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Spatial Ability, Visual Imagery, and Mathematical PerformanceAuthor(s): Glen Lean and M. A. (Ken) ClementsSource: Educational Studies in Mathematics, Vol. 12, No. 3 (Aug., 1981), pp. 267-299Published by: SpringerStable URL: http://www.jstor.org/stable/3482331Accessed: 09/04/2010 16:47

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GLEN LEAN AND M.A.(KEN) CLEMENTS

SPATIAL ABILITY, VISUAL IMAGERY, AND

MATHEMATICAL 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 Jacques Hadamard, Albert Einstein stated that he always thought about anything in terms of mental pictures and that he used words in a secondary capacity only (see Einstein's letter in Hadamard, 1954). In the field of

mathematics, some mathematicians have claimed that all mathematical tasks

require spatial thinking (see Fennema, 1979). Indeed, as early as 1935 H. R. Hamley, an Australian mathematician and psychologist, wrote that mathematical ability is a compound of general intelligence, visual imagery, and ability to perceive number and space configurations and to retain such configurations as mental pictures (McGee, 1979). Given statements such as these, it is not

surprising that there is a substantial literature in which relationships between

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 present paper is 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 recognized that no agreement on the definitions of each of these terms is evident in the litera-

ture). By 'spatial ability' we shall mean the ability to formulate mental images and to manipulate these images in the mind (see McGee 1979, for a review of definitions of spatial factors; see also Guay, McDaniel and Angelo, 1978). By 'imagery' we shall mean, following Hebb (1972), 'the occurrence of mental

activity corresponding to 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.

GLEN LEAN AND M.A. (KEN) CLEMENTS

not presented to the sense organ' and by 'visual imagery' we shall mean imagery 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 learning associated with the subject 'mathematics', as it is studied in schools and tertiary institutions throughout the world.

In addition to this introduction the present paper contains six sections. In the first a list of mathematical topics in which spatial ability and visual imagery are needed is provided. The next section reviews the literature concerned with the relationship between mathematical performance, spatial ability, and visual

imagery. Then follows a description of an investigation which we carried out 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 mathematics education in Papua New Guinea, are then discussed, and, finally, a summary of the main points arising from the investigation is 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 mathematical performance might be expected to be related, the following examples are offered.

Consider a 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 graph of 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 superimpose the line y = i on the graph of sin 3x and then find the values of x corresponding to 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 graphs of 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 considerable use of visual imagery and spatial ability in planning his solution procedure and in deducing the graph of y = sin 3x from that of y = sin x.

While it is true that some mathematical topics require greater use of spatial ability than others, the following list of topics, compiled from senior secondary and lower tertiary courses in Papua New Guinea and Australia, draws attention to the difficulties which students with poorly developed spatial abilities might experience in their mathematics programs.

1. Sketch graphs.

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SPATIAL ABILITY, IMAGERY AND MATHEMATICS

E.g. y = -x3 +x

y = 1-2 sin 3 x +

y = sin-'x (reflected y = sin x in the line y = x).

2. Conic sections. E.g. focus-directrix definitions, normals to curves. 3. Interpreting or drawing two-dimensional representations of three-

dimensional situations. E.g. the angle between two planes, geometry of the earth, engineering drawing.

4. Linear programming. 5. Geometrical transformations (translations, reflections, rotations, dilations,

expansions). 6. The Calculus. E.g. concept of a limit, areas under curves, solids of revo-

lution. 7. Probability. E.g. the normal curve (Find Pr (z 0). 8. Circular function. 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. drawing force diagrams.

There are many other topics in senior secondary mathematics and in primary and junior secondary mathematics which, depending on the individual, might involve the use of spatial abilities.

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 finding the value of 3 -7, a junior secondary pupil might, as a result of instruction, imagine someone walking forwards and backwards along a number line (see Figure 1).

walk turn around and forward walk 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 around and face to the left; the '7' tells you to walk backwards seven units. By this kind of

thinking, and not necessarily with the aid of a diagram, a junior secondary

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GLEN LEAN AND M.A. (KEN) CLEMENTS

pupil may determine that 3 -7 =+10. Similarly, many children confronted with the problem of finding the time three hours before 2.15 pJn. will attempt to work out the answer from an imagined circular clockface; and some children asked to find the value of i --, for example, think in terms of pictorial representations of fractions (see Figure 2).

w@_ 1e= ? -5 4-

Fig. 2.

Of course, not all secondary pupils, or even most secondary pupils, 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 subtracting directed numbers, add the opposite number'. Thus, 3 -7 = 3 ++7 = 10. for the 'I - i' task, many pupils would use an algorithm involving equivalent fractions. The fact that different children respond to the same written stimulus in different ways raises a number of questions which are of interest to the classroom teacher and the educational psychologist.

For a given task, is one form of response preferable to another? Which form of response is most widely used? To what extent is a person's preferred mode of response attributable to 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-logical mode for the same problems? Which is the best form of instruction for a person who prefers a visual mode of response (or, similarly, a verbal-logical mode)?

While there is a considerable body of research pertaining to most, perhaps all, of the above questions (see Gagn6 and White, 1978), the present paper describes an investigation into three issues which have not been the subject of much research:

1. Can the construct, 'preferred mode of processing mathematical information' be operationalised to the extent that reliable measures of the construct can be obtained for individuals?

2. Are 'preferred mode of processing mathematical information' and spatial ability related to mathematical performance?

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-logical modes of processing mathematical information?

