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Incorporating learning study in ateacher education program in

Hong Kong: a case studyMun Yee Lai

School of Teacher Education, Charles Sturt Universtiy, Bathurst, Australia, and Yin Wah Priscilla Lo-Fu

The Hong Kong Institute of Education, Hong Kong, China

AbstractPurpose The purpose of this paper is to report a case study of how learning study was incorporatedin teacher education programs in Hong Kong. It aims to share the success of the program and todisseminate how pre-service teachers enhanced their mathematical content knowledge andpedagogical content knowledge by practising learning study. Building on the work, this papersuggests incorporating the framework of learning study, a theory-guided pedagogical principle, as anintegrated subject of mathematics pedagogy and teaching practice in teacher education programs.Design/methodology/approach In total 32 pre-service teachers learning journals of theirreflections of learning processes were analyzed. The analysis of data and reporting of findings arelinked tightly to how pre-service teachers enhanced their mathematical content knowledge andpedagogical content knowledge by practising learning study.Findings The 32 pre-service teachers noted that the learning study subject fostered theirunderstanding of relationship between theory and practice and their understanding of transformingknowledge into action. In particular, they came to understand that knowledge of pupils and contentinvolves a particular mathematical idea or procedure and familiarity with students prior knowledgeand misconceptions. They also reported that they understood better what mathematics pedagogycontent knowledge means and what components it includes.

Originality/value The suggestions of incorporating the framework of learning study in teachereducation programs is supported and manifested by the positive feedback and comments of the32 pre-service teachers who underwent the entire learning process of learning study in Hong Kong.The findings demonstrate how pre-service teachers mathematical content knowledge and pedagogicalcontent knowledge were enhanced by practising learning study.Keywords Mathematics, Learning studies, Teachers, Education colleges, Hong Kong,Mathematics content knowledge, Pedagogical content knowledge, Theory of variation,Pre-service teacher educationPaper type Case study

1. IntroductionTraditional teacher education programs are often criticized for their failure to preparepre-service teachers for the realities of the classroom (Eilam and Poyas, 2009; Goodlad,1990; Korthagen and Kessels, 1999; Korthagen et al., 2006). Researchers (Eilam andPoyas, 2009; Ethel and McMeniman, 2000; Jaworski, 2006; Korthagen and Kessels,1999) explain that this failure is a result of the disconnection between theory andpractice in teacher education programs. The cause of the problem is related to issuesabout what is the most relevant knowledge for teaching (Korthagen and Kessels, 1999)as pre-service teachers often perceive that much of the formal university work appears

The current issue and full text archive of this journal is available atwww.emeraldinsight.com/2046-8253.htm

International Journal for Lesson andLearning StudiesVol. 2 No. 1, 2013pp. 72-89r Emerald Group Publishing Limited2046-8253DOI 10.1108/20468251311290141

The authors would like to acknowledge Professor Jo-Anne Reid and Dr Sandie Wong for theirinsightful comments and feedback on earlier versions of this paper.

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irrelevant when viewed from the perspective of classroom practice (Bromme andTillma, 1995; Russell, 1989). Zeichner and Tabachnick (1981), for instance, found intheir study that educational concepts developed during teacher education werediminished during field experience. One way to reconcile the dilemma of the theory-

practice nexus is to make an explicit integration of theory and practice, from whichpre-service teachers can utilize and transfer their knowledge of content and educationalprinciples as they become more knowledgeable about teaching. Researchers such as Jaworski (2006) propose the idea of Teaching as learning, in practice (Lave, 1996),which describes the process of growth in teaching as a continuous participation inpractice ( Jaworski, 2006). Likewise, Reid (2011) argues that it is time to (re)turn to afocus on practice in initial teacher education (p. 294). In this paper, we argue that newera teacher education programs need to link closely to professional developmentwithin schools (Bullough and Kauchak, 1997; Darling-Hammond, 1994) and to engagepre-service teachers in continuous participation in practice (Jaworski, 2006). From thisposition, questions arise as to what is the entry point for practice and what should bepractised, all of which are relevant to actual classroom teaching by pre-serviceteachers.

Research (e.g. Borko and Livingston, 1989) indicates that there are qualitativedifferences between the thinking and actions of expert teachers, novice teachers andpre-service teachers. To explain the differences, Borko and Livingston (1989) employthe framework of complex cognitive skill (Leinhardt and Greeno, 1986) which ischaracterized by the process of transforming subject matter knowledge into formsthat are pedagogically powerful and yet adaptive to the variations in ability andbackground presented by the students (Shulman, 1987, p. 15). Among the differentcomponents of complex cognitive skill, Borko and Livingston (1989) point out thatpropositional structures for pedagogical content knowledge seem to be virtuallynonexistent in novice and pre-service teachers knowledge systems, even though theseare major components of learning to teach. Propositional structures refer to teachersfactual knowledge about components of the teaching-learning situation such as thestudents in their classroom, subject matter and pedagogical strategies (Borko andLivingston, 1989). Additionally, some researchers (such as Wilson et al., 1987) haveargued that the transformation of teaching knowledge has to occur within teachingpractice as this process of transformation is associated with the planning and design of instructional activities, evaluation and reflection (Leikin and Zazkis, 2010).

Relating this work to mathematics teacher education, we are working with thepresumption that practising mathematical content knowledge and pedagogical contentknowledge is the entry point for practice in teacher education programs. In order tofacilitate pre-service teachers learning to teach mathematics, and to foster theirtransformation of teaching knowledge, this paper suggests the incorporation of theframework of learning study, a theory-guided pedagogical principle, as an integratedsubject of mathematics pedagogy and teaching practice, in teacher educationprograms. A learning study framework, grounded in the theory of variation (seeMarton and Booth, 1997), is speculated to provide a platform for pre-service teachers tounderstand, assimilate and practise the different components of mathematical contentknowledge and pedagogical content knowledge specialized content knowledge;knowledge of content and students; and knowledge of content and teaching allof which are found lacking in novice and pre-service teachers knowledge system(Borko and Livingston, 1989). Prior to a genuine classroom teaching experience, use of learning study allows pre-service teachers to be engaged in teaching activities such as

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analyzing and assessing student work; and to create lesson plans that meet bothcurriculum requirements and students needs in a relatively safe environment free from judgment by a supervising teacher or a classroom of students (Koirala et al., 2008).

