utilising a construct of teacher capacity to examine national curriculum reform in mathematics

ORIGINAL ARTICLE

Utilising a construct of teacher capacity to examinenational curriculum reform in mathematics

Qinqiong Zhang & Max Stephens

Received: 23 April 2012 /Revised: 25 September 2012 /Accepted: 5 February 2013# Mathematics Education Research Group of Australasia, Inc. 2013

Abstract This study involving 120 Australian and Chinese teachers introduces aconstruct of teacher capacity to analyse how teachers help students connect arithmeticlearning and emerging algebraic thinking. Four criteria formed the basis of ourconstruct of teacher capacity: knowledge of mathematics, interpretation of the inten-tions of official curriculum documents, understanding of students’ thinking, anddesign of teaching. While these key elements connect to what other researchers referto as mathematical knowledge for teaching, several differences are made clear.Qualitative and quantitative analyses show that our construct was robust and effectivein distinguishing between different levels of teacher capacity.

Keywords Algebra . Cross-cultural studies . Curriculum development . Numberconcepts . Teacher capacity . Mathematical knowledge for teaching

Curriculum reform focus

In many countries, official curriculum documents now endorse the building of closerrelationships between the study of number in the primary school and the developmentof algebraic thinking. Algebraic thinking is not the same as the use of algebraicsymbols. It is about identifying generalisations and structural relations in numbersentences and operations. This is very different to what in the past was seen as thestudy of arithmetic.

Math Ed Res JDOI 10.1007/s13394-013-0072-9

Q. Zhang (*)College of Mathematics and Information Science, Wenzhou University, Wenzhou,Zhejiang 325035, Chinae-mail: [email protected]

M. StephensGraduate School of Education, The University of Melbourne, Victoria 3010, Australiae-mail: [email protected]

The Australian Curriculum: Mathematics (Australian Curriculum, Assessment andReporting Authority (ACARA), 2010) presents Number and Algebra as a singlecontent strand for the compulsory years of school. In its overview statement for thisstrand, ACARA (2010, p. 2) states that:

Number and algebra are developed together since each enriches the study of theother. They [students] understand the connections between operations. Theyrecognise pattern and … build on their understanding of the number system todescribe relationships and formulate generalisations. They recognise equiva-lence and solve equations and inequalities… and communicate their reasoning.

This statement echoes important ideas that have been present for at least 5 years inrelated state curriculum documents, for example, in linking Number, Structure, andWorking Mathematically in the Victorian Essential Learning Standards (VictorianCurriculum and Assessment Authority 2008) and in other official curriculum docu-ments such as the Mathematics Developmental Continuum (Department of Educationand Early Childhood Development 2006).

China’s Mathematics Curriculum Standards for Compulsory Education (Ministryof Education of PRC 2001, 2011) also present a single strand entitled Number andAlgebra. In Stage 2, which covers Years 4 to 6, two “teaching objectives” refer to theimportance of considering the inverse properties of calculation and to investigatingthe properties of equivalent sentences. Objective 5 on “operation of numbers” is “toexperience the inverse relation between addition and subtraction, as well as that ofmultiplication and division in the process of concrete operation and solution onsimple practical problems” (p. 21). Objective 3 on “sentences and equations” (p. 21)is “to understand the property of equal sentences and enable to solve easy equations withthe property of equal sentences (e.g., 3x+2=5, 2x−x=3)”. Chinese researchers, such asXu (2003), emphasise that a closer alignment is needed between the study of numberand number relationships in the primary school and the study of algebra in the secondaryschool in this curriculum reform.

Official documents in both countries clearly endorse a more coherent treatment ofnumber sentences and operations and the development of algebraic thinking in theprimary and early years of secondary school; and we argue that teacher capacity is akey dimension in realising that goal. However, the implementation of curriculumchange is never simply from the top down. Teachers’ interpretations and responses atthe level of practice are never simple reflections of what is contained in officialcurriculum documents. While curriculum documents set out broad directions forchange, any successful implementation of these “big ideas” depends on teachers’capacity to apply subtle interpretations and careful local adaptations (Datnow andCastellano 2001). Teachers’ professional insight and agency in translating these ideasinto practice must frame any definition of teacher capacity (Smyth 1995).

The following sections set out our rationale for choosing teacher capacity as a keyorganising idea for this research. As we will show, our use of this term is not intendedto imply some generic notion of teaching ability. We define teacher capacity asprofessionally informed judgement and disposition to act; and in the context of thisstudy we apply this construct in terms of national curriculum reform in mathematics.

In the context of national curriculum reform in mathematics in both China andAustralia, this paper will focus on fostering links between students’ learning in

Q. Zhang, M. Stephens

number and the development of early algebraic thinking. A comparative study betweenChina and Australia will be conducted–both qualitatively and quantitatively–to find thesimilarities and differences in teachers’ capacity to understand these changes in theirrespective national curricula and to translate these changes into practice. The two mainresearch questions to be investigated in this paper are: (1) What are key components ofmathematics teacher capacity; and (2) What is the relationship between the componentsthemselves?

Teacher capacity and mathematical knowledge for teaching

Research on teacher capacity

The term “teacher capacity” comes out of the literature of school improvement,school leadership, and system reform (McDiarmid 2006; Katz and Raths 1985;Fullan 2010, 2011). In Australia, for example, the NSW Smarter Schools NationalPartnerships (2010) claimed to “focus on building capacity in classroom practiceresulting in improved student outcomes.” The two components of Victoria’s SmarterSchools National Partnerships (DEECD 2009a) had a first focus relating to buildingthe capacity of classroom teachers to effectively assess and monitor student progressand to deliver programs that are differentiated according to student need, and asecond focus relating to developing teacher capacity through professional learningopportunities.

