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International Journal of Mathematical Education inScience and Technology, Vol. 39, No. 7, 15 October 2008, 905–924

A revised theorization of the relationship between teachers’ conceptions

of 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 of‘official’ (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: [email protected]

ISSN 0020–739X print/ISSN 1464–5211 online

� 2008 Taylor & Francis

DOI: 10.1080/00207390802136602

http://www.informaworld.com

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

explanations, understanding and knowledge (e.g. [2–9]).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. [10–13], ‘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’ and‘socially 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. extrovert–introvert, analytic–global, love–hatred for mathematics,capable–incapable).

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

teacher’s focusMathematics teaching considers the child’s

needs and characteristics as the primaryfactors in instructional decision making

Mathematics teaching aims at and depends onthe mastery of concepts and procedures

The teacher believes in the ability and skill of thestudent to exhibit original thinking

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

The student constructs her or his knowledgeactively, she or he is doing mathematicsStudents engage in purposeful activities that

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

Mathematics teaching consists of passing on abody of knowledge, lecturing and explaining,communicating the structure of mathematicsmeaningfully

Mathematics teaching is an act of passinginformation on to others, it emphasises thesyllabus and curricular principles to guidetheir instruction

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

Mathematics instruction emphasises the trans-mission of knowledge and predilection ofpure and abstract mathematics

Mathematics teaching stresses the learners’construction of mathematical knowledgethrough social interaction; mathematicsteaching emphasises conceptualunderstanding

Learning is a personal-social processLearning is based mainly on personal-socialexperience and involvement and on discus-sions that evolved during problem solving

The student perceives or accepts knowledgepassively

Learning is based mainly on practice and drill oftechniques and it focuses on algorithmicperformance

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

Study procedures and higher order thinkingskills are within the reach of only the talentedstudents

All the students partake in and benefit from thestudy, discussions, testing, confirmation andrefutation of hypotheses

The ability to think mathematically is inborn

The ability to think mathematically is acquiredand depends on the quality of instruction

There is scepticism regarding the student abilityto exhibit thinking in original ways

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

Knowledge is assessed mainly by its retrieval orby solving problems that are similar topreviously solved ones

The teacher is a companion, supporter andknowledge source who poses the studentsactivities that give rise to non routine problemsolving

Teaching is well defined and does not raisedoubts in the students’ minds regardingmathematical contents and procedures

(continued )


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 students’perspectives. 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 of‘conception of an entity’ [3,19–21].

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 ‘good’or ‘desirable’, by their parents as ‘bad’ or ‘unacceptable’, or by society at large as‘educating for bad citizenship’. Those who wish to evaluate the teaching need to do sowith due consideration of both their purpose and the specific circumstances of theclassroom and persons involved.

Table 2. Continued.

Teaching consists of encouragement, facilitationand arrangement of carefully structuredsituations for investigations

Posing open questions and inquiry problems isdesigned for strong students whereas theweaker one will engage in less complexalgorithmic problems

Teaching raises conflicts and doubts in thestudents’ minds with regards to mathematicalcontents and procedures

Mathematics teaching includes genuine discus-sion, cooperative groupwork, project workand problem solving for confidence, engage-ment and mastery

The teacher encourages individual work in ageneral sense, but discussions are rare, andcontent progression and coverage is her or hismain concern

The teacher encourages individual work butattributes central importance to interactionsand dialogue among peers, which enablesstudents to experience alternative viewpoints


Studies on the dispersion of the official conceptions of mathematics and its teaching

in several countries found that generally the static official conception of mathematics

and the closed official conception of mathematics teaching prevail among prospective

and active elementary school teachers, and to a lesser extent among high school

mathematics teachers, and that in-service teachers have more diversified mathematical

beliefs than prospective teachers [22]. Specifically, elementary teachers and prospective

teachers view mathematics as a discipline that is based on rules and procedures to be

memorized [23,24], elementary and high school teachers see mathematics as highly

sequential and static [25], and elementary student teachers think that in mathematics

there is usually one best way to arrive at an answer, and that mathematics suits only

those with innate mathematical ability and mathematical minds that use rules and logic

rather than intuition [26–28].Large populations of elementary and high school prospective teachers hold the closed

conceptions of mathematics teaching [22], and the majority of elementary school teachers

adopt a traditional view of mathematics teaching, namely, that pupils learn in a passive

manner with great emphasis on practice and memorization, whole class discussion, and

teacher’s modeling and use of manipulatives [22–24,26–31], with the only exception in

which elementary and high school teachers thought that students are capable of

constructing their own mathematical knowledge in an atmosphere of negotiation and

relevance [32].The dispersion of the official conceptions of mathematics and its teaching in Israel

have not been studied so that our first research question is: Do the Israeli high school

mathematics teachers hold similar official conceptions?