2. Mathematical content knowledge and pedagogical content knowledgefor mathematics and its structureNo one would argue the fact that teachers mathematical content knowledge andpedagogical knowledge are crucial to their capacity for providing high-qualityteaching (Lowery, 2002; Ball et al., 2005; Cheang et al., 2007; Silverman and Thompson,2008; Ball et al., 2008). However, the actual nature of innovative preparation programsfor enhancing pre-service teachers mathematics and pedagogical knowledge and theextent of that knowledge as applied to their classroom teaching are still unknown.Particularly striking is the lack of understanding of the dimensions of intersection andconnection between mathematics content knowledge and pedagogical knowledge foreffective teaching.

Shulman (1986) proposed a special domain of teacher knowledge, pedagogicalcontent knowledge, which bridges subject matter knowledge, knowledge of pedagogyand classroom teaching. Shulman (1986) defined pedagogical content knowledge asan understanding of what makes the learning of specific topics easy or difficult: theconceptions and preconceptions that students of different ages and backgrounds bringwith them to the learning of those most frequently taught topics and lessons (p. 9).In other words, pedagogical content knowledge includes familiarity with topicschildren find interesting or difficult, the representations most useful for teaching anidea, and learners typical errors and misconceptions (Hill et al., 2004, p. 12). ThoughShulmans work could provide a conceptual orientation and a set of analyticdistinctions on the nature and types of knowledge needed for teaching a subject (Ballet al., 2008), the question about what exact professional knowledge of mathematics forteaching, tailored to the work teachers do with curriculum materials, instruction, andstudents (Ball et al., 2005, p. 16) is still unsolved. Ball et al. (2008) extend Shulmansnotion of subject content knowledge and pedagogical content knowledge to pursue abetter understanding of his idea on the relationship between them. The analysis of themathematical demands of teaching (Ball et al., 2008) further divided Shulmans contentknowledge into common content knowledge and specialized content knowledge;and his pedagogical content knowledge into knowledge of content and students andknowledge of content and teaching. We explain these concepts below.

2.1 Common content knowledgeBall et al. (2008) define this as the mathematical knowledge and skill used in settingsother than teaching. It is the common mathematical knowledge that is not unique toteachers but is used by teachers to recognize and judge wrong answers and definitions.By common, Ball et al. do not mean that everyone should have this knowledge.Rather, they see it as unspecialized understandings. For example, a teacher, a nurseand an electrical engineer should all know what decimals are between 1.1 and 1.2. Theyare also able to tell 0.5 is a half; and to do addition of fractions, multiplication of decimals etcetera.

2.2 Specialized content knowledgeThis is the mathematical knowledge and skill unique to teaching, such as a conceptualunderstanding of mathematics (Ball et al., 2008). Conceptual understanding of

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mathematics consists of logical relationships constructed internally and existing in themind as a part of a network of ideas (Van De Walle, 2004). It reflects an understandingof why a procedure works (Hiebert and Wearne, 1986) or of whether a procedure islegitimate (Bisanz and Lefevre, 1992). Scholars such as Ball and Bass (2003), Hill et al.

(2004), Ball et al. (2005) and Hill et al. (2005) provide a further definition of conceptualunderstanding by pointing out the demand of interplay between mathematicalprocedures and the specific mathematical concepts for interacting productively withstudents. Thus, specialized content knowledge involves the knowledge of decompressedmathematical knowledge that can make features of particular content visible to, andlearnable by, students (Ball et al., 2008). For example, teachers are required to understandthe difference between measurement and partitive models of division or themathematical development of informal units to formal units in length. Another example,most people are able to tell others to add a zero to the number when you multiply byten and move the decimal point one digit to the left when you divide by ten. However,teachers are required to hold unpacked mathematical knowledge which goes beyond thekind of tacit understanding of place value needed by most people (Ball et al., 2008). Thus,specialized content knowledge refers to the knowledge that requires conceptual demandsof mathematics different from the mathematical understandings needed by otherpractitioners of mathematics (Silverman and Thompson, 2008).

2.3 Knowledge of content and studentsThis is the combination of knowing about students and knowing about mathematics.Teachers should know the aspects of mathematics which are sensitive to studentsunderstanding and which must be made visible to students during classroominstructions (Silverman and Thompson, 2008). In order to provide visible mathematicsinstruction, teachers should be able to anticipate their students thinking,understanding, confusions and misconceptions (Ball et al., 2008). Developed fromthis idea, knowledge of content and students includes anticipating students responsesand motivation for learning, and interpreting students emerging and incompletethinking (Ball et al., 2008). In short, it is the knowledge of common student conceptionsand misconceptions about particular mathematical content (Ball et al., 2008). Using atask in which a child is asked to order decimal numbers is a good example to illustratethis point. Knowing that students often choose the decimal numbers with more decimalplaces as the larger one (Resnick et al., 1989) means that a teacher can anticipate this asa common student response. Thus, knowledge of content and students requires theteachers cognitive interaction across specific mathematical content understanding,familiarity with students and students mathematical thinking (Ball et al., 2008).

2.4 Knowledge of content and teaching This combines knowing about teaching and knowing about mathematics. Ball andForzani (2009) point out that helping students to learn mathematics content knowledgeand skills not only requires teachers strong content knowledge but also their capacityto make mathematics accessible to students. Knowledge of content and teaching refersto the knowledge that requires teachers to decide how to use time in each lesson,determine the key learning points (i.e. the object of learning (OL)), choose appropriateexamples, models and materials for instructional purposes, sequence the learningactivities that fit students learning path, ask appropriate questions in an appropriateorder for scaffolding learning; and choose appropriate precise or unambiguouslanguage that is pedagogically preferable for constructing concepts. Careful advance

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thought about such factors can make mathematics more sensible to, visible to andlearnable by students. Knowledge for teaching should be shaped as structures of mathematics as it is learned and not only in its finished logical form (Ball et al., 2005).Accordingly, teachers can structure the next step in the students development; and

oversee and assess the learners progress (Ball and Forzani, 2009).

3. Conceptual framework of learning studyIn this paper, the framework of learning study is suggested as a means for empoweringpre-service teachers mathematical content knowledge and pedagogical contentknowledge. This framework was developed by a Hong Kong research team (Lo et al.,2005). It is in part related to the systemic efforts of Japanese teachers in conducting in-depth research into their own lessons through lesson study (Lo and Ko, 2002; Pangand Marton, 2003; Lo et al., 2004) and also bears some relation to teaching study inmainland China. However, the Hong Kong model, learning study is theoreticallygrounded in variation theory (Marton and Booth, 1997; Pang and Marton, 2003; Martonand Tsui, 2004), which was originally developed at the University of Gothenburg,Sweden by Ference Marton, as part of what he calls phenomenography. Thistheoretical framework has the following characteristics:

. it focusses on the OL;

. it is premised on three types of variations. (i.e. variation in students priorunderstandings (V1); variation in teachers conceptualization of the subject topic(V2); and using variation theory as a pedagogical tool (V3); and

. the patterns of variation (Lo and Marton, 2012) facilitate the discernment of thecritical features.

