8
REFLECTIONS ON EDUCATIONAL STUDIES IN MATHEMATICS THE RISE OF RESEARCH IN MATHEMATICS EDUCATION As ESM was completing its tenth volume, Davis and Hersh (1980) were calculating that the mathematics then known would fill perhaps 100,000 such volumes. Against some 20 mathematics education research papers then appearing annually in ESM, mathematical results were being proved and published at an estimated rate of 200,000 a year. It is worth reminding ourselves of the recency and modesty of research in mathematics education when placed alongside an established field such as mathematics. However, twenty-something years on, the appearance of the fiftieth volume of ESM reflects the gradual acceleration of its publication schedule from one to three volumes per year, paralleled by a similar growth in the size and number of fellow international journals in the field. To take just one further – and broader – index of scale, the ERIC database of educational resources now holds over 50,000 items identified by the keyword ‘mathematics’, with around 2,000 such items being added every year. The expansion, and particularly the diversification, of research in math- ematics education has been seen by some as creating a fragmented field, vulnerable to proliferating paradigms (Sierpinska and Kilpatrick, 1998). However, the account which Davis and Hersh offer of the evolution of mathematics as a professional field is pertinent. They point to the break- down over the course of the twentieth century of the expectation that any one mathematician would be familiar with all – or even many – branches of the subject, as the growing field became increasingly differentiated into Educational Studies in Mathematics 50: 251–257, 2002. © 2002 Kluwer Academic Publishers. Printed in the Netherlands.

Reflections on Educational Studies in Mathematics

  • View
    212

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Reflections on Educational Studies in Mathematics

REFLECTIONS ON EDUCATIONAL STUDIES IN MATHEMATICS

THE RISE OF RESEARCH IN MATHEMATICS EDUCATION

As ESM was completing its tenth volume, Davis and Hersh (1980) werecalculating that the mathematics then known would fill perhaps 100,000such volumes. Against some 20 mathematics education research papersthen appearing annually in ESM, mathematical results were being provedand published at an estimated rate of 200,000 a year. It is worth remindingourselves of the recency and modesty of research in mathematics educationwhen placed alongside an established field such as mathematics. However,twenty-something years on, the appearance of the fiftieth volume of ESMreflects the gradual acceleration of its publication schedule from one tothree volumes per year, paralleled by a similar growth in the size andnumber of fellow international journals in the field. To take just one further– and broader – index of scale, the ERIC database of educational resourcesnow holds over 50,000 items identified by the keyword ‘mathematics’,with around 2,000 such items being added every year.

The expansion, and particularly the diversification, of research in math-ematics education has been seen by some as creating a fragmented field,vulnerable to proliferating paradigms (Sierpinska and Kilpatrick, 1998).However, the account which Davis and Hersh offer of the evolution ofmathematics as a professional field is pertinent. They point to the break-down over the course of the twentieth century of the expectation that anyone mathematician would be familiar with all – or even many – branchesof the subject, as the growing field became increasingly differentiated into

Educational Studies in Mathematics 50: 251–257, 2002.© 2002 Kluwer Academic Publishers. Printed in the Netherlands.

Page 2: Reflections on Educational Studies in Mathematics

252 REFLECTIONS ON EDUCATIONAL STUDIES IN MATHEMATICS

subspecialties, and socially organised around them. Challenging and ad-apting Courant’s metaphorical proposition that “the river of mathematics,if separated from physics, might break up into many separate little rivuletsand finally dry up altogether”, Davis and Hersh suggest that “the variousstreams of mathematics have overflowed their banks, run together, andflooded a vast plain, so that we see countless currents, separating andmerging, some of them quite shallow and aimless”, while “the channelsthat are still deep and swift-flowing are easy to lose in the general chaos”(p. 22).

To extend this metaphor to the field of mathematics education, the wa-ters of research may be rising, but much territory remains barely touched.Equally, in those areas that are already awash, it has often been the conflu-ence of several tributary sources which has scoured out underlying relief,which remains invisible on the turbulent surface. We view different stylesof research as acting together to lay bare the many aspects of mathemat-ics education that may too readily be taken for granted, and as providingpotentially complementary flows of ideas capable of uncovering and ad-dressing the complexities of policy and practice in the field. As a conduitfor research, ESM remains committed to capturing different currents andtapping their depths, pursuing a policy of openness as regards topic, the-ory and method. The metaphor of the conduit is appropriate, for a journallargely serves to channel the outflow of research which has been initiated,sponsored and conducted by others, even if editors and editorial boardsmay prime or staunch, direct or divert, the flow of publications.

