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University mathematics students'conceptions of mathematicsK. Crawford a , S. Gordon a , J. Nicholas a & M. Prosser ba University of Sydney, Australiab La Trobe University, AustraliaPublished online: 05 Aug 2006.

To cite this article: K. Crawford , S. Gordon , J. Nicholas & M. Prosser (1998) Universitymathematics students' conceptions of mathematics, Studies in Higher Education, 23:1, 87-94, DOI:10.1080/03075079812331380512

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Studies in Higher Education Volume 23, No. 1, 1998

R E S E A R C H N O T E

University Mathematics Students' Conceptions of Mathematics K. CRAWFORD, S. GORDON & J. NICHOLAS Univers#y of Sydney, Australia

M. PROSSER La Trobe University, Australia

87

ABSTRACT A Conceptions of Mathematics Questionnaire was developed, designed to provide an indicator of the nature of students' conceptions of the subject matter they were studying. The scales of the questionnaire were based upon a set of conceptions of mathematics identified in an earlier phenomenographic study of students studying mathematics. The scales represented fragmented and cohesive conceptions of mathematics. Evidence was found supporting both the validity and reliability of the scales. A fragmented conception was found to be associated with a surface approach and a cohesive conception was found to be associated with a deep approach to studying mathematics.

Introduction

Recent research on student learning in higher education has emphasised the importance of the interaction of, among other things, students' prior experiences of studying; prior concep- tions of leaming and the way students perceive their learning environment; how they approach their studies within that environment; and ultimately the quality of their learning outcomes (TrigweU & Prosser, 1991; Ramsden, 1992). One area that has received little attention is that of students' prior conceptions of the nature of the subject matter they are studying, as distinct to their prior knowledge of key concepts and ideas.

In this article we describe an approach to looking at students' conceptions of the subject matter they are studying. Specifically we describe the development of a Conceptions of Mathematics Questionnaire which we are using in an ongoing study of student learning in a first year university mathematics course. The approach is based upon using (1) the results of a phenomenographically-based study of the qualitative variation in students' conceptions of mathematics (Crawford et al., 1994), followed by (2) the development of a quantitatively- based questionnaire with scales based upon the categories of description identified in the phenomenographicaUy-based study.

Phenomenographically-based Qualitative Study

In an earlier investigation the conceptions of mathematics held by students enrolled in a first year university mathematics course were identified from an analysis of students' written statements. These were open-ended responses to the question: Think about the maths you've done so far. What do you think mathematics is? That study used a phenomenographic approach to identify the structure of the qualitative variation in those conceptions (Marton, 1988;

0307-5079/98/010087-08 © 1998 Society for Research into Higher Education

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Crawford et al., 1994). The conceptions, and a representative quotation for each, are summarised as follows.

Fragmented Conceptions: A. Maths is numbers, rules and formulae.

Mat_hs is the study of numbers and the application of various methods of changing numbers.

B. Maths is numbers, rules and formulae which can be applied to solve problems.

Mathematics is the study of numbers and their applications in other subjects and the physical world.

Cohesive Conceptions: C. Maths is a complex logical system and way of thinking.

Mathematics is the study of logic. Numbers and symbols are used to study life in a systematic perspective and requires [sic] the mind to think in a logical and often precise manner.

D. Maths is a complex logical system which can be used to solve complex problems.

Maths is an abstract reasoning process which can be utilised to explore and solve

problems.

E. Maths is a complex logical system which can be used to solve complex problems and provides insights used for understanding the world.

Techniques for thinking about observable, physical phenomena in a quantitative way and also for thinking more abstractly with little or no relation to the directly observable universe.

These conceptions, and their structural relationships, are described in detail in Crawford et aL (1994). Here it is important to note the structural distinction between Conceptions A and B on the one hand and Conceptions C, D and E on the other. Conceptions A and B present mathematics as a fragmented body of knowledge, while Conceptions C, D and E describe a cohesive view of knowledge. It should also be noted that the conceptions are logically and empirically inclusive, forming a hierarchy, with Conception A at the bottom of the hierarchy

and Conception E at the top. While that work was based upon detailed qualitative analyses of students' written

statements, our ongoing research required us to develop a way of obtaining information about students' conceptions of mathematics without the detailed qualitative work. As a result, it was decided to develop a Conceptions of Mathematics Questionnaire, based upon the results of the qualitative study.