Learning study is a staged, researchful process that involves (pre-service) teachers asactive researchers into their students learning. In what follows, we explain the model

in detail before turning to our discussion of its use in pre-service teacher education anda case studyof incorporating learning study in teacher education programs in Hong Kong.

3.1 What is OL? Central to variation theory is the philosophical notion of intentionality (Brentano,1874). According to Brentanos principle of intentionality, one cannot think withoutthinking about something; one cannot be conscious without being conscious of something. Thus to say I learnt without identifying what has been learnt ismeaningless. Learning must have an object and in the context of learning study, this iscalled the OL. When learning occurs, it will therefore always have an object. For anygiven lesson, on any given topic, it should therefore be possible to quite closely definethe intended OL. Lo and Marton (2012) point out that to see or experience an OL in acertain way requires the learner to be aware of certain features which are critical to theintended way of seeing the object. With any given topic, there will be a number of critical features that need to be identified, if the OL is to be discerned. Lo and Marton(2012) comment that when students do not learn, it may not be due to a lack of abilitybut due to a focus on aspects other than the critical aspects.

3.2 What are the three types of variations? 3.2.1 Variation in students prior understandings what is to be taught (V1) . It has longbeen recognized that students prior conceptions and misconceptions can have a major

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impact on what they learn, especially in mathematics (Nesher, 1987; Clement et al.,1989; Smith et al., 1994). As prior understanding is so influential on learning, teachersneed to be fully alert to this as a phenomenon and take it into account when planning,conducting and analyzing lessons. Students prior knowledge is captured within the

process of learning study in the concept of variation (V1), where there are three mainways of identifying students prior conceptions: student interviews, pre-test andconstant observation during the lesson. Any topic can have different objects of learning, depending on particular students prior understandings from which aprecise focus can be chosen. Identifying prior understandings also has a major bearingon the process of identifying the critical features for any given lesson.

Students vary in their prior understandings and it is the aim of the learning studymodel to identify these and act on them in planning and teaching a given lesson. Inthe process of planning a lesson, students are interviewed. The interviews take twodifferent forms. Teachers or collaborating researchers interview students who havelearnt this topic in the previous year, to find out what they retained and alsomisconceptions that they appear to retain. Teachers or collaborating researchers alsoconduct an interview with a random selection of students from classes which will betaught the topic, to try to find out some of the prior understandings that they will bringto their learning. As this only represents a small sample it cannot be definitive, andteachers therefore need to be continually alert when teaching the lesson, as otherconceptions and misconceptions may emerge. The results of these two forms of interview feed into discussions about the precise OL, the critical features that need tobe addressed in the lesson and ultimately the overall sequence of the lesson content, sothat the process of learning is progressive.

The pre-test is also part of the overall process of gathering information aboutstudents prior understanding. The group interviews provide a sample guide tounderstand students thinking and they assist in the design of the pre-test. As thepre-test is given to all students participating in the learning study, it is able to revealconceptions and misconceptions at a more individual level, to aid information gatheredfrom the group student interviews. The pre-test is diagnostic in nature, so that it opensup space for students to freely express themselves and their own ideas.

3.2.2 Variation in teachers conceptualization of the subject topic and in their ways of dealing with the particular OL (V2) . There is often a major problem with teachersconceptualization of the subject topic in mathematics as some teachers do not have astrong mathematics background. Usually, it is not appropriate to give mathematicsteachers a pre-test, as is done with students. However, if students are to gain in theirlearning, it is imperative that they are taught properly and accurately. The process of identifying teachers misconceptions and reconstructing teachers conceptualunderstanding of the topics being taught has to be handled with subtlety within thelearning study process. This is accomplished over the many discussions that form partof the overall process of a learning study, many of which occur before the practicalteaching cycles begin. They may arise, for instance, in the context of discussingstudents misconceptions. Other occasions when teachers are able to examine their ownconceptual understandings include the marking and analysis of the pre- and post-test,engaging in the lesson observation and the post-lesson analysis. For example, readingthrough and analyzing the results of the pre- and post-tests involves an attempt to tryto get into the minds of the students, to work out what they are actually thinking whenproviding their answers. What students say in their answers may not reflect what theymean and teachers therefore have to act as detectives. In doing so, teachers own

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understandings of the topic being discussed are up for scrutiny. Through engaging inthis process of analysis, teachers are thereby gaining practice in reflective professionaldevelopment. Open discussions between teachers are encouraged in planning meetingsand in the post-lesson conferences. Through in-depth discussion, teachers are able to

shape and/or re-shape their own understandings and re-conceptualize the relatedmathematics knowledge in the process of interactive dialogue. The topic of theconversation is invariably focussed on the OL, its critical features and the overalllesson plan. In the process, teachers not only advance their personal understandingof the topic. Through listening to each other, they also apply their ongoingre-conceptualizations to the lesson plan.

A second form of teachers variation (V2) relates to their previous experience of teaching the same or a similar topic. This is brought into the discussions in recognitionof the fact that teachers vary considerably in the way they teach a lesson, even whenstaying close to the textbook. The tacit, personal, practical knowledge (Elbaz, 1983;Connelly and Clandinin, 1995) that teachers have developed about their students, theirpedagogical content knowledge (Shulman, 1986), their understanding of the content,etcetera, are valued, sharpened and made explicit when the teachers and researchersshare their ideas in meetings before the research lesson, when observing team membersteaching and when engaging in professional dialogue in post-lesson conferences.

3.2.3 Using variation as a pedagogical tool (V3) . In addition to variations instudents and teachers conceptions, and in the different approaches to pedagogy thatteachers have previously employed, theory of variation is used as a pedagogical tool inlearning study. As noted previously, in the framework of learning study, considerableemphasis is placed on inducing discernment of the OL through exposing students tovariations in the way they experience phenomena. According to the theory of variation,a key feature of learning involves experiencing a phenomenon in a new light (Marton,1999). In other words, learning amounts to being able to discern certain aspects of thephenomenon that one previously did not focus on or which one took for granted,and simultaneously bring them into ones focal awareness (Lo et al., 2006, p. 3).Variation theory is posited on the view that when certain aspects of a phenomenonvary while its other aspects are kept constant, those aspects that vary are discerned(Lo et al., 2006, p. 3). This means that in order for students to discern the OL they needwto experience how the critical features of the OL are varied and not varied.Accordingly, certain invariances characterize certain ways of experiencing aphenomenon, and bring about particular ways of experiencing a particularphenomenon (Marton and Booth, 1997). This is how different students learningoutcomes came about (Marton and Morris, 2002; Marton and Tsui, 2004).