ENCOURAGING A CRITICAL CONSOLIDATION OF CORE THEORY

Over the past twenty or so years, mathematics education has grown intoa scientific discipline incorporating a wealth of ideas, approaches, the-oretical frameworks and links to neighboring disciplines that is steadilygrowing. The disciplines of psychology, sociology and ethnography havesuccessively begun to influence the thinking of researchers in mathematicseducation. The effects of these influences are not passing but cumulativeand have led to a fruitful but challenging expansion of theoretical perspect-ives in our domain. This expansion has been fruitful because it introducescomplementary ways of understanding the phenomena we are studying.It has been challenging because it has broadened the range of theoret-ical frameworks within which mathematics education researchers carry outtheir investigations.

Indeed, individually and collectively, researchers in the field may nowbe struggling to meet this challenge. One symptom is the proliferation of

Page 3: Reflections on Educational Studies in Mathematics

REFLECTIONS ON EDUCATIONAL STUDIES IN MATHEMATICS 253

theory as it has become the norm rather than the exception for researchersto propose their own conceptual framework rather than adopting or refiningan existing one in an explicit and disciplined way. This prolific theorisingmight be represented as the sign of a young and healthy scientific discip-line. But it may also mean that theories are not being sufficiently examined,tested, refined and expanded. A theory may be used mainly by its thecreators and their students rather than by a large number of independentand experienced researchers. It may be used for only one particular typeof research study, of population, of methodology or of context. Equally,essentially the same issues and research questions may be being describedand analyzed by a multiplying array of parallel theories. One concernmust be that such proliferation of theories, influences and frameworks maylead to mathematics education becoming a tower of Babylon, where manystrive, with excellent intentions, to provide light for their colleagues, butfew listen, read and take into account their colleagues’ ideas and work.

One of the trademarks of a mature science is that it strives for unity; thatit directs its collective thought toward unifying theories and frameworks.This is neither a quick nor an easy process. Physics is a case in point. Thisvery well established science has been fighting for much of the twentiethcentury to unify the well established theories of its sub domains, includingrelativity theory, quantum mechanics, and gravitation. While string theorystill counts as a candidate for unification, it has not, so far, been confirmedand ‘grand unification’ remains a dream of many physicists in the currentcentury. There is neither right nor reason to compare mathematics edu-cation to physics, but this should not prevent us from learning from itsdevelopment. We might learn that we should in the near future increaseour efforts

• To carry out research studies within frameworks determined by exist-ing theories with the intention to establish the range of applicabilityor validity or usefulness of these theories.

• To carry out comparative surveys of several theories, in particular oftheories that purport to provide frameworks for dealing with the sameor related areas, topics and questions.

• To compare the terminologies used by different theories in order toidentify cases where different terms are used for essentially the sameidea or where the same term is used to designate ideas that are essen-tially different.

• To attempt to see the common ideas between different theories andwork toward their partial unification; this might be particularly prom-ising in cases where the theories deal with different but closely relatedissues or areas.

Page 4: Reflections on Educational Studies in Mathematics

254 REFLECTIONS ON EDUCATIONAL STUDIES IN MATHEMATICS

We see it as one of the tasks of ESM over the next decade to supportefforts at theoretical solidification and consolidation in the sense pointedout above.

REVALUING THE CONTRIBUTION OF REPLICATION,RECONTEXTUALISATION AND REVIEW

In some respects, the influence of journals such as ESM may have contrib-uted to the proliferation of local theories. We sympathise with the concernwhich experienced researchers articulated to Silver and Kilpatrick (1994:737) regarding “the apparently common perception that every study needspersuasive justification that it is both important and original”. Such a pre-conception may inhibit referees from appreciating the distinctive contri-bution of certain forms of research and scholarship to the field, and sodiscourage researchers from proposing, undertaking and submitting suchwork. In particular, Silver and Kilpatrick mention the underestimated con-tribution and correspondingly low incidence of replication studies.

