.00
0.70
0.48
0.10
1.00
0.56
1.00
RPM
Elmath
ANA-VIS
0.37
0.26
-0.10
0.12
0.29
0.29
-0.21
0.15
0.09
0.12
-0.04
0.06
-0.04
0.20
-0.23
1.00
-0.23
1.00
-0.08
1.00
1.00
290
GLEN LEAN AND M.A.(KEN) CLEMENTS
From Table 2 it can be seen that the correlation between NFER (EG) and
NFER (II) was 0.70. To avoid possible multicollinearity problems it was
decided to create a new spatial ability (NFER (SP)) variable,defined by summing each individual'sscores on NFER (EG) and NFER (II). Also, in view of
the fact that all correlationsbetween ANA-VIS and the other possible predictors were negative,a new variableANA-VIS*was createdso that each ANAVIS* score had the same magnitudebut the opposite sign of the corresponding
ANA-VIS score. (This meant that a person gaining a high ANA-VIS* score
should tend to use verbal-logicalmethods when processing mathematical
problems, and someone with a negative ANA-VIS* score should tend to use
visual methods). Table 3 shows the product-momentcorrelationsbetween the
six possible predictor variables,now to be used, and the two dependent variables PMand AM.
TABLE3
PearsonProduct-moment
CorrelationsbetweenPairsof PredictorandDependentVariables.
NFER
(SP)
NFER (SP) 1.00
3D
Gest.
RPM
Elmath
ANA-VIS*
3D
Gest.
RPM
Elmath
ANA-VIS* PM
AM
0.57
1.00
0.11
0.15
1.00
0.35
0.09
-0.08
1.00
0.28
0.12
0.06
0.20
1.00
0.15
0.04
0.04
0.23
0.23
1.00
0.35
0.28
0.12
0.21
0.21
0.31
0.31
0.20
0.10
0.30
0.21
0.24
1.00
0.58
1.00
PM
AM
Table 4 shows means and standard deviations for eight variables listed in
Table 3.
TABLE4
Meansand StandardDeviationsof Predictorand DependentVariables
Variable
Mean
Standard
Deviation
NFER (SP)
3D
Gest.
RPM
Elmath
ANA-VIS*
PM
AM
134.6
31.0
5.26
7.74
11.1
2.49
63.7
21.8
28.1
5.47
1.49
2.44
2.13
8.06
15.5
5.60
SPATIAL
ABILITY,
IMAGERY
AND
291
MATHEMATICS
The 'Pure'MathematicsMultipleRegressionAnalysis
When all six predictorvariableswere retained in a multiple regressionanalysis
with PM as the dependentvariable,the contributionsof the six variablesto the
varianceof PMcould be assessedand compared.Table 5 shows the proportions
of PM varianceexplained(multiple R2) at each step of the analysis,the computer having been programmedto select from remainingpredictor variables
the one which made the greatestcontributionto PMvariance.
TABLE5
Contributionsto PMVarianceof Six PredictorVariables
Variable
Multiple R2
R2 Change
F-Value
ANA-VIS*
0.09
0.09
5.71
3D
0.15
0.06
3.26
NFER (SP)
RPM
Gest.
Elmath
0.19
0.20
0.21
0.22
0.04
0.01
0.01
0.01
3.68
1.84
0.93
0.69
From Table 5, it can be seen that, altogether, the six predictor variablescontributed to only 22% of the variancein PM. If unique partialledcontributions
of predictor variablesto the varianceof the dependent variableare considered,
then the ANA-VIS* variable contributed most (9%), followed by Mitchelmore's 3D DrawingTest (6%) and the N.F.E.R. Spatial Tests (4%). If a standardizedregressionequation for the relationshipbetween PMand the possible
predictor variables were formed containing only those predictor variables
whose estimated standardizedcoefficients (Beta values) differed significantly
from zero, then ANA-VIS* would be the only possible predictorvariableto
enter the equation.
The 'Applied'MathematicsMultipleRegressionAnalysis
For the regressionanalysiswith AM ('Applied'Mathematics)as the dependent
variablePM ('Pure' Mathematics)was added to the list of possible predictor
variables.(This was because it was thought that the solutions of problemsin
elementary mechanics often require the skills tested on the 'Pure' Mathematics test, namely, standardmanipulationsof algebraic, trigonometric,and
vector expressions).