In planning a lesson, it therefore becomes important to use variation in designingteaching episodes and classroom activities. A crucial point in the use of variation isthat it should be controlled and systematic in every case. What is varied and whatremains invariant is intended to have direct impact on the students discernment of theOL. Therefore, it is necessary to pay close attention to what varies and what isinvariant in a learning situation, in order to understand what it is possible to learn inthat situation and what it is not (Lo and Marton, 2012). When using different patternsof variation in classroom teaching, Marton cautioned about the haphazard wayin which teachers vary too many things at one time (Lo and Marton, 2012). Bysystematically keeping some things invariant while others are varied and thenchanging what is varied and what remains invariant, students are able to see(to directly perceive or intuit) the OL.

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Because of the limitation of space in this paper, the functions of four patterns of variation will not be explained in detail. To guide teachers in the possible ways of using of variation, the patterns of variation are referred to Lo and Martons (2012)study.

3.3 Steps in a learning study cycleUsing this framework, a learning study group usually comprises two or more pre-service teachers (i.e. researchers) teaching the same subject at the same level in thesame school. These comprise the research team, where each member contributes his/her own expertise, and each member is accorded equal status. Usually each week, astimetabling permits, the group meets for about an hour to work on the research lesson.The whole cycle takes about 10-12 weeks/meetings. Figure 1 is a flow chart of procedures involved in conducting a typical learning study.

A learning study goes through a number of steps, though these are not necessarilyin a fixed sequence. Some steps may occur simultaneously and there may be iterationcycles when certain steps are revisited. Nevertheless, the diagram above provides asummary of the processes usually adopted. This is the cycle that is usually employed inthe context of pre-service teacher education in Hong Kong.

The following section presents a case study of incorporating learning study inteacher education programs in Hong Kong. It aims at sharing the success of theprogram and to disseminate how pre-service teachers enhanced their mathematicalcontent knowledge and pedagogical content knowledge by practising learning study.

The use of three types of variation in a learning study cycle

Identifyingobject of

learning, OL(V2)

Selection oftopic(V2)

Lessonplanning(V2, V3)

Lessonimplementationand observation

(V1, V2, V3)

Finalevaluation

(V2)

Report anddissemination

(V2)

Another cycle oflearning study

Cycles ofteaching

Pre-test,student

interview(V1)

Post-test,student

interview(V1)

Finalizing OLand critical

aspects/ features

Source: Wong and Lo (2008, p. 21)

Figure 1Flow chart of learnin

study procedur

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4. A case study of incorporating learning study in teacher educationprograms in Hong Kong4.1 Methods and findings4.1.1 Background . Learning Study was first introduced in Hong Kong in 1999

(Lo, 2009). In Lo and Martons (2012) study, it is reported that over 300 learning studieshave been developed through various projects of the Hong Kong Institute of Education, and afterwards many schools developed learning studies on their own.As many studies such as Lo et al. (2005), Elliott and Yu (2008) and Cheng (2009)consistently found learning study contributes to teachers professional developmentand the learning of researchers. Since 2008, the Hong Kong Institute of Educationhas incorporated learning study as a core subject in the second year of allprimary school teacher education programs. The authors of this study weremembers of a team teaching learning study in mathematics in the Institute at the timeof the study.

4.1.2 The research process . The learning study subject was delivered in two stagesin a semester. In the first stage, the framework of learning study and theory of variation were taught and discussed thoroughly in one three-hour lecture each weekfor six weeks. The theory of OL and the theoretical groundings to understandingsome of the necessary conditions for learning (Lo and Marton, 2012) were alsoexamined in the lecture. In the second stage, pre-service teachers worked in groups of four on implementing two cycles of learning study in primary schools. With thelecturers guidance, pre-service teachers first identified the OL and the critical featuresof a given topic. Second, according to the identified OL and critical features, theydesigned a pre-test and administered it to two classes of primary students. They theninterviewed some of the students in order to better understand students priorknowledge of the topic being taught. Third, based on their findings and grounded inthe theory of variation, they planned a lesson in which sequenced classroom activitiesthat could address the identified OL and critical features of the given topic were thefocus. Finally, they gave a lesson to a class of primary students in the first cycle. Afterthe lesson, there was one post-lesson meeting. In the meeting, pre-service teachersevaluated their teaching and refined the teaching where they considered it necessaryfor the second cycle. They then gave the refined lesson to another class of students.The subject matter topic was Introduction to Fractions. We choose this topic as thepre-service teachers believed that the topic was well suited for them to practise theircontent knowledge and pedagogical content knowledge because fractions is anabstract topic which demands adequate mathematical knowledge from teachers andmany pupils experience conceptual difficulties related to fractions such as mixing up of concepts of a group of objects and a continuous object; and mixing up of concepts of whole number and decimals. Pupils of Grade 3 aged eight were invited to participate inthe research lessons. To evaluate the pre-service teachers teaching and to reflect ontheir own learning, pre-service teachers were required to disseminate their results in aPowerPoint presentation and to write a three-page learning journal to summarize theirlearning processes. In this paper, our purpose of presenting the case is, throughanalyzing pre-service teachers learning journals, to share the success of this programand to disseminate how pre-service teachers enhanced their mathematical contentknowledge and pedagogical content knowledge by practising learning study.

4.1.3 Participants . About 60 pre-service teachers who enrolled in the primary schoolteacher education programs and who took primary mathematics as their major teachertraining subject enrolled in learning study in mathematics in 2009. The pre-service

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teachers were well informed that their PowerPoint presentations and learning journalswould be used for research purposes at the beginning of the semester. A total of 32 pre-service teachers signed and returned their consent forms for allowing theauthors to use their work for this study. The PowerPoints and learning journals of

those pre-service teachers who indicated that they did not want to participate in thisstudy were either returned or destroyed. In this study, 32 pre-service teachers learning journals were analyzed and reported.