Arguably, taking replication and synthesis seriously -which includesproblematising them- could play a significant part in the development ofour field. From a technical perspective, replication of a study across variedsites not only makes it possible to address issues of generalisability andcontextual influence more rigorously, but provides an important mechan-ism through which theoretical ideas and research tools can be sharpenedand refined in action, particularly in response to the operational challengesand cultural differences which arise in translating them between educa-tional sites, phases and systems and between research teams. From a socialperspective, the diffusion of research design and instrumentation from onegroup to others through replication or extension studies not only mediatesthe development of more strongly shared systems of language and method,but also directs attention to the degree to which carrying through such workcalls for recontextualisation rather than straight replication, illuminatingcontextual influences and cultural differences which tend to be glossedover in current discussion, evaluation and synthesis of research in the field.Perhaps the international composition of the ESM editorial board, and thediffering perspectives its members bring to the appraisal of submissionsfrom a wide range of countries, helps to make us particularly sensitive tosuch issues. Critical reviews of research on particular topics, informed byappreciation of such contextual influences and cultural differences, couldplay a further important part in the development of the field. At present, thesynthesis of research receives insufficient attention, perhaps on account ofa popular perception of review studies as ‘secondary’ rather than ‘primary’

Page 5: Reflections on Educational Studies in Mathematics

REFLECTIONS ON EDUCATIONAL STUDIES IN MATHEMATICS 255

research, but also because of the challenges of carrying through such workrigorously and reflexively. To summarise, then, we believe that the value ofreplication, recontextualisation and review studies needs to be recognisedand their status raised. In the light of our experience, we also suspect thata degree of cross-national collaboration can often contribute to the qualityof such studies, particularly if this collaboration involves identifying andcrystallising key differences of interpretation.

REVIEWING THE CHARACTER OF EDUCATIONAL STUDIES IN

MATHEMATICS

The search for a defining scientific identity remains a central issue formathematics education (Sierpinska and Kilpatrick, 1998). Following anearly period characterised by a clear focus on finding – and appraising –different ways of elementarizing mathematical knowledge for the class-room, there has been a diversification of approaches and an increasingopenness to interdisciplinarity. This broadening of the research perspect-ive has not just enlarged the range of theory and method in use, but –in encompassing the kinds of issues and questions regarded as significantin disciplines from which ideas are being borrowed – has resulted in theguiding concerns of research becoming much more varied. Whereas at itsstart, mathematics education had a rather clear, direct object of research –namely, mathematical knowledge that had to be elementarized for purposesof teaching and learning – the focus of research in the field has now becomemuch more diffuse. Unification of theories and terminologies cannot besuccessful without a clearer understanding of the object of research; or, putanother way, the attempt to adjust theories and terminologies must be at thesame time an attempt to clarify the object of research. This object of re-search in mathematics education is not simply something like ‘the processof teaching and learning mathematical knowledge’ – in the sense only of aconcretely observable phenomenon – but rather must extend to theorisingrelations and constraints behind those processes so as to constitute and‘define’ the proper research object; conditions that are in the beginningsomehow invisible and that have to be constructed and introduced into theobservable learning and teaching processes.

The field should not neglect its original focus on the elementarizationof mathematical knowledge, even if it has now found new ways of framingsuch matters. This points to another preconception which may be undulynarrowing the kinds of work perceived as making a research contribution tothe field. Boero, Dapueto and Parenti (1996: 1108) have noted a decreasinginvolvement of researchers in work concerned with innovative approaches

Page 6: Reflections on Educational Studies in Mathematics

256 REFLECTIONS ON EDUCATIONAL STUDIES IN MATHEMATICS

to the teaching of specific mathematical topics or the development of par-ticular mathematical capabilities, and – possibly as much consequenceas cause – to the marginalisation of such work in research journals andconferences. While the title of the journal is often interpreted simply asa paraphrase for [Studies in [Mathematics Education]], it also featureswork which might be parsed as [[Educational Studies] in Mathematics],involving the (re)contextualisation of more generic educational ideas andconcerns in the special circumstances of mathematics. More significantly,however, ESM has continued to publish what might be parsed as [Edu-cational [Studies in Mathematics]], by which we mean systematic studiesof mathematical ideas and processes, in which an educational intent andrationale is explicit. Such studies have never been in a majority in thejournal, and their presence has been progressively diminishing over time(as Boero and colleagues demonstrate in their paper). At the same time,those appearing more recently have evolved to take more explicit accountof educational as well as mathematical analysis and argument. Broadlywe would characterise this type of study as involving careful didacticalanalysis of particular mathematical ideas and processes, attending to theirvernacular and scholarly forms, informed by mathematical and educationalperspectives, and aiming at the reasoned development and appraisal ofteaching designs. As Boero and his co-authors argued, high quality workof this type remains central to development of the field, but is currentlyunder-represented in research journals and conferences.

