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Text of PoMEnews 8

Number 8, May 1995 Issue Editor: Paul Ernest

    This Newsletter is the publication of the

    PHILOSOPHY OF MATHEMATICS EDUCATION NETWORK

    conferences, publications, and also books for ORGANISING GROUP

    Raffaella Borasi (USA), Stephen I. Brown (USA), review. Leone Burton (UK), Paul Cobb (USA), Jere OF CONTENTS TABLEConfrey (USA), Thomas S. Cooney (USA), Kathryn

    FUTURE OF THIS NEWSLETTER ................... 2 Crawford (Australia), Ubiratan D‘Ambrosio

    (Brazil), Philip J. Davis (USA), Sandy Dawson EDITORIAL: CRITICISM AND THE GROWTH (Canada), Ernst von Glasersfeld (USA), David OF KNOWLEDGE ................................................ 2 Henderson (USA), Reuben Hersh (USA), Christine DISCUSSION THEME: THE DISCIPLINE OF Keitel-Kreidt (Germany), Stephen Lerman (UK), NOTICING ............................................................ 5 John Mason (UK), Marilyn Nickson (UK), David

    Pimm (UK), Sal Restivo (USA), Leo Rogers (UK), IS ‘THE DISCIPLINE OF NOTICING’ A NEW Anna Sfard (Israel), Ole Skovsmose (Denmark), PARADIGM FOR RESEARCH IN Francesco Speranza (Italy), Leslie P. Steffe (USA), MATHEMATICS EDUCATION? ........................ 5 Hans-Georg Steiner (Germany), John Volmink RESPONSE TO PAUL’S COMMENTS ON THE (South Africa), Yuxin Zheng (P. R. China). DISCIPLINE OF NOTICING .............................. 8 AIMS OF THE NEWSLETTER WORKING WITH ‘THE DISCIPLINE OF The aims of this newsletter are: to foster awareness NOTICING’: AN AUTHENTICATING of philosophical aspects of mathematics education EXPERIENCE? .................................................. 11 and mathematics, understood broadly to include

    THE PRINCIPLE OF ECONOMY IN THE most theoretical reflection; to disseminate news of

    LEARNING AND TEACHING OF events and new thinking in these topics to

    MATHEMATICS, BY DAVID P. HEWITT ...... 12 interested persons; and to encourage informal communication, dialogue and international co-RESPONSE TO PAUL'S COMMENTS ON THE operation between teachers, scholars and others PRINCIPLE OF ECONOMY IN THE engaged in such research or thought. LEARNING AND TEACHING OF MATHS .... 14 COPYRIGHT AND DISTRIBUTION DIALOGUE ......................................................... 16 The copyright of all contributions to the newsletter PHILOSOPHY OF MATHEMATICS remain the property of the authors. (Unattributed EDUCATION: POM(E), PO(ME) OR POE(M) ?16 material is written by the issue editor.) However

    A REJOINDER TO OTTE BY ERNST VON permission is granted to copy all or part of the

    GLASERSFELD ................................................. 18 newsletter provided that the source is fully

    acknowledged and that it is for educational or other THE PEIRCEAN INTERPRETATION OF non-profit purposes. Indeed copying and MATHEMATICS BY CHRISTOPHER distributing the newsletter to interested persons is ORMELL ............................................................. 20 strongly encouraged with the aim of extending the ON CONFORMITY - A SEARCH FOR A dialogue in keeping with the above aims. POWER BASE? UNA HANLEY AND OLWEN COMMUNICATIONS MCNAMARA ...................................................... 21 The POME Group Chair is Paul Ernest, University THESE BOOTS WERE MADE FOR TALKING of Exeter, School of Education, Exeter EX1 2LU, TONY BROWN.................................................... 22 U.K. Phone: (0)1392-264857, Fax: (0)1392-

    BOOK REVIEWS ................................................ 24 264736, E-mail: PErnest@ex.ac.uk, and

    http://www.ex.ac.uk/~PErnest/ ASPECTS OF PROOF: SPECIAL ISSUE OF Please send any items for inclusion to him EDUCATIONAL STUDIES IN including a copy on disc or E-mail. These may MATHEMATICS ................................................ 24 include short contributions, discussions,

    provocations, reactions, notices of research groups,

POME Newsletter 8 (May 1995) 2

    EXPLORATIONS IN ETHNOMATHEMATICS justification which is central to Lakatos‘ AND SIPATSI, PAULUS GERDES ET AL. ..... 26 philosophy of mathematics (and science).

    Conversation or dialogue is increasingly a central PUBLICATION ANNOUNCEMENTS ............. 26 metaphor for inquiry, knowledge growth, and even MATHEMATICS, EDUCATION AND of mind, as the works of Bakhtin, Gergen, PHILOSOPHY: AN INTERNATIONAL Habermas, Harré, Lakatos, Mead, Rorty, Shotter, PERSPECTIVE .................................................. 26 Volosinov, Vygotsky, Wertsch, Wittgenstein and

    other thinkers show. But of course philosophical CONSTRUCTING MATHEMATICAL

    dialogue already reached a near timeless form in KNOWLEDGE: EPISTEMOLOGY AND

    the work of Plato. If we adopt dialectical reasoning MATHEMATICS EDUCATION ....................... 27

    (understood broadly, not strictly Hegelian) as STUDIES IN MATHEMATICS EDUCATION providing a model of conversation there three SERIES ............................................................... 27 moments forming a cycle: Thesis ? Antithesis ?