1.2. The relationship between conceptions of mathematics and its teaching

The surveyed studies employed a dichotomy of the conceptions of mathematics and its

teaching and further presumed that every teacher has two corresponding conceptions

which led to a 4-pair relationship between the conceptions, namely, the dynamic-open,

dynamic-closed, static-open and static-closed. This comprehensive framework has not

been formulated and the relationship has been expressed as particular pairs of

conceptions. These pairs have been inferred by philosophical-logical analyses of the

nature of mathematics and practice (e.g. [33,34], from analyses of teachers’ practice and

interviews about it that yielded models of teaching [5] or teaching types [35], or

textbook use [35]. The combination of philosophical analyses and inferences from

research findings unveiled educational ideologies of mathematics education that were

formulated as teacher prototypes, which implicitly embody certain pairs of the polar

official conceptions [3,36–38].These pairs of conceptions have been either by-products or served for other

purposes and their existence and prevalence has been tested in only a few empirical

studies.The prevalence of the pair of static-closed and the rarity of the pair of dynamic-open

among mathematics teachers were implicitly reflected by the teachers’ use of textbooks

[39]. A study in a sub-sample of nine (of 42) teachers holding the static or dynamic

conceptions of mathematics yielded inconclusive results [37]. In three small samples of four

mathematics teachers in three countries it has been found (in terms of Ernest’s teacher

types [36]) that the static-closed pair prevail in UK, the static-closed and the static-open


pairs abound in France, and the static-closed pair prevails in Germany [16]. Among 140mathematics teachers and student teachers, those who held the dynamic conception ofmathematics (as inferred from the NCTM standards) tended to hold the open conception

of mathematics teaching [40].Two studies tested the intercorrelations between the teachers’ conceptions

statistically. Factor analysis of questionnaire responses of 577 British secondaryschool mathematics teachers yielded four conceptions of mathematics as an area ofhuman activity (addressing external aspects of mathematics rather than its internalcomposition and features) and five conceptions of mathematics teaching. Of the 20possible intercorrelations between the two sets of conceptions about half (9) weresignificant but low (5 under 0.2 and 4 between 0.3 and 0.4, p5 0.05) (e.g. the view of

mathematics as a service to other areas of activity correlates with the view of openmathematics teaching) [8]. In a sample of university assistants who were notschoolteachers, the conception of relative knowledge tended to go with the student-oriented conception of instruction and the conception of absolute knowledge tended togo with the teacher-oriented conception of instruction (r¼ 0.28 and 0.38, p5 0.01,respectively) [3], and the latter findings may be relevant for mathematics and itsinstruction too.

Half of the few studies which were designed to test the nature of the relationshipbetween the conceptions among teachers or student teachers yielded inconclusive or

equivocal findings. We suggest that the inconsistent results are due to methodologicaland theoretical differences in the (pre-service and acting) teacher populations, theprobes which were applied to tap the conceptions, and the statistical techniques.Particularly, the correlational studies only indicate tendencies in aggregates of teachersand cannot specify which pair of conceptions is held by every teacher.

From the theoretical point of view, most of the inferential and empirical studiespresumed the dichotomization of conceptions and that each teacher holds one definitepolar conception in each domain and therefore were underlain (knowingly or

unconsciously) by the fourfold relationship between the conceptions. However, thesetheoretical assumptions were stated or implied but have not been substantiated by orrelated to any psychological theory or conceptual framework, and may well be flawedor practically unrealistic. Therefore, our study addressed the three following questions:Do all, or most Israeli high school mathematics teachers adopt the official conceptionsof mathematics and its instruction? Are the official conceptions of mathematics and its

teaching related among these teachers? And if so, do the static-close and dynamic-openpairs of conceptions prevail?