Table 6 shows the proportionsof AM varianceexplained (multiple R2) at
each step of the regressionanalysis.
292
GLEN LEAN AND M.A.(KEN) CLEMENTS
TABLE6
Contributionsto AMVarianceof SevenPredictorVariables
Variable
MultipleR2
R2 Change
F-Value
PM
RPM
Elmath
NFER (SP)
Gest.
ANA-VIS*
3D
0.29
0.33
0.35
0.37
0.38
0.39
0.39
0.29
0.04
0.02
0.02
0.01
0.01
0.00
31.2
3.35
1.02
0.96
0.62
0.54
0.32
From Table 6 it can be seen that, altogether, the seven predictor variables
contributed to only 39% of the variance in AM. The 'Pure' Mathematics
variable contributed most (29%o),with Raven's ProgressiveMatrices,with 4%
only, next. If a standardizedregressionequation for the relationshipbetween
AM and the possible predictor variableswere formed containing only those
predictor variables whose estimated standardized coefficients (Beta values)
differed significantly from zero, then 'Pure' Mathematicswould be the only
possible predictorvariableto enter the equation.
Factor Analysis
In order to explore more fully any relationshipsbetween the variablesused in
the present study a factor analysiswas done on the data set arisingfrom nine
of the variablesused (namely NFER (EG), NFER (II), 3D, Gest., RPM,Elmath,
ANA-VIS*, PM and AM). The principal diagonal method of factorization
(Harman, 1970, pp. 135-186) was used to obtain the initial factor matrix, the
communalitiesof the variables,calculated by an iterativeprocedure,appearing
in the leading diagonalof the final correlationmatrix. The final factor matrix
was obtained using Varimax rotation of axes, with a four-factormatrix being
deemed appropriate according to the Scree test criterion (Child, 1979, pp.
44-45). With n = 116, Burt and Banks' formula for determiningstatistical
significance of a factor loading indicated that a loading with magnitude0.30
or more was significant at the 0.01 level of confidence (see Child, 1979, pp.
45-46, 97-100). Table 7 shows the Varimax rotated factor matrixwhich was
obtained, the fifth column indicating the communalities of each of the nine
variables. Only loadings significant at the 0.01 level are given.
It would seem to be reasonableto identify Factor I as a 'spatial'factor and
Factor II as a 'mathematics'factor. Factor III, on which ANA-VIS*loaded
heavily, and Elmath also loaded, could be described as a 'mathematical
SPATIAL ABILITY, IMAGERY AND MATHEMATICS
293
TABLE7
RotatedFactorAnalysis
Varimax
II
Variable
I
NFER(EG)
NFER(I)
3D
PM
AM
ANA-VIS*
Elmath
RPM
Gest.
0.78
0.84
0.64
III
IV
0.72
0.71
0.66
0.31
0.69
Communality
0.69
0.77
0.46
0.64
0.56
0.48
0.16
0.62
0.07
processing' factor. Factor IV, for which the only substantial loading was
Raven's ProgressiveMatrices, might tentatively be regardedas a 'reasoning'
factor. Interestingly, no variablehad a loading of magnitudemore than 0.30
on more than one factor.
6. DISCUSSION
In view of the substantial and growing literature on relationshipsbetween
spatial ability and mathematical performance, an interesting aspect of the
present study is that spatial ability and knowledge of spatial conventions
had only a small influence on the mathematical performance of the 116
Engineeringstudents in the sample. Multiple regressionanalysis revealedthat
the unique contribution of the N.F.E.R. EG and II tests, and Mitchelmore's
3D DrawingTest totalled only about 10%of the varianceof the 'Pure'Mathematics Cumulativetest scores, and only about 2% to the varianceof 'Applied'
Mathematics test scores once the influence of 'Pure' Mathematicshad been
partialledout. Factor analysisalso drew attention to the lack of any substantive relationship between the spatial ability variablesand mathematicalvariables. The N.F.E.R. Tests and Mitchelmore's3D DrawingTest loaded strongly
on one factor, but did not load on the factor of which 'Pure'and 'Applied'
Mathematicsloaded strongly.