When used in this context, teacher capacity usually relates to teachers’ability to understand and act on the reforms that policy makers are seeking toimplement (Christie 2001; Snow-Renner 1998; Spillane 1999; Spillane andJennings 1997.) It is consistent with our definition of professionally informedjudgement and disposition to act. Researchers such as O’Day et al. (1995), andFloden et al. (1996) emphasise that teacher capacity is multidimensional andevolving. First, they argue that teachers’ ability to assist students in learning isdependent on teachers’ own knowledge, which includes knowledge of thesubject matter, knowledge of curriculum, knowledge about students, and knowl-edge about general and subject-specific pedagogy; second, they point to skills,arguing that while knowledge interacts with skills, there is a considerable gapbetween what teachers believe they should be doing in the classroom and theirability to teach in the desired ways (see Cohen and Ball 1990); and third, theypoint to the importance of dispositions, since enacting reform requires having thedispositions to meet new standards for student learning and to make the neces-sary changes in practice (Katz and Raths 1985).

The term “teacher capacity” is not widely used in mathematics education research,but as used in this study it has connections with Shulman’s (1986, 1987) construct ofpedagogical content knowledge; and with mathematical knowledge for teaching usedby Ball et al. (2008) and Hill et al. (2008). The principal difficulty of situating amodel of teacher capacity in the literature of school system and curriculum reform isthat the unit of analysis in that literature is the school or the school system; that is,what changes have taken place in schools or across schools. There, the term “teachercapacity” is used generically to identify one of the conditions for change, not as a

Utilising a construct of teacher capacity

comprehensive model to discuss the kinds of knowledge, skills, and dispositions thatteachers have or need to have in order for change to occur. For that reason, ourconstruct needs to retain closer links to the models of teacher knowledge that follow.

Different conceptualisations of teachers’ mathematical knowledge for teaching

Tirosh and Even (2007) point out that there is no universal agreement or widely-accepted framework for describing teachers’ mathematical knowledge for teaching.Three important models–that of Shulman (1986, 1987), of Ball et al. (2008), and ofRowland (2005) and Ruthven (2011)–illustrate the changes and complexities inwhich mathematical knowledge for teaching is now embedded.

Shulman’s model

Shulman (1987) identified pedagogical content knowledge (PCK) as the categorymost likely to distinguish the understanding of the content specialist from that of theexpert teacher. The importance given to PCK suggests that what is needed inmathematics teaching is not just knowledge of the subject, or general knowledge ofpedagogy, but rather a combination of both. However, after 25 years of exposure toShulman’s seminal thinking, Petrou and Goulding (2011) conclude that:

Although Shulman’s work was ground-breaking and his ideas continue toinfluence the majority of research in the field, later researchers in the sametradition argue that it is not sufficiently developed to be operationalised inresearch on teacher knowledge and teacher education. (p. 12)

It needs also to be pointed out that Shulman did not write specifically formathematics teaching; and that his categories tend to reflect the educational contextof the USA which has a very diffuse national curriculum.

In a review of Shulman’s (1986, 1987) original model, Grossman et al. (1989)emphasised that, while knowledge of the subject matter occupies a central place in theknowledge base of teaching, effective teachers not only know their content, but knowthings about their content that make effective instruction possible. They also reor-ganised Shulman’s (1986, 1987) categories into four main ones: subject-matterknowledge, general pedagogical knowledge, pedagogical content knowledge (includ-ing knowledge of students’ understanding, curriculum, and instructional strategies),and knowledge of context.

Michigan model

Ball et al. (2008), while sympathetic to Shulman (1986, 1987), prefer to use the termmathematical knowledge for teaching to represent a category within which theyidentify four constituent domains: (1) common content knowledge, defined as themathematical knowledge and skill used in settings other than teaching; (2) specialisedcontent knowledge, defined as the mathematical knowledge and skill unique toteaching specific topics; (3) knowledge of content and students, defined as knowl-edge that combines knowing about students and knowing about mathematics; and (4)knowledge of content and teaching, which combines knowing about teaching and


knowing about mathematics. Their map of mathematical knowledge for teaching,shown in Fig. 1, comprises these four categories, to which is added horizon contentknowledge and knowledge of content and curriculum (Ball et al. 2008).

Among these four domains discussed by Ball et al. (2008), common contentknowledge is a primary component of mathematical knowledge, and needs to becombined with a teacher’s specialised content knowledge, the subject matter knowl-edge needed for teaching specific mathematics content or topics. Knowledge ofcontent and students and knowledge of content and teaching are both intended todescribe distinct knowledge for teaching. However, the “content” component ofknowledge of content and teaching (KCT) may refer to today’s worksheet, or thisyear’s textbook, or what is contained in official curriculum documents. In this sense,KCT may not be too far removed from Shulman’s (1986, 1987) equally loosecategory of curriculum knowledge in which he includes teachers’ having a grasp ofrelevant materials and programs. While these knowledge domains are intended toanticipate classroom use, their instructional consequences are only implied.

Ball et al. (2008) use knowledge of content and students to build on Shulman’s(1986, 1987) construct of PCK by focusing on particular teaching topics in mathe-matics, where knowledge of specific mathematical topics is necessary for effectiveteaching. Their category of knowledge of content and curriculum does not explicitlyrefer to any official curriculum. Petrou and Goulding (2011) also argue that mathe-matical knowledge for teaching, as understood by Ball et al. (2008), does notacknowledge the importance of teachers’ beliefs in their teaching. This point issupported by Fennema et al. (1992) whose model of cognitively guided instructionexpressly includes the importance of teachers’ beliefs, as well as knowledge ofmathematics, pedagogical knowledge, content-specific knowledge, and knowledgeof learners’ cognitions in mathematics, as well.

