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 Education17 Trumpington Street

CambridgeCB2 1QA

UK

+44 (0) 1223 336290pra23@cam.ac.uk

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.

7. The six items of this factor, statements concerning what might be found in an individual teacher's classroom, describe the teaching and learning of mathematics as taking place in an open and cooperative environment. There is a sense that teachers who score positively on this factor express a commitment to the development of pupil autonomy through facilitation. It is argued that teachers who subscribe to such a perspective do not see themselves as representative of an authoritarian or instrumental tradition. Such a construct, with an explicit reference to mathematical processes, is more likely to derive from a fallibilist rather than absolutist perspective.

8. The four items of this factor offer a perspective on mathematics as an essential tool for successful participation in the world. It is a principle rather than a commitment to action and democratic rather than authoritarian. It is argued that this particular factor

presents mathematics as a means by which pupils might acquire empowerment - a lifetool without which the world becomes problematic. It is probably indeterminate in its underlying principles - both fallibilist and absolutist teachers might apportion it equal importance - although there may be an argument that such a perspective is less likely to be transformative than it is about mathematics as a set of well-defined rules and procedures.

9. The last of the factors comprised three items (one of which loaded on two other factors) concerned with mathematics as a service to other forms of activity. It is a principle which regards mathematics as an application rather than a process. We argue that such a perspective is unlikely to be other than absolutist.

On completion of the factor analyses mean scores were calculated for each teacher on each factor. At this stage correlations were calculated between the English teachers' scores on the constructs identified in this analysis and those of the earlier study. The earlier study yielded a ten factor solution of which nine were thought to be close to those identified here. Several comprised identical items and returned unit correlations. The results can be seen in Table 2. The figures yielded by this process, it is argued, indicate that despite differences in the items comprising them, the two sets of factors are remarkably similar constructs.

Table 2Correlations calculated on the English data between the factors of this study and the corresponding factors of the Andrews and Hatch (1999a) study.

Differences between nations

Differences in national perspectives were explored by means of t-tests. As can be seen from the figures of Table 3, both similarities and differences can be discerned between the two samples. In respect of the former both groups of teachers appeared equally positively disposed towards both mathematics teaching requiring pedagogic variety and personal enjoyment through an engagement with the subject. In respect of all other factors the scores of the national cohorts were significantly different.

The most obvious difference between the two groups concerned the creation of a mathematically enriched classroom environment. Hungarian teachers tended towards

a rejection, responding slightly on the negative side of neutral, whilst their English colleagues were clearly inclined to the positive. Indeed, a difference of almost 3.5 in respect of national means is likely to be indicative of irreconcilable perspectives. A similar, but slightly less marked, polarisation occurred in respect of mathematics teaching as a cooperative and facilitatory act - the English mean being on the positive side of neutral and the Hungarian on the negative. Thus it could be argued, and this is supported by the unusually strong significance levels, that English teachers may perceive environmental factors to have a pedagogic importance that Hungarians do not.

Table 3Mean scores on each factor for all teachers, both national cohorts and t-tests performed to determine the significance of any differences between the cohorts.

The Hungarian rejection of the idea of an enriched classroom is supported by the evidence of the picture below. Taken in the mid 1990s it shows a class of grade five pupils working on some early algebraic ideas. The classroom itself looks little different from the 1955 picture found between pages 86 and 87 of Hoffman's (1998) biography of Paul Erdös.

No other factors created such polarisation, although there were differences in emphases. Both groups tended towards denying that pleasure might be gained from mathematics as a personal economic tool although the Hungarians were clearly more negative than the English; similarly with teachers' professed use of task differentiation. Also, the Hungarian teachers were barely positive in respect of mathematics as a service to other areas of activity whilst the English were clearly so disposed.

Table 4Mean scores and t-tests comparing English teachers' scores by different qualification groups

Graduates only

Qualification

It is difficult to compare across nations in respect of qualification because the manner in which teachers are trained and their initial qualifications vary significantly. In Britain the prevailing mode of entry to secondary teaching is a subject degree followed by a year's teacher training. In Hungary teachers train to work in the primary phase (up to age 14) and then qualify as secondary teachers by means of attendance at award bearing in-service. However, an attempt has been made, not to compare across systems but to compare within systems. In respect of English teachers the results of tables 4 offer some interesting insights. Firstly, there appeared to be no difference between graduates and non graduates on any of the nine factors. There were small, albeit significant differences between those graduates with a post graduate teaching qualification (PGCE) and those without. The former appeared to gain greater pleasure from an engagement with mathematics and were more positive in respect of the creation of a facilitatory classroom environment. The latter were more positively disposed towards mathematics as a service to other areas of activity. However, with no differences between graduates in mathematics and those of other subjects, the most significant differences occurred between those teachers with higher degrees and those without. The former are more negatively inclined towards mathematics as a financial tool and less positive towards it as a service to other areas of activity. The same group seems less positive towards mathematics as skills-based.

