# A revised theorization of the relationship between teachers’ conceptions of mathematics and its teaching

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<ul><li><p>International Journal of Mathematical Education inScience and Technology, Vol. 39, No. 7, 15 October 2008, 905924</p><p>A revised theorization of the relationship between teachers conceptionsof mathematics and its teaching</p><p>Ron Hoza* and Geula Weizmanb</p><p>aBen-Gurion University, Beer-Sheva, Israel; bHemdat Hadarom Teachers College,Netivot-Azata, Israel</p><p>(Received 19 June 2006)</p><p>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.</p><p>Keywords: conceptions; mathematics; mathematics teaching; teachers; theoriza-tion; relations</p><p>1. Introduction</p><p>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.</p><p>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</p><p>*Corresponding author. Email: hoz@bgu.ac.il</p><p>ISSN 0020739X print/ISSN 14645211 online</p><p> 2008 Taylor & FrancisDOI: 10.1080/00207390802136602</p><p>http://www.informaworld.com</p></li><li><p>knowledge, subjective theory, perspective, philosophy, ideology, value, system of</p><p>explanations, understanding and knowledge (e.g. [29]).These terms split into narrow-scope and wide-scope implicated meanings.</p><p>Perception, value, belief, opinion, notion, idea, image, attitude, thought, basic principleand perspective insinuate or connote rather restricted and transient nature, echoing a</p><p>few aspects of the entity. Ideology, personal knowledge, philosophy, world view,</p><p>portrait, subjective theory and understanding connote a comprehensive, connected/</p><p>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,</p><p>and look steadfast, yet not inflexible or unmodifiable. Conception seems distinct from</p><p>the other concepts in which it has been all too often dichotomized (e.g. mathematics is</p><p>either absolute or provisional; or it is either absolute or fallible), whereas the other</p><p>concepts usually appear as continua (e.g. our attitude towards mathematics rangesfrom positive to negative). In this article, we will show that this presumed</p><p>dichotomization is unjustified and responsible for difficulties in the treatment of</p><p>conceptions of mathematics and its teaching, and propose a revision of this dual</p><p>theorization</p><p>1.1. Conceptions of mathematics and its teaching</p><p>Various dichotomous or polar characterizations, opinions, ideas or conceptions</p><p>proliferate in the literature regarding the nature of mathematics and its teaching,which were experts either authentic statements or their inferences or abstractions of</p><p>philosophical or empirical studies. The nature of mathematics was described as</p><p>absolutist vs. fallibilist (e.g. [1013], Platonist vs. problems solving [10,14],</p><p>absolute vs. relativistic/relative [15], and stable or changeable [16]. The natureof mathematics instruction have been portrayed as organizing and presenting</p><p>information vs. constructivist teaching [7], transmission vs. discovery [17],</p><p>student-centred vs. teacher-centred [15], content-focused vs. learner-focused, with</p><p>emphasis on performance [5], school-knowledge oriented vs. child-centred oriented</p><p>[16], authoritarianism vs. utilitarian, mathematics centred vs. progressive andsocially aware [10], traditional (i.e. transmission of knowledge) vs. progressive (i.e.</p><p>socio-constructivist) [2], or behaviourist vs. constructivist [18]. These splits resemble,</p><p>yet differ substantially from some bifurcations of people on various dimensions-</p><p>characteristics (e.g. extrovertintrovert, analyticglobal, lovehatred for mathematics,capableincapable).</p><p>To establish a unitary framework in relating to these dichotomies we will first label</p><p>those regarding mathematics as either static-stable or dynamic-changeable and those of</p><p>mathematics teaching as either open-tolerant or closed-strict. The polar expressions in</p><p>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</p><p>mathematical achievement) and the actual or desirable mathematics teaching were</p><p>assembled and appear in parallel in Tables 1 and 2, with further breakdown according to</p><p>the different issues with which they deal. Second, we will relate to these literature-based</p><p>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</p><p>of these notions appeared in any single paper on teachers conceptions about mathematics</p><p>and its teaching.</p><p>906 R. Hoz and G. Weizman</p></li><li><p>Table 1. The official conceptions of mathematics.</p><p>The dynamic conception The static conceptionThe dynamic dimension of rationality is</p><p>emphasised in mathematicsThe mathematical systems develop continu-</p><p>ously and therefore its concepts are dynamicThe static dimension of rationality is empha-</p><p>sised in mathematicsMathematics is a priori and infallible</p><p>Mathematics is problem driven as a continu-ally expanding filed of human creation andinvention, in which patterns are generatedand then distilled into knowledge</p><p>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</p><p>Mathematics develops through conjectures,proofs and refutations</p><p>Mathematics is a clear body of knowledge andtechniques</p><p>The essence of mathematics is heuristics notthe outcomes</p><p>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</p><p>Mathematical knowledge is provisional, it isnot a finalized product but rather is open tore-examination and reconsideration</p><p>Mathematics is a social constructionMathematics is conceived as doing it and is a</p><p>product of human inventionMathematics is a process of enquiry</p><p>Mathematics is conceived as a crystallizedbody of knowledge that transcends thehuman mind</p><p>Mathematical knowledge is a finalized productthat rests on concepts, principles and inter-relating facts</p><p>Uncertainty is inherent in the discipline ofmathematics</p><p>Mathematics is a static but unified body ofcertain rules that are to be discovered andare not amenable of personal creation</p><p>The status of mathematical truth is determinedto some extent relative to its contexts anddepends, at least in part, on historicalcontingency</p><p>Mathematics has an independent existence, itis unobservable and thus is completelydifferent from the other sciences</p><p>Mathematics is pure, hierarchically structuredbody of objective knowledge</p><p>The universality, absoluteness and perfectibil-ity of mathematics and mathematicalknowledge is questionable</p><p>Mathematics and mathematical knowledge areuniversal, absolute and perfect</p><p>The concepts in the mathematical systems areabsolute, unambiguous and unchanging</p><p>Mathematics is based on universal and truefoundations, the paradigm of knowledge,certain, absolute, value free and abstract</p><p>The difficulty of mathematics is not unique,and it exists in other domains as well</p><p>Mathematics is objectively difficult</p><p>International Journal of Mathematical Education in Science and Technology 907</p></li><li><p>Table 2. The official conceptions of mathematics teaching.