    PHILOSOPHIA MATHEMATICA ................... 28 Synthesis (the new Thesis). The ‗positive‘ phase of

    Thesis concerns a knowledge proposal in its first IN MEMORIAM STIEG MELLIN-OLSEN ..... 28 form relative to this cycle, but of course all MATHS EDUCATION ON THE INTERNET . 28 knowledge claims are embedded in personal and

    cultural history, and come out of the never-ending ADULTS LEARNING MATHS: A RESEARCH

    conversation of humankind. The ‗negative‘ phase FORUM (ALM) .................................................. 28

    of Antithesis concerns a critical scrutiny of the FUTURE OF THIS NEWSLETTER knowledge proposal; its weaknesses outlined and In the future this newsletter will be published via maybe a counterproposal or progress to a the Internet. See the editor‘s homepage at: resolution implied. Completing the cycle, a new http://www.ex.ac.uk/~PErnest/ ‗positive‘ phase of Synthesis concerns a reformed This carries all editions of the newsletter (minus a knowledge proposal, which has been strengthened few missing bits of early numbers). It is also by overcoming and accommodating critique, and planned to include a file of reactions, comments, which now both embodies new knowledge claims discussion, etc. to be accessed via this homepage and is partly or wholly warranted by the so please send anything you would like to say or appropriate knowledge community. add to PErnest@ex.ac.uk. Colleagues unable to Popper underscored the importance of these access PoMEnews via the Internet are requested to phases in his title ‗Conjectures and Refutations‘ write to the editor enclosing self-adhesive (making up his ‗Logic of Scientific Discovery‘) (international) address label(s). The subscription and Lakatos extended this process to mathematics for such persons will be UK?5 per issue (Cheque in his ‗Proofs and Refutations‘ (making up his to P Ernest), but free to colleagues in East Europe, ‗Logic of Mathematical Discovery‘). In Criticism ex-USSR, and in developing countries. No mailing and the Growth of Knowledge Popper‘s logicist list will be maintained and you will need to request and internalist ‗Thesis‘ was challenged by Kuhn‘s each desired issue. relativist and historicist ‗Antithesis‘ and Lakatos

    EDITORIAL: CRITICISM AND THE offered a ‗Synthesis‘ in his Methodology of

    1GROWTH OF KNOWLEDGE Scientific Research Programmes.The title of this editorial is taken from the What has this to do with the philosophy of eponymous book edited by I. Lakatos and A. mathematics education? I want to make a Musgrave (CUP, 1970). That book represents the controversial claim. I believe that research in written version of a planned dialogue between mathematics education has come to over- focus on Thomas Kuhn and Karl Popper at the Bedford the ‗Thesis‘ or proposal stage in knowledge College seminar in 1965, at which Lakatos was generation, and currently underemphasises secretary. Three other edited volumes (Lakatos ‗Antithesis‘ or criticism. Yet the latter is just as editor) preceded this book‘s final publication in essential as the former, and is necessary to improve 1970, treating symposia on problems in inductive

     logic, philosophy of mathematics and the 1Even Kuhn‘s Structure of Scienctific Revolutions embodies philosophy of science. Each of the volumes the dialectic in the form Normal science ? Anomaly or embodied the dialogical format of the conference contradiction ? Revolutionary science (new Normal and the dialectical theory of knowledge genesis and science).

    and warrant knowledge. Let me elaborate my individual before the group, which of course is the claim. Three main reasons come to mind. invisible but all-pervasive ideology of modern First of all, there are epistemological and consumerist, capitalist society. (For a powerful methodological reasons. As a community we have POME Newsletter 8 (May 1995) critique of individualism in mathematics education 3

    to a large extent shifted from an objectivist see the important paper by McBride, 1994). epistemology to a subjectivist epistemology. A Third, and related to the above, mathematics couple of decades ago knowledge and mathematics educators are in the main nice persons who care were widely conceived in the mathematics about each other and don‘t want to hurt each education community in absolutist, objectivist others‘ feelings. This is partly our stock-in-trade.