What appears to be a common feature of models of both MKT by Ball et al. (2008)and PCK by Shulman (1986, 1987) is a weak interpretation of “curriculum” and

Fig. 1 Domains of mathematical knowledge for teaching


“curriculum knowledge” which may be based too closely on their USA experience,where curriculum knowledge can be interpreted simply as “the particular grasp of thematerials and programs that serve as ‘tools of the trade’ for teachers” (Shulman 1987,p. 8). Ball et al. (2008) do not seem to have moved beyond this.

In their chapter “Conceptualising teachers’ mathematical knowledge in teaching,”Petrou and Goulding (2011) concluded that Shulman’s (1986, 1987) description ofcontent knowledge “conjures up a view of a loose curriculum frame with a degree ofchoice about materials and approaches, which may not be applicable in differentcontexts” (p. 11). While the USA does not have a national (official) curriculum,Petrou and Goulding (2011) argue that:

contemporary teaching practice in countries such as the United Kingdom–andwe would also suggest other countries such as China and Australia–is stronglyconstrained by official curriculum guidance and assessment systems; so theteachers’ curriculum knowledge needs not only to include knowledge of mate-rials and resources from which they can draw, but also include mandatedaspects of the curriculum frame in which they are working. (p. 11)

In countries where the official curriculum plays a more pivotal role, research intoteacher capacity needs a stronger and clearer reference to knowledge of the officialcurriculum.

We agree that a category of specialised content knowledge, as used by Ballet al. (2008), is a necessary corrective to the Shulman (1986, 1987) model. Bycontrast, Watson’s (2001) Australian study considered the official curriculum asvery important when looking at teacher competence and confidence, but did notexamine closely how teachers need to translate and interpret the intentionsand/or expressions of official curriculum into the design of specific mathematicstasks and/or sequences of teaching. This aspect is addressed in our model ofteacher capacity.

Cambridge model

A University of Cambridge project titled “Subject Knowledge in Mathematics”(Rowland et al. 2003, 2005; Rowland 2005, 2007) investigated the relationshipbetween pre-service teachers’ subject matter knowledge and their pedagogical contentknowledge in mathematics. This project resulted in the identification of a frameworkcalled “the knowledge quartet” consisting of four dimensions: foundation, transfor-mation, connection, and contingency. This knowledge quartet framework has beenapplied to support teaching development for early-career teachers in England andother countries. Petrou and Goulding (2011), following Petrou (2009), agree that, ingeneral, this framework provides a comprehensive classification of teaching situa-tions in which participants’ mathematical knowledge surfaces in teaching. However,we also agree with their observation that the knowledge quartet is focussed primarilyon the pre-service training of teachers, not on the work of experienced teachers, andstill less on curriculum change.

Working within the Cambridge model (Rowland et al. 2003, 2005; Rowland 2005,2007), Petrou and Goulding (2011) positioned themselves with Fennema and Franke(1992) in believing that teacher knowledge can be understood only in the terms of


specific contexts in which teachers work. Accordingly, they offered the synthesis ofthe Cambridge model of mathematical knowledge for teaching shown in Fig. 2:

Their model gives a more prominent place to curriculum knowledge than that impliedby the Shulman (1986, 1987) model or by theMichiganmodel (Ball et al. 2008; Hill et al.2008). It implies that curriculum knowledge is central in understanding what teachersneed to know in order to teach mathematics effectively. In addition, their model impliesthat teachers’ SMK and PCK transform how teachers understand, interpret, and use themathematics curriculum and its associated materials (Ball andCohen 1996; Petrou 2009).

Diverging perspectives within the Cambridge model

Ruthven (2011) further elaborates the Cambridge model by presenting four distinctconceptualisations, each of which is intended to move forward debate about andresearch on mathematical knowledge for teaching–but in different directions. Thefirst can be described as subject knowledge differentiated, which, according to Petrouand Goulding (2011), gives more priority to knowledge of the official curriculum andinterpretation of its intentions than does either the Michigan model (Ball et al. 2008)or the Cambridge model (Rowland et al. 2003, 2005). A second line of thinking isdescribed as subject knowledge contextualised (Hodgen 2011). The fundamentalthrust of this approach is that the use and development of subject-related knowledgein teaching is strongly influenced by teaching materials and social context. A thirdline of thinking was described by Steinbring (2011) as subject knowledge interacti-vated. The fundamental thrust of this conception is that mathematical knowledge isonly indirectly communicable and locally constructible through social interaction.The final line of thinking developed has been described by Watson and Barton (2011)as subject knowledge mathematised. Its fundamental thrust is that teachers must actmathematically in order to enact mathematics with their students, and that doing socalls for a kind of knowledge rather different from that which normally receivesemphasis in discussions of mathematical knowledge for teaching.

Fig. 2 Petrou and Goulding synthesis of models of mathematical knowledge for teaching


These four lines of thinking on subject knowledge–differentiated, contextualised,interactivated, and mathematised–show that mathematical knowledge for teaching isno longer a single unified idea. Researchers also need to be aware of the limitations ofsome or all of these four approaches: (1) All four have a strong focus on how toimprove pre-service teachers’ mathematical knowledge needed for their teaching inthe future; (2) apart from the first framework adopted by Petrou and Goulding (2011),the approaches do not appear to place a strong emphasis on the way in which officialmathematics curriculum documents are intended to guide teaching; (3) apart from thefirst framework, the approaches tend to view knowledge for teaching mathematics ingeneral terms, whereas our study has its focus on specific areas of mathematicalcontent important for curriculum reform; (4) none of four theoretical frameworks iseasy to conceptualise into empirical research. However, with the goal of investigatingthe relationship between teacher capacity and national curriculum reform, our posi-tion is closest to that of Petrou and Goulding (2011) in the Cambridge model and tothat of Ball et al. (2008) in the Michigan model. We strongly support the point madeby Ball et al. (2008) that any definition of mathematical knowledge for teachingshould begin with teaching, not teachers. Any such definition must be:

concerned with the tasks involved in teaching and the mathematical demands ofthese tasks. Because teaching involves showing students how to solve prob-lems, answering students’ questions, and checking students’ work, it demandsan understanding of the content of the school curriculum. (p. 395)