Another important observation is that the modified form of Suwarsono's
mathematical processinginstrumentwhich was used would seem to provide
a promising method for measuringa person's 'preferredmode of processing
mathematical information'. An examination of the correlationmatrix arising
from the variablesused in the present study, and the multiple regressionand
factor analyses, reveals that the ANA-VIS* variable clearly measuresa nontrivial component of cognition which is distinct from any of the other
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GLEN LEAN AND M.A.(KEN) CLEMENTS
componentsmeasured.From the correlationmatrix shown in Table 3, it can be
seen that ANA-VIS* has correlationswith Raven's ProgressiveMatricesElementary Mathematics,'Pure' Mathematicsand 'Applied' Mathematics,which
are statistically significantlydifferent from zero. The multiple regressionanalysis with 'Pure' Mathematicsas the dependent variable(see Table 5) indicates
that ANA-VIS* was the only predictor variableto make a significant contribution to the variance of PM. Factor analysis (see Table 7) confirmed the
view that ANA-VIS* measureda distinct component of cognition: ANA-VIS*
loaded strongly on one of the four factors which was extracted,with 'elementary mathematics' being the only other variable to load on this factor (the
Elmath loading being much smallerthan the ANA-VIS*loading).This factorization suggests that the Suwarsono instrument measures a 'mathematical
processing' trait. Further research, aimed at clarifying the characteristicsof
this trait, is needed.
The nature of the relationshipbetween ANA-VIS* and certain other variables used in the study is worthy of further comment. From the correlation
matrix, shown as Table 3, it can be seen that ANA-VIS* correlatespositively
with all other variables,includingthe mathematicaland spatialability variables.
Thus, there was a tendency for studentswho preferredto processmathematical
information by verbal-logicalmeans to out-perform other students on both
mathematical and spatial tests. So far as mathematical performanceis concerned, this interpretation is supported by the multiple regressionanalysis
with 'Pure'Mathematicsas the dependentvariable.
The relationships between ANA-VIS* and the mathematical and spatial
variablesin the present study are not easily reconciledwith the existing literature. In particular,our results, might appearto be in direct conflict with those
of Moses (1977, 1980) and Webb(1979), who reportedthat studentswho prefer visual solution processes when attempting mathematicalproblems tend to
outperform those who prefer less visual processes. A possible explanation for
the apparent conflict is that in the present study the mathematicalvariables
were measured by tests which did not require the solution of difficult, unfamiliarword problems whereas this was the case in both the Mosesand Webb
studies. We would recommend that future researchers should distinguish
between processes preferredby persons attempting routine and non-routine
mathematicalword problems.
So far as the relationship between preferredmathematicalprocessingand
mathematical performance found in the present study, we would offer the
following tentative interpretation of our results. Since the modified form of
Suwarsono's instrument (Elmath) mostly contained relatively simple word
problems only, a person who displayed a definite preference for a visual
SPATIAL ABILITY,
IMAGERY AND MATHEMATICS
295
processingmode when attempting them would appearto be unable,or unwilling, to abstractin situationswhere abstractingwould providethe most efficient
methods of solution. Such a person tends, in our view, to retainas part of his
thinking, unnecessary'concrete' details. By contrast, the person who uses a
more verbal-logicalmode demonstratesan ability to cast away suchunnecessary
'concrete' details. In the languageof the developmentalpsychologistthe latter
person is more likely to be at the stage of 'formaloperations'than the former.
When confronted with more difficult word problemsthe latter personis likely
to do better because his thinking will not be clutteredwith unnecessaryvisual
images. We would emphasizethat our results do not indicate that a personwho
prefers a less visual processingmode is likely to be weak spatially.Indeed,the
student who obtained the highest total on the N.F.E.R. spatial tests showed
a strong preferencefor solvingmathematicalproblemsby verbal-logicalmeans.
(This was the student whose solutions to three problemsare shown in Figure
5(a).)