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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 previous section have, in fact, been the subject of two pioneering studies by Moses (1977, 1980). Her first study involved 145 fifth-grade students 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-solving inventory consisting of 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-solving inventory, 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-solving inventory. Moses found that although the correlations of spatial ability with problem-solving performance and 'degree of visuality' were significantly different from zero, the correlation between problem-solving performance and 'degree of visuality' was not significantly different from zero. She concluded that spatial ability is a

good predicator of problem-solving performance, and that although individuals with high spatial ability usually do well on pencil-and-paper problem-solving exercises, their written solutions do not always give a proper indication of the extent to which visual solution processes have been used.

In the second study, Moses (1980) investigated sex and age-related differences on spatial visualization, reasoning and mathematical problem-solving tasks, and the effects that a sequence of visual thinking exercises had on these differences. An experimental and a control group, each containing middle-class students at the fifth-grade, ninth-grade and university levels were defined, and both groups were given seven pencil-and-paper tests as a pre-test and 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-solving test containing ten non- routine mathematical problems. It should be noted that in this second study Moses employed a slightly different set of spatial tasks and a slightly different problem-solving inventory. Her results tended to confirm those of her earlier study. The correlations between scores on the problem-solving inventory and the measures of spatial ability, reasoning, and 'degree of visuality' were all significantly different from zero. Once again, 'degree of visuality' was measured by analysing students' written responses to the problem-solving tasks. In this second study Moses found that instruction in visual thinking affected spatial ability and reasoning ability, but not problem-solving performance or 'degree of visuality'.

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In our view Moses' interpretation of her results which relate to her 'degree of visuality' construct are of doubtful validity. There are 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-solving inventories. With respect to the first criticism, Moses measured a student's 'degree of visuality' by analysing written responses to the problems on the problem-solving inventories 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 procedure is that some students might not have expressed in their written solutions the visual imagery they used when solving problems. Moses admitted that this point caused her difficulty, and interviews which she conducted with students confirmed that many written solutions gave no hint of the large amount of visual imagery used. Another difficulty with the procedure arose in the first study because one of the questions on the problem-solving inventory actually asked respondents to draw diagrams. Given the manner in which Moses measured 'degree of visuality' it is not surprising that she found that more students used a visual processing mode with this question than any of the other nine questions; the problem-solving inventory for the second study, however, contained no question which specifically asked for diagrams to be drawn.

The second major criticism we would make of Moses' studies is that the

problem-solving inventories were 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-grade obtained 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-grade and university students respectively. Given the difficulty of the tests it is almost certain that Moses was forced to attach 'degree of visuality' measures to solution attempts by children who probably had little idea how to solve the problems. Thus, the validity of her procedure for measuring 'degree of 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 measurement of the 'degree of visuality' construct, and that other aspects of the studies, and especially the attempts to improve spatial performance by means of spatial training programs, are worthy of careful attention. Furthermore, the criticisms we have offered of Moses's attempt to operationalize the 'degree of visuality' construct draw attention to several issues which must be considered by any person intending to use this, or a similar construct, in future research. In particular:

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1. It needs to be recognized that persons who use visual imagery in solving mathematical problems do not always give any indication of this when setting out written solutions;

2. Questions whose formats involve diagrams, and questions which indicate that diagrams should be drawn, should be avoided in problem-solving inventories constructed for the purpose of 'degree of visuality' research;

3. The matter of how incorrect solution attempts should be scored for

'degree of visuality' measures needs to be considered.

Visual Imagery and Mathematical Performance

There have been a number of studies of the importance of visual imagery for

solving questions which appear on spatial tests. In an early correlation study 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 performance on 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 perhaps kinaesthetic) 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). Barratt individually questioned undergraduate students after they had taken each test of a battery of twelve tests which included a number of spatial tests (Thurstone's P.M.A. Space test, Flags, Spatial Equations, Cube Surfaces, Raven's Progressive Matrices, Minnesota Paper Form Board); he asked the students to indicate the extent to which they had used visual imagery when attempting the questions on the test, how vivid their imagery had been, and whether they had difficulty in manipu- lating visual images whenever such manipulations had been needed. Barratt found that on all twelve tests subjects who had used visual imagery extensively in their solution attempts tended to do better than those who had made little use of imagery. Further, those who had used visual imagery extensively tended to do especially well on those tests with high loadings on a spatial manipulation factor, but their performances on tests with high loadings on a reasoning factor, but not a spatial factor, were no better than those by students who had not made much use of imagery. Thus Barratt's conclusions were not in agreement with those of Carey.

Smith (1972), in reviewing the literature concerning the relationship between

spatial ability and visual imagery, commented that although many psychologists who have worked with spatial tests have been convinced that persons who are endowed with good visual imagery have a considerable advantage in doing

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these tests, work by Haber and Haber (1964) on children possessing eidetic

imagery suggests that such high imagery children are, typically, no more intel-

ligent 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 disadvantage to someone attempting a mathematical

problem would surprise those mathematics educators who hold, as an article of pedagogical faith, that children's conceptual understanding is enhanced by their use of visual imagery (Lin, 1979). On this point, Twyman (1972) dis-

tinguished between the ability to form 'memory' images and the ability to form 'abstract' images, and commented that if both abilities exist then flexi-

bility in moving from one to another, and not being bound by the level of

imagery being used, could be an important factor in human abilities. Twyman added that if one is trying to correlate imagery with ability in some task situ- ation one may have to introduce another variable, namely the use of imagery, and that it is possible that a good reasoner with poor imagery may do better than a bad reasoner with good imagery. He also posed the question whether there is any actual barrier to being a good reasoner with good imagery. On this question of whether strong visual imagery can interfere with 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 formulate necessary abstractions (see also McKellar, 1968; Krutetskii, 1976).