Teacher capacity as professionally informed dispositions to act

Our construct of teacher capacity, as professionally informed judgements and dispo-sitions to act, is intended to capture a common ground between movements for schoolsystem and curriculum reform, discussed above, and the construct of MKT elaboratedby Ball et al. (2008) and by Hill et al. (2008). As a result, our construct of teachercapacity is not an entirely new research model. It takes seriously two elements–knowledge of mathematics and knowledge of students’ mathematical thinking–thatare central to the MKT model. On the other hand, our construct develops from theMKT model two aspects–knowledge of the official mathematics curriculum andcapacity to design instruction–that are not strongly emphasised in that model. Whatis more, we shift the emphases away from “knowledge” in MKT to give greaterattention to “how teachers utilise that knowledge in their practical teaching”. That iswhy the term “capacity”, is used in this paper.

All four components of teacher capacity have connections and/or parallels withfour categories of MKT. Our criterion A, knowledge of mathematics, is intended tocapture their category of specialised content knowledge; our criterion B, interpreta-tion of the intentions of official mathematics curriculum, and criterion C, understand-ing of students’ mathematical thinking, can be related to their category of knowledgeof content and students; and our criterion D, design of teaching, connects to theircategory of knowledge of content and teaching.

Criterion A–knowledge of mathematics–is intended to be applied to the tasks thatthe students have completed or are being asked to complete. However, we do not


propose to separate out CCK and SCK, for the same reason as Ball et al. (2008) whopoint out: “it can be difficult to discern common from specialized knowledge inparticular cases” (p. 403). Knowledge of mathematics is intended to capture the keymathematical ideas for teaching specific content.

Criterion B–interpretation of the intentions of the official mathematicscurriculum–is concerned with how teachers relate what is mandated or recommen-ded in official curriculum documents of China and Australia to what their students arebeing asked to learn. Criteria B and C are linked to the work of Hill et al. (2008) andof Ball et al. (2008), especially to their map of MKT, in particular their category ofknowledge of content and students. However, this criterion differs from their constructof knowledge of content and students in giving a greater emphasis to knowledge ofofficial curriculum documents and willingness to use them in planning instruction.

Criterion C–understanding of students’ mathematical thinking–is directlyconcerned with teachers’ capacity to interpret and differentiate between what studentsactually do (or did) and to anticipate what they are likely to do. It implies that teachersare able to demonstrate knowledge of how students learn particular subject matter,and students’ typical mathematical thinking in relation to that specific subject matter,including whether they are able to recognise the typical errors that students make andwhat mathematical thinking led to these errors (see Hill et al. 2008, p. 376).

Consequently, criterion D–design of teaching–places a clear emphasis on teachers’capacity to design teaching in order to move students’ thinking forward and to respondto specific examples of students’ thinking in the light of official curriculum documents.Criterion D connects to category of knowledge of content and teaching, which,according to Ball et al. (2008), embodies “an interaction between specific mathematicalunderstanding and an understanding of pedagogical issues that affect student learning”(p. 401). However, criterion D is intended to give greater emphasis to how teachers usetheir professionally informed judgement to design practical teaching on specific topics.Our model of teacher capacity is shown in the Fig. 3.

Fig. 3 Our model of teacher capacity


Each of the four criteria is focussed on teachers’ professionally informed judge-ments and dispositions to act, distinguishing them from the four knowledge domainsof Ball et al. (2008). Criterion D (design of teaching) is put forward as the centralcomponent in our model. As the instructional embodiment of teacher capacity, designof teaching rests on strong connections with criterion A (knowledge of mathematics),criterion B (interpretation of the intentions of the official mathematics curriculum)and criterion C (understanding of students’mathematical thinking). However, we alsoanticipate that, when separated from criterion D, the inter-relationships betweenthe other three criteria will not be as strong as their relationship to criterion D(design of teaching).

Methodological position

In this study, our mathematical focus is on students’ ability to read and interpretnumber sentences as expressions of mathematical relationships, rather than seeingthem exclusively as calculations to be performed. Specifically, we draw attention tothe importance of assisting students to use ideas of equivalence and compensation tosolve number sentences involving subtraction. These methods, Irwin and Britt (2005)have argued, may provide a foundation for algebraic thinking (p. 169). Jacobs et al.(2007) use the term relational thinking to refer to these kinds of strategies. They agreethat there is still room for debate regarding whether relational thinking in arithmeticrepresents a way of thinking about arithmetic that provides a foundation for learningalgebra or is itself a form of algebraic reasoning; however, they argue strongly that“one fundamental goal of integrating relational thinking into the elementary curriculumis to facilitate students’ transition to the formal study of algebra in the later grades so thatno distinct boundary exists between arithmetic and algebra” (p. 261).

The research instrument

Teachers in both countries were invited to complete a written questionnaire based ona “scenario” where some researchers had visited their school and given students(either in Year 6 or Year 7) the following number sentences, asking them to write anumber in the box to make a true statement, and in each case to explain their workingbriefly. These two questions, according to the scenario, had been accompanied bysimilar questions dealing with addition, and were intended to see how studentsinterpret and solve number sentences involving different operations:

For each of the following number sentences, write a number in the box to makea true statement. Explain your working briefly.