4.2 Findings the learning journal Demonstrating how the learning study framework is grounded on the theory of variation provides pre-service teachers with a platform for understanding,assimilating and practising the different components of mathematical contentknowledge and pedagogical content knowledge specialized content knowledge;knowledge of content and students; and knowledge of content and teaching. Theanalysis of data and reporting of findings set out below is linked tightly to the threecategories of knowledge. The argument is constructed from the perspective of how pre-

service teachers changed in belief in teaching and learning; and how they acquiredthese three categories of knowledge.4.2.1 Specialized content knowledge. 4.2.1.1 Pre-service teachers awareness of

including conceptual understanding in teaching. One of the most frequently occurringreflections from pre-service teachers was that before doing the learning study subject,they perceived mathematics teaching as teaching for numeracy and proficiency, eachof which are procedural understandings. Further, they did not know what conceptualunderstanding and mathematical reasoning were required in teaching mathematics.After practising learning study, they became aware of their na ve understanding of specialized content knowledge especially the conceptual understanding of mathematics and came to recognize its impacts on students learning. They alsocame to realize that high-quality teaching requires teaching both procedural and

conceptual understandings. One pre-service teacher commented:Mathematics always includes different conceptual topics. If students cannot fully grasp theconcepts, they will find it very difficult in keeping up with progress. So mathematics teachershave responsibility to teach students the correct mathematical concepts with conceptualunderstanding and should not only focus on proficiency in algorithms. In the two cycles of teaching, we observed that the majority of pupils followed closely their teachers teachingon problem solving methods and calculations. If a teachers own subject knowledge isincompetent, it will directly affect students learning. So being a pre-service teacher, I will payextra effort on enhancing my mathematics content knowledge especially those related toconceptual understandings.

4.2.1.2 Pre-service teachers understanding of specialized content knowledge. Throughthe lectures and two cycles of teaching, the pre-service teachers acknowledged thatthey had had the opportunity to simulate and practice the process of identifyingthe OL and its critical features of a given topic. They claimed that they came to havebetter understanding of specialized content knowledge in mathematics. For example,specialized content knowledge of fractions requires understanding differentinterpretations of a/b for a group of objects and a continuous object two pre-service teachers noted:

I learnt that, before planning a lesson, we should divide a topic into numbers of object of learning, for example, the introduction of fractions involves two sub-concepts and they arepart-whole of a group of objects and part-whole of a continuous object. So, the outcome of a/b

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of a group of objects is not the same as that of a/b of a continuous object. Thus thebreakdown of a topic into small objects of learning allows teachers to identify easilythe critical features of the given concept and in turn make the knowledge morecomprehensible to and learnable by students.

We believe that fractions are one of the most difficult topics in primary school mathematicscurriculum. It is because the meaning of fractions and the size of fractions, all of whichinclude the part-whole concept, are not easy to be understood by Grade 3 students. Forexample, what do the numbers of numerators and denominators represent, how toknow a whole is divided equally, why a proper fraction has to be less than one and so onare very abstract concepts and all these are what our pre-service teachers need to learnand practice.

4.2.2 Knowledge of content and students. 4.2.2.1 Intersecting knowledge of content andknowledge about pupils. Another frequently occurring reflection from pre-serviceteachers was that they had not previously realized that they needed to anticipatepupils misconceptions about particular mathematical content in designing theirlessons. Because of the hierarchical structure of mathematics, they had previouslyperceived that mathematics teaching needed to closely follow this hierarchicalstructure. That was, they had not recognized that mathematics teaching requiredknowledge at the intersection of content and students (Ball et al., 2008). Through doingthe learning study subject, they came to understand that knowledge of pupils andcontent is an amalgam, involving a particular mathematical idea or procedureand familiarity with what pupils often think or do (Ball et al., 2008, p. 401). Three pre-service teachers made the following notes:

Before doing the Learning Study subject, I perceived that teaching was to transmit pieces of knowledge to students. I was meant to plan the sequence of teachings in the way that theknowledge was structured. After the two cycles of teaching, I realized that teaching is toenable students to learn, to help students to cope with and overcome their learningdifficulties, To achieve this goal, teaching should be structured in the way that it meetspupils learning needs and ways of thinking.

The planning of a lesson is to make an appropriate connection between students priorknowledge and the learning content. The assessment of students prior knowledge is tomeasure, firstly, how far they had already mastered the content knowledge, secondly howmuch they do not know, and thirdly what misconceptions have already influenced theirlearning. All these kinds of information give clear direction to the design of teaching. So,teaching is not only a matter of how to shape a lesson to match the structure of knowledgebut also a matter of how to make the best use of students prior knowledge in order to meettheir learning needs.

4.2.2.2 Addressing students difficulties and misconceptions in teaching. The pre-service teachers acknowledged that by analyzing students pre-test, they learnt what

understanding and misconceptions students held. They noted that this knowledgeincreased their ability to interpret students mathematical ideas and to examinestudents work. They also commented that it enhanced their ability to addressstudents misconceptions and to build on students understanding in their instructionaldecisions and actions. Furthermore, they said that they became aware of their use of mathematical language in teaching. The following are the typical comments in theirlearning journals:

The results of pre-test showed that in general, students had a clear concept of equipartitioning a group of objects. But their understanding of equally dividing a

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continuous object was very vague. They did not know what factors, other than the identicalshape of each of the divided parts, also determined that the object was divided equally.Therefore, we set equally partitioning a continuous object as the object of learning for thefirst teaching cycle. In the lesson, activities which included judging intuitively an object being

divided equally and overlapping divided parts for judging an object being divided equally,were the starting points for classroom discussion. To develop the concept of equipartitioing acontinuous object, students were required to divide a rectangle in three different ways.Through doing so, students learnt that identical in shape of divided parts was not the onlydeterminant of an object being divided equally.

We had great difficulties in using proper mathematical language, which students of Grade 3could understand, to write up questions of pre-test. We also did not know how to use childrenlanguage to give instructions during teaching. Through the two cycles of teaching and withlecturers guidance, we had chance to practise and finally came to realize that simple sentencewith least adjectives and conjunctives was easier to be understood by young children.

4.2.3 Knowledge of content and teaching . The pre-service teachers experience duringthe two cycles of teaching contributed to their understanding that there was more toteaching than what they perceived and imagined previously. After their teaching, theyreported that they understood better what mathematics pedagogical contentknowledge means and what components it includes. The following notedemonstrates one students growing awareness:

Previously, I thought that pedagogy was just knowledge of how subject content knowledgewas transmitted and how to control classroom discipline. After the teaching, I becamerealize that teaching is a complex body of knowledge which includes identifying object of learning, choosing appropriate examples and manipulatives, sequencing questions and usingproper language. All these components have positive impacts on pupils learning if they areused appropriately.

The pre-service teachers were previously unaware of how manipulatives, examples

and models were operated in effective classroom practice until they had theopportunities to practice in the two cycles of teaching. Two pre-service teachers noted:

In the past, I did not value the benefit of using manipulatives in teaching and even did notbother to learn how to make use of them in my teaching. But whenwe did the actual teaching,I found that students learnt effectively when manipulating concrete objects. Especially whenthere was a situation where we needed to explain an abstract concept but did not know theunambiguous terms which the Grade 3 students could understand; concrete objects playeda significant role at this point.