According to Neisser (1967), an individual never uses only mental imagery when performing a task, because mental images are rarely very clear and other processing modes 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 varies from task to task and from individual to individual. He points to three variables which influence the amount of visual imagery a person uses when performing a task. First, there are stimulus attributes, that is to say, the characteristics of the task; usually a task which requires thinking about familiar physical objects evokes more visual imagery than one which does not involve physical objects. 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 approach which does not make much use of visual imagery then 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:

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Johnson-Laird (1972), and Wood, Shotter, and Godden (1974) point out, for

example, that a person who is familiar with a task tends to use linguistic processing more than visual imagery because the former processing mode requires a minimum amount of information to be stored in the short-term memory 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 Western countries, perform memory tasks with little or no verbal mediation, that less acculturated students have better visual memories than students who are more acculturated, and that students coming from areas where the local language contains no easy conditional mood will tend towards a greater use of visual memory and ikonic processing. Swanson (1978) has reported that children who verbally encode visual stimuli outperform children who do not use verbal codes on visual memory tasks involving the same stimuli. By contrast, Clements and Lean (1980), in an investigation involving community school and international primary school children in Papua New Guinea, have reported that the use of verbal codes depresses performance on 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 visual imagery or 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', consists of individuals who do not have a tendency to prefer either a verbal or visual processing mode. According to Walter (1963) most people belong to the last group, but there appears to be some difficulty in obtaining an instrument which will enable people to be classified reliably into the groups. Indeed, researchers have not been able to agree on the processing modes individuals use when attempting well-defined tasks. For example, Lunzer (1965), Huttenlocher (1968), Huttenlocher and Higgins (1971), Clark

(1969a, 1969b, 1971), Johnson-Laird (1972), and Rosenthal (1977), who have examined the processing modes used by children attempting three-term series problems (e.g. "Sara is taller than Jane; Jane is shorter than Mary. Who is the

shortest?"), have not been able to agree on which processing mode, visual or verbal, children tend to prefer with such problems.

In the area of mathematics learning, V. A. Krutetskii, the Russian psychologist and mathematics educator, has also concluded that individuals can be divided into three categories so far as the processing of mathematical information is concerned (Krutetskii, 1979). First, there is the 'analytic' type who,

275


according to Krutetskii, prefers verbal-logical modes to visual-pictorial modes; second, there is the 'geometric' type, who prefer visual-pictorial modes; and third, there is the 'harmonic' type, who uses both verbal-logical and visual-

pictorial modes freely. Given the similar research findings of linguists, psychologists, and mathematics educators it would appear to be important that mathematics educators conduct research which clarifies the implications of information processing theories for mathematics teaching and learning. Needless to say, care must be exercised in the design of such research. Mathematics educators can learn from A. R. Jensen (1971) who demonstrated that although, for over a decade, many educational psychologists had been conducting research which was based on the assumption that 'auditory' and 'visual' learners could be identified, there was no unambiguous evidence for the existence of these kinds of learners. Subsequent research has failed to provide such evidence (see DeBoth and Dominowski, 1978).

In a recent paper, Webb (1979) analyzed the problem-solving processes and

performances of forty high school students (from four schools), and found that of thirteen component variables considered, the three which accounted for the most variance in performances on a problem-solving inventory were Math Achievement, Pictorial Representation, and Verbal Reasoning. According to Webb, Math Achievement and Verbal Reasoning were conceptual knowledge factors but Pictorial Representation, which was interpreted to represent processes related to drawing or using pictures, was a process factor. Webb found that students who drew and used pictures when attempting mathematical problems tended to obtain higher scores on the problem-solving inventory, and concluded that the fact that the heuristic components, in particular Pictorial Representation, accounted for a sizeable proportion of the variance in scores in addition to what was accounted for by the pretest components, suggests that the use of such processes are important in solving problems (Webb, 1979, p. 92). Such a conclusion should encourage teachers who believe they influence the thought processes their students use, and should provide incentive for researchers interested in investigating the extent to which process variables influence problem-solving performance.

Spatial Ability and Mathematical Performance

After analyzing the spatial ability literature Smith (1964) concluded that while spatial ability is positively related to high-level mathematical conceptualization it may have little to do with the acquisition of low-level mathematical concepts and skills (such as those required for simple calculations). Guay and McDaniel (1977), however, have reported data which not only suggest that a positive

276


relationship exists between mathematical and spatial thinking among elementary school children, but also that this relationship holds for low-level as well as high-level spatial abilities (where low-level spatial abilities were defined as

requiring the visualization of two-dimensional configurations but no mental transformations of these visual images, and high-level spatial abilities as requiring the visualization of three-dimensional configurations, and the mental

manipulations of these visual images). In a large longitudinal study involving senior high school students, Sherman (1979), after careful analysis in which the effect of spatial ability on mathematical performance was considered, with a number of other cognitive and affective variables controlled, reported that the spatial ability factor was one of the main factors which significantly affected mathematical performance.

Most writers who have reported data pertaining to a relationship between

spatial ability and mathematical performance (see Fennema, 1974, 1979; McGee, 1979) have based their discussion mainly on the patterns of correlation coefficients which they calculated. While this method is appropriate for explora- tory 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 variables has priority over the other in the

learning process, or that any causal relationship can be legitimately inferred.

While, for example, a Pearson product moment coefficient of 0.64 suggests that about 40% of the variance of either one of the variables can be attributed to variance in the other, there is always the additional question of why that should be the case.