There were fewer significant difference between the different constituent Hungarian groups (Table 5). Teachers initially trained for upper primary teaching were more negative towards the creation of an enriched classroom environment than those not. Those who were initially trained for lower primary teaching were less negative than other teachers in respect of mathematics as a personal financial tool, while those who had achieved a mathematics degree (usually in-service) were more negative than other teachers in terms of the creation of a facilitatory classroom environment.

Gender

For both cohorts tests were run to determine the influences, if any, of gender and age. The results for gender can be seen in Table 6. In respect of Hungarian teachers, women appeared more certain than men that little pleasure might be gained from mathematics as a means of maintaining one's finances. Hungarian women were also less positive than men in respect of mathematics teaching as a skills-oriented activity and substantially more negative in terms of creating a mathematically enriched classroom environment. English teachers appeared to have a different set of gender-influenced issues. Women were more convinced than men that mathematics teaching

required pedagogic variety, the creation of a mathematically challenging classroom environment and were more positive in respect of mathematics as an essential lifetool.

Table 5Mean scores and t-tests comparing Hungarian teachers scores by different qualification groups

Age

In respect of age there were several significant results for the English cohort. The sense was that increasing age led to less positive responses with respect to mathematics both as a service to other areas of activity and as a lifetool. Similar relationships were noted with respect to mathematics teaching as requiring task differentiation, being a facilitatory act and the creation of a challenging classroom environment. That is, there is a sense that as far as English teachers are concerned, the younger the teacher the more likely he or she is to adopt an instrumental perspective

on the nature of the subject whilst acknowledging the need for ways of working in the classroom that maximise extrinsic motivation and acknowledge the individual. There were no age-related differences in respect of Hungarian colleagues' conceptions of mathematics and its teaching.

Table 6Means and comparisons by gender and nationality.

Discussion

It is of interest that on not one factor did the Hungarians score more positively than the English or the English score more negatively than the Hungarians. Indeed, the English scored negatively on just two (barely) compared with the Hungarians' four. In essence the English seem to place a greater curricular emphasis on the applications of mathematics than do the Hungarians; scoring significantly more positively in respect of skills-oriented teaching, mathematics as a lifetool and mathematics as a service to other areas of activity. The English also appear disposed towards a view that mathematics teaching is enhanced through the fulfilment of specific environmental considerations - that classrooms are enriched with challenging and informative materials (something the Hungarians reject emphatically) and that teachers encourage informal forms of classroom management - small groups of pupils working together while their teachers support individuals (something the Hungarians are minded to reject). These latter ideas evoke an image of mathematics as process rather than product, fallible rather than absolute, democratic rather than authoritarian and which support a conjecture that the English tend to see their role as teaching a subject justified in absolutist terms but presented in ways commensurate with a fallibilist perspective. That is, there may be a dissonance between the ways English teachers

profess to operate in the classroom and their conception of the subject they teach. This lack of coherence seems less profound in respect of Hungarian teachers who seem to exhibit a pedagogic resonance rather than the pedagogic dissonance of the English.

Table 7Mean scores and analyses of variance (F ratio) on national cohorts by age.

Possible explanations for these disparities could be derived from an examination of some of the educational traditions of the two countries. One substantial difference lies with the societal view held of mathematics. In Hungary great value, at all levels of society, is placed on mathematics as an intellectual problem-solving activity. Applications are subordinated to an engagement with the three fundamental areas of elementary mathematics - geometry, algebra and number theory - coupled with an explicit and ever-present emphasis on justification and proof. This contrasts markedly with the English tradition where mathematics is presented as an application which necessitates the acquisition of techniques rather than an engagement with concepts, proof and justification. This is coupled with a popular view that a fascination with mathematics at a conceptual rather than instrumental level is the preserve of the

eccentric and indicative of someone to be ridiculed rather than admired. A second tradition of Hungarian classrooms has been the acceptance that it is the class, usually mixed-ability, rather than the individual that forms the teaching unit. Teachers create environments in which mathematical problems are posed, solutions shared publicly, and understandings discussed and analysed. There is little sense that teachers see their roles as imparters of knowledge. English teachers, for whatever reasons, tend to develop curricula and pedagogies for individual pupils rather than work with whole classes and are generally reluctant to engage with mixed ability classes - a perspective sanctioned by official directives based on a common-sense obviousness. Their role is predominately one of transmitting a limited collection of skills and techniques which are consolidated by extensive repetitive practice. In essence the distinction lies between the Hungarian teacher's uniform pedagogy delivering a single curriculum and the English teacher's plurality of pedagogies delivering many curricula.