Work on the elementarisation of mathematical knowledge has tendedto focus on its epistemological and cognitive dimensions. Arguably, how-ever, the aesthetic and pragmatic dimensions deserve greater attention.With regard to the former, Davis and Hersh (1980) wrote insightfully andeloquently about the aesthetic appeal of mathematics, “both in passivecontemplation and in actual research pursuit” (p. 168). However, they alsorecognized that in attempts to portray the beauty of mathematics, as inart and music, despite efforts to make the form explicit, the underlyingaesthetic qualities remain elusive. Perhaps this is why some learners findthe subject “as dry as dust” (p. 169). These issues are manifestly per-tinent to issues of teaching and learning mathematics. Yet this, too, isan underresearched area in mathematics education, as witnessed by thesmall number of papers appearing in ESM that address the related fieldof affective issues in teaching and learning mathematics. Appreciation ofbeauty is personal (Davis and Hersh, 1980). As the field develops andthe appreciation of a mathematical aesthetic becomes acknowledged asboth personal and social, the way may open to using aesthetic sense moreexplicitly in mathematics education at all levels, as in art and music. The

Page 7: Reflections on Educational Studies in Mathematics

REFLECTIONS ON EDUCATIONAL STUDIES IN MATHEMATICS 257

attempt to identify what it means to have an ‘aesthetic of mathematics’in teaching mathematics is a move in this direction. Another importantarea that is gaining increasing attention is the recognition that mathem-atics is learned in context (e.g., Brenner and Moschkovich, 2002). Thisrecognition embraces pragmatic aspects in its attempts to bridge the gapbetween ‘everyday’ and ‘academic’ mathematics – although it is fruitfulto problematise these terms (ibid.). The recognition that issues of power,values, and the nature of discourse are implicated in these processes re-quires a continued openness to new theoretical areas as the need arises inthis developing field.

On present trends, the hundredth volume of ESM will arrive all toosoon. Already, we are starting to become accustomed to an on-line editionof the journal. By then, we expect that hypermedia will have become muchmore central to the dissemination of research, permitting forward as wellas backward referencing of work, active linking not only to such referencesbut to research material related to each publication in the form of sourcematter, data sets and analysis records, as well as an on-line discussionarchive, perhaps including the suitably edited reviews of referees. Suchopening up of the research process is likely to be highly beneficial to thefield, in encouraging greater transparency, fuller debate, and more exten-ded analysis of important and influential studies, so contributing to thekinds of substantive development in the field that we have signalled above.

The editors and advisory editors of ESM have contributed to the abovethoughts in order to provoke a discussion in the pages of the coming issuesof ESM. We invite readers to send short reaction papers to the Editor-in-Chief with their own thoughts about the future of ESM and the future ofmathematics education.

REFERENCES

Boero, P., Dapueto, C. and Parenti, L.: 1996, ‘Didactics of mathematics and the profes-sional knowledge of teachers’, in A. Bishop et al. (eds.), International Handbook ofMathematics Education, Kluwer, Dordrecht, pp. 1097–1121.

Brenner, M.E. and Moschkowitz, J.N. (eds.): 2002, Everyday and Academic Mathemat-ics in the Classroom, Reston, Virginia: National Council of Teachers of Mathematics,Monograph 11.

Davis, P.J. and Hersh, R.: 1980, The Mathematical Experience, Penguin, Harmondsworth.Sierpinska, A. and Kilpatrick, J. (eds.): 1998, Mathematics Education as a Research

Domain: A Search for Identity, Kluwer, Dordrecht.Silver, E.A. and Kilpatrick, J.: 1994, ‘E pluribus unum: Challenges of diversity in the future

of mathematics education research’, Journal for Research in Mathematics Education(25th Anniversary Special Issue) 25(6), 734–754.

Page 8: Reflections on Educational Studies in Mathematics