39 – 15 = 41 –

104 – 45 = – 46

Australian and Chinese teachers were then presented with seven responses selectedfrom actual responses by Australian students and Chinese students based on a study


reported by Stephens (2008). The seven responses of Chinese and Australian studentswere matched as closely as possible. In the Australian sample (see Appendix 1), twoAustralian students, A and B, correctly found the missing number by calculating theresult of the subtractions 39–15, and 104–45, and then used these results to calculatethe value of the missing numbers on the right hand side. Student C refrained fromcalculating, attempting to use equivalence, but compensated in the wrong direction toget answers of 13 and 103 respectively. Two students, D and E, successfully arguedthat since 41 is two more than 39 the missing number has to be two more than 15 tokeep both sides equivalent. They applied similar reasoning to the second problem.Student F used arrows connecting the two related numbers (e.g., 39 and 41), and alsoconnecting the other number (15) to the unknown number. Above the arrows StudentF wrote +2 for the first problem and +1 for the second problem, obtaining correctanswers. Finally, student G placed the letters A and A1 beneath 39 and 41, and B andB1 beneath 15 and the unknown number, and found correct values for the unknownnumbers using an explanation based on equivalence and compensation.While the answerto the first problem is correct, Student G’s explanation contained a small written error.

The questionnaire contained three key questions, with each question on a separateA4 page. In the first question, teachers were asked to comment briefly on each of theseven samples. In a second question, teachers were asked how they would respondspecifically to Students A, B, and C if they were students in their class. In a thirdquestion, teachers were asked:

In planning your teaching program, how do you want to move students’thinking forward in regard to these and related questions? How will youdevelop the kind of mathematical thinking that students need to solve thesekinds of number sentences? You can talk about a short design of one or severallessons, or a longer plan over the year.

The participants

Both samples used in this exploratory study were convenience samples. TheAustralian sample consisted of 60 numeracy coaches working in Victorian govern-ment schools who were participating in several extended professional developmentprograms. Forty-five were based in primary schools, and 15 were in secondaryschools. Several of the coaches were not mathematics specialists, although all wereteaching mathematics. The Chinese sample of 60 teachers–45 primary and 15 sec-ondary to match the Australian sample–was randomly selected from a larger group ofmore than 100 specialist mathematics teachers who had agreed to complete thequestionnaire (Chinese version) during several teacher professional developmentprograms in Nanjing, Whenzhou, and Chongqing. All Chinese teachers were teachingmathematics across several grades.

Qualitative analysis

Each criterion of our analytical framework was elaborated in terms of four specificindicators (see Table 1). In the case of the first criterion, these indicators expressed


how well teachers’ responses indicated a clear understanding of the mathematicalthinking that the two problems were intended to examine; and in the case of thesecond criterion how teachers’ thinking reflected key ideas of current official curric-ulum documents in the respective countries. Indicators associated with the thirdcriterion looked specifically at how well teachers could describe and interpret keyfeatures of performance expressed by the seven students, and how well they wouldrespond to what these students had done. Those for the fourth criterion looked at howwell teachers could plan for teaching that fostered a deeper appreciation of the

Table 1 Four criteria and associated indicators

Criterion A–Knowledge of Mathematics:

(1) Does the teacher recognise that there are two mathematical approaches to solving these kinds of problems–using calculation; or using equivalence and compensation for the operations of subtraction or difference?

(2) Does the teacher recognise that students need to attend specifically to subtraction or “difference” whenusing equivalence?

(3) Does the teacher refer specifically to mathematical terms such as “equivalent difference” or “differenceunchanged”?

(4) Does the teacher understand that equivalence using subtraction is compensated differently fromaddition, and/or that the key idea of equivalence also applies to the other operations?

Criterion B–Interpretation of the Intentions of Official Mathematics Curriculum:

(1) Does the teacher realise that “mathematical thinking” should be treated as an important considerationwhilst calculation remains valued?

(2) Does the teacher understand and support the intention of the curriculum to link number learning andalgebraic thinking?

(3) Does the teacher show, in his/her descriptions of children’s responses, an awareness of the keycurriculum goal of moving students from reliance on calculation to using equivalence in numbersentences, here with respect to “difference” or subtraction?

(4) Does the teacher’s response use terms, words, or expressions that are found in official curriculumdocuments?

Criterion C–Understanding of Students’ Mathematical Thinking:

(1) Does the teacher recognise that Australian students D, E, F & G (or Chinese students C, D & E) werecorrectly using relational thinking although expressed in different ways?

(2) Does the teacher identify the typical error (compensating in the wrong direction) shown in solution C ofAustralia sample (or solutions F(2) and G(1) of China sample)?

(3) Does the teacher recognise the importance of identifying those students who can only use calculation?

(4) Do Chinese teachers see that solution G(2) suggests a deeper misunderstanding; or do Australianteachers recognise that student G has a clear understanding of equivalence although makes a small errorin the explanation for question 1?

Criterion D–Design of Teaching:

(1) In designing teaching, does the teacher focus on the important aspects of mathematics to be taught andfostering mathematical thinking, not on general strategies?

(2) Does the teacher have a short-term teaching plan to respond to selected students in the next lesson?Does the teacher recognise that it is more important to let students who can think relationally explain theirthinking to the whole class than it is to let those who used calculations explain their thinking?

(3) Does the teacher have a longer-term teaching plan to move students’ relational thinking forward? Howwell does this plan reflect knowledge of students’ thinking (criterion C)?

(4) Does the teacher give teaching examples or use teaching with variation to help students’ learning andthinking?


mathematical thinking embodied in these and related tasks, especially in fosteringideas of equivalence and compensation.

Qualitative evidence of demonstrated teacher capacity

The following examples provide evidence of Chinese and Australian teachers’(coded either as Teacher n CH or Teacher n AU) responses with respect to eachof the four criteria.