We believe that more clinical investigations, which concentrate on the extent to which spatial ability is used by persons attempting different kinds of mathematical problems, are necessary before relationships between spatial ability and mathematical performance can be clarified. Interestingly, Krutetskii, who used clinical methods extensively in his study of mathematical ability, has concluded that gifted mathematicians do not always possess above-average spatial abilities and often prefer solution methods which make little use of spatial ability (Krutetskii, 1976). Radatz (1979), in discussing mathematical errors which can arise because of spatial weaknesses in pupils, has commented that the ikonic representation of mathematical situations can involve great difficulties in information processing, and that perceptual analysis and synthesis of mathematical information presented implicitly in a diagram often make

greater demands on a pupil than any other aspect of a problem. From the preceding review of literature it is clear that although there have

been many investigations into relationships between spatial ability, the use of

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visual imagery, and mathematical performance, very few, if any, definite statements can be made as a result of the investigations. Research has not thrown much light, for example, on the question of whether persons who prefer to use visual imagery, with little verbal coding, when processing mathematical information are likely to do better on certain mathematical tasks than persons who prefer a verbal-logic processing mode. In the investigation which will now be described, a battery of spatial and mathematical tests, and a mathematical

processing instrument, were administered to a sample of tertiary students in

Papua New Guinea, and analyses were carried out which sought to clarify relationships between spatial ability, preferences for certain modes of processing mathematical information, and mathematical performance.

4. THE EXPERIMENTAL STUDY

The subjects were 116 entrants into the Engineering foundation year at the

University of Technology, Lae, Papua New Guinea. (Hereafter this University will be refered to as 'Unitech'.) Of these, 111 were Papua New Guineans from 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 Papua New Guinea's four National High Schools; 57 of the sub-

jects were in this category. Alternatively, students may be selected to enter the

University after completing Grade 10 at one of the Provincial High Schools; they must then complete a preliminary year at the University before entering the Foundation Year; 34 subjects were in this category. The remaining subjects were 'overseas' students, or mature Papua New Guineans who had had work experience and had completed a certificate-level course 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.

The Instruments and their Administration

A battery of five spatial tests was administered to the subjects during the first two weeks of their course. The tests, in order of administration were:

1. Spatial Test EG by I. MacFarlane Smith, published by the National Foundation for Educational Research in England and Wales (N.F.E.R.).

2. Spatial Test II by A. F. Watts, D. A. Pidgeon and M. K. B. Richards (also published by N.F.E.R.).

3. Gestalt Completion Test by R. F. Street (1931), published by Teachers' College, Columbia University, New York.

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SPATIAL ABILITY, IMAGERY AND MATHEMATICS 279

4. Standard Progressive Matrices, Set D, by J. C. Raven, published by the Australian Council for Educational Research (Raven, 1938).

5. Three-Dimensional Drawing Test by M. C. Mitchelmore (1974). The NFER Spatial Test EG, which deals with two-dimensional material, has

six sub-tests each preceded by 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-dimensional material, has five sub-tests each preceded by a practice test; the sub-tests are: matchbox corers, shapes and models, square completion, paper folding, and block building, The total working time is approximately 45 minutes.

Street's Gestalt Completion Test comprises twelve 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 projected onto a screen. Subjects were required to complete the pictures mentally, and to indicate in written responses what they thought the pictures represented. The first two items presented were practice examples. The exposure time for each item was 10 seconds and the total time for the test was approximately 5 minutes.

Raven's Standard Progressive Matrices Set 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 missing from the original matrix are given, and subjects are required to select one. The total time for the test was 5 minutes.

The Three-Dimensional Drawing Test developed by M. C. Mitchelmore com-

prises four separate tasks which share a common feature in that subjects are

required to represent parallel lines in space using the conventions appropriate to representing three-dimensional objects two-dimensionally. In the first exercise, subjects are given a diagram of 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, subjects are shown an up- right bottle half full of liquid and are then shown how to represent the liquid on a diagram of the bottle. Diagrams of the bottle in various orientations are then presented to the subjects, who then draw the liquid surfaces. Two minutes were allowed for this exercise. In the third exercise the subjects were shown a cuboid made from small wooden cubes together with a diagram representing the cuboid. Subjects were then given seven minutes to complete the drawings of four other blocks (no models shown) to make them appear as if they were constructed from small cubes. In the final exercise, subjects are shown a clear

plastic cube together with a diagram representing the cube which uses the con- vention that drawn dotted lines represent edges of the cube hidden from view.


Subjects are then required to complete the diagrams of 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 administered to each group of sub-

jects during a two-hour session in the first week of their course. The three

remaining spatial tests were administered during a two-hour session the follow-

ing 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. Subse-

quently, ten students were interviewed in order to determine their preferred methods of solving the problems in the mathematics test. The results obtained

by interview were then compared with those obtained by the questionnaire. During their course the subjects sat for two further mathematics tests as part of their course assessment. The first was a 'Pure' Mathematics test with 24 items assessing routine mathematical techniques. The second was an 'Applied' Mathematics test with 27 items assessing the understanding of physical and mechanical concepts. Data from both of these tests were used in the subsequent statistical analysis. Two typical items from the 'Pure' Mathematics test and two from the 'Applied' Mathematics are shown in Figure 3.

Suwarsono's Mathematical Processing Instrument

This instrument, which was developed in 1979 by S. Suwarsono, a doctoral student at Monash University, Melbourne,2 has two parts: the first consists of

thirty mathematical word problems which 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 problems in Part I. Usually three to five possible methods are described for each problem.

Pupils are asked to attempt the problems in Part I and then to indicate which (if any) of the methods described in Part II they used. If a pupil believes that his method for solving any problem was unlike any of those described in Part II then he is instructed to say so, and to describe his method in writing, giving as many details as possible.

To illustrate the use of the instrument we give an example of one of the

questions in Part I, and the corresponding section in Part II.

Question 13 (in Part 1)

At each of the two ends of a straight path 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?

280


Two typical items from the 'pure' mathematics test

Question l(b) An aeroplane headed on a true bearing of 225?

at a speed of 850 km/h is being blown off course by a wind

coming from the northwest at 120 km/h. Find the resultant

speed and direction of the aeroplane.