Development of the Questionnaire

Conceptions of Mathematics Questionnaire

In the previous qualitative study, the written statements made by the students to the open-ended question of what they thought mathematics was about were classified in terms of the conceptions of mathematics identified in the study. These statements were then used as the basis for developing a pool of questionnaire items representing two broad conceptions-- fragmented and cohesive--representing the major qualitative divide.

The face validity of the items was examined by each author classifying each item

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Research Note 89

according to whether it represented a fragmented or a cohesive conception. After comparing the classifications of each item among the authors, the items were revised. This resulted in a pilot version of the questionnaire. This version was piloted on a group of upper secondary mathematics students. After a series of internal consistency reliability and factor analyses, the questionnaire was filrther revised and used with a group of about 300 first year university mathematics students. These students were enrolled in a mathematics course designed for students intending to major in mathematics and who had a relatively high level of achieve- ment in mathematics in their last 2 years of high school. The questionnaire was administered twice to these students--at the beginning of their first semester of studying mathematics at university and, 5 months later, at the beginning of their second semester of studying mathematics.

Analysis of Data

The results from the questionnaires were analysed to help establish the reliability and further establish the validity of the questionnaire (the face validity of each item had previously been established). The data collected were analysed as follows.

Individual item analysis. The means and standard deviations of each item and each scale were analysed in order to ensure that each item had a satisfactory mean (i.e. there was no ceiling effect) and standard deviation (i.e. that each item had a reasonable dispersion).

Item validity: item factor analyses. Item factor analyses were conducted on both the pre-test and the post-test data with the aim of testing whether the factor structure agreed with our previous classification of the items; that is, that the items which were previously classified as representing fragmented conceptions loaded on one factor, and items classified as repre- senting cohesive conceptions loaded on another, separate, factor.

Scale reliabilities. The internal consistency of the scales on both the pre-test and the post-test was examined using Cronbach's alpha. An estimate of the test-retest reliability was made conducting a Pearson correlation analysis between the pre-test and the post-test data. This should be treated with some caution as the students received instruction in mathematics between the pre-test and the post-test. Nevertheless, unless the intervention had differential effects on students so that students who scored highly on the pre-test also scored lowly on the post-test, substantial and statistically significant correlations would provide support for test-retest reliability.

Scale validity. Support for the criterion-related validity of the scales (see Gronlund, 1976) was provided by correlating the scale scores with scale scores on an approaches to study questionnaire. Our previous qualitative study had established a relationship between students' conceptions of mathematics and their approach to studying it. In that study- it was shown that students with fragmented conceptions tended to adopt a surface approach to studying mathematics, while students with cohesive conceptions adopted a deep approach. Such a relationship would be expected if in adopting surface approaches students focus on reproducing parts, whereas in adopting deep approaches students focus on understanding wholes. A surface approach would be expected to be related to fragmented conceptions, whereas a deep approach would be expected to be related to cohesive conceptions. A modified version of the Study Process Questionnaire (Biggs, 1987) was used in order to obtain data about the students' approaches to studying mathematics. The modified question- naire, Approaches to Learning Mathematics Questionnaire, is composed of two scales; one measuring a surface approach to studying mathematics and the other a deep approach to studying mathematics. Students who adopt a surface approach to learning mathematics do so

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with an intention to reproduce, while those who adopt a deep approach do so with an intention to understand.

Table I shows means and standard deviations for each item and scale on both the pre-test and post-test. The item and scale means ranged from 1.75 for I tem 4 on the pre-test to 4.07 for I tem 3 on the pre-test. The individual item standard deviations ranged from 0.82 for I tem 4 on the pre-test to 1.07 for Item 11 on the pre-test. These standard deviations represent reasonable dispersions for all items. It should be noted that for eight items there was a statistically significant difference in means between the pre-test and the post-test. Six of these were items representing Fragmented Conceptions scale, with five showing a statisti- cally significant reduction in the mean. The remaining two items represented Cohesive Conceptions, and in both cases the means also declined.

The pre-test and post-test means of the Fragmented Conceptions Scale were 3.17 and 3.09 respectively. This represented a statistically significant decline from pre-test to post-test. The Means for the Cohesive Conceptions scales were 3.62 and 3.58 respectively, with no statistically significant difference from pre-test to post-test. It should be noted that the means for the Cohesive Conceptions scales were substantially higher than the means for the Fragmented Conceptions scales.