2. Method

2.1. The participants

The participants in the study were 174 mathematics teachers (88% of the teachers) from allthe 33 junior and senior high schools in Southern Israel, of which 12 were excluded onmethodological grounds. Their average teaching seniority was 21 (range from 1 to 30)years. Seventy percent of the senior and 34% of the junior high school teachers hold atleast a first degree in mathematics (the rest do not have such degree). Eleven of these

teachers were interviewed individually to control for the effects of additional factors on theconceptions’ measurement.


2.2. The measurement of the conceptions

The four official conceptions of mathematics and its teaching were measured by aquestionnaire that covered all aspects in Tables 1 and 2. Its pilot version included 85 itemsthat were derived from Ernest [14] and Thompson [38] or developed by two mathematicseducation professors and the second author. Each item consisted of a statement onmathematics or its teaching (in separate parts), to which the response was made on a fivelevels Likert scale (from ‘agree’ to ‘do not agree’). The responses to this questionnaire from16 mathematics teachers helped revise the questionnaire so to increase its validity, and toalleviate possible burden due to the need to relate to numerous inter-mathematics issues.

The revision contained (1) Addition of two items from Lerman’s questionnaire [38],(2) Deletion of non-discriminating items (on which over 80% of the teachers checked thesame alternative), and items that dealt with knowing and learning mathematics or withattitudes towards mathematics and its teaching (such as self confidence, fears, andprestige) and the role of mathematics in society or science (similar to [8]), (3) Formatchanges in items that expressed a causal relation between several ideas by breaking themdown into their component ideas, that were responded separately, and (4) Change inresponse mode from Likert scale to ‘agree’ or ‘disagree’, which was designed to tap theadoption of one official conception (and rejection of the opposite one). The statementswere listed randomly with equal positive and negative statements.

The final version of the questionnaire consisted of 23 items pertinent exclusively tomathematics and 46 pertinent to mathematics teaching (sample items are presented in theAppendix). These unequal numbers reflect the lesser reference to the nature ofmathematics. The questionnaire attained between judges agreement of over 95% andwas found a highly (construct) valid measurement of the four identified conceptionsthrough individual interview with a sample of 7% of the teachers to corroborate theirwritten answers [41].

Sample items of the final questionnaire with their scoring are the following.‘Mathematics is a crystallized body of knowledge that transcends the human mind andexists outside it’ (agreement points at the static-stable, and disagreement points to thedynamic-changeable conceptions of mathematics, respectively). ‘Mathematics teaching iswell defined and does not create doubts regarding the mathematical contents andprocedures’ (agreement points at the closed-strict, and disagreement points to the open-tolerant conceptions of mathematics teaching, respectively).

An individual interview was designed and structured to enable teachers to explain andelaborate the ideas that they checked in the questionnaire and to provide informationabout additional issues (which are not reported here).

2.3. The procedure

Israel Ministry of Education the school principals approval enabled us to contact themathematics department heads, all of whom agreed to serve as our proxies in their schools.They were given explanations on how to complete the questionnaire and handed itindividually to all the 220 teachers and collected them during one month. Total of 188teachers (85%) returned the questionnaire and the reasons for not returning thequestionnaires seem to be mainly logistic and motivational. Teachers’ identities werecoded for anonymity. Fourteen (7%) of the questionnaires were not processed because ofmissing data on the professional variables, and of the remaining 174 questionnaires, 78were from junior high school teachers and 96 from senior high school teachers.


Generally it appears that the option ‘I have no opinion’ minimized item skipping, as 9

(5%) of the 174 teachers skipped three items or more and their data were excluded in from

certain analyses (leaving 165 teachers in Table 3). Yet, to increase the number of teachers

in the analyses we included in each analysis the largest number of questionnaires without

missing data in the relevant variables and this practice resulted in small variability (at most

1%) in the number of teachers, which, we hope does not bias the results of these analyses.

2.4. The design

Our study comprised quantitative and qualitative parts but here we report only on the

quantitative part, which involved a large sample of teachers and positivist statistical

hypothesis testing. The relationship between the conceptions of mathematics and its

teaching was tested statistically and the qualitative part included the individual interviews

with 11 teachers whose interviews were text analysed in order to assess: if the teachers

understood certain of the questionnaire items, if and to what extent they were committed

to their questionnaire responses to these items (providing convergent evidence that bears

on the questionnaire reliability over time), and the construct validity of the ‘official

conceptions’ as measured by the questionnaire.