Question 5(c) The rectangular coordinates of two points are

given below. Find their polar coordinates.

i) (-5, -5 JE) ii) (-3.46, 2)

Two typical items from the 'applied' mathematics test

Question 5

Both of the boxes A and B are in equilibrium. Which box

weighs more? If both weigh the same, mark C.

Answer: A. B. C.

Question 11.

Which is the harder

way to carry the hammer?

If both are equally

difficult, mark C.

Answer: A. B. C.

A-

Fig. 3.

281


The Section in Part II Corresponding to Question 13 (on Part I)

Solution 1

I solved the problem this way: Every 5 m along the path a tree was planted. This means that the path was divided into I = 5 equal parts. Every path corresponded to one tree. But at one of the two ends of the path the part corresponded to 2 trees. There- fore, the number of trees was:

= (4 1) + (1 X 2)

=4+2

=6

Solution 2

I solved the problem by imagining the path and the trees, and then counting the trees in my mind. I found there were 6 trees on the path.

Solution 3

I solved the problem by drawing a diagram representing the path and the trees, and then counting the trees.

I found 6 trees.

I did not use any of the above methods.

I attempted the problem in 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.

282


Four questions from Suwarsono's

Mathematical Processing Instrument

Question 4 On one side of a scale there is a 1-kg mass

and half a brick. On the other side there is one full brick.

The scale is balanced. What is the mass of the brick?

Question 9 Only four football teams took part in a foot-

ball competition. Each team played against each of the other

teams once. How many football matches were there in the

competition.

Question 11 A mother is seven times as old as her daughter.

The difference between their ages is 24 years. How old are

they.

Question 14 A balloon first rose 200 m from the ground, then

moved 100 m to the east, then dropped 100 m. It then travelled

50 m to the east, and finally dropped straight to the ground.

How far was the balloon from its starting 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 dia-

gram (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 dia-

gram or on some ikonic visual image;


0 if no answer was 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-logical method which did not involve a diagram or the construction of an ikonic visual image;

- 2 if a correct answer was obtained and reasoning was based on a verbal- logical method which did not involve a diagram or the construction of an ikonic visual image.

Thus, for Suwarsono's instrument 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 original instrument contain-

ing 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 persons who give incorrect responses are not confident that the methods they have used are appropriate. By contrast persons giving correct responses are more likely to be confident that the methods they have used are appropriate.

In December 1979 Suwarsono administered Parts I and II of his instrument to 200 grade 7 pupils in a Victorian High School. He scored each Part II response made by the pupils and then applied Andrich's multiplicative binomial extension of the Rasch model (Andrich, 1975) to the set of scores. This enabled him to place his word problems on an assumed 'analyticality-visuality' dimension and, further, to measure the extent to which a pupil preferred visual, or verbal-logical methods 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 'preferred mode of processing mathematical information' to be opera- tionalized to the extent that valid and reliable measures of 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 original thirty problems, to the 116 Foundation Year Engineering students at Unitech.

Suwarsono's method 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 Spearman rank-order correlation 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

284


backgrounds of the two groups this is an impressive result which suggests that the ANA-VIS scaling procedure is largely independent of the sample being used. (In one sense this is not surprising, for it will be recalled that Suwarsono used the Rasch model in establishing his instrument, and this model should

provide sample-free calibrations of items-see Andrich, 1975). Further evidence for the validity of the ANA-VIS scaling was obtained when

one of the present writers (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 interviewer being 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 interviewer as an 'analytic' or an 'harmonic' or a 'visual' thinker, according to the amount of ikonic visual imagery he seemed to use, or the number of pictorial representations he made when explaining his solutions. Five of the ten students were classified as 'analytic', four as 'harmonic', and one as 'visual'. The original written responses given by these ten students were then analyzed, and the ANA-VIS scaling procedure used to rank the students on an analyticality-visuality dimension. The following results were obtained: The five 'analytic' students were ranked 1, 2, 3, 4, and 7; The four 'harmonic' students were ranked 5, 6, 8 and 9; The one 'visual' student was ranked 10. These data also provided impressive support for the Suwarsono instrument.

Finally, we provide some evidence for the reliability of the ANA-VIS scal-

ing procedure. Six Unitech Engineering students who had completed the fifteen questions were selected for further consideration; according to ANA- VIS results, two of the students strongly preferred to use non-visual methods when processing mathematical problems, two strongly preferred to use visual

methods, and two showed no definite preference. The six students were asked to attempt the following three problems, (which, although not included among the fifteen problems given to the Unitech students were among the thirty problems used by Suwarsono).

PROBLEM 1: Tau has more money than Dilli and Mike has less money than Dilli. Who has the most money?

PROBLEM 2: 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?

285

286 GLEN LEAN AND M.A.(KEN) CLEMENTS

(1) Tau 3x

Dilli 2x

Mike x . . Tau has more money.

(2) John 10 m to Peter m

Tom 4m to Jim : m

Jim 3m to Peter : m

John to Tom ?

Jo P Jo - P 10m to 0 10 - 0 = 10

Jim P Jo - Jim

3 to 0 = 10 =3=7

Tom 4m to 5 = 10 - 3 - 4 = 3

. John 3 metres ahead of Tom.

13) Kuni

5x + 2x + x

If Kuni = 10 yr old.

. 5x - 2x = 3x

:'. 10x - 3x = 7

. Jack is 7 years old.

Fig. 5(a). Solutions by a non-visual student (unedited).

PROBLEM 3: Jack, Luke and Kuni all have birthdays on 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 appropriate boxes 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-visual students 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. Figure 5(a) and Figure 5(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-visual and visual processing modes.

(1)


T D M

'rau has the most money.