Criterion A (knowledge of mathematics)

Teacher 1 CH commented: “If the same number is added to both minuend andsubtrahend, the difference represented by the number sentence will be un-changed……this is also called the law of difference unchanged.” This was a typicalresponse of Chinese teachers, many more of whom were able to give a name to thespecific law used in this scenario compared to their Australian counterparts.

Similarly, Teacher 11 AU realised that: “Although the process is the same withboth addition and subtraction, students often misunderstand whether they have to addor subtract to get the equivalent value on both sides of the equals sign.” Showing theunderstanding of the meaning of equals sign and equivalence, it was also evident thathe/she could classify the difference between the addition and subtraction.

Criterion B (interpretation of the intentions of official mathematics curriculum)

Teachers 3 and 4 AU referred to Key Characteristics of Effective Numeracy TeachingP-6 (DEECD 2009b). Teacher 4 AU pointed to the need to:

engage students in identifying and using arithmetic relationships within numbersentences to solve problems without calculating and teach a repertoire ofstrategies–(using) guess-guess-check (systematic trial and error), logicalarithmetic reasoning and inverse operations to solve a wider range of numbersentences.

These teachers were clear that the focus of this teaching topic should not be onstudents’ calculating proficiency, but on their understanding of relationships withinnumber sentences and operations. This is consistent with the emphasis, given byVELS (VCAA 2008), on developing students’ understanding of structure.

Teacher 15 CH said: “In the elementary teaching of number and algebra, integrity andcoherence [of number and algebra] need to be embodied.” This teacher had deeplyinterpreted the key intentions of the content area “number and algebra” in the ChineseNational Curriculum in Mathematics (Ministry of Education of PRC 2001, 2011).

Criterion C (understanding of students’ mathematical thinking)

Teacher 45 AU pointed out that:

Despite their different responses, students C, D, E, F & G were all usingrelations to solve the questions which is different from students A & B. This


is a better method and to be encouraged because it is closer to the structuralthinking that students need when learning algebra. These number sentenceshave been carefully chosen to make this method better. Student C spotted therelationships between the numbers being used in both algorithms (addition andsubtraction) …. s/he has added to one of the numbers, (whereas) s/he needed tosubtract from the other.

This teacher clearly understood similarities and differences in students’mathematicalthinking shown in the seven sample responses.

Teacher 13 CH said, “It is not easy to judge whether A and B solve it throughcalculation, or through the reverse principle between addition and subtraction.” Thisteacher was one of few teachers who noted the importance of distinguishing betweenthose students who only use calculation and those who can do both (relational andcalculation) but choose to use calculation. Teacher 8 CH said that “After students’agreement on Type 2 (relational thinking), it is further necessary to explain the rationaleof Type 2 to help students understand it, in order to move students’ thinking forward.”

Criterion D (design of teaching)

Teacher 36 AU gave a well-designed five-stage plan to move students’ thinkingforward with each stage reflecting a different level of mathematical thinking. Inpresenting a specific teaching plan, this teacher also included a long-term plan todevelop students’ mathematical thinking.

Teacher 37 CH suggested: “Explore variations, change the ‘−’ in both sides into ‘+’or have the change in one side and leave the other unchanged.” Teaching with variationis used effectively in the following teaching examples:

1. Fill in “>,” “<” or “=” in ○. 2. Fill in “+” or “−” in ○, and numbers in □.

45� 36 ○ 45þ 3ð Þ � 36þ 3ð Þ 87� 45 ¼ 87 ○□ð Þ � 45� 3ð Þ45� 36 ○ 45� 3ð Þ � 36� 3ð Þ 98� 36 ¼ 98� 5ð Þ � 36 ○□ð Þ198� 42 ○ 198þ 2ð Þ � 42þ 2ð Þ 184� 56 ¼ 184 ○□ð Þ � 56þ 8ð Þ

Through these carefully developed questions, teachers are able to help studentsthink about what has changed and what remains unchanged in different sentences.This is a typical illustration of “teaching with variation” which has enjoyed a longtradition in China’s mathematics classrooms.

Weak or inappropriate responses to criteria B and D. Teacher 7 AU said: “I am notfamiliar with working in this area of the school I would need to consult the Maths(Developmental) Continuum ...... I need further help as I was probably looking in thewrong progression point.” This teacher clearly showed a lack of familiarity with someimportant ideas in the current mathematics curriculum.

In addition, when talking about the design of teaching, some Australian teachers,like Teacher 10 AU, responded in very general terms, such as “I would give studentsa word problem… I would provide lots of concrete examples for students tomanipulate.” Similarly, some Chinese teachers, like Teacher 52 CH, were disposed“to let students appreciate different ways of solutions and then to optimize thembased on the situations of their own.” These teachers neither refer to specific


mathematical thinking, nor do they have a plan to guide students to make thenecessary optimisation.

A dissenting response to criteria A and B but strong on D. Teacher 37 CH, whoshowed a clear understanding of the mathematical elements of the questions anddesigned very clear instructional examples (as shown above) to help students developrelational thinking, had a very strong resistance to fostering relational approaches.Teacher 37 CH took the position that:

The solutions of (China sample) Students A & B (who used calculation and gotthe correct answers) need to be energetically popularized (to the whole class),because most students can master them… The deep thinking [of those whocorrectly used relational thinking] deserves praise, but it shouldn’t be intro-duced, because … some students may be confused by it and so cause mistakes.

This dissenting response points to the importance of including values, beliefs, anddispositions in the construct of teacher capacity.

Quantitative analysis

By assigning a score of 1 if one of the four indicators was evident in a teacher’sresponse, and 0 if it was omitted from their response or answered inappropriately, itwas possible to construct a score of 0 to 4 for each criterion, and hence a maximumscore of 16 across the four criteria. We allowed for the possibility that teachers mightprovide convincing alternative indicators to the four indicators listed.