As most of the mathematical concepts are very abstract to primary students, how to chooseappropriate examples to illustrate the concepts is important in teaching. In the past, I did notrealise the benefit of using daily life examples and models. I just followed what the textbookpresented. But now, I not only recognise the advantages of using daily examples and models,I also value the counter examples. By comparing example and its counter example, studentscan identify easily the critical features of the concept being taught.

The pre-service teachers also noted that after the two cycles of teaching, they came torecognize a critical importance of pedagogy is being able to formulate a sequence of questions:

No one would argue that questioning technique is crucial to effective teaching. The LearningStudy subject provided us opportunities to practise and learn from each other. Now, we aremore aware of how we sequence our questions and how to formulate questions.

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Each of the 32 pre-service teachers noted that the learning study subject fostered theirunderstanding of the relationship between theory and practice and theirunderstanding of transforming knowledge into action.

5. Discussion practising mathematical content knowledge andpedagogical content knowledge by using learning studyAs noted previously, traditional teacher education programs are criticized for thefailure to prepare pre-service teachers for the realities of the classroom (Eilam andPoyas, 2009; Goodlad, 1990; Korthagen and Kessels, 1999; Korthagen et al., 2006)and the meager transfer from theory presented during teacher education to practice inschools (Wideen et al., 1998). In this paper, we argue that one of the reasons forproposing to incorporate learning study as an integrated subject of mathematicspedagogy and teaching practice in teacher education programs is its authenticity(Walker, 2007), a genuine teacher-driven way of learning to teach. Walker (2007) statesthat the conceptualization of learning study is to allow teachers intellectual space totake charge of their own professional development by dealing with authentic learningproblems more relevantly (p. 103). We believe that the feedback and comments of the32 pre-service teachers, as reported in the previous section, have manifested this point.In addition to its authenticity as a teaching-learning situation, we speculate thatlearning study can provide a platform for pre-service teachers to develop and rehearsethe mathematical content knowledge and pedagogical content knowledge, and inparticular, the specialized content knowledge of their MCK. They also practice usingthe knowledge of content and students and knowledge of content and teaching of theirPCK. Thus, we believe that learning study can link theoretical insights about teachingand learning with practice in teacher education. In this way, learning study in teachereducation becomes a means by which novice teachers can acquire the experience theyneed to be able to continue to learn as beginners in the classroom (Reid, 2011).We conclude this paper with a descriptive attempt to associate the three forms of variation in the model with specialized content knowledge, knowledge of content andstudents and knowledge of content and teaching.

Following Ball and Forzani (2009), we argue that teaching requires teachers specialattention to what they are helping students to learn, and it requires teachers to seethe content from their students perspectives. Such knowledge affects teachers abilityto assess their own students, which in turn affect directly how and what teacherswill do in classrooms. Hence, an appropriate knowledge of content and students willallow teachers to better match classroom instruction to facilitate students learning.The first type of variation in the framework of learning study, variation in studentsprior understandings (V1), focussing on unfolding students prior conceptions andmisconceptions allows pre-service teachers to investigate, analyze and understandstudents thinking in a specific content domain. This paper hypothesizes that byenacting the V1, pre-service teachers on the one hand can develop, apply and rehearsetheir knowledge of content and students, and on the other hand focus their attention onthe learning process of students, instead of on the issue of maintaining classroom order(Vedder and Bannink, 1987). As a result, pre-service teachers learn not so much bybeing taught by their teacher educators, but by structured reflection on theirexperiences and discussions with peers (Korthagen et al., 2006) about their studentsand student learning. They then become aware of their own learning processes andbegin to create their own professional knowledge of content and students that feedsback into that knowledge.

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As early as 1985, Leinhardt and Smith highlighted the importance of teachersconceptual understandings of mathematics in teaching. Thompson et al. (2007) alsonote that teaching for understanding requires teachers to have strong conceptualunderstanding:

If a teachers conceptual structures comprise disconnected facts and procedures theirinstruction is likely to focus on disconnected facts and procedures. In contrast, if a teachersconceptual structures comprise a web of mathematical ideas and compatible ways of thinking, it will at least be possible that she attempts to develop these same conceptualstructures in her students. We believe that it is mathematical understandings of the lattertype that serve as a necessary condition for teachers to teach for students high-qualityunderstanding (pp. 416-7).

The second variation, variation in teachers conceptualization of the subject topic andin their ways of dealing with the particular OL (V2), provides pre-service teachers witha channel to clarify what they understand conceptually and what they do not, whenthey are engaging in in-depth discussion on a mathematics topic. They can consolidateideas that are conceptually correct and re-construct those that are not, for carrying outthe tasks of teaching. As a result, pre-service teachers can use appropriate pictures ordiagrams to represent mathematics concepts and procedures to students, providestudents with explanations for common rules and mathematical procedures, andanalyze students solutions and explanations (Hill et al., 2005). Thus, we propose thatby enacting V2, pre-service teachers can enhance their specific content knowledge of mathematics.

Although high-quality teaching requires teachers deep knowledge of subjectmatter, content knowledge alone does not suffice for good teaching (Kahan et al., 2003).To ensure effective teaching performance, what matters most is teachers pedagogicalcontent knowledge, in particular, knowledge of content and teaching. Korthagen et al.(2006) stated that to fully illuminate the dynamics of a teaching situation, pre-service

teachers need opportunities to understand what is involved in planning the teaching,doing the teaching, and reflecting on the teaching (p. 1029). By enacting the V3, usingvariation as a pedagogical tool, pre-service teachers can genuinely engage in a learningenvironment where such lesson planning and instructional design for discerning theOL and its critical features are the focus, rather than in an environment wherecontrolling students is their dominant concern. Pre-service teachers have opportunitiesto access the thought and actions of colleagues in ways that help to illuminatethe reasons for particular teaching actions (Korthagen et al., 2006). It means thatpre-service teachers not only practise skills of teaching such as questioning, wait-time,listening, structuring content and timing, but also learn to provide pedagogical reasonsfor certain teaching actions. By doing so, pre-service teachers gain insights into howthey may come to better understand the teaching-learning situation and how to act on

it with regard to their OL. The V3 allows pre-service teachers to develop a moreprocess-oriented view of knowledge of teaching (Korthagen et al., 2006), and onto construct their own recipes for how to teach. We believe that by enacting V3,pre-service teachers can create their own professional knowledge of content andteaching.