    terms, although there were, of course, notable As ex-teachers and teacher educators, in the main, exceptions. Now knowledge and learned we have to be careful not to crush our charges: to mathematics are widely conceived in subjectivist let them grow in confidence and capability. This is terms, in the mathematics education community. in part a reaction to the competitive world of Constructivism and the interpretative research mathematics, where a fierce pecking order exists paradigm stress the subjectivity of knowledge and and error is rooted out harshly. So at mathematics of all knowing. But such a focus can over-education conferences even at the highest emphasise the generative act in knowing: international level, criticism is mostly mild, and conjecturing, theorising, imagining, intuiting, etc. harsh critiques are viewed as bad manners. Where Whilst this is a vital ingredient, it nevertheless criticism is built into the field (and without it, it represents, in my view, an overemphasis on the would immediately fall into disrepute) is in the ‗Thesis‘ stage, with a neglect of ‗Antithesis‘. critical scrutiny of submissions for journal and Constructivism does include the accommodation of conferences. This is essential for the maintenance schema by interaction with the world (the subject‘s of quality (not according to any fixed notions of experiential world), but the emphasis is on the quality, but according to the evolving conceptions subjective knowledge of the individual which of active members and leaders in the field). progresses by an internal dialectical logic. This However even here mutterings of discontent have helps improve the knowledge, but does not been heard: too much is being accepted for publicly warrant it. My claim is that that conference presentation that is the product of mathematics education community has not sloppy thinking and does not add to knowledge. sufficiently made the shift to a third philosophical Another reason that we are too nice to each other is position, a social epistemology. Here public that we have banded together in adversity. criticism and warranting play the role of dual to Education has often been viewed with partial genesis and formulation of knowledge (public or contempt by other university departments, private). There is increasing lip service paid to including mathematics. interpersonal negotiation, but the subjectivist Having offered three points in support of my assumptions remain deeply entrenched. controversial claim, I will deviate here to describe Second, and related to the above epistemological a recent and relevant controversy.

    story, is one of ideology. Mathematics education Mathematics Versus Mathematics has typically in the last couple of decades been tied Education: An Epistemological Battle in with progressive reform. We want to respect the Recently a dispute erupted in the British national child or learner as epistemological agent, press over problems in mathematics teaching. emphasising her informal knowledge, discoveries, Mathematicians claimed that standards have fallen problem solving and posing, activities, feelings etc. and that much of the blame rests with the The philosophy of progressive education wants to mathematics education community and its shield the child from hurtful influences and let her misguided progressivism and mistaken grow and realise her potential creatively from constructivist and relativist epistemology. within, like a flower. Now this ideology, for all its Epistemology in mathematics education hits the strengths, again focuses on the moment of genesis national press? Who would have thought it? On 28 and minimises the moment of criticism. December 1994 The Guardian Newspaper carried Progressive teachers have said ―I don‘t like to mark the front page banner headline ‗School Maths in children‘s work wrong‖ and ‗assessment‘ can be a Crisis‘. It also carried an article by T. Barnard and

    dirty word from this perspective. Underpinning it is P. Saunders (1984), two mathematicians from a pervasive ideology of individualism, the

POME Newsletter 8 (May 1995) 4

    King‘s College, London, from which the following the content on which these are supposed to be extracts are taken. operating, and if you reduce the latter you

    inevitably limit the former. There is a very curious situation in mathematics

    at the moment. There are on the one hand, clear The mathematicians‘ ‗thesis‘ was rapidly answered signs of a crisis. There is a strong consensus in by an ‗antithesis‘ claiming that the problem was the university mathematics world that the not that mathematics educators had let the side mathematical awareness, skills and down with their irrelevant theories, but that understanding of pupils completing secondary mathematicians were as usual taking the narrow education have deteriorated rapidly in recent view and blaming others for the fact that years. Students know less, understand less, mathematics is a decreasingly popular subject for have little facility with simple operations, have university study in the modern world. Worse, it is little idea what is meant by proof, find difficulty the antiquated, instrumental and content-centred in solving any but the shortest of problems. approaches in further and higher education which The problem is not simply the schools being were the real culprits, and which are driving perhaps too complacent and the universities potential students away . (I must admit to adding perhaps too demanding. There are fundamental my voice to this ‗antithesis‘ group, in Ernest 1995)

    disagreements about what mathematics is, what It took the calmer voices of Margaret Brown pupils should learn about mathematics at (THES) and Sue Burns (LMS Newsletter) to offer school, what skills they should learn within a synthesis, pointing out the fact that although an mathematics both to be used later on in the increasing number of students take ‗A‘ Level subject and elsewhere, what else they should be exams in the UK at age 18, the number taking ‗A‘ learning while they are learning these and what Level Mathematics declines annually in absolute the priorities are. They go right to the heart of terms, and those who do succeed at it may be how pupils learn anything. opting for medicine, economics, computer science, Central to the issue lies the epistemological etc. The number opting for mathematics degree divide between a traditional emphasis on course declines annually. So mathematicians may knowledge as an external landscape, and a post-be having to recruit from the lower strata of a Piagetian emphasis on knowledge as an shrinking pool of applicants, and hence observing a individually constructed internal picture. The decline in competencies. situation is sometimes regarded as if there were Conclusion two diametrically opposite positions. Firstly, The point that I have been making is that criticism the Absolutists, cultural restorationists who see plays an essential role in the growth of knowledge. teaching as presenting a corpus of facts and This newsletter is dedicated to dialogue on learning as a passive process of receiving this. reflective and philosophical aspects of mathematics Opposed to them are the Relativists, liberal education. It has tried to counterpose conflicting progressives, for whom teaching means views, embraced all criticism as part of its presenting learning activities and learning is an conversation, and the present issue can be seen to active process of building a mental network of sit well in this tradition. However not all forms of relations. critique are the same, and it is as well to remember But is this a true picture? In mathematics that there is a moral dimension to dialogue. In the teaching, the changes are sometimes seen as a Platonic dialogues, Socrates uses the method of jettisoning of content - Euclidean geometry, the elenchus or refutation, to lead people to see that arithmetic of fractions, algebra - to allow time their views are perceived by others as wrong. This for a careful development of process skills: has been contrasted by feminist philosophers with interpreting, communicating, selecting, 'the adversary method', which is "a model of applying, and so on. But where an instrumental philosphic methodology which accepts a positive understanding of content is merely replaced by view of aggressive behaviour and then uses it as an instrumental understanding of process, there the paradigm of philosophic reasoning. ...The is no improvement. On the other hand, a adversary method requires that all beliefs and relational understanding of content carries with claims be evaluated only by subjecting them to the it a relational understanding of process. The strongest, most extreme opposition." (Harding and converse is not true. Pupils cannot genuinely Hintikka, 1983: xv). Getting the balance right understand processes if they do not understand