Each of the two researchers operated independently to score teachers’ responses;then a careful confirmative check took place in order to resolve any differencesbetween the two assignments. A high degree of consistency was present in the initialgrading by the two graders, where, in less than 40 cases of 0/1 grading, only minordifferences occurred. Any differences could be resolved by consensus, leaving nodisagreement on grading the 120 responses across the four criteria.

A summary for Chinese and Australian samples

For the 60 Chinese teachers, the highest score was 15 and the lowest score was 3,with a median score of 9. For the 60 Australian teachers, the highest score was 14 andthe lowest was 2, with a median score of 8.5. The respective mean scores were 8.7(Chinese) and 8.57 (Australian) with standard deviations of 2.07 and 3.06 respec-tively. In the Australian sample, 10 teachers scored less than 5, whereas only 5Chinese teachers scored less than 5. Table 2 shows means and deviations that were

Table 2 Means for each criterion and global means and deviations

Sample Criterion A Criterion B Criterion C Criterion D Total

Chinese(60) 2.73(0.63) 1.98(0.68) 1.92(0.65) 2.07(0.84) 8.7(2.07)

Australian(60) 2.25(0.79) 2.18(1.01) 2.18(0.77) 1.97(1.15) 8.58(3.06)

CH&AU(120) 2.49(0.76) 2.08(0.87) 2.05(0.72) 2.01(0.94) 8.64(2.60)


calculated for each of the criteria, and a global mean score and standard deviationcalculated across all four criteria.

Of the four criteria, Criterion D (Design of Teaching) had the lowest mean (2.01)followed by Criterion C (Understanding of Students’ Mathematical Thinking) withthe mean of 2.05, followed by Criterion B (Interpretation of the Official MathematicsCurriculum) with the mean of 2.08. Criterion A (Knowledge of Mathematics) had thehighest mean at 2.65.

Three classifications of teacher capacity

Three sub-categories of our construct of teacher capacity were created. Theboundaries were set on the basis of a qualitative analysis of teachers’ responsesas discussed earlier. These were high capacity (score 11–16), medium capacity(score 6–10) and low capacity (score 0–5). An initial classification of the two samplesis shown in Table 3.

There were more high-capacity teachers in the Australian sample than in theChinese sample (respectively 13 and 10); but more Australian teachers were classi-fied as low capacity than Chinese teachers (respectively 10 and 5). The scores ofChinese teachers were more centred, with 75 % in the medium-capacity group.Table 2 also shows the standard deviations on all four criteria and the total to begreater in the case of the Australian teachers.

A high capacity to make an effective bridge between the teaching of number andfostering of algebraic thinking was demonstrated by a clear understanding of themathematical nature of the tasks in which students had been engaged; by a capacity torelate these tasks to relevant curriculum documents; by high interpretative skills whenapplied to each of the seven samples of students’ work; and by an extensive range ofideas for designing and implementing a teaching program to support the developmentof students’ mathematical thinking. Medium capacity was shown by teachers who,while possessing knowledge and skills supportive of these directions, clearly need toincrease their current levels of professional knowledge and skills. In both samples,Low capacity was evident in a minority of teachers who appeared unable to clearlyarticulate the mathematical nature of the tasks, or what differentiated the sevenresponses used in the questionnaire. These teachers were unable to point with anyconfidence to how the tasks related to what is contained in official curriculumdocuments, and found it difficult to describe how they would plan a program ofteaching to foster these and related mathematical ideas.

Chinese and Australian teachers in the sample appeared to perform similarly onCriteria B (Interpretation of the Intentions of Official Mathematics Curriculum) and D(Design of Teaching). However, Chinese teachers sometimes used more formalexpressions than their Australian counterparts in elaborating the mathematical

Table 3 Classifications of teachercapacity

Capacity Chinese Australian

High 10(16.7 %) 13(21.6 %)

Medium 45(75 %) 37(61.7 %)

Low 5(8.3 %) 10(16.7 %)


thinking embedded in the tasks on which the students were asked to work. On theother hand, Australian teachers were more likely than their Chinese counterparts touse officially sanctioned terms such as “equivalence” in responding to Criterion C(Understanding of Students’Mathematical Thinking). However, the scoring profile ofboth groups was such that the whole sample was used in the summary statistics forthe three groups (low, medium and high capacity) as shown in Table 4.

As can be seen from Table 4, all four criteria were effective in discriminatingbetween teachers of high, medium and low capacity, but Design of Teaching wasby far the most powerful criterion in distinguishing between teachers. Thisfinding is confirmed by an analysis of the quantitative scoring where 22 of the23 high-capacity teachers scored 3 or 4 on Criterion D (Design of Teaching),with more than one third (8/22) scoring 4. For the 82 medium-capacity teachers,none scored 4 on Criterion D and only 10 medium-capacity teachers scored 3,representing only 12 % of the medium-capacity group. On the other hand, onCriterion D, no medium-capacity teacher scored 0, and only a few scored 1. Bycontrast, among the 15 low-capacity teachers, no-one scored more than 1 onCriterion D, and 8 scored 0.

Correlations between the criteria

Table 5 shows the bivariate correlation between any two of the four criteria.Correlations were calculated using SPSS 19.0 (English version) for the 120Chinese and Australian samples.

All pairings of the four criteria have a significant correlation at the 0.01 level (2-tailed), supporting our theory that all four components of teacher capacity areinterrelated. However, we cannot know whether the correlation between any twocriteria was direct or was influenced, at least to some extent, by a third variable(Pallant 2001, p. 130). After statistically controlling for one of the four criteria, wethen carried out four partial correlation analyses between any two of the other threecriteria. When Criterion A (Knowledge of Mathematics) was controlled, the partialcorrelations between the other three criteria–B/C (.318), B/D (.529), and C/D(.384)–were all significant at 0.01 level. When Criterion B (Interpretation of theIntentions of the Official Mathematics Curriculum) was controlled, partial correla-tions between A and D (.440) and between C and D (.337) were significant at the0.01 level, while A/C (.233) was significant at 0.05 level. Similarly when CriterionC (Understanding of Students’ Mathematical Thinking) was controlled, all threepartial correlations–A/B (.318), A/D (.486), and B/D (.539)–were significant at the0.01 level.