In respect of qualification it seems that there may be a significant message to be drawn from the nature, purpose and traditions of in-service training in the two countries. The evidence above indicated that English teachers with higher degrees - around one in six of the sample - were less susceptible towards an instrumental perspective on mathematics than their less well-qualified colleagues. In some cases this may be a consequence of higher degree holders having more sophisticated perspectives on education in general - viewed as more than just the acquisition of employment-related skills - and mathematics in particular than other colleagues. In other cases it may be a consequence of existing conceptions of mathematics having been challenged by courses geared towards such change - there is evidence from work done in the United States and Israel that such courses can shift teachers' from the transmission of an immutable body of knowledge to an open, collaborative and problem-solving approach to mathematics (Amit and Hillman, 1999). Whatever teachers' personal reasons, the evidence suggests that those with higher degrees are less likely to privilege the applications of mathematics than other colleagues and may, therefore, be closer to their Hungarian counterparts in how they view the subject and its teaching. This would tend to suggest, also, that they are more likely to be effective teachers of mathematics in accord, not only with the evidence of the TIMSS data (Beaton et al 1996) but also work undertaken in the UK on effective teachers of numeracy (Askew et al, 1997).

The Hungarian tradition whereby teachers qualify as primary and then become secondary through a combination of in-service training and work in the field seems not to lead to the same range of conceptions. It is not without interest that Stokes

(1995) describes a similar situation in France (Stokes 1995) which is another country with higher levels of mathematical attainment than England. There are minor differences which appear more concerned with environmental issues than fundamental perspectives on the nature of mathematics and how it might be conceptualised for teaching and learning. That is, the further training received by secondary teachers in Hungary seems to reinforce rather than challenge existing conceptions. This seems not unrelated to the evidence reported above that age was no indicator of a Hungarian teacher's perspectives on mathematics and its teaching - there seemed, by and large, an agreed and recognisable Hungarian pedagogy. The was not true of English teachers for whom increasing age suggested a shift from attempts to mediate an applications-based subject through the creation of open and collaborative classrooms towards a greater use of whole-class teaching. However, whether there was a shift with age towards the teaching of mathematics as a discipline independent of its applications it is impossible to infer from the data reported here.

In summary there would seem to be evidence emerging from this study to suggest that the British expectation of recruiting subject graduates to work in secondary schools may be less productive than might be expected. If, for the sake of this particular argument, Hungarian teachers are taken as the benchmark then it would seem that effective teachers are more likely to be those whose higher education experience of mathematics has been as trainee teachers. Such training traditions, it can be argued, are more likely to encourage the development of a unified and effective pedagogy - as in the Hungarian tradition - than would the arbitrary recruitment of engineers, scientists, accountants and others who perpetuate a view of mathematics as a collection of techniques and who develop idiosyncratic and (possibly) ineffective pedagogies as a consequence of their successful, but frequently ungeneralisable, experiences as learners of mathematics as a skills-based subject.

The results pertaining to the influence of gender are not as easily interpretable. In both countries there were discernible gender differences with, it seems, Hungarian women being less inclined towards mathematics as an applicable subject than men. They were also more of the opinion than men that teaching has little need of enriched classrooms, pedagogic variety or collaboration. English women tended towards the other extreme. They saw greater value than men in enriched classrooms and the need for pedagogic variety whilst being more inclined than men to acknowledge the applications of mathematics other than as a personal economic tool. So strong are these differences that further tests were run comparing English women against Hungarian women and English men against Hungarian men. The results of these can

be seen in Table 8. The evidence indicates a greater sense of conformity amongst men than women - eight of the nine constructs yielded significant differences for women compared with just four for men. That is, men, despite their nationalities, agreed on five of the nine constructs while women disagreed on eight. This is a difficult finding to interpret. It seems to suggest that the differences between the sets of teachers are more likely to be a consequence of the disparity between women rather than as a consequence of clear national characteristics. Why women should polarise like this is something of an enigma and will require further investigation.