(2) J' .4

,i 1 4 4gon

J3' < to0

T

p - 41 Pr. ( i

If Jim is 3m ahead of Peter then Tom is 7m ahead

of Peter, and if John is 10m ahead of Peter then

John is 3m ahead of Tom.

93 \ I YEAR (3) _ . I

10 yArtS

Jack ?

- Jack is 7 years old.

Fig. 5(b). Solutions by a visual student (unedited).

Hypotheses, Results and Preliminary Analyses

A multiple regression analysis was planned in order to investigate the influence on mathematical performance of the cognitive abilities and preferences measured by N.F.E.R. Spatial Tests E.G. and II, the Mitchelmore 3-D Drawing test, Street's Gestalt Completion Test, the twelve items from Set D of Raven's

Progressive Matrices, and an Elementary Mathematics Test (the fifteen-problem modified version of Suwarsono's mathematical processing instrument). Although

287


it was recognized that the sample of 116 students in the present study might not be representative of senior secondary or lower tertiary mathematics students in other places, it was decided, nevertheless to employ inferential statistical procedures, it being understood that any suggested inferences should be

regarded as tentative hypotheses suitable for further investigation with other

samples. For the regression analysis scores on the two cumulative Engineering Mathe-

matics tests (the 'Pure' Mathematics test requiring manipulation of algebraic, trigonometric, and vector expressions, and the 'Applied' Mathematics Test containing problems in elementary mechanics) would constitute dependent variables, and the other variables possible predictor variables. For the regression analysis with 'Applied' Mathematics as the dependent variable, 'Pure' Mathe- matics would also be included as a possible predictor variable, but 'Applied' Mathematics would not be included as a possible predictor variable for 'Pure' Mathematics. It was hypothesized that either or both of the dependent vari-

ables, 'Pure' and 'Applied' Mathematics, should be expressed as a linear com- bination of some or all of the possible predictor variables. For each possible predictor variable a standardized regression coefficient (Beta value) would be estimated and the probability calculated that an estimated standardized coefficient of at least this magnitude could be obtained by chance if, in fact, the population coefficient were zero. The proportions of variance in the dependent variables arising from each of the possible predictor variables would also be calculated. All calculations would be performed by a computer, a non-causal multiple regression model in the computer subroutine REGRESSION of the Statistical Package for the Social Sciences (Nie et al., 1975, pp. 333-350) being used. The variable names, variable labels, and respective ranges of possible scores were 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 applied to the set of scores defining each predictor variable to check whether the distribution of scores was suffici- ently 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 variables were calculated in order that any difficulty due to multicollinearity, which can arise with highly correlated predictor 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 product- moment coefficients obtained are shown in Table 2.

288


TABLE I

Variable Names, Labels, and Ranges of Possible Scores

Label R SI

Name Lange of Possible cores

1. Unitech 'Pure' PM 0 to 100 Mathematics Test

2. Unitech 'Applied' AM 0 to 30 Mathematics Test

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 3D 0 to 40

Drawing Test

6. Street's Gestalt Gest. 0 to 10

Completion Test

7. Raven's Progressive RPM 0 to 12 Matrices (12 items)

8. Suwarsono's elementary Elmath 0 to 15 Maths. problems (15 problems)

9. Suwarsono's mathematical ANA-VIS - 30 to + 30

processing instrument

TABLE 2

Pearson Product-moment Correlation Coefficients between Possible Predictor Variables

NFER NFER 3D Gest. RPM Elmath ANA-VIS (EG) (II)

NFER 1.00 0.70 0.48 0.10 0.37 0.26 -0.10 (EG)

NFER 1.00 0.56 0.12 0.29 0.29 -0.21 (I)

3D 1.00 0.15 0.09 0.12 -0.04

Gest. 1.00 -0.08 0.06 -0.04

RPM 1.00 0.20 -0.23

Elmath 1.00 -0.23

ANA-VIS 1.00


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 sum-

ming each individual's scores on NFER (EG) and NFER (II). Also, in view of the fact that all correlations between ANA-VIS and the other possible pre- dictors were negative, a new variable ANA-VIS* was created so that each ANA- VIS* score had the same magnitude but 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-logical methods when processing mathematical

problems, and someone with a negative ANA-VIS* score should tend to use visual methods). Table 3 shows the product-moment correlations between the six possible predictor variables, now to be used, and the two dependent variables PM and AM.

TABLE 3 Pearson Product-moment Correlations between Pairs of Predictor and Dependent Variables.

NFER 3D Gest. RPM Elmath ANA-VIS* PM AM (SP)

NFER (SP) 1.00 0.57 0.11 0.35 0.28 0.15 0.35 0.31 3D 1.00 0.15 0.09 0.12 0.04 0.28 0.20 Gest. 1.00 -0.08 0.06 0.04 0.12 0.10 RPM 1.00 0.20 0.23 0.21 0.30 Elmath 1.00 0.23 0.21 0.21 ANA-VIS* 1.00 0.31 0.24

PM 1.00 0.58 AM 1.00

Table 4 shows means and standard deviations for Table 3.

eight variables listed in

TABLE 4 Means and Standard Deviations of Predictor and Dependent Variables

Variable Mean Standard Deviation

NFER (SP) 134.6 28.1 3D 31.0 5.47 Gest. 5.26 1.49 RPM 7.74 2.44 Elmath 11.1 2.13 ANA-VIS* 2.49 8.06 PM 63.7 15.5 AM 21.8 5.60

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The 'Pure'Mathematics Multiple Regression Analysis

When all six predictor variables were retained in a multiple regression analysis with PM as the dependent variable, the contributions of the six variables to the variance of PM could be assessed and compared. Table 5 shows the proportions of PM variance explained (multiple R2) at each step of the analysis, the com-

puter having been programmed to select from remaining predictor variables the one which made the greatest contribution to PM variance.