Table 4 Summary statistics for the three categories (N=120)

Category A B C D Sum

Low (n=15) Mean (SD) 1.47 (.52) .87 (.64) 1.07 (.59) .60 (.51) 4.00 (1.25)

Med (n=82) Mean (SD) 2.45 (.59) 2.05 (.63) 2.06 (.55) 1.91 (.57) 8.48 (1.17)

High (n=23) Mean (SD) 3.30 (.47) 3.00 (.67) 2.65 (.65) 3.30 (.56) 12.26 (1.45)


However, as Table 6 shows, when Criterion D (Design of Teaching) was con-trolled, there were no statistically significant correlations between the other threepairs: A/B (.091), A/C (.112), B/C (.155).

After statistically controlling any one of the other three criteria, Criterion D stillhad significant correlation with the other two at 0.01 level. However; when CriterionD was statistically controlled, there was no statistically significant relationshipbetween any two of the other criteria. A multi-step linear regression analysis con-firmed the importance of Criterion D: On its own, Criterion D accounted for 78 % ofthe variance of our model. This also endorses our placing of Design of Teaching at thecentre of the model shown in Fig. 3, where it is informed by the other three criteria,which are only weakly related when dissociated from Design of Teaching.

Issues relating to validity and reliability

On the basis of our model, three hypotheses were formed, and each was confirmed bythe data: (1) The four criteria and associated indicators are effective in discriminatingbetween teachers on the basis of capacity; (2) Teachers of high capacity performstronger on each of the four criteria than teachers of medium and low capacity, andteachers of medium capacity perform stronger on each of the four criteria thanteachers of low capacity, making it easy to predict individual teachers’ capacity fromtheir performance on any specific criterion; (3) While all four criteria together defineteachers’ capacity, each criterion is expected to measure different facets of teachercapacity.

Table 5 Bivariate correlations between any two of four criteria (N=120)

A B C D

A Pearson Correlation 1 .425a .371a .577a

Sig. (2-tailed) .000 .000 .000

B Pearson Correlation .425a 1 .425a .636a

Sig. (2-tailed) .000 .000 .000

C Pearson Correlation .371a .425a 1 .506a

Sig. (2-tailed) .000 .000 .000

D Pearson Correlation .577a .636a .506a 1

Sig. (2-tailed) .000 .000 .000

a Correlation is significant at the 0.01 level (2-tailed)

Table 6 Partial correlations (criterion D as controlled variable)

Control variables A B C

D A Correlation 1.000 .091 .112

B Correlation .091 1.000 .155

C Correlation .112 .155 1.000


The following tests, conducted with SPSS 19 (English version), were intendedto explore four components of validity and reliability of our construct in thisstudy:

First, to have a valid measure, reasonable discriminations should exist among thethree categories of teacher capacity based on scores on the four criteria. The standarddeviations of the four criteria and the total among the 120 samples represent anacceptable range of values distinguishing between individual performances on eachof the four criteria and the global score.

Second, the means of the four criteria for each sub-category show that eachcriterion effectively discriminates between the three classifications of high, medium,and low capacity. Table 5 above shows that the correlations between the four criteriaare not high enough to suggest that any pair is measuring the same thing, or lowenough to suggest that they are completely unrelated.

Third, a criterion-total statistic suggests that each criterion is measuring someaspect of teacher capacity; and inter-criterion correlations between the four criteriasuggest an acceptable consistency between the criteria, while still justifying the use ofall four. Finally, Cronbach’s alpha coefficient of the scale is above 0.7, so the scalecan be considered reliable with the sample (Pallant 2001, p. 87). Cronbach’s alphawas found to be 0.736.

Conclusion

Teacher capacity was investigated in terms of four components: knowledge ofmathematics, interpretation of the intentions of official curriculum documents, un-derstanding of students’ thinking (the ability to analyse and interpret students’responses and to frame appropriate responses to individual students), and design ofteaching to foster the underlying mathematical ideas. Performance on each criterionwas ascertained using a precise set of indicators that were related to the specificmathematical tasks, students’ expected thinking in relation to those tasks, the rela-tionship between the tasks and official curriculum documents, and teachers’ ability todesign explicit teaching sequences to support students’ learning.

Design of teaching, informed by the other three criteria, has emerged as the criticaldimension for the implementation of curriculum reform; and the criterion that mostclearly distinguishes between different levels of teacher capacity to enact reform. Ourconstruct of teacher capacity strongly reflects the view, stated earlier, that effectiveimplementation of any curriculum reform depends on teachers’ subtle interpretationsof official curriculum documents and their professional dispositions to act on thoseideas, which go well beyond general descriptions or statements of intent that areusually embodied in official curriculum advice.

Our construct of teacher capacity, interpreted here as teachers’ professionallyinformed judgements and dispositions to act, builds on earlier research into mathe-matical knowledge for teaching by Ball et al. (2008), and Hill et al. (2008). However,to make the conclusions of this study compatible to their key idea of mathematicalknowledge for teaching would appear to require a major re-framing of their category,knowledge of content and teaching. In light of this study, that category appears to betoo static and less suited to direct attention to design of teaching, which we have


interpreted as understanding key aspects of national curriculum reform and knowinghow to enact these aspects in practice. Our construct seems more suited to capture thiskey feature of curriculum reform.

Appendix 1

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