POME Newsletter 8 (May 1995) 5

    between constructive and destructive criticism is ? a concern with the authentic nature of lived an important issue of judgement. Readers will have experience, which strikes a deep chord with to decide if I have got the balance right in this many progressive educators, issue. ? an emphasis on an interpretative research

    methodology, focusing on case study and References concrete particulars of life and educational Barnard, T. and Saunders, P. (1994) ‗Superior sums

    practice. that don‘t add up to much: How mathematics is taught

    These strengths make this a seductively attractive is in dispute, and our children‘s knowledge of it is

    worse than ever‘, The Guardian, 28 December. approach to educational research, one which is Ernest, P. (1995) Number Crunch, The Guardian proving popular with a significant sector of those (Education Supplement), 3 January. in British mathematics education who might Harding, S. and Hintikka, M. B. Eds (1983) loosely be described as drawn to the teacher-as-Discovering Reality: Feminist Perspectives, researcher movement. Dordrecht: Reidel. (Re: Moulton, J.: ‗A Paradigm of Because of this burgeoning popularity, in this Philosophy: The Adversary Method‘, 149-164.) short piece I plan to look critically at DN, rather McBride, M. (1994) The Theme of Individualism in than to celebrate its strengths. Critique is an Mathematics Education: An Examination of

    important feature of any community of researchers, Mathematics Textbooks, For the Learning of

    although it is a feature that is too often neglected in Mathematics, 14(3) 36-42.

    mathematics education, as I discuss in the editorial. DISCUSSION THEME: THE DISCIPLINE OF I believe it is incumbent on the community of NOTICING researchers in mathematics education to examine As in previous issues of POMENEWS the discussion critically any perspective that is proposed or section aims to provide a location for in-depth adopted as a novel educational research paradigm, comment around a particular theme. This issue the especially when it is used by a growing number of theme to be discussed is an emergent research young researchers as the basis for PhD research. perspective in mathematics education: the Such is the position of DN. Anyway, DN has been ‗Discipline of Noticing‘. This is primarily due to widely honoured in many research presentations, the work of John Mason (although he would be the including a prestigious plenary talk at the PME first to credit Caleb Gattegno) who is one of the conference in Lisbon (Mason 1994). I shall take founding members of the Organising Group of the this most recent exposition of DN as a Philosophy Of Mathematics Education Network. representative statement of the position. The

    characteristics of the approach of DN appear to me IS ‘THE DISCIPLINE OF NOTICING’ A NEW to be the following. There is: PARADIGM FOR RESEARCH IN ? A concern with the subjective experience of the MATHEMATICS EDUCATION? learner and teacher (especially with acts of

    Paul Ernest construal, awareness and changes in awareness,

    the noticing of significance), For at least a decade John Mason has been

    ? The direction of another‘s or others‘ attention to elaborating what he terms ‗The Discipline of

    what has been noticed as significant: selecting Noticing‘ (in brief: ‗DN‘) (Mason, 1994: 183).

    and emphasising certain events or experiences This is a perspective or stance with regard to

    and directing own or others attention to accounts research in mathematics education, and represents

    of them, or trying to enable others to recreate the John‘s personal research program in mathematics

    experience, education over the past decade. This includes a

    ? An attempt to develop an authentic voice to number of strengths (as they strike me) including

    describe the researcher‘s experience (whilst at the following:

    the same time regarding such accounts as ? the valuing of individual sense-making, which is

    secondary to the experiences they describe), so central to the dominant constructivist

