The influence of context on teachers' conceptions
of mathematics and its teaching
Paper presented at the European Conference on Educational Research, Edinburgh, 20-23 September 2000
Paul Andrews
University of Cambridge School of Education
17 Trumpington Street
Cambridge
CB2 1QA
UK
+44 (0) 1223 336290
Abstract
Much of the research into teachers' conceptions of mathematics and its teaching has implicitly assumed the generalisability of its results to national contexts beyond those in which the studied teachers work. This paper challenges such assumptions by comparing the perspectives on mathematics and its teaching held by teachers from Hungary and England. A questionnaire was developed and sent to all teachers in around 200 secondary schools in England and 40 in Budapest. Responses were received from almost 600 English teachers of mathematics and more than 100 Hungarian. Factor analyses yielded nine constructs - four of mathematics and five of mathematics teaching. Statistical analyses identified several differences in national conceptions indicating the existence of both global and national perspectives which are informed by national pedagogic traditions. English teachers' perspectives indicated an applications-oriented subject mediated by means of open ways of classroom working. Hungarian teachers appeared to hold more coherent perspectives and, significantly, minded to reject the English view. Age, gender and qualification appeared to influence both cohorts' perspectives although substantially greater variation was found amongst the English and which supported further the notion that Hungarians operate within a more closely defined pedagogic tradition.
Introduction
The systematic study of teachers' conceptions of mathematics and its teaching has been of continuing interest to researchers for several years for, as Thompson (1992) notes, they "play a significant role in shaping teachers' characteristic patterns of instructional behavior" (Thompson, 1992, pp.130/131). For some, the relationship between belief and practice has been a philosophical enquiry with, inevitably, conjectural outcomes (Lerman, 1983, 1990; Steiner, 1987; Ernest, 1989a, 1989b, 1995) whilst others have sought confirmatory evidence (Thompson, 1984; Frank, 1990; Ball, 1990; Foss and Kleinsasser, 1996, Andrews and Hatch 1999a). Despite this, beliefs and their impact on teaching have not been substantially researched because those involved "have assumed that readers know what beliefs are" and "the difficulty of distinguishing between beliefs and knowledge" (Thompson, 1992, p.129).
This is not the place for a lengthy discussion on the nature of beliefs, knowledge and conceptions - this can be found in, for example, Pajares (1992) although a summary would be appropriate. The literature indicates that beliefs operate at two levels. At the lowest level, although Green (1971) has doubts that such things exist, are single beliefs which may be characterised in four ways. They may pertain to the existence of entities outside the believer's control, they may represent an idealistic alternative world, they may have both affective and evaluative components and they may derive from a person's episodic experiences (Abelson, 1979; Nespor, 1987). They are "deeply personal, rather than universal, and unaffected by persuasion. They can be formed by chance, an intense experience, or a succession of events, and they include beliefs about what oneself and others are like" (Pajares, 1992, p.309). At the second level, which Green argues is where they are overtly manifested, beliefs are clustered into systems which Thompson (1992) describes as organising structures. A belief system may be held in isolation of others, making it possible for individuals to hold apparently conflicting beliefs (Green, 1971). They do not require social consensus or even internal consistency (Da Ponte, 1994). Within a belief system may be found both primary and derivative, central or peripheral beliefs which indicate that the beliefs within a system are neither entirely independent nor equally susceptible to external influence. Nespor (1987) suggests that beliefs are distinguishable from knowledge primarily because belief systems are non-consensual and, consequently, disputable. Knowledge, despite systematic change is, generally, verifiable. Beliefs tend to change according to gestalt shifts rather than in response to reason with linkages between beliefs and real-world events being bound up with the episodic experiences of the believer and unpredictable in their manifestation. Knowledge tends to be better defined in its application.
The literature in respect of conceptions is less well-defined. Da Ponte (1994) writes that conceptions are the underlying organising frames of concepts and, essentially, cognitive in nature. Ernest (1989b) argues that a conception is a belief system and, in essence, affective. Thompson adopts a more eclectic perspective and describes conceptions as "conscious or subconscious beliefs, concepts, meanings, rules, mental images, and preferences" (Thompson, 1992, p.132). That is, conceptions may have both cognitive and affective dimensions. All write that the totality of an individual's conceptions and belief systems form the basis of a philosophy, though not necessarily articulated, of mathematics which in turn inform a philosophy of mathematics education.
