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  • International Journal of Mathematical Education inScience and Technology, Vol. 39, No. 7, 15 October 2008, 905924

    A revised theorization of the relationship between teachers conceptionsof mathematics and its teaching

    Ron Hoza* and Geula Weizmanb

    aBen-Gurion University, Beer-Sheva, Israel; bHemdat Hadarom Teachers College,Netivot-Azata, Israel

    (Received 19 June 2006)

    We assembled the ideas about mathematics and about its teaching which wereexpressed by mathematicians and mathematics educators into two pairs ofofficial (collective) conceptions: mathematics is either static or dynamic, andmathematics teaching is either closed or open. These polar conceptions produce a4-pair relationship between the conceptions of mathematics and its teaching. Theadherence to official conceptions was tapped by a questionnaire encompassing176 Israeli high school mathematics teachers, aimed at examining the relationshipbetween their conceptions of mathematics and its teaching. The majority of theseteachers either hold a single conception in one of the domains or do not adhere toany conception, and a quarter of them hold either the static-closed or dynamic-open pairs of conceptions that prevail among teachers in other countries.Consequently, we define a conception of an entity as a comprehensive andhomogenous set of ideas about a particular characteristic or feature of that entity.Reality is that teachers practice their profession without adhering to any officialconception, and perhaps are (/to be?/) praised for their reluctance to blindly adoptthe clear-cut rigid official conceptions of mathematics and its teaching whilemaintaining their individual and independent blends of ideas.

    Keywords: conceptions; mathematics; mathematics teaching; teachers; theoriza-tion; relations

    1. Introduction

    Teachers activities in classes and schools are but a small part of teaching whose largerinvisible part comprises the cognitive entities and the ideas that pertain and navigate-guidethe instructional behaviours (e.g. [1]). In this article, we deal with a small part of the largeinvisible ideational realm of mathematics teachers, namely, their conceptions aboutmathematics and its teaching, and the relationship between them.

    The term conception (of any entity) appears in the philosophical, educational andpsychological literature conjointly with a host of other terms that have been treated asequivalent, alternative or interchangeable with conception. Yet, the meanings of theseterms have neither been clarified or defined nor were they anchored in a theoreticalestablished framework. Examples are belief, thought, idea, attitude, perception, opinion,notion, basic principle, portrait, world view, image, epistemological belief, personal

    *Corresponding author. Email: hoz@bgu.ac.il

    ISSN 0020739X print/ISSN 14645211 online

    2008 Taylor & FrancisDOI: 10.1080/00207390802136602

    http://www.informaworld.com

  • knowledge, subjective theory, perspective, philosophy, ideology, value, system of

    explanations, understanding and knowledge (e.g. [29]).These terms split into narrow-scope and wide-scope implicated meanings.

    Perception, value, belief, opinion, notion, idea, image, attitude, thought, basic principleand perspective insinuate or connote rather restricted and transient nature, echoing a

    few aspects of the entity. Ideology, personal knowledge, philosophy, world view,

    portrait, subjective theory and understanding connote a comprehensive, connected/

    integrated, coherent and organized nature, referring to several aspects and perspectivesof the subject entity, they are justifiable and defensible by the person(s) holding them,

    and look steadfast, yet not inflexible or unmodifiable. Conception seems distinct from

    the other concepts in which it has been all too often dichotomized (e.g. mathematics is

    either absolute or provisional; or it is either absolute or fallible), whereas the other

    concepts usually appear as continua (e.g. our attitude towards mathematics rangesfrom positive to negative). In this article, we will show that this presumed

    dichotomization is unjustified and responsible for difficulties in the treatment of

    conceptions of mathematics and its teaching, and propose a revision of this dual

    theorization

    1.1. Conceptions of mathematics and its teaching

    Various dichotomous or polar characterizations, opinions, ideas or conceptions

    proliferate in the literature regarding the nature of mathematics and its teaching,which were experts either authentic statements or their inferences or abstractions of

    philosophical or empirical studies. The nature of mathematics was described as

    absolutist vs. fallibilist (e.g. [1013], Platonist vs. problems solving [10,14],

    absolute vs. relativistic/relative [15], and stable or changeable [16]. The natureof mathematics instruction have been portrayed as organizing and presenting

    information vs. constructivist teaching [7], transmission vs. discovery [17],

    student-centred vs. teacher-centred [15], content-focused vs. learner-focused, with

    emphasis on performance [5], school-knowledge oriented vs. child-centred oriented

