mathematics‐teachers’ knowledge bases: implications for teacher education

14

Click here to load reader

Upload: steven

Post on 14-Apr-2017

214 views

Category:

Documents


2 download

TRANSCRIPT

Page 1: Mathematics‐teachers’ Knowledge Bases: implications for teacher education

This article was downloaded by: [Newcastle University]On: 21 December 2014, At: 02:47Publisher: RoutledgeInforma Ltd Registered in England and Wales Registered Number: 1072954Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Asia-Pacific Journal of TeacherEducationPublication details, including instructions for authors andsubscription information:http://www.tandfonline.com/loi/capj20

Mathematics‐teachers’ KnowledgeBases: implications for teachereducationClive Kanes a & Steven Nisbet aa Griffith UniversityPublished online: 02 Jun 2006.

To cite this article: Clive Kanes & Steven Nisbet (1996) Mathematics‐teachers’ KnowledgeBases: implications for teacher education, Asia-Pacific Journal of Teacher Education, 24:2,159-171, DOI: 10.1080/1359866960240205

To link to this article: http://dx.doi.org/10.1080/1359866960240205

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the“Content”) contained in the publications on our platform. However, Taylor & Francis,our agents, and our licensors make no representations or warranties whatsoeveras to the accuracy, completeness, or suitability for any purpose of the Content. Anyopinions and views expressed in this publication are the opinions and views of theauthors, and are not the views of or endorsed by Taylor & Francis. The accuracyof the Content should not be relied upon and should be independently verifiedwith primary sources of information. Taylor and Francis shall not be liable for anylosses, actions, claims, proceedings, demands, costs, expenses, damages, and otherliabilities whatsoever or howsoever caused arising directly or indirectly in connectionwith, in relation to or arising out of the use of the Content.

This article may be used for research, teaching, and private study purposes. Anysubstantial or systematic reproduction, redistribution, reselling, loan, sub-licensing,systematic supply, or distribution in any form to anyone is expressly forbidden.Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Page 2: Mathematics‐teachers’ Knowledge Bases: implications for teacher education

Asia-Pacific Journal of Teacher Education, Vol. 24, No. 2, 1996 159

Mathematics-teachers' Knowledge Bases:implications for teacher education

CLIVE KANES & STEVEN NISBET, Griffith University

ABSTRACT Understanding the knowledge bases of mathematics teachers is an important taskin working towards the construction of adequate models for: (i) teacher education anddevelopment, and (ii) teacher operations in the classroom. To date, little systematic attentionhas been focused on this task. The primary aim of this study is to obtain a view from the fieldof mathematics teacher knowledge with respect to content knowledge in mathematics, content-specific pedagogical knowledge in mathematics and curriculum knowledge relevant to teachingtasks. This study has used data obtained from a survey of primary teachers and secondarymathematics teachers. Analysis of the results has indicated that less than half of the teachersin the study believed that they were sufficiently prepared in mathematics content, and thatalmost two-thirds of the teachers in the sample believed that their level of knowledge incontemporary teaching methodologies in mathematics is not sufficient for their role as schoolteachers. Key differences emerge between the primary and secondary sectors and also within thesecondary sector. Implications for preservice and in-service mathematics teacher education aredrawn.

Background

The question of what makes an effective teacher of mathematics has been the subjectof research and debate for some time. Some models have concentrated on the observa-tion and analysis of teacher behaviour and control of the classroom climate. Researchof this kind has tended to focus on the access teachers have to standardised proceduresin order to achieve denned learning outcomes (Tobin & Fraser, 1987, 1988). Analternative approach has been to emphasise the cognitive aspects of a teacher's oper-ation (Resnick, 1983; Clark & Peterson, 1986; Armour-Thomas, 1989; Peterson et ah,1989; Leinhardt, 1990; Evans, 1994). Studies such as these would highlight thesignificance of establishing a more complete understanding of the structures andelements of teachers' knowledge bases.

A focus on teachers' knowledge bases also contributes to current debates in Australiaand other parts of the world on models for mathematics teacher education. Apart fromsome exceptions (for instance models derived from critical theory), traditional teacher-education courses have, in the main, been constructed on assumptions derived fromconventional teacher practice rather than on cogent structures derived from researchand theory (Hatton, 1989; Russell, 1993). The authors believe that the enrichment ofunderstanding of teacher knowledge enhances prospects for teacher-education pro-grammes which are more theoretically sound and more effective in practice than mightotherwise have been the case.

1359-866X/96/020159-13 © 1996 Australian Teacher Education Association

Dow

nloa

ded

by [

New

cast

le U

nive

rsity

] at

02:

47 2

1 D

ecem

ber

2014

Page 3: Mathematics‐teachers’ Knowledge Bases: implications for teacher education

160 C. Kanes&S. Nisbet

Effective Teaching and the Structure and Content of Mathematics-teachers' Knowledge Bases

Effective teaching in mathematics depends largely on the extent and richness of ateacher's pedagogic content knowledge (Fenstermacher, 1986; Shulman, 1986, 1987;Fawns & Nance, 1993). Shulman denotes by pedagogic content knowledge the fusionof general pedagogic knowledge with mathematics content knowledge. Knowledge ofthis kind consists of conceptual (domain-specific pedagogical representations, orderingof content, etc.) and procedural elements, and leads to teacher actions which assiststudents in the development of their understanding of concepts and procedures inmathematics. Other significant categories of teacher knowledge have also beenidentified by Shulman. These may be set out as follows: mathematics contentknowledge (knowledge of the subject matter), curriculum knowledge (knowledge ofresources and materials), general pedagogic knowledge (knowledge of classroommanagement and generic teaching strategies), knowledge of learners and their charac-teristics, knowledge of educational contexts, knowledge of educational purposes andvalues.