In this paper the word conception - whether of mathematics itself or of mathematics teaching - is used in accord with Thompson's (1992) broader interpretation which includes components of both knowledge and belief. It is argued that both are inextricably linked in that each continuously influences the other. It is acknowledged, also, that some writers prefer the word perception to conception. Others, despite indicating that they pertain to different constructs, appear to use the words conception, perception and belief synonymously.
It is well documented that teachers begin their careers with previously constructed, naive, idiosyncratic, and subconscious theories about teaching (Clark, 1988; Powell, 1992). These theories, frequently modelled on teachers who taught them (Feiman-Nemser and Buchmann, 1986; Calderhead and Robson; 1991; Harel, 1994), are not easily changed - either during training (Harel, 1994; Foss and Kleinsasser, 1996, Sinkinson 1996) or after (Clark 1988). They are, quite naturally, prone to their perspectives on the nature of mathematics (Ernest, 1989; Frank, 1990; Ball, 1990; Foss and Kleinsasser, 1996; Andrews and Hatch, 1999a). Indeed, as Thompson (1984) notes;
"...the observed consistency between the teachers' professed conceptions of mathematics and the manner in which they presented the content strongly suggests that the teachers' views, beliefs and preferences about mathematics do influence their instructional practice" (Thompson, 1984, p.124/125)
The relationship between belief and practice appears to be generally consistent (Andrews and Hatch, 1999a) although there may be inconsistencies (Thompson, 1984) related to the depth and consciousness of a teacher's beliefs and the particular schools in which they operate (Ernest, 1989). However, the evidence indicates that little explicit account has been taken of either the wider context - national - in which teachers operate or biographical details concerning age, gender and qualification. This paper outlines the results of a questionnaire study undertaken in both England and Hungary.
Method
One of the difficulties of research of this nature is that an individual's beliefs "may lurk beyond ready articulation" (Munby 1982, p.217) to the extent that they may be "accessible only by inference" (Fenstermacher 1978, p.103). In acknowledgement of this a five section questionnaire was devised to explore respectively, teachers' conceptions of mathematics, conceptions of mathematics teaching, self-reports of their own classroom behaviours, and topic preferences in respect of their teaching. The fifth section sought details pertaining to qualification, gender and teaching experience. A pilot was developed, in consultation with mathematics education colleagues, trialed on an opportunity sample of 54 teachers following award bearing in-service education courses in the university where the author once taught, and revised in the light of feedback. In respect of the final version, the three sections of the questionnaire which form the basis of this paper (the first three of the five sections described above) comprised 55 items. Each item was placed against, effectively, a nine point Likert-like scale - one equating to a positive response and nine negative. The final version was translated by English-speaking colleagues in the mathematics education department at the Eötvös Loránd Tudományegyetem (ELTE) in Budapest. In presenting this report it is acknowledged, despite attempts to make the translation as true to the original as possible, that practices which seem self-evident in one system may not be in another - one recent study found that assumptions made about the transferability of educational vocabulary across systems were frequently unfounded (Schmidt et al 1996). Also, when we discuss English and Hungarian teachers we are referring to their professional locations and not their nationalities.
Sufficient copies of the questionnaire and stamped addressed envelopes were posted to all teachers in more than 200 schools from three regions of England and almost 40 in Budapest. Details of the procedures adopted in England, and subsequently in Hungary, can be found in Andrews and Hatch (1999a). The numbers of responses received - 577 English and 108 Hungarian - were representative of similar mean responses per school and, fortuitously, proportionate to the size of the countries' populations. Responses were manually coded and analysed using SPSS 9.0 for Windows. The English sample was obtained, effectively, from four regions of the country in order to determine whether teachers' conceptions were prone to regional variation. Details of the regional analysis can be found elsewhere (Andrews and Hatch 1999b).