Table 8The figures below show the results of t-tests comparing by gender across nations

Conclusion

The research reported here has highlighted some of the ways in which English teachers of mathematics and Hungarian teachers of mathematics both differ and agree in their thinking and points to the existence of both universal and national conceptions of mathematics and its teaching - the latter contributing to the characteristic pedagogical flow of the country concerned (Schmidt et al, 1996). The evidence also indicates that Hungarian teachers, unlike their English colleagues, are more consistent with both age and qualification appearing to have only minimal impact on their perspectives on mathematics and its teaching. One explanation derives from evidence showing that Hungarian teachers operate within a single pedagogic tradition (Andrews, 1997, 1999; Hatch 1994, 1999) which is unlike the situation found in England (Boaler 1996, Leung 1999) and which, according to Simon (1981, 1999) is one of the major cause of English educational underachievement. However, evidence of nationally formed conceptions do not account for the gender disparities identified

above which indicate that males, irrespective of nationality, tend to conform to a middle ground whereas females polarise according to nationality.

Table 9Mean percentage scores for grade 8 children on the TIMSS mathematics items for England and Hungary compared with the international mean. The figures in bold indicate a score above the international mean.

The main significance of this work, however, must lie in its implications for what happens in classrooms. That is, what might be inferred from this work in respect of insights into more effective teaching and learning? The evidence of the TIMSS was that Hungarian children achieve significantly higher levels of mathematical attainment than English. At age 13 (grade 8), as can be seen in table 9, they achieved above the international average on all six topic areas compared with the English's one. So great is the attainment shortfall of English children that Prais (1999) equates it to the equivalent of two or three years' education at age fourteen.

It is proposed here that the conceptual differences between the two nations are more likely to explain differences in children's attainment than similarities. Hungarian teachers are less enthusiastic than English in justifying mathematics and its teaching on the basis of its utility. They also categorically reject, something the English seem to hold dear, the notion that mathematics teaching requires the creation of a challenging and enriched classroom environment. Should this be the case then it is unlikely that mathematics education in England will develop beyond its current impoverished state. Recent innovations like the national numeracy strategy have attempted to impose a uniform pedagogy - at least as far as its encouragement to teachers to engage with whole classes in interactive ways - but this is only one element of a wider set of issues. English teachers are still expected to engage with an applications-justified subject and to expend considerable resources on the cosmetics

of classroom decoration. It is argued here that until the issue of curricular ambivalence is addressed - until mathematics is presented as both process and cultural artefact rather than application - the attainment of English children is unlikely to improve beyond its current state of mediocrity.

References

Abelson, R. (1979) 'Differences between belief systems and knowledge systems', Cognitive Science 3, 355-366.

Amit, M. and Hillman, S. (1999) Changing mathematics instruction and assessment: challenging teachers' conceptions, in B. Jaworski, T. Wood and S. Dawson (Eds) Mathematics teacher education, London, Falmer.

Andrews, P. (1997) A Hungarian perspective on mathematics education: the results of a conversation with Sari Palfalvi, Eva Szeredi, Vera Sztrokay and Judit Torok, Mathematics Teaching, 161, pp 14-17.

Andrews, P. (1999) 'Looking behind the rhetoric: some new insights from Hungary', Mathematics Teaching 167, 6-10.

Andrews, P. and Hatch, G. (1999a) A new look at secondary teachers' conceptions of mathematics and its teaching', British Educational Research journal 25: 3, 203-223.

Andrews, P. and Hatch, G. (1999b) Some expected and unexpected influences on secondary mathematics teachers' conceptions of mathematics and its teaching, Paper presented to the fourth British Congress for Mathematics Education, Northampton, Nene University College.

Askew, M., Brown, M., Rhodes, V., Johnson, D. and Wiliam, D. (1997) Effective teachers of numeracy: final report, London, King's College London

Ball, D.L. (1990) The mathematical understandings that prospective teachers bring to teacher education, The Elementary School Journal 90: 4, 449-466.

Beaton, A.E., Mullis, I.V.S., Martin, M.O., Gonzalez, E.J., Kelly, D.L. and Smith, T.A. (1996) Mathematics in the middle school years: IEA's third international mathematics and science study (TIMSS), Boston College, Boston MA.

Boaler, J. (1996) Learning to lose in the mathematics classroom: a critique of traditional schooling practices in the UK, International Journal of Qulaitative Studies in Education, 9 (1), pp 17-33.