TABLE 5 Contributions to PM Variance of Six Predictor Variables

Variable Multiple R2 R2 Change F-Value

ANA-VIS* 0.09 0.09 5.71 3D 0.15 0.06 3.26 NFER (SP) 0.19 0.04 3.68 RPM 0.20 0.01 1.84 Gest. 0.21 0.01 0.93 Elmath 0.22 0.01 0.69

From Table 5, it can be seen that, altogether, the six predictor variables contributed to only 22% of the variance in PM. If unique partialled contributions of predictor variables to the variance of the dependent variable are considered, then the ANA-VIS* variable contributed most (9%), followed by Mitchel- more's 3D Drawing Test (6%) and the N.F.E.R. Spatial Tests (4%). If a standardized regression equation for the relationship between PM and the possible predictor variables were formed containing only those predictor variables whose estimated standardized coefficients (Beta values) differed significantly from zero, then ANA-VIS* would be the only possible predictor variable to enter the equation.

The 'Applied 'Mathematics Multiple Regression Analysis

For the regression analysis with AM ('Applied' Mathematics) as the dependent variable PM ('Pure' Mathematics) was added to the list of possible predictor variables. (This was because it was thought that the solutions of problems in

elementary mechanics often require the skills tested on the 'Pure' Mathe- matics test, namely, standard manipulations of algebraic, trigonometric, and vector expressions).

Table 6 shows the proportions of AM variance explained (multiple R2) at each step of the regression analysis.

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TABLE 6 Contributions to AM Variance of Seven Predictor Variables

Variable Multiple R2 R2 Change F-Value

PM 0.29 0.29 31.2 RPM 0.33 0.04 3.35 Elmath 0.35 0.02 1.02 NFER (SP) 0.37 0.02 0.96 Gest. 0.38 0.01 0.62 ANA-VIS* 0.39 0.01 0.54 3D 0.39 0.00 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 Progressive Matrices, with 4% only, next. If a standardized regression equation for the relationship between AM and the possible predictor variables were formed containing only those

predictor variables whose estimated standardized coefficients (Beta values) differed significantly from zero, then 'Pure' Mathematics would be the only possible predictor variable to enter the equation.

Factor Analysis

In order to explore more fully any relationships between the variables used in the present study a factor analysis was done on the data set arising from nine of the variables used (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 communalities of the variables, calculated by an iterative procedure, appearing in the leading diagonal of the final correlation matrix. The final factor matrix was obtained using Varimax rotation of axes, with a four-factor matrix being deemed appropriate according to the Scree test criterion (Child, 1979, pp. 44-45). With n = 116, Burt and Banks' formula for determining statistical

significance of a factor loading indicated that a loading with magnitude 0.30 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 matrix which 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 reasonable to 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

292


TABLE 7 Varimax Rotated Factor Analysis

Variable I II III IV Communality

NFER (EG) 0.78 0.69 NFER (I) 0.84 0.77 3D 0.64 0.46 PM 0.72 0.64 AM 0.71 0.56 ANA-VIS* 0.66 0.48 Elmath 0.31 0.16 RPM 0.69 0.62 Gest. 0.07

processing' factor. Factor IV, for which the only substantial loading was Raven's Progressive Matrices, might tentatively be regarded as a 'reasoning' factor. Interestingly, no variable had a loading of magnitude more than 0.30 on more than one factor.

6. DISCUSSION

In view of the substantial and growing literature on relationships between

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

Engineering students in the sample. Multiple regression analysis revealed that the unique contribution of the N.F.E.R. EG and II tests, and Mitchelmore's 3D Drawing Test totalled only about 10% of the variance of the 'Pure' Mathe- matics Cumulative test scores, and only about 2% to the variance of 'Applied' Mathematics test scores once the influence of 'Pure' Mathematics had been partialled out. Factor analysis also drew attention to the lack of any substan- tive relationship between the spatial ability variables and mathematical variables. The N.F.E.R. Tests and Mitchelmore's 3D Drawing Test loaded strongly on one factor, but did not load on the factor of which 'Pure' and 'Applied' Mathematics loaded strongly.

Another important observation is that the modified form of Suwarsono's mathematical processing instrument which was used would seem to provide a promising method for measuring a person's 'preferred mode of processing mathematical information'. An examination of the correlation matrix arising from the variables used in the present study, and the multiple regression and factor analyses, reveals that the ANA-VIS* variable clearly measures a non- trivial component of cognition which is distinct from any of the other


components measured. From the correlation matrix shown in Table 3, it can be seen that ANA-VIS* has correlations with Raven's Progressive Matrices Ele- mentary Mathematics, 'Pure' Mathematics and 'Applied' Mathematics, which are statistically significantly different from zero. The multiple regression analysis with 'Pure' Mathematics as the dependent variable (see Table 5) indicates that ANA-VIS* was the only predictor variable to make a significant contribution to the variance of PM. Factor analysis (see Table 7) confirmed the view that ANA-VIS* measured a 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 smaller than the ANA-VIS* loading). This factorization suggests that the Suwarsono instrument measures a 'mathematical processing' trait. Further research, aimed at clarifying the characteristics of this trait, is needed.

The nature of the relationship between 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* correlates positively with all other variables, including the mathematical and spatial ability variables. Thus, there was a tendency for students who preferred to process mathematical information by verbal-logical means to out-perform other students on both mathematical and spatial tests. So far as mathematical performance is concerned, this interpretation is supported by the multiple regression analysis with 'Pure' Mathematics as the dependent variable.