2.5. Data processing

The test items were scored as follows: A ‘match’ to an official conception was recorded if

the teacher checked either agreement with a statement that represents that conception or

disagreement with a statement that reflects the opposite conception. The ‘match’ responses

to the static conception of mathematics or the closed conception of mathematics teaching

were scored 1, those to the dynamic conception of mathematics or the open conception of

mathematics teaching were scored 2, and the response ‘I have no opinion’ was not scored,

because it is not on the agreement continuum as the other two. This scoring is different

from other practices (e.g. [15,37]) where the lack of opinion and omission of response

could be assigned the middle value on the conceptions’ Likert scales. The purposes of

offering the latter option were first to enable any teacher whose conceptions differ

completely than the official ones to bypass our official suggestions, and second to

minimize the number of missing answers which is a major impediment to the processing

and interpretation of the data.The scores on each domain (mathematics and its teaching) were averaged for every

teacher to obtain two individual mean conceptions scores ranging between 1 and 2, and

Table 3. Teachers’ adherence to the official conceptions of mathematics and its teaching.

Official conception of mathematics

Dynamic Static No adherence Total43 (26.1%) 39 (23.6%) 83 (50.3%) 165 (100%)

Official conception of mathematics teaching

Closed Open No adherence Total40 (24.3%) 38 (23.0%) 87 (52.7%) 165 (100%)

Notes: In parentheses is the percentage in the teacher sample. ‘No adherence’ indicates ‘not adheringto any official conception’.


their distributions were graphed. In these distributions the mean conception scores that

are close to 1 signify the acceptance of many statements in the static conception of

mathematics or closed conception of mathematics teaching, and those that are close to 2

signify the acceptance of many statements in the dynamic conception of mathematics or

open conception of mathematics teaching.We set two cut-off points on each mean score distribution which created in each

distribution two tails each of which contained about a quarter of the scores and a

middle region between them. We used that tri-partition to classify every teacher as

adherent to an official conception if her or his mean conception score lied in a tail or

else as not adherent.

2.6. The hypotheses

(1) The official conceptions of static mathematics and the closed mathematics teaching

are adopted by the majority and the dynamic mathematics and open

mathematics teaching are held by the minority of the Israeli high school

mathematics teachers.(2) Most of Israeli high school mathematics teachers adopt the pair of static

conception of mathematics and the closed conception of mathematics teaching,

and their minority adopt the pair of dynamic conception of mathematics and the

open conception of mathematics teaching.

3. Results

3.1. The distribution of conceptions

The distribution of all the teachers’ mean mathematics conception scores are presented in

Figure 1. It represents well the teachers’ mean mathematics teaching conception scores

(not shown): Both distributions are nearly symmetrical, with small proportions of the

teachers lying near the endpoints 1 and 2.The choice of the cut-off points was based on the two-fold rationale: relatively large

jumps in frequency (‘elbows’) occur at or near these points, and the tails contain more

than just a few teachers. These cut-off points were 1.35 and 1.75 for mathematics and

1.35 and 1.70, for mathematics teachings and the classifications of teachers by their

adherence to the official conceptions are presented in Table 3.According to this table about half the teachers do not adhere to any official

conception, a quarter of the teachers adhere to each of the official static and dynamic

conceptions of mathematics and a quarter of the teachers adhere to each of the official

closed and open conceptions of mathematics teaching. These results refute the first

hypothesis that most teachers would adhere to the official static and closed conceptions

of mathematics and its teaching and their minority to the official dynamic mathematics

and the open conceptions of mathematics teaching.

3.2. The relationship between the conceptions

The joint distribution of the mean conception scores for the whole sample of teachers is

presented in Table 4, and similar tables were constructed for the junior and senior high

school teachers (not shown).


Table 4. The relation between all teachers’ adherence to the official conceptions of mathematics andits teaching.

Official conception of mathematics

Official conception ofmathematics teaching Dynamic Static No adherence Total

Closed 10.0 47.5 42.5 100%4 19 17 40

9.3 48.7 20.5 24.3

Open 44.7 0.0 55.3 100%17 0 21 3839.5 0.0 25.3 23.0

No adherence 25.3 23.0 51.7 100%22 20 45 8751.2 51.3 54.2 52.7

Total 26.1 23.6 50.343 39 83 165

100% 100% 100%

Notes: Frequencies are in boldface; percentages within rows and within columns appear above andbelow the frequencies, respectively. ‘No adherence’ indicates ‘not adhering to any officialconception’.