Table II shows the results of the item factor analyses, using principal components with varimax rotation, on both the pre-test and the post-test data. In the analysis of the pre-test data, four factors with eigenvalues greater than one were identified (4.t3, 3.64, 1.16 and 1.14). The analysis of the post-test data identified three factors with eigenvalues greater than one (5.03, 4.28 and 1.08). In both analyses, an examination of the scree plots suggested that there were only two factors. Given Cattell's (1978) suggestion that the 'eigenvalue = 1' criterion can overestimate the number of factors in analyses with a relatively large number of variables, we report the two factor solutions.

The factor analyses of both the pre-test and the post-test data confirms the factor structure of the questionnaire developed in the pilot study. That is, the items designed to represent the Fragmented Conceptions scale loaded on one factor, while the items designed to measure the Cohesive Conceptions scale loaded on the other factor. The only item to be a little inconsistent was CMQ15 , which was an item which on face validity represented a cohesive conception, but which on the pre-test had a reasonably high loading on the factor representing the Fragmented Concept ion scale. This item was subsequently deleted from the further analysis.

A defining item from each of the two scales and scale internal consistency reliabilities are shown in Table III. Table III shows that the two scales have high internal consistency on both the pre-test and the post-test administrations. This suggest that the items within each scale are empirically measuring the same or a similar construct.

While the questionnaire was not administered to check the test-retest reliability (there was an intervention between the pre-test and the post-test), nevertheless it would be expected that there would be substantial stability over time and that those who scored highly on the pre-test would also score highly on the post-test (Gronlund, 1976). Table IV shows Pearson correlation coefficients between the two scales on both the pre-test and the post-test.

This analysis shows substantial, positive mad statistically significant correlations between the pre-test and post-test scales (r = 0.59, p < 0 . 0 1 for the Fragmented Conception scale; r = 0.60, p < 0.01 for the Cohesive Concept ion scale), indicating that students who score highly on a pre-test scale also score highly on the same post-test scale. These correlations are consistent with the questionnaire having substantial test-retest reliability (and also that the conceptions are quite stable over time). Table IV also shows negative, small, but statistically significant, correlations between the Fragmented and Cohesive scales. This suggests that

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TABLE I. Conceptions of Mathematics Questionnaire items, scales, means and standard deviations

Pre-test Post-test

Item Mean SD Mean SD

1. For me, mathematics is the study of numbers 2.73 0.97 2.71 0.96 2. Mathematics is a lot of rules and equations 3.17 1.01 3.22 0.99 3. By using mathematics we can generate new knowledge 4.07t 0.82 3.91t 0.83 4. Mathematics is simply an overcomplication of

addition and subtraction 1.75t 0.80 2.05t 0.92 5. Mathematics is about calculations 3.47 t 0.96 3.23t 0.93 6. Mathematics is a set of logical systems which have been

developed to explain the world and relationships in it 3.74 0.92 3.81 0.83 7. What mathematics is about is finding answers through

the use of numbers and formulae 3.63t 0.92 3.44t 0.88 8. I think mathematics provides an insight into the

complexities of our reality 3.27 1.01 3.30 1.04 9. Mathematics is figuring out problems involving numbers 3.43t 0.90 3.22 t 0.87

10. Mathematics is a theoretical framework describing reality with the aim of helping us understand the world 3.33 1.00 3.35 1.03

11. Mathematics is like a universal language which allows people to communicate and understand the universe 3.42 1.07 3.38 1.01

12. The subject of mathematics deals with numbers, figures and formulae 3.70t 0.81 3.48t 0.85

13. Mathematics is about playing around with numbers and working out numerical problems 3.15 0.96 3.16 0.94

14. Mathematics uses logical structures to solve and explain real life problems 3.79* 0.85 3.68* 0.88

15. ~rhat mathematics is about is formulae and applying them to everyday life and situations 3.31 0.92 3.26 0.92

16. Mathematics is a subject where you manipulate numbers to solve problems 3.32 0.91 3.25 0.82

17. Mathematics is a logical system which helps explain the things around us 3.59 0.85 3.51 0.88

I8. Mathematics is the study of the number system and solving numerical problems 3.41 t 0.83 3.26t 0.84

19. Mathematics is models which have been devised over years to help explain, answer and investigate matters in the world

Fragmented Scale Items (1,2,4,5,7,9,12,13,16,18) Cohesive Scale Items (3,6,8,10,11,14,15,17,19)

3.70 0.83 3.66 0.94

3.17" 0.53 3.09* 0.58 3.62 0.63 3.58 0.68

Notes: *difference between pre-test and post-test means is significant at p < 0.05; ]'difference between pre-test and post-test means is significant at p < 0.01.