Within the literature there is evidence of two distinctive approaches to the questionof how these categories are structured. One is more task oriented and views pedagogiccontent knowledge, for instance, as a series of teaching techniques relating to specificsubject-matter material (see, for example, Sobel & Maletsky, 1975; Lovitt & Clark,1988). The other view is more fluid in that teachers are encouraged to 'reflect on theirown views of mathematics and mathematics teaching while actively exploring importantmathematical concepts and processes that they will be required to teach' (Wilson,1994, p. 369). In general, however, the literature contains little specific informationconcerning details of these components parts for a given body of practising teachers.Previous studies have tended to be limited in various ways; in some, for instance, moreattention has been placed on teachers' formal qualifications rather than their active oroperationalised mathematical knowledge base (Queensland Board of Teacher Edu-cation, 1985; Peard, 1987). More recent studies such as that of Even (1993) havefocused on the relationship between the subject-matter knowledge and the pedagogicalcontent knowledge of prospective mathematics teachers. However, little analysis hasbeen undertaken of the constituent elements of a typical mathematics-teachers' opera-tionalised knowledge bases.

Moreover, the current literature is marked by a lack of conceptual clarity concerningthese knowledge bases. For instance, Ball & McDiarmid (1990), in their discussion ofthe importance of subject-matter knowledge, tend not to distinguish between ateacher's knowledge of content and pedagogic content knowledge. They assert (p. 439)that 'teachers' intellectual resources and dispositions largely determine their capacity toengage students' minds and hearts in learning.' As Garter (1990) points out, thisapproach downplays the substance of the knowledge that teachers actually possess orneed to possess about classrooms, content, and pedagogy, and the way such knowledgeis organised. She concludes (p. 307) that 'more discussion needs to be directed to whatit means to teach, rather than simply to what is learned in which setting. Undueemphasis on content knowledge therefore oversimplifies the nature of teaching, putsat risk the development of effective models for teacher education and gives misleadingsignals to policy makers and legislators (Goodson, 1993; Kennedy, 1993). Within theAustralian context, for instance, the recommendations of the Discipline Review ofTeacher Education in Mathematics and Science (Department of Education, Employ-ment and Training, 1989) downplay references to the pedagogic content element in

Dow

nloa

ded

by [

New

cast

le U

nive

rsity

] at

02:

47 2

1 D

ecem

ber

2014

Page 4: Mathematics‐teachers’ Knowledge Bases: implications for teacher education

Mathematics-teachers' Knowledge Bases 161

structures for teacher-education programmes, and insufficient attention is given toevidence that these elements are of critical importance.

The Critical Role of Knowledge Bases in Mathematics-teacher's Education

In considering the formation of expertise in teaching, Shulman suggests (1987, p. 13)that teacher knowledge needs to be considered first and foremost in any move to correct'the resoluteness with which research and policy have so blatantly ignored . . . aspectsof teaching in the past'. As indicated above, a key feature of the knowledge typology heproposes is the distinction between knowledge of the subject matter (content knowl-edge), knowledge of pedagogical structures and procedures adequate to the task ofteaching this material (pedagogical content knowledge), and curriculum knowledge.Essential to effective teaching, in Shulman's view, is the development of sophisticatedconceptual and procedural knowledge structures within each of these knowledgedomains. Shulman argues that typical teacher-education programmes focus too muchattention on content knowledge at the expense of pedagogic content knowledge, andtherefore they fail to equip beginning teachers adequately.

Review of the research literature on the knowledge bases of mathematics teachers(consider, for example, McNamara, 1991) reveals that a significant weight is indeedgiven to content knowledge over pedagogic content knowledge. This supports Shul-man's general contention concerning the imbalance between these kinds of knowledgebases. We believe that studies of teacher knowledge might usefully consider pedagogicalcontent knowledge in relationship to both content knowledge and general pedagogicconstructs.

In summary, recent research suggests that mathematics content knowledge is anecessary but not a sufficient condition for good mathematics teaching. An importantimmediate task for mathematics education (teaching and curriculum, research, policy)must therefore be to make pedagogic content knowledge available for scrutiny, criticalappraisal and validation. New methods and conceptual models (for instance, Cochranet al., 1993) need to be developed in order to accomplish these tasks. It is suggestedthat the increased visibility of this component of teacher expertise might not only raisethe status of expert teaching, but may also help to demystify it in the eyes of beginningteachers and the community at large. Rather than being seen as an unstructuredcollection of instructional 'methods', mathematics teacher knowledge, in our view,should become more recognisable as a body of rich and conceptually rigorous knowl-edge.