25

20

15

10

5

0

Num

ber

of te

ache

rs

Mean score of conception about mathematics

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2

Figure 1. The means scores of the conception of mathematics in the whole sample.


The relation between the conceptions of mathematics and its teaching was tested by�2 tests of independence which show that the two conceptions are statisticallysignificantly and moderately related in the whole sample and for the senior high schoolteachers (�2¼ 28.4, p5 0.0001 and �2¼ 16.1, p5 0.003, and Contingency coefficients

of 0.37 and 0.38, respectively), but they are not related for the junior high schoolteachers (�2¼ 6.3, p¼ 0.17). The hypothesized 4-pair relationship appears is the upperleft corner of Table 4 (the teachers who adopted two official conceptions), and therelationship is significant and strong (�2¼ 40.7, p5 0.0001, contingencycoefficient 0.98).

The relationship between the official conceptions of mathematics and its teachingamong all the teachers (and likewise among the senior high school teachers) is expressed bythe partition of the teachers into two different groups.

The first group comprises 75% of the teachers: A quarter of the teachers who adheredto two official conceptions (of mathematics and its teaching) and split evenly betweenadopting the dynamic-open and the static-closed pairs of conceptions, and a half of theteachers who adhered to one official conception (of either mathematics or its teaching) and

split evenly between adopting an official conception in the other domain and not adoptingany official conception.

The second group comprises a quarter of the teachers who adhere to none of theofficial conceptions.

Two aspects of that relationship are noteworthy. The marginal distributions are highlysimilar to those of the teachers who adhere to no conception, and are markedly differentfrom those of the teachers who adhered to two official conceptions.

4. Discussion

Our major finding was that the distributions of the mean conceptions’ scores weresymmetrical. This counter evidence to the polar portrayal of the conceptions ofmathematics and its teaching shows that the strict dichotomization of teachers’conceptions of mathematics and its teaching and their relationship have beenoversimplifications of both the nature of the conceptions and their relationship. Thatrecognition led to a tri-partition theorization of the conceptions in a manner similar to

the psychological and educational theory and research that acknowledge individualswho respond inconsistently to the tasks and do not belong to either extreme, which ledto the replacement of the 4-fold with an expanded 9-fold relationship between theconceptions of mathematics and its teaching.

In the following sections we will address the adherence to the official conceptionsand their relationship and propose possible reasons for those findings, and discusssome theoretical and practical implications of the revised theorization.

4.1. The adherence to the official conceptions by the Israeli schoolteachers

The official conceptions of mathematics and its teaching have not been adopted by morethan half the Israeli junior and high school teachers, sharply contrasting with the

predominance of the static conception of mathematics among mathematics teacherselsewhere [22–24,26,29] and the closed conception of mathematics teaching [22,29]. We willfirst show that these findings are not artifacts of our methodology of theorizing and


measuring teachers’ adherence to the official conceptions, and then will propose several

reasons for those findings.

4.1.1. Methodological considerations

The rationale in compiling the questionnaire by assembling all the experts’ one-

dimensional clear-cut, unequivocal, succinct, general and all-inclusive ideas is that any

expert expressed reliably her or his undisclosed larger set of cohesive and consistent ideas.

Additionally, the variety of ideas in the official conceptions represent their multifaceted

nature on the one hand, yet it is rather difficult to be consistent on numerous single items

without either seeing the whole picture or remembering the previous responses.

Consequently, it is difficult to assure that teachers who were identified as non-adherents

actually did not adhere to any conception. Such considerable consistency among the

responses was not required from the experts, so that to reduce the risk of misclassifications

we allowed for wide margins at the second and third quadrilles. Admittedly, this drawback

of classifications may be valid for almost all instruments and response modes where the

respondent can neither see her or his answers nor can form the whole picture (which in

many cases can be obtained after the analysis) and better alternatives are graphical

instruments such as concept mapping (e.g. [42,43]).The forced choice response mode did not distort the teacher’s adoption or rejection of

any official idea compared to the use of standard 3-, 5-, or 7-point Likert scale, which

would not affect substantially the rather symmetric shape of our score distribution. To the

contrary, it would bias the scores of teachers who checked the non-extreme alternatives

towards the centre and shrink the tails.Individual follow up interviews were conducted with a stratified (by abundance)

sample of 11 teachers who represent all the pairs of conceptions. Their interviews

corroborated the individual conceptions and the particular pair that we attributed to them,

of which they were totally unaware, since no additional or new ideas about mathematics

and its teaching have been expressed.There seem to be four reasons for teachers’ non-adherence to the official conceptions

of mathematics.