6. ConclusionThis paper attempted to associate the three forms of variation (V1, V2 and V3) in themodel of learning study with specialized content knowledge, knowledge of content and

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students and knowledge of content and teaching. We strongly believe that the casestudy presented has already provided solid evidence to support our proposition:incorporating the framework of learning study in the mathematics teacher educationprogram can enhance pre-service teachers mathematical content knowledge and

pedagogical content knowledge. We speculate that practising learning study is oneway to reconcile the dilemma of the theory-practice nexus and is to make an explicitintegration of theory and practice. To conclude, the learning of student teachers isonly meaningful and powerful when it is embedded in the experience of learning toteach (Korthagen et al., 2006, p. 1030). We speculate that learning study can create thissituation for pre-service teachers.

To further investigate how broad and deep the pre-service teachers have enhancedtheir specialized content knowledge, knowledge of content and students and knowledgeof content and teaching through practising learning study, some other quantitative andpsychometric measures such as survey, written test and oral test, etc., may be required.

ReferencesBall, D.L. and Bass, H. (2003), Making mathematics reasonable in school, in Martin, G. (Ed.),

Research Compendium for the Principles and Standards for School Mathematics , NationalCouncil of Teachers of Mathematics, Reston, VA, pp. 27-44.

Ball, D.L. and Forzani, F.M. (2009), The work of teaching and the challenge for teachereducation, Journal of Teacher Education , Vol. 60 No. 5, pp. 497-511.

Ball, D.L., Hill, H.C. and Bass, H. (2005), Knowing mathematics for teaching, American Educator , Vol. 29 No. 3, pp. 14-46.

Ball, D.L., Thames, M.H. and Phelps, G. (2008), Content knowledge for teaching: what makes itspecial?, Journal of Teacher Education , Vol. 59 No. 5, pp. 389-407.

Bisanz, J. and LeFevre, J. (1992), Understanding elementary mathematics, in Campbell, J.I.D.(Ed.), The Nature and Origins of Mathematical Skills , North Holland, Amsterdam,

pp. 113-36.Borko, H. and Livingston, C. (1989), Cognition and improvisation: differences in mathematics

instruction by expert and novice teachers, American Educational Research Journal ,Vol. 26 No. 4, pp. 473-98.

Brentano, F. (1874), Psychology from an Empirical Standpoint , Routledge and Kegan Paul,London.

Bromme, R. and Tillema, H. (1995), Fusing experience and theory: the structure of professionalknowledge, Learning and Instruction , Vol. 5 No. 4, pp. 261-7.

Bullough, R.V. Jr and Kauchak, D. (1997), Partnership between higher education and secondaryschools: some problems, Journal of Education for Teaching , Vol. 23 No. 3, pp. 215-33.

Cheang, W.K., Yeo, K.K.J., Chan, C.M.E., Lim-Teo, S.K., Chua, K.G. and Ng, L.E. (2007),Development of mathematics pedagogical content knowledge in student teachers, The Mathematics Educator , Vol. 10 No. 2, pp. 27-54.

Cheng, C.K. (2009), Cultivating communities of practice via learning study for enhancingteacher learning, KEDI Journal of Education Policy , Vol. 6 No. 1, pp. 81-104.

Clement, J., Brown, D.E. and Zietsman, A. (1989), Not all preconceptions are misconceptions:finding anchoring conceptions for grounding instruction on students intuitions, paperpresented at the Annual Meeting of the American Educational Research Association,San Francisco, CA, March.

Connelly, F.M. and Clandinin, D.J. (1995), Teachers Professional Knowledge Landscapes ,Teachers College Press, New York, NY.

86

IJLLS2,1

8/13/2019 17073398

17/19

Darling-Hammond, L. (1994), Professional Development Schools: Schools for Developing a Profession , Teachers College Press, New York, NY.

Eilam, B. and Poyas, Y. (2009), Learning to teach: enhancing pre-service teachers awareness of the complexity of teaching-learning processes, Teachers and Teaching: Theory and

Practice , Vol. 15 No. 1, pp. 87-107.Elbaz, F. (1983), Teacher Thinking: A Study of Practice Knowledge , Croom Helm, London and

New York, NY.

Elliott, J. and Yu, C. (2008), Learning Studies as an Educational changes Strategy in Hong Kong: An Independent Evaluation of the Variation for the Improvement of Teaching and Learning (VITAL) Project , Centre for School Experience and Partnership, Hong KongInstitute of Education, Hong Kong.

Ethel, R.G. and McMeniman, M.M. (2000), Unlocking the knowledge in action of an expertpractitioner, Journal of Teacher Education , Vol. 51 No. 2, pp. 87-101.

Goodlad, J.I. (1990), Places Where Teachers are Taught , Jossey Bass, San Francisco, CA.Hiebert, J. and Wearne, J. (1986), Procedures over concepts: the acquisition of decimal number

knowledge, in Heibert, J. (Ed.), Conceptual and Procedural Knowledge: The case of Mathematics , L. Erlbaum Associates, Hillsdale, NJ, pp. 199-223.

Hill, H.C., Rowan, B. and Ball, D.L. (2005), Effect of teachers mathematical knowledge forteaching on student achievement, American Educational Research Journal , Vol. 42 No. 2,pp. 371-406.

Hill, H.C., Schilling, S.G. and Ball, D.L. (2004), Development measures of teachers mathematicsknowledge for teaching, The Elementary School Journal , Vol. 105 No. 1, pp. 11-30.

Jaworski, B. (2006), Theory and practice in mathematics teaching development: critical inquiryas a mode of learning in teaching, Journal of Mathematics Teacher Education , Vol. 9 No. 2,pp. 187-211.

Kahan, J.A., Cooper, D.A. and Bethea, K.A. (2003), The role of mathematics teachers contentknowledge in their teaching: a framework for research applied to a study of student

teachers, Journal of Mathematics Teacher Education , Vol. 6 No. 3, pp. 223-52.Koirala, H., Davis, M. and Johnson, P. (2008), Development of a performance assessment taskand rubric to measure prospective secondary school mathematics teachers pedagogicalcontent knowledge and skills, Journal of Mathematics Teacher Education , Vol. 11 No. 2,pp. 127-38.

Korthagen, F., Loughran, J. and Russell, T. (2006), Developing fundamental principles forteacher education programs and practices, Teaching and Teacher Education , Vol. 22 No. 8,pp. 1020-41.

Korthagen, F.A.J. and Kessels, J.P.A.M. (1999), Linking theory and practice: changing thepedagogy of teacher education, Educational Researcher , Vol. 28 No. 4, pp. 4-17.

Lave, J. (1996), Teaching as learning, in practice, Mind Culture and Activity , Vol. 3 No. 3,pp. 149-64.