The Purpose of this Study

The purpose of this study was to undertake initial research aimed at documentingthe knowledge bases of mathematics teachers in the field. Conclusions were drawnconcerning the nature of mathematics teacher knowledge, the implications for mathe-matics teacher education at both the preservice and the in-service levels, and theappropriateness and effectiveness of research tools used in studying these issues. Theresults are being further investigated by the researchers in a large-scale study.

Dow

nloa

ded

by [

New

cast

le U

nive

rsity

] at

02:

47 2

1 D

ecem

ber

2014

Page 5: Mathematics‐teachers’ Knowledge Bases: implications for teacher education

162 C. Kanes & S. Nisbet

Method

Studies of teacher knowledge bases are heavily constrained by both conceptual andlogistical difficulties. In the first place, a large body of literature indicates that not all thesalient components of a teacher's knowledge base are explicit and thus, even inprinciple, immediately available for investigation. Tacit knowledge, taking the form ofhighly automated procedures, plays a crucial role in teachers' day-to-day operations.Moreover, the literature indicates that these dimensions of teacher knowledge (explicitand tacit) are not always congruent in their structure or consistent in their content.Thus, a problem for research of this kind is the development of instruments which arecapable of eliciting not only the structure and content of a teacher's explicit and avowedapproach to teaching mathematics but also their actual modus operandi. An additionalconstraint concerns the purely logistic question of gathering valid qualitative data froma representative sample. Typically, research into the tacit knowledge structures ofclassroom teachers has been heavily reliant on the videoing of classroom teaching,student and teacher interviews, and stimulated recall in order to probe the actualclassroom decision-making processes of teachers and students. However, such tech-niques are not feasible for the scale of the activities conducted in this investigation.

The technique devised for this study engaged a survey instrument consisting ofopen-ended questions, this gave teachers the opportunity to express views on teacherfunctions and student learning (see the Appendix). This indirect-questioning techniquecaptures data of significant complexity and richness, whilst not presenting insurmount-able logistical problems. For instance, particular items probed teachers' mathematicscontent knowledge, mathematics pedagogic content knowledge and mathematics cur-riculum knowledge. For pedagogic content knowledge this was indirectly achieved byasking teachers to describe a recent mathematics lesson in which they used a successfulteaching technique, and why it was successful. A further item sought an example ofanother technique which reflected their teaching style. The teachers' curriculum knowl-edge was examined by asking them to acknowledge the sources of their teaching ideasand their involvement in mathematics in-service seminars. The mathematics contentknowledge bases were investigated by asking teachers to describe the most interestingmathematics topics and the most challenging mathematics topics they had encounteredand studies. A further indication of their knowledge bases was sought through self-evaluation of their own teacher preparation and their perceived needs for upgrading oftheir skills and knowledge. Data was also sought in relation to formal qualifications inmathematics, mathematics pedagogy and years of experience. The researchers hoped touse data of this kind to gain insight into the nature and content of teacher knowledgeimplicit within the response of teachers to particular items.

A pilot investigation for this study was conducted. This consisted in a trial of thequestionnaire instrument in two schools, one primary and one secondary; the infor-mation obtained was used to revise items in order to improve the levels of validity andthe reliability of the survey instrument.

Concerning the sample, a number of primary and secondary schools in Brisbane wereapproached to seek their cooperation in gaining access to teachers of mathematics at alllevels. A total of 44 teachers in 10 schools (13 teachers in 4 primary and 31 teachersin 6 secondary) responded to the request. In this sample, the years of teachingexperience ranged from 2 to 32 years, 10 teachers had served up to 15 years and 15teachers had more than 15 years of experience.

Data obtained in this study were analysed in the following way. First, straightforward

Dow

nloa

ded

by [

New

cast

le U

nive

rsity

] at

02:

47 2

1 D

ecem

ber

2014

Page 6: Mathematics‐teachers’ Knowledge Bases: implications for teacher education

Mathematics-teachers' Knowledge Bases 163

responses, such as years of experience and qualifications, were tabulated. Secondly,qualitative responses were analysed. Responses of this kind were generally of two levelsof complexity. At the first level, a simple categorisation process was adopted. Forinstance, responses to Q3(c) (question 3(c) in the Appendix) "What was the mostchallenging mathematics you have studied?' were grouped by standard mathematical-topic descriptors (for example, chaos theory). At the second level of complexity, a moredetailed process was adopted. For instance, in Ql(a) teachers were asked to describe arecent mathematics lesson which they believed to be successful. Teachers wereprompted in Ql(b) to explain why in their judgement this technique had been sosuccessful. In analysing the responses obtained, the full set of individual scripts werefirst read and a set of descriptors characterising the data developed. Next, thesedescriptors were grouped according to the categories to which they seemed to belong(for example, knowledge, understanding, teacher resources, affect, etc.). Finally, indi-vidual responses were assessed against these categories; this facilitated the tabulation ofresults within individual schools and across sectors (primary and secondary).