Da Ponte, J.P. (1994) Mathematics teachers' professional knowledge, in J.P. da Ponte and J.F. Matos (eds.): proceedings of the eighteenth international conference for the psychology of mathematics education, Lisbon, July 29 - August 3.

Ernest, P. (1989) 'The impact of beliefs on the teaching of mathematics, in Ernest, P., (ed.): Mathematics teaching: the state of the art, Falmer, Lewes.

Ernest, P. (1989b) The knowledge, beliefs and attitudes of the mathematics teacher: a model, Journal of Education for Teaching 15: 1, 13-33.

Ernest, P. (1995) Values, gender and images of mathematics: a philosophical perspective, International Journal of Mathematical Education in Science and Technology 26: 3, 449-462.

Fenstermacher, G.D. (1978) A philosophical consideration of recent research on teacher effectiveness Review of Research in Education 6, 157-185.

Foss, D. and Kleinsasser, R.C. (1996) Preservice elementary teachers' views of pedagogical and mathematical content knowledge, Teaching and Teacher Education 12: 4, 429-442.

Frank, M.L. (1990) What myths about mathematics are held and conveyed by teachers?, Arithmetic Teacher 38: 5, 10-12.

Green, T.F. (1971) The activities of teaching, McGraw Hill, London.

Harel, G. (1994) On teacher education programmes in mathematics, International Journal of Mathematical Education in Science and Technology, 25 (1), p 113-119.

Hatch, G. (1994) Coming up to Russian expectations, Mathematics Teaching 146, 29-32.

Hatch, G.(1999) It wouldn't be like that here, Mathematics Teaching 168, 26-31.

Hoffman, P. (1998) The man who loved only numbers, London, Fourth Estate.

Lerman, S. (1983) Problem-solving or knowledge-centred: the influence of philosophy on mathematics teaching, International Journal of Mathematical Education for Science and Technology 14: 1, 59-66.

Lerman, S. (1990) Alternative perspectives of the nature of mathematics and their influence on the teaching of mathematics, British Educational Research Journal 16: 1, 53-61.

Leung, F. (1999) The traditional chinese views of mathematics and education: implications for mathematics education in the new millenium, in C. Hoyles, C. Morgan and G. Woodhouse (eds) Rethinking the mathematics curriculum, London, Falmer.

Litwin, M.S. (1995) How to measure survey reliability and validity, Sage, London.

Munby, H. (1982) The place ot teachers' beliefs in research on teacher thinking and decision making: an alternative methodology, Instructional Science 11, 201-225.

Nespor, J. (1987) The role of beliefs in the practice of teaching, Journal of Curriculum Studies 19: 4, 317-328.

Pajares, M.F. (1992) Teachers' beliefs and educational research: cleaning up a messy construct, Review of Educational Research 62: 3, 307-332.

Prais, S: (1999) How did English schools and pupils really perform in the 1995 international comparisons in mathematic?, in B. Jaworski and D. Phillips (eds): Comparing standards internationally: research and practice in mathematics and beyond, Symposium Books, Oxford, 79-118.

Schmidt, W.H., Jorde, D., Cogan, L.S., Barrier, E., Gonzalo, I., Moser, U., Shimizu, K., Sawada, T., Valverde, G.A., McKnight, C., Prawat, R.S., Wiley, D.E., Raizen, S.A., Britton, E.D. and Wolfe, R. (1996) Characterizing pedagogical flow, Kluwer, Dordrecht.

Simon, B. (1981) Why no pedagogy in England? in B. Simon and W Taylor (eds.): Education in the eighties, Batsford, London.

Simon, B. (1999) Why no pedagogy in England? in J. Leach and B. Moon (eds.): Learners and pedagogy, Paul Chapman, London.

Steiner, H-G. (1987) Philosophical and epistemological aspects of mathematics and their interaction with theory and practice in mathematics education, For the Learning of Mathematics 7: 1, 7-13.

Stokes, P. (1995) Education in England and Wales: out of step with Europe?, Oxford Studies in Comparative Education, 5 (2), pp 47-62.

Thompson, A.G. (1984) The relationship of teachers' conceptions of mathematics and mathematics teaching instructional practice, Educational Studies in Mathematics 15: 2, 105-127.

Thompson, A.G. (1992) Teachers' beliefs and conceptions: a synthesis of the research, in Grouws, A.D. (ed.): Handbook of research on mathematics teaching and learning, Macmillan, New York, 127-146.