    ? The rejection of the epistemology and perspectives in mathematics education,

    methodology of the scientific research paradigm ? a concern to relate this to the actions of the

    and the endorsement of a methodology akin to teacher, which is too often missing in

    that of the interpretative research paradigm. constructivist accounts,

    My commentary on DN follows the division of its

    principles or themes into four aspects. In the words

POME Newsletter 8 (May 1995) 6

    of Mason ―The Discipline of Noticing ...[has] four The ‗language of identification‘ used in [1] aspects:‖ suggests that there are pre-existing patterns in the

    experiences. This makes the claim either (i) [1] Systematic Reflection; retrospective re-entry

    resemble the inductivism that Popper in his Logic through brief-but-vivid accounts of incidents,

    of Scientific Discovery critiques decisively (the without judgement or explanation; once a

    claim that patterns are in the world and can be corpus accumulates, it is possible to identify

    induced by finite experiences), or (ii) some common threads. (Mason 1994: 183-184):

    phenomenological or rationalist style language of This assumes that it is unproblematic to provide

    experience (the patterns are derived by a priori ‗brief-but-vivid‘ accounts of experiences or events

    necessity or some other force within the mind). without judgemental aspects: that it is feasible to

    do this. A poet provides ‗brief-but-vivid‘ accounts, [2] Recognising and Labelling Choices:

    but they are not of an underlying experience. locating alternative strategies and gambits for Whilst an individual may possibly construct use in particular situations in the future, either expressions which symbolise and help re-evoke a from reading, from sharing incidents, or from personal experience, the assumptions implicated in observing others; labelling typical incidents and claiming so cannot easily be validated. The corresponding gambits so as to trigger memory against which the ‗re-evoked experience‘ recollection of possible actions spectively [sic].

    2 is judged, the very means of judging validity, may (Mason 1994: 183-184)

    This is an unobjectionable form, suitably be altered by the process of constructing a‘ brief-

    elaborated, of the injunction to learn from but-vivid‘ account. The assumption that such

    experience. It ties in through references in Mason expressions may serve to give a proxy a matching

    (1994) with work on teacher‘s decision making. experience is very problematic. It either assumes

    However, the research on teacher-decision making that (i) there is an essential meaning in a signifier,

    is sometimes regarded as having a positivistic or in what may indeed be an arbitrary signifier, which

    anti-constructivist tinge: based on the assumption is communicable to another (which is refutable); or

    that teacher‘s perceptions, choices and behaviours (ii) that there is no essential meaning in a signifier

    can be described accurately by an outsider. The and the proxy does not have a matching

    language in [2] is that of injunction, not that of experience, but has some other experience which is

    knowing, so this criticism can be side-stepped. But deemed valuable in some unspecified way. The

    there are epistemological issues raised. ‗How can I problem with case (ii) is that it does not justify the

    know about my state and position?‘ is only term ‗re-evoke‘ for any other than the original

    partially problematic, concerning the difference experiencing subject. Yet [4], below, claims that

    between knowing and believing. But ‗How can I ―refining task-exercises which permit colleagues to

    know about another‘s mental states and processes, experience (directly or by analogy) something

    and about their potential and actual behaviours?‘ is proposed as worthy of noticing‖ is both possible

    more problematic. It is very problematic from the and feasible. So a problem that arises for me is

    perspective of a subjectivist epistemology. If this that if one takes a wholesale subjectivist position

    is the wrong epistemology to interrogate, what is on the priority of meaning and experience over

    the appropriate one? expression (as DN does), how is communication of

    a selected meaning possible? [3] Preparing and Noticing: entering recent

    The second claim is that ―once a corpus experiences post-spectively, then imagining accumulates, it is possible to identify common making an alternative choice pro-spectively. It threads‖. This is in part the claim of Grounded takes considerable effort to move the moment of Theory (Glaser and Strauss 1968), except that GT

    is very clear that the researcher‘s perspective constructs the pattern (‗common threads‘) and that 2The term ‗spective‘ (and ‗spection‘ and ‗spectively‘) seem to many such patterns are possible. GT then concerns be derived from ‗perspective‘ in the sense of view-point or

    the validation of bottom-up theories through the world-view. Mason (1994: 182) distinguishes extraspection,

    intraspection and interspection which appear to mean an involvement of others. This comes up in [4] external objectivist viewpoint, an internal, subjective, self-below, but it is not made clear that the aspects of observing viewpoint, and an intersubjectively agreed or DN make up a cyclic pattern. For invalidation negotiated perspective, respectively. I am not persuaded that

    a new terminology is called for, especially one that is not through others (always a possible outcome) would completely defined. necessitate ‗identifying other common threads‘.

interspective exchange of brief-but-vivid recognition from the retrospective to the

    incidents and observations; refining task-spective. (ibid.: 183-84)

    exercises which permit colleagues to experience Many of the problems raised for me by the POME Newsletter 8 (May 1995) 7 (directly or by analogy) something proposed as injunctions in this section have already been

    worthy of noticing. (Mason 1994: 183-184) articulated, above. An issue which arises here and

    above is that of an unspoken set of values and It is central to research in the interpretative

    paradigm in education that subjectivity is guarded model of good practice. What values underpin the

    against, and that the equivalent of validity and DN, both spoken and unspoken. Is not the

    reliability can be attained for the results of emphasis on individual choice an expression of an

    research. This fourth aspect stresses this ideological individualism, which has long

    dimension of research, as an antidote to the dominated Western thought, and is emerging most

    possible subjectivity of the insights derived from sharply in the ‗free market‘ or ‗market-place‘