Results
A reliability analysis was performed on the entire sample of 685 teachers (577 English and 108 Hungarian) and yielded a Cronbach Alpha of 0.809 which was higher than Litwin's (1995) 0.7 threshold for acceptability. However, the systematic removal of eight items raised the coefficient to a much more acceptable 0.863.
Initial factor analyses, also performed on the full sample, proved problematic due to the substantially greater number of English teachers appearing to mask the perspectives of the relatively few Hungarians. Consequently a decision was made to re-run these initial analyses with the full Hungarian cohort and a randomly selected set of 108 English teachers.
The revised factor analyses - principal components with varimax rotation - were performed on the remaining 47 items and a reduced sample of 216 teachers. The number of factors extracted was determined by recourse to several criteria. A consideration of those factors with Eigen values in excess of one yielded a possible fourteen factors which accorded with the principle of accepting only those which accounted for more than the variance associated with a single item (2.13%). However, a scree diagram (fig 1) showed, by factor 11 at the latest, a uniform tailing-off. Thus, it was decided to explore solutions of eight, nine and ten factors. The former and latter returned factors which proved difficult to interpret. The nine factor solution proved satisfactory for two reasons. Not only were the factors more obviously interpretable but they matched almost exactly those yielded by the earlier analysis undertaken on the full English sample (Andrews and Hatch 1999a). The accuracy of this matching is discussed below and full details of the nine factor solution can be seen in Table 1. Four of these factors were thought to be indicative of teachers' perspectives on mathematics and five on mathematics teaching. It was decided, in accord with the approach adopted by others, that factor loadings of 0.4 or higher would be included.
Factor descriptions
1. The five items comprising this factor represent a perspective on mathematics which emphasises the satisfaction gained from mathematics as a means of maintaining one's domestic finances. Although there is a clear stress on enjoyment the key aspect concerns the use of mathematics as a personal economic or financial tool. The construct is believed to be more absolutist than fallibilist in its representing a highly utilitarian perspective on the subject.
2. The seven (one of which loaded more significantly on factor eight) items of this factor emphasise mathematics teaching as concerned with the formal teaching of skills and fluency through regular practice of routine procedures. The factor, which is a measure of principle rather than a commitment to action, also alludes to mathematics as precise with applications to other subjects. Our perception is that it is related more to notions of instrumental understanding than relational, absolute rather than fallible.
Figure 1
Below can be seen the scree graph showing the eigen values of the first twenty factors extracted by the principal components analysis.
3. The eight items of this factor represent mathematics as a pleasurable and worthwhile activity. It is concerned with the fascination and enjoyment which an individual gains from an engagement with mathematics and mathematical knowledge. It could be argued that the overt stress on mathematical investigation and the processes of proof and model-building is indicative of a conception more fallible than absolute.
Table 1
Nine factor principal components with varimax rotation solution.
4. The seven (two of which loaded more significantly on factors two and three respectively) items of this factor offer a perspective on mathematics teaching which reflects an emphasis on pedagogic variety, one which acknowledges the importance of a range of approaches to the teaching of the subject. It is related to principles rather than behaviours - effective mathematics teaching might involve, for example, discussion but an agreement is no commitment to action or a guarantee that the teacher concerned promotes such classroom activity. There is a more that implicit sense that the factor is concerned more with relational understanding and mathematical processes than it is with skills acquisition and instrumental understanding. It is argued that this is a fallible perspective on the teaching of mathematics.
5. The five items here, statements pertaining to what might be found in an individual teacher's classroom, offer a perspective on mathematics teaching which stresses task variety and notions of differentiation. Our perception is that the factor is relates to a teacher's willingness to engage with task differentiation. Despite its representing a perspective on mathematics teaching which acknowledges the individual it is probably indeterminate in its underlying principles - both fallibilist and absolutist teachers might apportion it equal importance.
6. The three items of this factor are unambiguously concerned with the nature of the classroom environment and the individual teacher's expressed commitment to the creation of a mathematically enriched and challenging classroom. As with the previous factor its underpinnings are likely to be indeterminate and dependent upon the nature of the materials teachers choose to place on their walls.