Responses were made using a five point Likert scale ranging from Strongly Disagree (1) to Strongly Agree (5).

t h e r e is a sma l l t e n d e n c y for s t u d e n t s w h o score h igh ly o n o n e scale to score lowly o n t he

o the r , a n d t h a t the two scales are re la t ively i n d e p e n d e n t o f e a c h o t h e r a n d are m e a s u r i n g two

qua l i t a t ive ly d i f f e ren t c o n c e p t i o n s . T h i s m a y s e e m a l i t t le i n c o n s i s t e n t w i t h t h e ear l ie r

s u g g e s t i o n t h a t t he c o n c e p t i o n s are h i e r a r ch i ca l l y re la ted . H o w e v e r , i t s h o u l d b e r eca l l ed t h a t

t h e r e is a m a j o r b r e a k b e t w e e n t h e first two c o n c e p t i o n s ( m a k i n g u p t he F r a g m e n t e d

C o n c e p t i o n s scale) a n d t he las t t h r e e ( m a k i n g u p t he C o h e s i v e C o n c e p t i o n s scale) . I t is th i s

m a j o r b r e a k t h a t t h e p r e s e n t analys is is h igh l igh t ing .

As a n i n d i c a t o r o f t he c r i t e r i o n - r e l a t e d va l id i ty o f t h e q u e s t i o n n a i r e , T a b l e V

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TABLE IL Conceptions of Mathematics Questionnaire item factor analyses

Pre-test Post-test

Scale items Factor 1 Factor 2 Factor I Factor 2

Cohesive CMQ3 48 61 CMQ6 74 72 CMQ8 69 73 CMQ10 76 77 CMQ11 66 62 CMQ14 68 80 CMQ 15 34 48 56 CMQ17 78 83 CMQ 19 64 81

Fragmented CMQt 54 CMQ2 54 CMQ4 48 CMQ5 57 CMQ7 58 CMQ9 72 CMQ12 64 CMQ13 66 CMQ16 45 CMQ18 70

Explained variance (%) 22 19 27

53 56 44 67 63 77 77 70 68 74

23

Decimal points omitted. Items with loadings tess than 0.40 omitted.

shows the Pearson correlation coefficients between the Conceptions of Mathematics scales

and Approaches to Study as measured by a modified version of Biggs's Study Process Questionnaire (Biggs, 1987; Crawford et al., 1995) which is composed of scales measuring

deep and surface approaches to study. Table V shows substantial, positive and statistically significant correlations between the Fragmented Concept ion scale and the Surface Approach

scales on both the pre-test (r = 0.37, p < 0.01) and the post-test (r = 0.36, p < 0.01), suggest- ing that students with high scores on the Fragmented Conceptions scale tend to have high scores on the Surface Approaches to Study scale and vice versa. It also shows large, positive

and statistically significant correlations between the Cohesive Conception scale and the Deep

TABLE III. Conceptions of mathematics scales items and internal consistency reliabilities

Cronbach Alpha

Scale and defining item Pre-test Post-test

Fragmented (10 items) CMQ9 Mathematics is figuring out problems involving numbers

Cohesive Conception (8 items) CMQ17 Mathematics is a logical system which helps explain the things around us

0.79 0.85

0.84 0.88

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TABLE IV. Correlations between the pre-test and post-test administrations of the Conceptions of Mathematics Questionnaire

Conceptions Scale

Fragmented Conception s c a l e Pre-test Post-test

Cohesive Pre-test Post-test

Fragmented Pre-test Post-test

Cohesive Pre-test Post-test

0.59t -0.13" -0.07* -0.13" -0.11"

0.60t

*p < 0.05; t p< 0.01.