Leikin, R. and Zazkis, R. (2010), Teachers opportunities to learn mathematics through teaching, inLeikin, R. and Zazkis, R. (Eds), Learning through Teaching Mathematics: Development of Teachers Knowledge and Expertise in Practice , Springer, New York, NY.

Leinhardt, G. and Greeno, J.G. (1986), The cognitive skill of teaching, Journal of Educational Psychology, Vol. 78 No. 2, pp. 75-95.

Leinhardt, G. and Smith, D.A. (1985), Expertise in mathematics instruction: subject matterknowledge, Journal of Educational Psychology , Vol. 77 No. 3, pp. 247-71.

Lo, M.L. (2009), The development of the learning study approach in classroom research in HongKong, Educational Research Journal , Vol. 24 No. 1.

8

Learning studyin teacheeducatio

8/13/2019 17073398

18/19

Lo, M.L. and Ko, P.Y. (2002), The enacted object of learning, in Marton, F. and Morris, P. (Eds),What Matters? Discovering Critical Conditions of Classroom Learning , Acta UniversitatisGothoburgenis, Goteborg.

Lo, M.L. and Marton, F. (2012), Towards a science of the art of teaching: using variation theory

as a guiding principle of pedagogical design, International Journal of Lesson and Learning Studies , Vol. 1 No. 1, pp. 7-22.

Lo, M.L., Chik, P. and Pang, M.F. (2006), Patterns of variation in teaching the colour of light toPrimary 3 students, Instructional Science , Vol. 34 No. 1, pp. 1-19.

Lo, M.L., Pong, W.Y. and Chik, P.M. (2005), For Each and Everyone: Catering for Individual Differences through Learning Studies , Hong Kong University Press,Hong Kong.

Lo, M.L., Marton, F., Pang, M.F. and Pong, W.Y. (2004), Towards a pedagogy of learning, inMarton, F. and Tsui, A.B.M. (Eds), Classroom Discourse and the Space of Learning ,Lawrence Erlbaum Associates Inc., Publishers, Mahwah, NJ.

Lowery, N.V. (2002), Construction of teacher knowledge in context: preparing elementaryteacher to teach mathematics and science, School Science and Mathematics , Vol. 10 No. 2,pp. 68-83.

Marton, F. (1999), Variatio est mater Studiorum, Opening address delivered to the 8thEuropean Association for Research on Learning and Instruction Biennial Conference,Goteborg, August 24-28.

Marton, F. and Booth, S. (1997), Learning and Awareness , Lawrence Erlbaum Associates,Mahwah, NJ.

Marton, F. and Morris, P. (2002), What Matters? Discovering Critical Conditions of Classroom Learning , Acta Universitatis Gothoburgensis, Goteborg.

Marton, F. and Tsui, A.B.M. (2004), Classroom Discourse and the Space of Learning , LawrenceErlbaum, Mahwah, NJ.

Nesher, P. (1987), Towards an instructional theory: the role of students misconceptions, For the

Learning of Mathematics , Vol. 7 No. 3, pp. 33-40.Pang, M.F. and Marton, F. (2003), Beyond lesson study comparing two ways of facilitating thegrasp of economics concepts, Instructional Sciences , Vol. 31 No. 3, pp. 175-94.

Reid, J. (2011), A practice turn for teacher education?, Asia-Pacific Journal of Teacher Education , Vol. 39 No. 4, pp. 293-310.

Resnick, L.B., Nesher, P., Leonard, F., Magone, M., Omanson, S. and Peled, I. (1989), Conceptualbases of arithmetic errors: the case of decimal fractions, Journal for Research in Mathematics Education , Vol. 20 No. 1, pp. 8-27.

Russell, T. (1989), The roles of research knowledge and knowing-in-action in teachersdevelopment of professional knowledge, paper presented at the Annual Meeting of theAmerican Educational Research Association, San Francisco, CA.

Shulman, L.S. (1986), Those who understand: knowledge growth in teaching, Educational Researcher , Vol. 15 No. 2, pp. 4-14.

Shulman, L.S. (1987), Knowledge and teaching: foundations of the new reform, Harvard Educational Review , Vol. 57 No. 1, pp. 1-23.

Silverman, J. and Thompson, P.W. (2008), Toward a framework for the development of mathematical knowledge for teaching, Journal of Mathematics Teacher Education , Vol. 11No. 6, pp. 499-511.

Smith, J.P., DiSessa, A.A. and Roschelle., J. (1994), Misconceptions reconceived: a constructivistanalysis of knowledge in transition, Journal of the Learning Science , Vol. 3 No. 2,pp. 115-63.

88

IJLLS2,1

8/13/2019 17073398

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Thompson, P.W., Carlson, M.P. and Silverman, J. (2007), The design of tasks in support of teachers development of coherent mathematical meanings, Journal of MathematicsTeacher Education , Vol. 10 Nos 4-6, pp. 415-32.

Van De Walle, J.A. (2004), Elementary and Middle School Mathematics: Teaching Developmentally ,

5th ed., Allyn and Bacon, Boston.Vedder, J. and Bannink, P. (1987), The development of practical skills and reflection at the

beginning of teacher training, paper presented at the Meeting of the Association of Teacher Education in Europe, Berlin.

Walker, E. (2007), A teacher-educators role in an Asia-derived learning study, Studying Teacher Education: A Journal of Self-study of Teacher Education Practices , Vol. 3 No. 1,pp. 103-14.

Wideen, M., Mayer-Smith, J. and Moon, B. (1998), A critical analysis of the research on learningto teach: making the case for an ecological perspective on inquiry, Review of Educational Research, Vol. 68 No. 2, pp. 130-78.

Wilson, S.M., Shulman, L.S. and Richert, A.E. (1987), 150 different ways of knowing:representations of knowledge in teaching, in Calderhead, J. (Ed.), Exploring Teachers Thinking , Cassell, Philadelphia, PA.

Wong, C.Y. and Lo, M.L. (2008), The implementation of the project, in Lo, M.L., Pong, W.Y.,Kwok, W.Y. and Ko, P.Y. (Eds), Variation for the Improvement of Teaching and Learning Final Report. Centre for Learning-Study and School Partnership , The Hong Kong Instituteof Education, Hong Kong, pp. 18-28.

Zeichner, K. and Tabachnick, B.R. (1981), Are the effects of university teacher education washedout by school experiences?, Journal of Teacher Education , Vol. 32 No. 3, pp. 7-11.

Further readingLo-Fu, Y.W.P. (2007), An exploration of the use of a learning study in teaching evaporation and

condensation in a Hong Kong primary school, unpublished thesis of Doctor of Education,University of Durham, Durham.

Corresponding authorMun Yee Lai can be contacted at: mlai@csu.edu.au

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