(a) The nature of teachers’ conceptions. Teachers’ ideas and conceptions can be

dissimilar to the multifaceted official conceptions because they are vague and

incoherent [14] and that teachers’ implicit theories are eclectic sets of unclear ideas

that stem mainly from their personal experience and prejudices which are not

rooted in the academic literature [3,44].(b) The origin of the conceptions of mathematics and the acquaintance with the

dynamic aspects of mathematics. The official conceptions were formed by the

experts’ long professional career, whereas the teachers could get exposed to these

ideas in their university or college mathematics courses and during in-service

pedagogical courses. In any case, as students the teachers had relatively small

number of mathematics courses which dealt with quite few aspects of mathematics,

and their majority probably had no adequate access and exposure to the dynamic

aspects of mathematics and the views of the active mathematicians which have

been voiced to a lesser extent and relatively scantily in the media than in the

professional literature. This is especially true in Israel where public debates focused

on the nature of elementary school mathematics curriculum at the expense of high

school mathematics. These circumstances could not provide the teachers with due


opportunities to accommodate to any definite official conception which might leadthem to entertain ideas from both poles so to become non-adherents.

(c) ‘Academic’ and ‘school mathematics’. Errors in tapping teachers’ adherence to anyofficial conception could result from a possible mismatch between the meaningsthat experts and teachers assigned to ‘mathematics’ in our questionnaire.The official conceptions apparently related to the evolving and developingnature of ‘academic mathematics’ and it is unclear if the teachers referred to thatmeaning or to ‘school mathematics’, which is a stable/unchanged minute part ofthe former.

(d) Appreciation for the official views. There is ample evidence that prospectiveteachers resist the temptation to adopt the ideas presented to them in pre-serviceeducation (e.g. [44,45]). In light of the widespread attempts to reform the non-changing nature of instruction it is clear that teachers and student teachers tend todismiss the experts’ views as unrealistic and irrelevant to extant school teaching(e.g. [46,47]).

There seem to be two reasons for teachers’ non-adherence to the official conceptions ofmathematics teaching.

(a) The presentation of the official conceptions. Some of the official conceptions ofmathematics teaching in Israel were part of argumentation on the quality ofteachers and the outcomes of their instruction, namely, pupils’ achievement andtheir measurement. Largely, these discussions had not touched or elaborated onthe nature of actual or desirable teaching, and the debates still leave the teachers,like the rest of the public with contradictory ideas from both official conceptions ofmathematics teaching.

(b) The nature of the official conceptions. The official conceptions of mathematicsteaching reflect ideal, desirable or optimal notions. Teachers who struggle to findways to accommodate for the extant curriculum, classes and students cope with theclassroom realities and even if they wanted they cannot stick with one of theofficial conceptions and ignore or nullify the practical considerations.Consequently many teachers live with conflicting ideas from both officialconceptions.

4.1.2. Factors affecting the adoption of the official conceptions

Our attempts to propose reasons for the substantial non-adoption of the officialconceptions by the teachers began with the finding that the conceptions were relatedonly among the senior but not the junior high school mathematics teachers. Theseteachers differed on the following professional characteristics. Background inmathematics: 70% of the senior and 34% of the junior high school teachers have atleast a first degree in mathematics. Familiarity with mathematics: the senior highteachers deal with more advanced and highly formal mathematical topics than thejunior high teachers. Time, achievement and coverage constraints: the mainresponsibility of the senior high school teachers is to prepare their students for thenational exams, which drives them to more direct and tending towards more rigidteaching routines, whereas the junior high teachers have fewer constrains and can bemore flexible in both the contents and teaching methods.