    DN. My criticism here is not that it is unimportant metaphors of recent social policy? McBride (1994)

     it is overwhelmingly important but that it fails has offered a powerful critique of how

    to relate to the extensive and growing literature in individualism is the hidden ideology of much

    interpretative and qualitative research in education modern school mathematics. She argues how the

    and the social sciences. The validation of the emphasis on individual choice, and mathematics as

    methodology and results of interpretative and underpinning rational choice, permeates school

    qualitative research has been the major issue of mathematics texts, and represents a perspective

    discussion in scores of journals and dozens of whose strategy is to deny or conceal the historical,

    books for over a decade, and there is a rich and social, cultural, political nexus in which all

    well-tried literature to be drawn upon. It is largely knowledge and practices take place. On the cult of

    missing from DN. Is DN reinventing the wheel of the individual: Is the DN a version of the

    interpretative research methodology? philosophy of Nietzsche‘s ‗Superman‘, the person

    DN as communicated in its four aspects has some whose authentic experience and refined judgement

    interrelated and dominant features. The titles of is paramount; either in discerning what is

    the four aspects are reformulated injunctions. happening, or (more frighteningly) in pointing out

    Injunctions are commands, recommendations, to others what matters? This analogy is not

    orders or invitations to do something (which relates completely far-fetched. Consider the tradition of

    back to the guru-chela authoritarian power Zen masters which DN parallels: the guru or

    relationship mentioned above). It is therefore my enlightened one indicates to the chela or disciple

    presumption that DN is derived from the what is of import for the latter to attend to attain

    mathematical heuristics that Mason et al. (1982) wisdom. This is on the one hand a wonderful

    articulated and developed from the work of Polya tradition of mystical teaching, on the other a pure

    (1945) and others. There is a strong analogy authoritarian power relationship wholly dependent

    between Polya‘s (1945) heuristic problem solving on the integrity (or whim) of the master. The term

    cycle and the process of educational research, as is ‗Discipline of Noticing‘ is explicitly linked to

    widely noted (e.g. in Ernest 1994). Mason‘s Buddhist thought by Mason, and both ‗discipline‘

    earlier work on the teacher-as-researcher clearly and ‗noticing‘ are concepts that occur in

    3 links the active role of the student of mathematics Buddhism.

    as problem solver and the active role of the [4] Validating with Others: to guard against

    teacher-researcher. My conclusion is that DN is a solipsism and self-delusion, continued

    problem solving-based research methodology; a

    theory of how to undertake research. Whether my

    3genealogy is justified or not, it is clear that DN is On the guru-chela relation it is easy enough to identify

    persons potentially adopting the role of chela to Mason-as-primarily methodological in character. This is also guru, after all every successful teacher has students, and made clear in the text of Mason (1994, 189). Mason is a very successful teacher. It is interesting to This raises another problem. In my view DN is speculate who might be, if one exists, the guru to Mason-as-

    chela. A figure whose insights run through Mason‘s work, presented as more than a methodology, it is and who is an obvious candidate is Gattegno. He is the presented as a fully-fledged educational research author of such aphorisms as ―all I have to share is my paradigm: a new way of looking, being, teaching, awareness of my awareness‖ Mason (1994, 177). Perhaps

    DN arises from an interweaving, transformation and and doing research. But an educational research elaboration of the insights of Gattegno and Polya? paradigm is usually understood to have three

POME Newsletter 8 (May 1995) 8

    interrelated components: an ontology, an Polya, G. (1945) How to Solve it, Princeton, New

    Jersey: Princeton University Press. epistemology and following on from the former

    two, a research methodology, namely a theory of RESPONSE TO PAULS COMMENTS ON the means to generate knowledge in accordance THE DISCIPLINE OF NOTICING with the epistemology. DN on the face of it seems