Results

Mathematics Content Knowledge

Of the primary teachers surveyed, 54% said they judged themselves to be sufficientlywell prepared in mathematics content knowledge, 8% said they were not and 38% didnot respond to the question. Of the secondary teachers, 45% were satisfied with theircontent knowledge, 19% were not and a further 35% did not respond. When aggre-gated across the sample these results are follows: 48% were sufficiently prepared, 16%were insufficiently prepared and 36% made no response. These results show that justunder half of the teachers surveyed indicated they were sufficiently prepared inmathematics content for their classroom teaching. High levels of no response to thisquestion can either be interpreted as showing a lack of understanding of the questionby many respondents or a high-level of reluctance to disclose this kind of informationin the sample. Given that the survey instrument had been subjected to a pilot studywhose precise purpose was to improve clarity and correct ambiguities, it is arguable thatless weight should be given to the first of these alternatives.

Table I indicates the highest level of mathematics attainment. When asked toindicate the most interesting mathematical topic studied, 39% quoted a topic theyattributed to tertiary-level study. The topics mentioned most often were within discretemathematics and included operations research, applied linear algebra and optimisationtheory. Calculus and analytical mechanics also were frequently mentioned. However, ina small number of cases there was evidence that this material was not genuine tertiarycontent (percentages, 'numbers in nature'). Five per cent of respondents indicated atopic drawn from the secondary curriculum (symbolic equations, and probability andstatistics); and a further 27% of respondents indicated informal sources such as sourcebooks, television, science fiction and other materials (chaos theory and fractals, linearprogramming and complex numbers). Only one respondent (2%) indicated in-servicemathematics as a source of interesting mathematics knowledge.

Similarly, when asked to indicate a topic which they found to be the greatestmathematics challenge, 45% proposed a topic derived from tertiary study (calculus,complex variables, numerical techniques and computer programming), 14% referred toa topic in the secondary area (algebra, trigonometry, calculus) and 7% indicated

Dow

nloa

ded

by [

New

cast

le U

nive

rsity

] at

02:

47 2

1 D

ecem

ber

2014

Page 7: Mathematics‐teachers’ Knowledge Bases: implications for teacher education

164 C. Kanes & S. Nisbet

TABLE I. The highest level of mathematics attainment reached, by school sector

Highest level ofattainment Primary (%) Secondary (%) Total (%)

Mathematics majorSome tertiary level mathematics*Secondary mathematicsNo response

00

8515

13551616

9393616

* This excludes curriculum/methodology subjects.

informal sources—such non-specified personal resources as 'an old text book I boughtfor $1.50'. One respondent nominated the multiplication algorithm as the mostchallenging topic encountered. Only one respondent (2%) indicated mathematicsin-service as a source of challenging mathematics knowledge.

Mathematics Pedagogic Content Knowledge

When asked to consider the adequacy of teacher preparation in mathematics, 64%reported that in their view they had not been sufficiently prepared in important facetsof contemporary methodology, for example, in assessement strategies, the use oftechnology and the role of language. Eighteen per cent indicated that they had beensufficiently prepared, and a further 18% did not respond to this question.

For teachers at the primary level, it is assumed that the majority of subjectsundertook mathematics education appropriate to primary-school mathematics. Inthis survey 54% acknowledged their primary-teacher training, 15% acknowledgedsecondary-teacher training, 8% indicated that they had no training and 23% did notrespond to the question. At the secondary level, 68% had mathematics-teachingqualifications, 16% did not possess formal qualifications to teach mathematics at thislevel and the remaining 16% did not respond to the question.

Some subjects appeared not to appreciate the difference between mathematicsqualifications (Q5(b)) and mathematics-teaching qualifications (Q5(d)), or in, Shul-man's terms, the difference between content knowledge and pedagogic content knowl-edge.

Responses to Ql(a), l(b), l(c), 2(a), 2(b) provided an opportunity to examine andobtain corroborating evidence for the level of the respondents' pedagogic contentknowledge. The topics the teachers chose covered a wide range of syllabus items in theareas of number, measurement and algebra. The methods selected are given inTable II.

TABLE II. Illustrative teaching methods nominated by school sector

Method

Activity (concept formation)Activity (consolidation)ExpositionProblem solvingGamesDiscussion

Primary (%)

40201025

50

Secondary (%)

3816221653

Total (%)

3918181952

Dow

nloa

ded

by [

New

cast

le U

nive

rsity

] at

02:

47 2

1 D

ecem

ber

2014

Page 8: Mathematics‐teachers’ Knowledge Bases: implications for teacher education

Mathematics-teachers' Knowledge Bases 165

TABLE III. Factors to which teachers attributed teaching success

Teaching success attributed

Category

Establishing links betweenknowledge domains

Understanding

Teacher resources

Student autonomy

Affect

Non-scientificexplanation

to

Subcategory

Prior knowledge linkedto target knowledgeReal life linked to classroomConcrete linked to abstract

DeepSurface

SimplicityFeasibility (time)Rapid feedbackPersonal characteristics

Interest and motivationStudent participationStudent enjoyment of lessonIncentive to learnConfidenceRelationship between affect and cognition

Examples: 'it works!', 'I lovemy subject'

- Number ofresponses

21017

31

2111

8

1285211

7

Total

29

4

5

8

29

7

Teachers attributed the success of their nominated methods to factors which are setout in Table III. Responses to Q2(a) indicated that these methodologies were, however,not necessarily typical of normal teaching practice. Whilst 48% reported that thesestrategies were typical, 16% said they were typical to some degree, 27% indicated theywere atypical and 9% did not respond to the question.