The relationships between ANA-VIS* and the mathematical and spatial variables in the present study are not easily reconciled with the existing literature. In particular, our results, might appear to be in direct conflict with those of Moses (1977, 1980) and Webb (1979), who reported that students who prefer visual solution processes when attempting mathematical problems tend to outperform those who prefer less visual processes. A possible explanation for the apparent conflict is that in the present study the mathematical variables were measured by tests which did not require the solution of difficult, un- familiar word problems whereas this was the case in both the Moses and Webb studies. We would recommend that future researchers should distinguish between processes preferred by persons attempting routine and non-routine mathematical word problems.

So far as the relationship between preferred mathematical processing and 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

294


processing mode when attempting them would appear to be unable, or unwill-

ing, to abstract in situations where abstracting would provide the most efficient methods of solution. Such a person tends, in our view, to retain as part of his

thinking, unnecessary 'concrete' details. By contrast, the person who uses a more verbal-logical mode demonstrates an ability to cast away such unnecessary 'concrete' details. In the language of the developmental psychologist the latter

person is more likely to be at the stage of 'formal operations' than the former. When confronted with more difficult word problems the latter person is likely to do better because his thinking will not be cluttered with unnecessary visual

images. We would emphasize that our results do not indicate that a person who

prefers a less visual processing mode 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 preference for solving mathematical problems by verbal-logical means.

(This was the student whose solutions to three problems are 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, McDaniel and Angelo (1978) have argued that

good spatial tests must require Gestalt processing, and our results therefore raise several questions. Are the N.F.E.R. and Mitchelmore tests adequate tests of spatial ability? Is Street's Gestalt Completion Test a poor test of Gestalt

processing? Is it in fact true that Gestalt processing is an important factor in mathematical and spatial processing? These, and other possible questions, might be worthy of investigation by future researchers.

The multiple regression analysis with 'Pure' mathematics as the dependent variable indicate that only about 22% of the variance in PM was explained by the six predictor variables. While this analysis encouragingly revealed that the

processing variable ANA-VIS* contributed more than other predictor variable to the variance of PM, the analysis must, nevertheless, serve as a warning to those who stress the importance of spatial and processing variables for mathematical problem-solving. There are many non-mathematical variables, such as student motivation, work habits, teaching, and language competence, which are potentially important in explaining mathematical performance. In Papua New Guinea the language factor could be especially important because English, the language of instruction and the language in which mathematical problems are invariably posed, is usually the third or fourth language acquired by children. It is likely that even university students in Papua New Guinea are often not able to cope with the subtleties of English expression which can occur in the wording of mathematical problems.

It is stressed that the above conclusions arose from a study involving 116


first-year Engineering students in Papua New Guinea. Generalizations based on such a sample may not apply to mathematics learners at the same or different levels in other parts of the world. Further, the mathematical tasks used for the PM and AM tests were of a routine type, and the imagery variable was based on student's processing of elementary mathematical tasks. A different pattern of results may have been obtained if non-routine mathematical tasks had been used.

7. SUMMARY

In concluding this paper we summarize the seven points made in the previous section with respect to possible implications of the analyses which had been

presented. 1. Multiple regression analysis suggested that spatial ability and knowledge

of spatial conventions did not have a large influence on the mathematical

performances of the 116 Engineering students in the sample. 2. Suwarsono's mathematical processing instrument would appear to pro-

vide a promising method for measuring a person's 'preferred mode of processing mathematical information'. Also, the use of the instrument in the present study provided data which, when analyzed, suggested the existence of a distinct cognitive trait associated with mathematical processing.

3. There was a tendency for students who preferred to process mathematical information by verbal-logical means to outperform more visual students on both mathematical and spatial tests.

4. The results of the present study appear to be in conflict with other studies which suggest that it is desirable to use visual processes when 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' Mathematics tests, whereas in most other relevant studies difficult, non-routine mathematical word problems have been used.

5. The tendency towards superior performance on mathematical tests by students who preferred a verbal-logical mode of processing mathematical information might be due to a developed ability to abstract readily, and, therefore, to avoid the formation of unnecessary visual images.

6. The failure of Street's Gestalt Completion Test to correlate significantly with any of the mathematical and spatial tests needs to be explained.

7. Many non-mathematical variables, such as student motivation, work habits, teaching, and language competence, which could contribute significantly to mathematical performance, were not measured in the present study. The

language competence variable could be especially important in the Papua New Guinean context.

Papua New Guinea University Monash University of Technology

296


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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Andrich, D.: 1975, 'The Rasch multiplicative binomial model: Applications to attitude data'. Research report no. 1, Measurement and Statistics Library, Department of Education, University of Western Australia.

Barratt, P. E.: 1953, 'Imagery and thinking', Australian Journal ofPsychology 5, 154-164. Bishop, A. J.: 1973, 'Use of structural apparatus and spatial ability: a possible relation-

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Article Contentsp. [267]p. 268p. 269p. 270p. 271p. 272p. 273p. 274p. 275p. 276p. 277p. 278p. 279p. 280p. 281p. 282p. 283p. 284p. 285p. 286p. 287p. 288p. 289p. 290p. 291p. 292p. 293p. 294p. 295p. 296p. 297p. 298p. 299

Issue Table of ContentsEducational Studies in Mathematics, Vol. 12, No. 3 (Aug., 1981), pp. 267-398Spatial Ability, Visual Imagery, and Mathematical Performance [pp. 267-299]Cognitive Demand of Secondary School Mathematics Items [pp. 301-316]Concepts Associated with the Equality Symbol [pp. 317-326]An Investigation into Subtraction [pp. 327-338]Personality and the Learning of Mathematics [pp. 339-350]Instrumentalism as an Educational Concept [pp. 351-367]Busprongs [pp. 369-371]Undergraduate Investigations in Mathematics [pp. 373-387]The Complementary Roles of Intuitive and Reflective Thinking in Mathematics Teaching [pp. 389-397]

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