    to presuppose its ontology and epistemology. They John Mason are implicit. It is claimed that it is based on a One of the fascinating things about seeing other ‗common epistemology‘ (Mason, 1994, 189), but people discuss ideas I have worked on, is the way this seems only to make explicit the fact that it is individuals stress some aspects and ignore others. an implicit, assumed view of knowledge. Only the It provides a useful reminder of Prousts remark, methodology seems to be given explicitly. Of that ―every reader is a reader of themselves‖, which course one way to address this criticism is to reflects an Amerindian Weltenschaung expressed further elaborate the epistemology and ontology of by Hymenehosts Storm (1985) as ―The world is a DN. Like any other emergent research program mirror for the people‖. I have found myself not that continues to grow, critique can point to areas just assenting to that adage, but asserting and living that need extension. by it whenever it comes to mind. Finally, I should return to my initial question ‗Is The previous paragraph provides a potted version the ‗discipline of noticing‘ a new paradigm for of noticing. I was struck, when reading Storm, by research in mathematics education?‘ Have I his adage. It fitted with past experiences and gave answered it? The answer to this second question me a neat handle (label) to act as focal point for, must be no. Evidence can be adduced that DN is among other things, a variety of techniques to serving as a paradigm for research in mathematics employ when I catch myself reacting to or being education, in that it underpins a variety of current mystified by someone elses reading. Furthermore, research in Britain. But my question was not as I typed the word assent, the word assert was descriptively factual as this answer suggests: rather triggered into memory. The reason is that assent-it was critical and evaluative. In this latter sense I assert is a framework which Joy Davis and I cannot claim to have offered anything like a worked on some years ago as a label for decisive critique. I can only claim to have distinguishing between intensities of participation indicated some problem areas, and there is no and between kinds of learning. Indeed it finds doubt that DN is continually being elaborated and correspondence in the not-noticingnoticingstrengthened. markingrecording framework which lies at the At this point I invited John to respond heart of the Discipline of Noticing. Labels act as (acknowledging that justice cannot be done to any focal points for triggers (usually metonymic), elaborated approach like John‘s in a few pages). resonances (usually metaphoric), and strategies to References choose to enact if it seems appropriate. I desist Ernest, P. (1994) An Introduction to Educational from inflicting upon readers of PoME any Research Methodology and Paradigms, Exeter: exercises designed to highlight similar experiences University of Exeter School of Education. in them, but exercises are also a critical part of the Glaser, B. G. and Strauss, A. L. (1968) The Discovery methodology of DN. of Grounded Theory, London: Weidenfeld and

    Nicholson. Brief-but-vivid accounts McBride, M. (1994) The Theme of Individualism in I find it useful to locate the phenomenon being Mathematics Education, For the Learning of analysed, and to distinguish it from the analysis, as Mathematics, 14(3) 36-42. far as possible. Paul points out that the notion of Mason, J. with Burton, L. and Stacey, K. (1982) brief-but-vivid accounts, which form a core Thinking Mathematically, London: Addison-Wesley, element of DN, is problematic. This is London undoubtedly the case, for accounts are based on Mason, J. (1994) ‗Researching from the Inside in language, and language is problematic. However, Mathematics Education Locating an I-You

    people who have worked on generating brief-but-Relationship‘ in Ponte, J. P. da, and Matos, J. F. Eds

    vivid-accounts, especially with colleagues, know Proceedings of the 18th Annual Conference of the

    International Group for the Psychology of that it is indeed feasible to produce effective Mathematics Education, Lisbon, Portugal: University BBVs. In fact I maintain that we are all aware of of Lisbon, Vol. 1, 176-194. BBVs, for when we launch into a story and we see

POME Newsletter 8 (May 1995) 9

    other peoples eyes glaze over, we know we have disciplined approach to negotiating technical strayed! It does not take long to discover what is terms, along the lines suggested for the growth of effective and what not. An account of an incident labelled frameworks, would not go amiss in the may be brief, may even be vivid, yet not community! necessarily be brief-but-vivid. That classification Paul appears to seek structure, in that he sees a is reserved for accounts which others readily contrast between the cyclicity of Grounded Theory recognise. There is not room here to elaborate, but and the apparent lack of structure in DN. DN is it is one of the key elements of the experience of not usually presented as a cyclic process simply noticing, as distinct from theories about noticing because as soon as noticing becomes a mechanical conducted solely in and through language. sequence of actions, it is no longer noticing. Paul opens his account of DN with a mixture of Unfortunately the urge to simplify and routinize account-of-like summaries in his list of strengths, has struck other action research methodologies in and accounting-for, when he uses expressions like which cycles have become unnecessarily rigid. All seductively attractive. It is as if he is warning a aspects of DN are potentially present at any younger generation against the immoral diversions moment, depending on circumstances. DN is not of a wicked world. I am intrigued by the multitude merely cyclic, but mutually co-evolving, to use of possibilities which could be read into this Varela, Thompson, & Roschs (1991) phrase.

    phrase. DN is at pains to reduce the emotive I am mystified by Pauls assertion that DN

    associations of language in order to focus attention depends on pre-existing patterns of experience. I

    on phenomena and to engage in critique and suspect it is because of our collective uncertainty questioning. about how to speak sensibly and fittingly about