It is interesting to observe that scores obtained in the present study on
Street's Gestalt Completion Test do not correlate significantly with scores
on any of the other tests. Guay, McDanieland Angelo (1978) havearguedthat
good spatial tests must require Gestalt processing, and our results therefore
raise severalquestions. Are the N.F.E.R. and Mitchelmoretests adequatetests
of spatial ability? Is Street's Gestalt Completion Test a poor test of Gestalt
processing?Is it in fact true that Gestalt processingis an important factor in
mathematical and spatial processing? These, and other possible questions,
might be worthy of investigationby future researchers.
The multiple regressionanalysis with 'Pure' mathematicsas the dependent
variableindicate that only about 22% of the variancein PM was explainedby
the six predictor variables.While this analysis encouraginglyrevealedthat the
processingvariableANA-VIS* contributed more than other predictorvariable
to the variance of PM, the analysis must, nevertheless,serve as a warningto
those who stress the importanceof spatial and processingvariablesfor mathematical problem-solving.There are many non-mathematicalvariables,such
as student motivation, work habits, teaching, and languagecompetence,which
are potentially important in explaining mathematical performance.In Papua
New Guinea the languagefactor could be especially importantbecauseEnglish,
the languageof instructionand the languagein which mathematicalproblems
are invariablyposed, is usually the third or fourth languageacquiredby children. It is likely that even universitystudents in Papua New Guineaare often
not able to cope with the subtleties of English expressionwhich can occur in
the wordingof mathematicalproblems.
It is stressed that the above conclusions arose from a study involving 116
296
GLEN LEAN AND M.A.(KEN) CLEMENTS
first-yearEngineeringstudents in PapuaNew Guinea.Generalizationsbased on
such a sample may not apply to mathematics learnersat the same or different
levels in other parts of the world. Further,the mathematicaltasks used for the
PM and AM tests were of a routine type, and the imagery variablewas based
on student'sprocessingof elementarymathematicaltasks.A differentpatternof
resultsmay havebeen obtainedif non-routinemathematicaltaskshadbeen used.
7. SUMMARY
In concluding this paper we summarizethe seven points made in the previous
section with respect to possible implications of the analyseswhich had been
presented.
1. Multipleregressionanalysis suggested that spatial ability and knowledge
of spatial conventions did not have a large influence on the mathematical
performancesof the 116 Engineeringstudents in the sample.
2. Suwarsono'smathematical processing instrument would appearto provide a promisingmethod for measuringa person's 'preferredmode of processing mathematicalinformation'. Also, the use of the instrument in the present
study provided data which, when analyzed, suggested the existence of a distinct cognitivetrait associatedwith mathematicalprocessing.
3. There was a tendency for students who preferredto processmathematical informationby verbal-logicalmeans to outperformmore visualstudents on
both mathematicaland spatialtests.
4. The results of the present study appear to be in conflict with other
studies which suggest that it is desirableto use visualprocesseswhen attempting mathematical problems. However, this apparent conflict could be due to
the use, in the present study, of straightforward,routine tasks on the 'Pure'
and 'Applied' Mathematicstests, whereas in most other relevant studies difficult, non-routinemathematicalword problemshave been used.
5. The tendency towards superior performance on mathematical tests by
students who preferreda verbal-logicalmode of processingmathematicalinformation might be due to a developed ability to abstractreadily,and, therefore,
to avoid the formation of unnecessaryvisualimages.
6. The failure of Street's Gestalt Completion Test to correlatesignificantly
with any of the mathematicaland spatial tests needs to be explained.
7. Many non-mathematical variables, such as student motivation, work
habits, teaching,and languagecompetence, which could contributesignificantly
to mathematical performance,were not measured in the present study. The
languagecompetence variablecould be especially important in the PapuaNew
Guineancontext.
PapuaNew GuineaUniversity
of Technology
Monash University
SPATIAL
ABILITY,
IMAGERY
AND
MATHEMATICS
297
NOTES
1 The authors wish to thank Ms A.
McDougall, of Monash University and Dr. C. Wilkins,
of Papua New Guinea University of Technology, Lae, for assistance with the computer
and statistical analyses.
2 S. Suwarsono to M. A.
Clements, personal communication, 1980. During 1979 M. A.
Clements assisted Suwarsono in the development and trialling of the mathematical processing instrument.
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