In order to obtain an indication of possible institutional influences on teachers'concepts of successful teaching, the factors in Table III were reclassified in terms ofstudent learning (establishing knowledge links, understanding, student autonomy andstudent affect), the teacher's management (teacher resources), and non-scientific factors.The number of attributional statements made per teacher was then tabulated and themeans and standard deviations across the primary and secondary levels were calculated(Table IV). The results showed a significant difference between primary and secondaryteachers at the 0.01 level for both student learning (t= 12.87) and non-scientific factors(t = 5.66). Further analysis of the data (cross tabulation of the attributional factors byschool) indicated that for the secondary schools in the sample there was a clustering ofteachers with similar pedagogic-content-knowledge profiles within schools.

Mathematics Curriculum Knowledge

In response to Ql(c), in which information relating to the source of the teachingtechnique is sought, 41% of respondents claimed that the successful methods wereoriginal ideas; 16% made use of textbook and sourcebook materials; 13% made use ofkits and teacher-journal articles; 18% responded that they did not know the source of

Dow

nloa

ded

by [

New

cast

le U

nive

rsity

] at

02:

47 2

1 D

ecem

ber

2014

Page 9: Mathematics‐teachers’ Knowledge Bases: implications for teacher education

166 C. Katies & S. Nisbet

TABLE IV. Mean number of attributed factors ascribed to teacher success (per teacher)

Means and standard deviations (in parentheses) for the number ofattributed factors ascribed to teaching success (per teacher)

School type Student learning The teacher's management Non-scientific factors

Primary 2.3 0.2 0(0.7) (0.3) (0.0)

Secondary 1.4 0.1 0.4(1.0) (0.1) (0.4)

the material; and the remainder, 14%, did not respond to this item. In response to thequestion about the modifications of teaching techniques (Ql(d)), 55% reported somelevel of modification, 34% reported no modification and 11% did not respond to thisquestion. These results indicate that whilst the teachers in the sample were active ingenerating and modifying teaching ideas (pedagogic content knowledge) relating tomathematical topics, use of curriculum material found in published sources did notpredominate. This raises issues for further research on the degree of visibility andavailability of teaching materials, and the extent to which teachers find these materialsto be relevant in the routine tasks of teaching mathematics (curriculum knowledge).

Results gathered from Q3(b) and 3(d) enable us to categorise the sources ofmathematics-content-knowledge development applied in the curriculum context. Whenquestioned about the most interesting mathematics topic read about or studied(Q3(b)), 43% referred to formal education (secondary or tertiary) as a source, 30%referred to personal sources such as television or general reading and only 5% referredto professional-development programmes such as in-service experiences or conferences.Similar results were obtained for Q3(d), a question about the most challengingmathematics topic.

When asked to describe a mathematics in-service experience which was of benefit,52% were able to nominate a topic; however, 48% reported that no in-service had beenof benefit; 7% did not respond to this question. The topics identified were wide rangingand included problem solving, assessment and the use of technology.

Forty-eight per cent of the total sample thought that in-service courses should becredited towards formal qualifications at tertiary level, a further 16% offered con-ditional support for this suggestion and 20% did not agree with the suggestion. Sixteenper cent of the sample did not respond. Of those who thought that formal qualificationswould be desirable, most thought that Masters degrees (47%) or Graduate Diplomas/Certificates (37%) would be the most appropriate award for credit.

Discussion

Mathematics Content Knowledge

The results indicated that less than half of the teachers sampled were prepared to saythat in their own judgement they are sufficiently prepared in mathematics content forthe tasks of classroom teaching; the secondary teachers appeared to be less satisfiedthan the primary teachers. The high level of no response to this question is of concernto the authors. As previously discussed, we believe that this may indicate a hesitancy todisclose information, and it is suggested that this may be related to teachers' self-per-

Dow

nloa

ded

by [

New

cast

le U

nive

rsity

] at

02:

47 2

1 D

ecem

ber

2014

Page 10: Mathematics‐teachers’ Knowledge Bases: implications for teacher education

Mathematics-teachers' Knowledge Bases 167

ception of the level of appropriateness and/or adequacy of their personal mathematical-content-knowledge base. One factor contributing to this result may relate to levels ofteacher affect (motivation and confidence). Whilst the study reported here did notdirectly explore this, it is nevertheless accepted that this might be a fertile direction forfurther investigation. Another factor that may be critical to this result is the actual levelof mathematics content knowledge of the teachers in the sample studied. Since 55% ofsecondary teachers in the sample nominated tertiary-level topics in mathematics asproviding interest, and 65% nominated tertiary-level topics in mathematics as providinga challenge, we infer that at least one-third of teachers at this level may feel morecomfortable at a pre-tertiary level of mathematics. Moreover, the finding that thehighest level of mathematics-content-knowledge attainment of approximately one-thirdof the secondary teachers in the sample was at a pre-tertiary level is consistent with thisinference. Given that the level of mathematics content knowledge is an importantindicator of overall teacher effectiveness, the results suggest that the current level ofcontent-knowledge attainments among teachers may be an issue that requires furtherscrutiny.