    experience that he resorts to logic, for example by Communication identifying pre-existing patterns of experience At the heart of any linguistic enterprise there must with philosophical positions criticised surely reside the constant refrain of Paul Cobb: (convincingly for him) by Popper. In the spirit of taken-as-shared. As long as we seem to be DN, I invite Paul to work at recounting brief-but-agreeing, we keep talking; when we start vivid descriptions of incidents, and only then to ask conflicting, we have to negotiate or even part himself whether his ability to relate to the company. DN makes no claims to produce experience of others is due solely to parallel certainty or truth in absolute terms, or to study discursive practices or whether there is also a sense objective events. Indeed I find it informative to in which people in similar situations have similar think of an event as constituted by the collection of experiences. How else indeed could apparent accounts of and for, together with associated communication be possible? Paul asks how he can psychological states and inarticulable images, not know about states and processes, about potential only from direct participants, but also from second and actual behaviours, of others. I submit that and third hand participants. Of course memory is within some discourses, this is possible. I suggest fickle and malleable. But DN does not take events that it actually happens, quite frequently, otherwise as having an objective ontology to which accounts his students would remain distanced and untouched approximate. by being in his presence. It is precisely because he Pauls fundamental question for DN seems to be: can relate, can speak meaningfully to students and If one takes a wholesale subjectivist position on the colleagues, that I know that he can know about priority of meaning and experience over states and processes of others. But this is of course expression, how is communication of selected just one discourse among many, an account-of that meaning possible? he may be able to reject by invoking a different DN depends on negotiation of meaning between discourse. colleagues, and a constant testing of significance

    and sensitising to notice in the moment. The social Ontology and Epistemology is not denied, and is important, but it is not the I am disappointed that Paul does not find the totality either. That taken-as-shared epistemology and ontology he seeks, because I communication is possible is self evident; that have endeavoured to address these. They are mathematics educators do not on the whole essential components of any paradigm. DN communicate well could also be concluded from addresses directly the problem of descriptions and experience of conferences and teacher-training. A labels bringing into existence the qualities they

POME Newsletter 8 (May 1995) 10

    purport to describe, by emphasising the ongoing DN. DN is concerned with the experience of the necessity of testing in experience, communicating researcher. It is intraspective in origins and both with a wider community, and using the criterion of intraspective and interspective in validation. It future practice being informed. The epistemology draws upon extraspective research where it is of DN is based on the proposal that you cannot do informative and relevant. Most accounts of better than that, no matter what methods you interpretation that I have read immediately move employ, which cuts across traditional philosophical into the extraspective, in which the account, the approaches to base itself on lived psychological transcript, and the video, become the phenomenon experience. being studied. That some aspects of meaning are due to How do you decide what to notice? discursive practice seems to me to be as evident as The question I am most often asked is ―How do that individuals construe. But proposing discursive you decide what to notice?‖. In fact, noticing is practice as the sole answer to ontological questions not initially something most people have control concerning meaning does not fit with my over, and it is precisely in order to experience the experience. It does not assist me to make as much wonderful state of freedom, of choice, that the sense of my past or inform my future practice as Discipline of Noticing works to make noticing a my own eclectic view that the psyche and the matter of common intentional experience. I read in social are interwoven. Notice that I demonstrate various criticisms a concern that if I am led to my epistemology and criteria for validity within my notice some things, I may overlook or be led away response: I interrogate my past and test to see how from more important aspects. Yet any stressing future acts may be informed. I console myself with has consequent ignoring. As long as our stressings the assumption that Paul and I need to negotiate remain implicit and buried we are at the mercy of what is expected in a discourse on epistemology the habituation and enculturation that has taken and ontology as distinct from method. It may be place. By bringing our implicit frameworks and that what constitutes ontology and epistemology theories to the surface we can work at being able to may differ between different paradigms. choose to alter them. Paul refers to rejection of the epistemology and

    Roots and Sources methodology of the scientific research paradigm.

    DN is an elaboration of a way of working carried I am not convinced by the appropriateness of his

    out over many many years, derived and re-definite articles, and I certainly do not reject any

    articulated in my context from a few fragments I paradigm out of hand. Rather, I consider fitness-

    have encountered in writings from many cultures for-task. For the kinds of questions which interest

    reporting thousands of years of psychological and me (how do I develop my sensitivity to notice and

    sociological investigation. Mevlana Jalaluddin appreciate?, what is it like to be ...?, what is the

    Rumi (the mystic who initiated the whirling psychology of recognising a possibility?), I have

    Dervishes in the 11th century) said at one point found the DN more useful than any other

    ―Don‘t look at me, take what is in my hand‖ and I paradigm. But I draw extensively on any results

    recommend this injunction to anyone who obtained by any method, if they resonate with or

    concentrates on the person rather than the seeing. challenge my own experience. The results I ignore

    What matters to me is not pedigree but meaning, are the ones that pass me by, that do not seem

    not knowing-that but knowing-to. Part of what I relevant to my concerns. In this I suspect we are

    have learned from others is that what I can all rather alike. Indeed this observation is the basis

    reconstruct for myself is mine; what falls away for the epistemology on which the DN is founded.

    over time belonged to them. So I have deliberately Links to literature not emphasised my own sources, but rather dwelt It is true that writing about DN has not stressed the in what I have internalised, made my own, vast literature on interpretative and qualitative assimilated and accommodated, reconstructed for research, nor referenced all related philosophical myself, ... the list of euphemisms is long. It is up discussions or occurrences of similar ideas in other to others to do the same for themselves. And that domains. I look forward to specific references is part of the epistemology of DN: each individual which can be followed up and which shed light on must test assertions in their own experience, past, central issues. I want only to reinforce the present, and future. By proceeding in a fairly difference between what I have read elsewhere and

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