</p><p>The open conception The closed conceptionThe student and her or his development is the</p><p>first priority of instructionThe content is the major objective and is at the</p><p>teachers focusMathematics teaching considers the childs</p><p>needs and characteristics as the primaryfactors in instructional decision making</p><p>Mathematics teaching aims at and depends onthe mastery of concepts and procedures</p><p>The teacher believes in the ability and skill of thestudent to exhibit original thinking</p><p>The teacher is the knowledge authority and sheor he is obliged to transfer it to the students.She or he rigidifies knowledge in order topreserve her or his status</p><p>The student constructs her or his knowledgeactively, she or he is doing mathematicsStudents engage in purposeful activities that</p><p>grow out of problem situations, that requiresreasoning and creative thinking, gatheringand applying information, discovering,inventing and communicating ideas, andtesting those ideas through critical reflectionand argumentation</p><p>Mathematics teaching consists of passing on abody of knowledge, lecturing and explaining,communicating the structure of mathematicsmeaningfully</p><p>Mathematics teaching is an act of passinginformation on to others, it emphasises thesyllabus and curricular principles to guidetheir instruction</p><p>Students engage in purposeful activities thatgrow out of problem situations, requiringreasoning and creative thinking, gatheringand applying information, discovering,inventing and communicating ideas, andtesting those ideas through critical reflectionand argumentation</p><p>Mathematics instruction emphasises the trans-mission of knowledge and predilection ofpure and abstract mathematics</p><p>Mathematics teaching stresses the learnersconstruction of mathematical knowledgethrough social interaction; mathematicsteaching emphasises conceptualunderstanding</p><p>Learning is a personal-social processLearning is based mainly on personal-socialexperience and involvement and on discus-sions that evolved during problem solving</p><p>The student perceives or accepts knowledgepassively</p><p>Learning is based mainly on practice and drill oftechniques and it focuses on algorithmicperformance</p><p>All students must experience tackling problemsthat call for discovery, so that they can gainthe skills to execute such procedures. Thedifference between student levels will bereflected through the content complexity</p><p>Study procedures and higher order thinkingskills are within the reach of only the talentedstudents</p><p>All the students partake in and benefit from thestudy, discussions, testing, confirmation andrefutation of hypotheses</p><p>The ability to think mathematically is inborn</p><p>The ability to think mathematically is acquiredand depends on the quality of instruction</p><p>There is scepticism regarding the student abilityto exhibit thinking in original ways</p><p>Knowledge is assessed by its retrieval but also bythe abilities to explain and justify ideas and todefend and support conclusions or use themin new contexts</p><p>Knowledge is assessed mainly by its retrieval orby solving problems that are similar topreviously solved ones</p><p>The teacher is a companion, supporter andknowledge source who poses the studentsactivities that give rise to non routine problemsolving</p><p>Teaching is well defined and does not raisedoubts in the students minds regardingmathematical contents and procedures</p><p>(continued )</p><p>908 R. Hoz and G. Weizman</p></li><li><p>The official conceptions are more general constructs that encompass a variety ofbeliefs, concepts, meanings, rules, mental images and preferences of mathematics andits teaching that deal mostly with teachers and teaching but eschew the studentsperspectives. The static-stable and dynamic-changeable conceptions of mathematicsare clusters of all the experts characteristics of mathematics, respectively, and theofficial open-tolerant and closed-strict conceptions of mathematics teaching areclusters of all the experts views on mathematics teaching. These dichotomizationspresume that mathematicians, mathematics educators and teachers alike hold one ofthe polar conceptions in each domain, and it carries a strong resemblance to certainpsychological classifications, e.g. extroversion and introversion, analytic and global. Yetthere is a substantial difference in that the psychological theories acknowledge theexistence of individuals who respond inconsistently to the tasks, and are classified asmid-rangers or impure on the specific feature (and excluded from the analyses). Thegamut of ideas in each conception point that it seems difficult to agree with all ofthem, which led to another classification of teachers with respect to the officialconceptions: A teacher is an adherent to an official conception (and automatically is anon-adherent to the polar conception) if she or he accepts the majority (to bedetermined particularly for each research) of ideas in that official conception, or a non-adherent to the official conception if she or he does not comply with this requirement(i.e. accepts ideas from both conceptions). That formulation is congruent with both thepsychological tri-partition of individuals and other extending theorizations ofconception of an entity [3,1921].</p><p>Two cautions are due here. The labels of the conception carry no specific value,and the term official is not to be understood as a cultural or scientific consensus onthe content of the conception. Value judgment is legitimate and needs to be doneseparately and with reference to a specific context or framework. For example, thecharacterization of teaching as lenient does not justify judging it as bad or good,disruptive or encouraging, since it can be viewed differently by the students as goodor desirable, by their parents as bad or unacceptable, or by society at large aseducating for bad citizenship. Those who wish to evaluate the te...</p></li></ul>