When considering the source of mathematics knowledge, it is notable that the impactof professional-development programmes is negligible (only one respondent com-mented favourably in this respect). This raises questions about the adequacy of theprovision of such services in mathematics-content areas, and it points to the fact thatteachers rely on informal sources for extending their mathematical knowledge. Univer-sities and other higher-education providers may need to evaluate the suitability of themathematical topics of courses targeted at mathematics teachers (both primary andsecondary).

Mathematics Pedagogic Content Knowledge

Across both the primary and the secondary levels, almost two-thirds of teachers in thesample reported that in their view they had not been sufficiently prepared in importantfacets of contemporary teaching methodology. This result suggests that both thepreservice and the in-service modes of teacher education and development may need tobe examined in terms of their effectiveness and atunement to the needs of teachers incurrent classroom practice.

Whilst no significant differences were observed in the formal qualifications in themathematics teaching of teachers in the primary and secondary levels, a higher pro-portion of informal qualifications were noted at the secondary level (13% for secondaryteachers, none for primary teachers). However, differences between the primary and thesecondary levels were observed when the focus turned to the actual classroom use ofpedagogic content knowledge. These may be stated as follows. Our results demonstratethat a wide range of teaching strategies are used at times in mathematics classrooms(Table II) and that, overall, teachers are able to articulate a rationale for these practicesto a reasonable depth (Table III). Moreover, it is notable that the principal focus wasplaced on student learning rather than on conditions for teacher function, and that thisis generally consistent with the constructivist paradigm for mathematics educationwhich has dominated the mathematics-education literature in the last decade (Confrey,1987; Yackel et al., 1990; Ernest, 1994, von Glasersfeld, 1995). However, key differ-ences emerge (i) between primary and secondary teachers, and (ii) within the secondarysector. Primary-level teachers seemed to be more articulate when attributing factorsinvolved in successful teaching than their secondary counterparts. Within the secondary

Dow

nloa

ded

by [

New

cast

le U

nive

rsity

] at

02:

47 2

1 D

ecem

ber

2014

Page 11: Mathematics‐teachers’ Knowledge Bases: implications for teacher education

168 C. Kanes & S. Nisbet

sector, significant differences exist between schools in terms of teachers' abilities toarticulate a rationale for effective teaching. Moreover, the profile of factors mentionedby teachers varied significantly across secondary schools; those schools which scored asignificantly lower mean number of student-learning factors per teacher ranked non-scientific factors more highly.

It is inferred from these findings that mathematics educators involved in teachereducation need to raise the profile of mathematics pedagogic content knowledge as adistinctive and powerful domain of teacher knowledge. To this end, Marland (1993)has suggested that the promotion and dissemination of research in this area is bestaccomplished in the context of a partnership between teachers and researchers. Ata theoretical level, Cochran et al. (1993) have proposed a remodelling of aspectsof Shulman's teacher-knowledge taxonomy in the light of constructivist learningtheories.

Mathematics Curriculum Knowledge

Relatively few teachers in the sample used sourcebook and textbook materials informing lessons. This suggests either that these materials were not considered to beadequate or that they were not considered at all; in the latter case it is likely thatteachers are unaware of the range of materials available or that they usually do not havethese materials at their disposal in the school. In this study, evidence was found for bothof these alternatives. Professional-development experiences provided another source ofteaching ideas, and although half of the sample acknowledged that they were of somebenefit, it appears that ideas derived from this source are not routinely operationalisedin classroom teaching practice. The effectiveness of conventional professional-develop-ment programmes is therefore questioned in the context of this study. Considerableinterest was shown by the teachers sampled in professional-development models whichincorporated a credit for formal university qualifications. These findings should alertuniversities to the need for flexibility in teacher-education programmes generally.Articulation between formal study programmes conducted at universities and develop-ment programmes located within actual teaching situations needs to be facilitated.

Conclusion

Conclusions drawn from this study can be categorised under three headings: methodo-logical issues relating to the investigation of teacher knowledge, mathematics teacherknowledge itself, and implications for preservice initial and in-service education ofmathematics teachers. First, the results from this study suggest the effectiveness of anindirect method for eliciting relevant information. Items from the instrument usedprovided the researchers with data which could be analysed on several levels, and thisfacilitated the development of an understanding of both explicit and tacit knowledgestructures salient to the effective operation of teachers in the classroom. Teachereducators may find this method useful in monitoring the growth in teacher knowledgein student teachers.