Approach scale on both the pre-test (r = 0.59, p < 0.01) and the post-test (r = 0.43, p < 0.01),

suggesting that students who score highly on the Cohesive Conceptions scale also tend to score highly on the Deep Approaches to Study scale and vice versa. These correlations therefore suggest that a fragmented conception of mathematics tends to be associated with a

surface approach to studying mathematics, whereas a cohesive conception tends to be

associated with a deep approach to the study of mathematics. This result is consistent with the theory and our previous qualitative study, and as such, supports the validity of the Conceptions of Mathematics Questionnaire.

Discuss ion and Conclus ion

Whilst there has been substantial previous research into students ' prior knowledge of subject

matter, and how that relates to their approaches to study and subsequent learning outcomes, there has been little or no research into students ' prior unders tanding of the nature of the

subject matter they are studying. In this research note we have described an approach to investigating this issue. It involves the identification of the structure of the qualitative

variation of conceptions using a phenomenographic analysis of open-ended written state- ments by students and the subsequent development of a questionnaire based upon the

TABLE V. Correlations between the Conceptions of Mathematics scales and the Approaches to Study scales

Conceptions variables

Fragmented Cohesive Study process variables Pre-test Post-test Pre-test Post-test

Surface Pre-test (~ = 0.78) 0.37t 0.33 t - 0.21t 0.13" Post-test & = 0.77) 0.25t 0.36t - 0.15t - 0.04

Deep Pre-test (~ = 0.86) - 0.12" - 0.13" 0.59t 0.31 t Post-test (~ = 0.87) - 0.09 - O. 12" 0.40t 0.43t

*p < 0.05; tp < 0.01.

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categories of descr ipt ion and s tudent responses. This study has shown that such a quest ion- naire can be developed which has reasonable reliability and validity.

The study has also shown, as par t of an investigation of the cri ter ion-related validity of the questionnaire, that s tudents ' pr ior conceptions are systematically related to the way they approach their study. Given the previous research showing that approaches to study are systematically related to the quality of s tudents ' unders tanding of key concepts of the subject mat te r they are studying, an awareness of s tudents ' pr ior concept ions would be expected to be an impor tan t considerat ion in efforts to improve the quality of s tudent learning.

Our ongoing research looks at the relat ionships between s tudents ' pr ior conceptions, their percept ions of the learning context , their approaches to stud3, and eventually the quali ty of the learning outcomes. However , even on the basis of the work we have already comple ted we would argue that s tudents ' pr ior concept ions of the nature of the subject mat te r they are s tudying needs to be taken into account in the design and teaching of courses in higher educat ion, and that the deve lopment of questionnaires in different fields o f s tudy similar to the one developed in this s tudy would be an impor tan t first step in further unders tanding and deve lopment of s tudent learning within part icular subject fields.

Correspondence: M. Prosser, Academic Deve lopment Unit , La T r o b e University, Bundoora ,

Victoria, Austral ia 3083.

REFERENCES

BIGGS, J.B. (1987) Student Approaches to Learning and Studying (Hawthorn, Victoria, Australian Council for Educational Research).

CATTELL, R.B. (1978) The Scientific Use of Factor Analysis in the Behavioural and Life Sciences (New York, Plenum Press).

CRAWFORD, K., GORDON, S., NICHOLAS, J. & PROSSER, M. (1994) Conceptions of mathematics and how it is learned: the perspectives of students entering university, Learning and Instruction, 4, pp. 331-345.

CRAWFORD, K., GORDON, S., NICHOLAS, J. & PROSSER, M. (1995) Patterns of meaning of students' mathemat- ical experiences at university, Proceedings of the Eighteenth Annual Conference of the Mathematics Education Group of Australasia, Darwin.

GRONLUND, N.E. (1976) Measurement and Evaluation in Teaching (New York, Macmillan). MARTON, F. (1988) Describing and improving learning, in: R. SCHMEC (Ed.) Learning Strategies and Learning

Styles, pp. 53-82 (New York, Plenum Press). RAMSDEN, P. (1992) Learning to Teach in Higher Education (London, Routledge). TRIGWELL, K. & PROSSER, M. (1991) Improving the quality of student learning: the influence of learning

context and student approaches to learning on learning outcomes, Higher Education, 22, pp. 251-266.

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