Eight statistical �2(df¼ 1) tests were conducted for each of these groups. Table 5 showsthe outcomes of these tests: No relations were found (1) among the senior high school


teachers between the conceptions of mathematics and its teaching and both the holding ofa degree or length of experience in teaching, (2) among the junior and senior high schoolteachers between the conception of mathematics teaching and both the holding of a degreeand length of experience in teaching. Two significant relationships were found among thejunior high school teachers between the conception of mathematics and both the holdingof a degree and length of experience in teaching. Generally the teachers with a degree inmathematics tend to adhere to the static conception of mathematics, and those with longerseniority in teaching tend to abandon their adherence to an official conception.Specifically, (a) Of the teachers without a degree in mathematics, 2/3 do not adhere toany conception of mathematics, 1/4 adhere to the dynamic official conception and 1/12adhere to the static official conception; Of the teachers with a degree in mathematics2/5 adhere to the static and 2/5 to the dynamic conception of mathematics, and 1/5 did notadhere to any conception of mathematics. (b) Of the novice teachers (up to 5 years ofteaching) 3/5 adhere to one of the official conceptions of mathematics and 2/5 do notadhere to any official conception; Of the veteran teachers, 3/10 adhere to one of the officialconceptions of mathematics and 7/10 do not adhere to any official conception ofmathematics.

The effect of the degree in mathematics may be due to the fact that teachers without adegree had lesser acquaintance with formal aspects of mathematics and their schoolmathematics is mostly static. The teachers with a degree in mathematics who deal with lessformal junior high school mathematics either persist in their adherence to the static officialconception of mathematics that results from their mathematics education, or adopt ideasfrom both conceptions in order to reconcile the difference in the nature of their universityand the school mathematics.

A possible reason that teachers abandoned the conceptions of mathematics withlonger teaching experience is that during their neophyte years of teaching they couldstick to the official conception adopted in their pre-service education but they realizedthat the school mathematics they have been teaching for a long period of time includesboth dynamic and static aspects, and these aspects reflected in their conceptions.

In sum, the majority of Israeli mathematics teachers do not adhere to theofficial conceptions of mathematics and its teaching, the neophyte junior high schoolteachers are committed to the official conceptions of mathematics but tend toabandon them as their teaching experience increases and their mathematicaleducation expands, and the senior high school are affected by these factors innon-systematic ways.

Table 5. The relationships of the conceptions of mathematics and its teaching with school type,holding a mathematics degree and length of experience in teaching.

Senior high school Junior high school

Relatednessbetween

Conception ofmathematics

Conception ofmathematicsteaching

Conception ofmathematics

Conception ofmathematicsteaching

Mathematics degree unrelated unrelated related unrelatedSeniority in teaching unrelated unrelated related unrelatedConceptions ofmathematics andits teaching

related unrelated


4.2. The relationship between the conceptions of mathematics and its teaching

Our study yielded corroborating and refuting findings to the hypothesized 4-pairrelationship between the conceptions of mathematics and its teaching. The supportivefinding is that the relationship was long held by the mathematics education community hasbeen established substantially for the quarter of the teachers who held/adhered to pairs ofconceptions of both domains. The divergent evidence is that the tri-faceted conceptionsproduced an expanded 9-pair relationship of the majority of the teachers hold 5 pairs (witheither one conception or none). The validity of the expanded relationship is supported bythe similarity of our findings to the relatedness coefficients of 0.3–0.4 that were obtainedby quantitative methodologies [8,15] and since the traditional 4-pair relationship is part ofthe more comprehensive relationship.

We can propose several reasons why the expanded relationship has not beenexposed before or its existence was camouflaged. (1) For various reasons the expertsheld one pair of conceptions, either the static-closed or the dynamic-open, and thesewere expressed in their writing and disseminated in the professional communities,(2) the empirical findings have been unintentionally biased because they were based onsmall samples of teachers who happen to hold conceptions in both domains, (3) Theinference of the relationship by ‘logical’ analyses that presumed causality in that theconception of mathematics teaching follows from the conception of mathematics, and‘phenomenologically-based’ analyses that presumed polarity of the official conceptionsand (4) Our own inferences of the conceptions from the empirical models wereimprecise or erroneous.