The second major conclusion concerns teacher knowledge. The results from theteachers sampled in this study reveal that, while there are strengths in mathematicsteacher knowledge bases, particularly at primary school level, there are grounds forabout teachers' content knowledge, pedagogic content knowledge and curriculum

Dow

nloa

ded

by [

New

cast

le U

nive

rsity

] at

02:

47 2

1 D

ecem

ber

2014

Page 12: Mathematics‐teachers’ Knowledge Bases: implications for teacher education

Mathematics-teachers' Knowledge Bases 169

knowledge. Moreover, these concerns tend to be concentrated in particular schoolsrather than at the level of individual teachers. Therefore, any attempt at reforming thissituation may need to take account of teacher relationships and the formation of aprofessional ethos within each school. An additional concern relates to the significantproportion of mathematics teachers who do not have tertiary qualifications in eithermathematics or mathematics education. This study found that such qualifications werelinked with richer bases of operationalised knowledge for teaching. The teacher-education sector surely needs to be more pro-active in advancing the growth of teacherknowledge in these areas.

A third major conclusion concerns teacher education at both the preservice and thein-service level; reform processes need to balance needs of mathematics contentknowledge with those of mathematics pedagogic content knowledge. The case ofprimary-school teachers in the sample is instructive on this point. We found that theywere significantly less well equipped in mathematics content, yet they were considerablymore proficient with mathematics-content pedagogy. This finding correlates with thefact that almost all primary-school teachers in Australia undertake mathematics-methodology studies in their preservice teacher education, whereas almost half of thesecondary teachers surveyed had no qualifications in mathematics education. Teachersin this category have preservice teaching qualifications in other discipline areas (science,physical education, music, etc.). Further, the teachers in this study did not findprofessional development helpful to them in classroom situations. Grounds for there-evaluation of such programmes can therefore be established on two counts. How-ever, given the variability of the levels of teacher knowledge found in the schools in thisstudy, it is clear that effective teacher education may not be achievable within thecontext of individual schools.

Correspondence: Clive Kanes, Faculty of Education, Griffith University, Queensland,Australia 4111.

NOTE

The authors wish to acknowledge that this article is based on work presented in papers delivered tothe 17th Annual Conference of the Mathematics Education Research Group of Australasia, Lismore,July, 1994, and the ICMI-China Regional Conference on Mathematics Education, Shanghai, August1994.

REFERENCES

ARMOUR-THOMAS, E. (1989) The application of teacher cognition in the classroom: a new teachingcompetency, Journal of Research and Development in Education, 22, pp. 29-37.

BALL, D.L. & MCDIARMID, G.W. (1990) The subject-matter preparation of teachers, in: W. R.HOUSTON (Ed.) Handbook of Research on Teacher Education, pp. 437-449 (Macmillan PublishingCompany).

CARTER, K. (1990) Teacher's knowledge and learning to teach, in: W. R. HOUSTON (Ed.) Handbookof Research on Teacher Education, pp. 291-310 (Macmillan Publishing Company).

CLARK, C. & PETERSON, P. (1986) Teachers' thought processes, in: M. C. WITTROCK (Ed.) Handbookof Research on Teaching, 3rd edn, pp. 255-296 (New York, Macmillan).

COCHRAN, K., DE RUTTER, J. & KING, R. (1993) Pedagogical content knowing: an integrative model forteacher preparation, Journal of Teacher Education, 44, pp. 263-272.

Dow

nloa

ded

by [

New

cast

le U

nive

rsity

] at

02:

47 2

1 D

ecem

ber

2014

Page 13: Mathematics‐teachers’ Knowledge Bases: implications for teacher education

170 C. Kanes & S. Nisbet

CONFREY, J. (1987) The current state of constructivist thought in mathematics education, paperpresented to the Annual Meeting of the International Group for the Psychology of Mathematics Education,Montreal, July.

DEPARTMENT OF EMPLOYMENT, EDUCATION AND TRADING (1989) Discipline Review of Teacher Edu-cation in Mathematics and Science (Canberra, Australia Government Publishing Service).

ERNEST, P. (Ed.) (1994) Mathematics, Education, and Philosophy: an international perspective (London,Falmer Press).

EVANS, G. (1994) The knowledge base of beginning teachers: competence and competencies, in:QUEENSLAND BOARD OF TEACHER REGISTRATION, Knowledge and Competence for Beginning Teaching,pp. 5-21 (Toowong, Brisbane, Queensland Board of Teacher Registration).

EVEN, H. (1993) Teacher competence: panacea, rhetoric, or professional challenge? Journal of Educationfor Teaching, 19, pp. 145-162.

FAWNS, R. & NANCE, D. (1993) Teacher knowledge, education studies and advanced skills credentials,Australian Journal of Education, 37, pp. 248-258.

FENSTERMACHER, G. (1986) Philosophy of research on teaching: three aspects, in: M. C. WITTROCK(Ed.) Handbook of Research on Teaching, 3rd edn, pp. 37-49 (New York, Macmillan).

GooDSON, I. (1993) Forms of knowledge and teacher education, Journal of Education for Teaching, 19,pp. 217-229.

HATTON, E. (1989) 'Bricolage' and theorizing teachers' work, Anthropology and Education Quarterly, 20,pp. 74-96.

KENNEDY, K. (1993) National standards in teacher education—why don't we have any? South PacificJournal of Teacher Education, 21, pp. 101-110.