    [16], authoritarianism vs. utilitarian, mathematics centred vs. progressive andsocially aware [10], traditional (i.e. transmission of knowledge) vs. progressive (i.e.

    socio-constructivist) [2], or behaviourist vs. constructivist [18]. These splits resemble,

    yet differ substantially from some bifurcations of people on various dimensions-

    characteristics (e.g. extrovertintrovert, analyticglobal, lovehatred for mathematics,capableincapable).

    To establish a unitary framework in relating to these dichotomies we will first label

    those regarding mathematics as either static-stable or dynamic-changeable and those of

    mathematics teaching as either open-tolerant or closed-strict. The polar expressions in

    the literature pertinent exclusively to the nature of mathematics (excluding references toother aspects of mathematics, such as its utility or value, or the factors that affect

    mathematical achievement) and the actual or desirable mathematics teaching were

    assembled and appear in parallel in Tables 1 and 2, with further breakdown according to

    the different issues with which they deal. Second, we will relate to these literature-based

    polar pairs as the official conceptions, keeping in mind that they are artifacts, whichneither have been stated, formulated or adopted by any academic community, nor have all

    of these notions appeared in any single paper on teachers conceptions about mathematics

    and its teaching.

    906 R. Hoz and G. Weizman

  • Table 1. The official conceptions of mathematics.

    The dynamic conception The static conceptionThe dynamic dimension of rationality is

    emphasised in mathematicsThe mathematical systems develop continu-

    ously and therefore its concepts are dynamicThe static dimension of rationality is empha-

    sised in mathematicsMathematics is a priori and infallible

    Mathematics is problem driven as a continu-ally expanding filed of human creation andinvention, in which patterns are generatedand then distilled into knowledge

    Mathematics is identified by the creativeactivity and heuristic processes and thus islike the other sciences; it is a growing andperpetuating problem solving in the domainof human creation

    Mathematics develops through conjectures,proofs and refutations

    Mathematics is a clear body of knowledge andtechniques

    The essence of mathematics is heuristics notthe outcomes

    Mathematics is a monolith, immutable pro-duct: unified body of knowledge, a crystal-line realm of interconnected structures andtruths, bound together by filaments of logicand meaning

    Mathematical knowledge is provisional, it isnot a finalized product but rather is open tore-examination and reconsideration

    Mathematics is a social constructionMathematics is conceived as doing it and is a

    product of human inventionMathematics is a process of enquiry

    Mathematics is conceived as a crystallizedbody of knowledge that transcends thehuman mind

    Mathematical knowledge is a finalized productthat rests on concepts, principles and inter-relating facts

    Uncertainty is inherent in the discipline ofmathematics

    Mathematics is a static but unified body ofcertain rules that are to be discovered andare not amenable of personal creation

    The status of mathematical truth is determinedto some extent relative to its contexts anddepends, at least in part, on historicalcontingency

    Mathematics has an independent existence, itis unobservable and thus is completelydifferent from the other sciences

    Mathematics is pure, hierarchically structuredbody of objective knowledge

    The universality, absoluteness and perfectibil-ity of mathematics and mathematicalknowledge is questionable

    Mathematics and mathematical knowledge areuniversal, absolute and perfect

    The concepts in the mathematical systems areabsolute, unambiguous and unchanging

    Mathematics is based on universal and truefoundations, the paradigm of knowledge,certain, absolute, value free and abstract

    The difficulty of mathematics is not unique,and it exists in other domains as well

    Mathematics is objectively difficult

    International Journal of Mathematical Education in Science and Technology 907

  • Table 2. The official conceptions of mathematics teaching.

    The open conception The closed conceptionThe student and her or his development is the

    first priority of instructionThe content is the major objective and is at the

    teachers focusMathematics teaching considers the childs

    needs and characteristics

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