The researchers whose classifications were based on the presumed dichotomies imposeon the teachers definite conceptions which some of them actually did not adhere to. Suchmisclassifications went undetected because the assignment of conceptions was achieved bythe implicitly use the centre of the symmetric conceptions’ distributions as the cut-offpoint, which produced mutually annulling errors in both directions. Such measurementerrors imply that the empirical studies have been conducted either among the teachers whoadhered with both official conceptions or among teachers who were misclassified into thefourfold matrix. In either case these errors explain some inconclusive findings (e.g. [37]) orthe low correlations in statistic-based studies (e.g. [8,15]) as compared with the strongestrelation among our teachers who adhered to two conceptions.

4.3. Implications and conclusions

We will conclude with theoretical and practical implications to our study and lessons tomathematics educators and mathematics teachers.

First, experts’ ideas that have been presented publicly were assembled to form the‘official conceptions’ of mathematics and its teaching. Their complex and elaborate naturenecessitate to revise the definition of conception (of mathematics and its teaching or anycognitive or physical entity) and formulate it as ‘a comprehensive and homogenous set ofideas on a particular characteristic or feature of that entity’. The probes needed to uncoverthese unconscious conceptions of individuals are necessarily multifaceted and demanding.

Second is that in the measurement of teachers’ conceptions their tri-partition is calledfor as adherents to a (official) conception at either pole or non-adherents with anyconception.

Third is that the relationship between the conceptions is expressed by an expandedmatrix with 9 pairs of conceptions or their absence.


Fourth, numerous teachers manage to teach and even excel in that who adhere to atmost one official conception. The message to teacher education institutes is that they needto reconsider their attempts to disseminate the official conceptions among both studentteachers and in-service teachers to incorporate them in the teachers’ professional identity.

The encouragement to the teachers is that they should not be discouraged by thefindings, since we need to appreciate and praise their reluctance to blindly adopt the clear-cut rigid official conceptions of mathematics and its teaching and their insistence onmaintaining their individual and independent blends of ideas, which can be an educationalmodel for their students.

At the end we are left with challenging questions that await further study. Amongthese are:

What is the composition and organization of the individual’s conceptions (eithersimilar or different from the official ones) which are part of her or his professionalknowledge?

Why are the teachers’ conceptions coupled the ways they were?Why, how and when do the conceptions of mathematics and its teaching form and

coupled?Can the conceptions be influenced, shaped or amended by the mathematical and

teacher education, and if so – to what extent and how?What roles do the conceptions or their lack play in the teachers’ professional thinking

and acting?Which cognitive processes are responsible for the adoption of the official conceptions

of mathematics and its teaching and the formation of larger cognitive units of couples ofconceptions?

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Appendix: Sample questionnaire items

The response options to each item and sub-item are Agree, disagree, I do not have an opinion.

(1) Teaching mathematics is a means by which the teacher transfers to the student knowledgethat has the form of principles, formulae, facts and rules.

(2) Mathematical thinking is not inborn but can be nurtured.(3) It is highly desirable that the most frequent terms in mathematics classes would be: Is that

so? Maybe, what can you deduce? Make a conjecture, Hypothesize.(4) In teaching mathematics it is a must to require students to cope with many unusual tasks

and problems.

(a) It is desirable that the teacher design the mathematical activities in her or his classes asa game that has regulations and rules.

(b) The main part of the game is the justification of these rules.

(5) The emphasis in teaching mathematics to the weaker students should be on:

(a) Practising the procedures to full mastery.(b) Memorization of facts and procedures.(c) The above two, even at the expense of understanding.

(6) Mathematics is a developing domain whose contents change with the scientific research inthat domain.


(7) Mathematics is a precise domain of knowledge that is free of vagueness and contrastinginterpretations.

(8) In mathematics teaching it is very important that the textbook and the teacher are the mainsource for answers.

(9) The mathematical body of knowledge is rigid and stable.(10) The main objective of mathematics teaching is to produce students who can carry out

definite mathematical tasks and procedures that are required by the curriculum.(11) The textbook problems should be used only for practice and establishment of the materials.

(a) Mathematics is a rigid and complete discipline.(b) Engagement in mathematics almost rules out emotional involvement.(c) Assertion a is the reason for assertion b.

(12) In mathematics classes the main emphasis should be on the precise execution of operationsaccording to rules and laws.

(a) Mathematics is different from the other sciences.(b) Only in mathematics one can be either right or wrong.(c) Assertion b is the reason for assertion a.

(13) It is important that the teacher designs clear and definite instructional procedures that donot elicit doubts in the students’ minds.


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