LEINHARDT, G. (1990) Capturing craft in knowledge in teaching, Educational Researcher, 19, pp. 18-25.Lovitt, C. & CLARK, D. (1988) Mathematics Curriculum and Teaching Program (MCTP) (Canberra,

Curriculum Development Centre).MARLAND, P. (1993) A review of the literature on implications of teacher thinking research for

preservice teacher education, South Pacific Journal of Teacher Education, 21, pp. 51-63.MCNAMARA, D. (1991) Subject knowledge and its application: problems and possibilities for teacher

educators, Journal of Education for Teaching, 17, pp. 113-128.PEARD, R. (1987) Statistics education in Queensland: a study of the qualifications and attitudes of

teachers of mathematics in the junior grades (8-10), Teaching Mathematics, 12, pp. 8-12.PETERSON, P., FENNEMA, E., CARPENTER, T. & LOEF, M. (1989) Teachers' pedagogical content beliefs

in mathematics, Cognition and Instruction, 6, pp. 1-40.QUEENSLAND BOARD OF TEACHER EDUCATION (1985) Teachers for Mathematics and Science (Toowong,

Brisbane, Queensland Board of Teacher Education).RESNICK, L.B. (1983) Towards a cognitive theory of instruction, in: S. G. PARIS, G. M. OLSON & H.

STEVENSON (Eds) Learning and Motivation in the Classroom, pp. 5-38 (Hillsdale, NJ, LawrenceErlbaum).

RUSSELL, T. (1993) Teachers' professional knowledge and the future of teacher education, Journal ofEducation for Teaching, 19, pp. 205-215.

SHULMAN, L. (1986) Those who understand: knowledge growth in teaching, Educational Researcher, 15,pp. 4-14.

SHULMAN, L. (1987) Knowledge and teaching: foundations of the new reform, Harvard EducationalReview, 57, pp. 1-22.

SOBEL, M. & MALETSKY, E. (1975) Teaching Mathematics: a sourcebook of aids, activities and strategies(New Jersey, Prentice Hall).

TOBIN, K. & FRASER, B. (Eds) (1987) Exemplary Practice Science and Mathematics Education (Perth,Curtin University of Technology).

TOBIN, K. & FRASER, B. (1988) Investigations of exemplary teaching in Australian mathematics classes,Australian Mathematics Teacher, 44, p. 5-8.

VON GLASERSFELD, E. (1995) Radical Constructivism: a way of knowing and learning (London, FalmerPress).

WILSON, M. (1994) On preservice secondary teacher's understanding of function: the impact of acourse integrating mathematical content and pedagogy, Journal of Research in Mathematics Education,25, pp. 346-370.

YACKEL, E., COBB, P., WOOD, T., WHEATLEY, G. & MERKEL, G. (1990) The importance of socialinteraction in children's construction of mathematical knowledge, in: T. COONEY & C. HIRSCH (Eds)Teaching and Learning Mathematics in the 1990s, pp. 12-21 (Reston, NCTM).

Dow

nloa

ded

by [

New

cast

le U

nive

rsity

] at

02:

47 2

1 D

ecem

ber

2014

Page 14: Mathematics‐teachers’ Knowledge Bases: implications for teacher education

Mathematics-teachers' Knowledge Bases 171

Appendix: survey instrument

1 (a) Give an example of a recent mathematics lesson in which you used a teaching technique whichyou believe was particularly successful. Please indicate the topic, the year level (and subject if secondarye.g. Maths A, B, or C). Provide as much detail of the lesson as possible.(b) In your judgement, why was this technique so successful?(c) What was the original source for this technique?(d) To what extent have you modified it to meet your own needs and situation?(e) What factors do you consider provide for high levels of learning in mathematics?

2 (a) When you reflect on your own teaching style, is this technique typical of the kinds ofmathematics teaching strategies you generally use?(b) Please give an example of another technique which reflects your teaching style.

3 (a) What is the most interesting mathematics topic you have read about or studied?(b) Where did you read about it or study it?(c) What is the most challenging mathematics topic you have studied?(d) Where did you read about it or study it?

4(a) Describe a mathematics in-service education experience which was of benefit to you as a teacherof mathematics? (Brief details of topics, provider and methods would be helpful.)(b) What in-service education courses, seminars or workshops would you like to attend in order tobenefit you as a teacher of mathematics?(c) Should such in-service courses be credited towards a formal university qualification in mathematicseducation, e.g. Graduate Certificate, Graduate Diploma, Masters degree, etc.?(d) Which formal university qualifications would attract you when considering and choosing anin-service course appropriate to your career?

5 (a) Do you feel that your education in mathematics has prepared you sufficiently for the classroom?(b) What topics would you appreciate further study in e.g. number, algebra, measurement, chance &data, calculus, geometry?(c) Describe your qualifications in the subject of mathematics itself (not teaching).(d) Do you feel that you have had sufficient preparation in the area of teaching mathematics, e.g.alternative assessment strategies, SPS (Student Performance Standards), use of new technology such asgraphical calculators and computer software, the role of language in learning mathematics?(e) Describe your formal qualifications in mathematics teaching.

6. How many years experience have you had in teaching mathematics at various levels?

Dow

nloa

ded

by [

New

cast

le U

nive

rsity

] at

02:

47 2

1 D

ecem

ber

2014