Using lesson study to support knowledge development in initial teacher education: Insights from early number classrooms

Teaching and Teacher Education 57 (2016) 161e175

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Using lesson study to support knowledge development in initial teacher education: Insights from early number classrooms ad Hourigan Aisling M. Leavy*, Maire Mary Immaculate College, University of Limerick, South Circular Road, Limerick, Ireland

h i g h l i g h t s  Lesson Study (LS) supported reflection on teaching and enhanced teacher noticing.  Improvements in pedagogical content knowledge (PCK) were observed.  Enhanced knowledge of content and students (KCS) and content and teaching (KCT) are reported.  Knowledge development occurred simultaneously across both knowledge subdomains.

a r t i c l e i n f o

a b s t r a c t

Article history: Received 17 June 2015 Received in revised form 28 March 2016 Accepted 1 April 2016

This paper describes Lesson Study research with 25 pre-service primary teachers. We focus on pedagogical content knowledge (PCK) development as participants design, teach and reflect upon early number lessons. Engaging in Lesson Study promoted mathematics PCK development, notably in Knowledge of Content and Students (KCS) and Knowledge of Content and Teaching (KCT) subdomains. Reflecting on classroom teaching facilitated growth across both knowledge subdomains and resulted in highly integrated and robust pedagogical understandings that transferred beyond the study context. This development of early number PCK is outlined and the features of LS that make it effective in initial teacher education identified. © 2016 Elsevier Ltd. All rights reserved.

Keywords: Initial teacher education (ITE) Lesson study (LS) Early number Mathematics education Teacher noticing Reflective practice Pedagogical content knowledge (PCK) Knowledge of content and students (KCS) Knowledge of content and teaching (KCT) One-to-one correspondence Rational counting Rote counting Cardinality Subitising

‘In my previous teaching my focus was way too much on coordinating an activity. Before I did lesson study I failed to recognize how prerequisite number concepts were needed to learn a number concept. Now that I understand this, my teaching will change e for example I will revise one-to-one correspondence before getting children to rational count. And I'll ensure they fully understand rational counting before I'd dream of getting

* Corresponding author. E-mail addresses: [email protected] (A.M. Leavy), mairead.hourigan@mic. ul.ie (M. Hourigan). http://dx.doi.org/10.1016/j.tate.2016.04.002 0742-051X/© 2016 Elsevier Ltd. All rights reserved.

them to make a set of objects! Now when I think of activities I think about what number concept I want to explore and how my activity is achieving this. I think about what questions I'll ask to help children explain their thinking’ Teacher #9, individual reflection

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1. Introduction The critical role of teacher reflective practice in enhancing the quality of teaching and learning in classrooms is well documented. Teachers who have the opportunity to critically reflect on their practice are more open to new pedagogical practices, are better able to respond to teaching dilemmas and are more willing to take risks (Darling-Hammond & Bransford, 2005; Zeichner & Liston, 2006). The ability of teachers to engage in genuine reflection has, however, been contested. Reflecting on practice is a complex task requiring observation and reflective skills. Researchers caution against the overly descriptive nature of reflection that may occur and a corresponding lack of critique, analysis and evaluation (Davis, 2006). Building reflective skills can be particularly challenging in the context of pre-service teacher education. The ability of preservice teachers to engage in systematic analyses of teaching is limited by the preconceptions they bring with them on entry to initial teacher education (Hammerness, Darling-Hammond, & Bransford, 2005). These preconceptions can cloud their judgements and influence how they reason about the effectiveness of their teaching (Star & Strickland, 2008). Thus the experiences provided to pre-service teachers during initial teacher education (ITE) are critical to challenging the assumptions they bring with them on entry to ITE. It is important that ITE provides opportunities to critically reflect on prior experiences while simultaneously providing support and guidance in terms of classroom teaching practices. A common critique of ITE is that the theories espoused and the content taught is far removed from the daily practices of classroom teachers thus leading to a ‘theory to practice gap’. This theory-topractice divide is a major factor, Tabachnick and Zeichner (1999) contend, contributing to the lack of success of many ITE programmes in influencing teacher learning. This has led to calls to bridge the divide between the content of ITE courses and the realities of classroom teaching (Darling-Hammond, 2000) in addition to a growing awareness of the importance of providing meaningful field experiences. Given this emphasis on pre-service teachers engaging in critical reflective practice concomitant with the need to forge closer links between university courses and school placement in ITE, we explored how Lesson Study (LS) can act as a vehicle to promote meaningful learning and knowledge development among preservice teachers. More specifically, we are interested in the types of pedagogical content knowledge (PCK) developed by pre-service teachers as a result of engaging in LS. 2. Theoretical framework 2.1. The importance of ‘noticing’ and reflection in teaching With so much happening in the complex environment of classrooms, such as attending to learning and responding to unexpected events as they emerge, monitoring is a critical skill for the classroom teacher. There has been a proliferation in interest over the past decade in ‘teacher noticing’, with particular attention to how expert teachers monitor a classroom. Teacher noticing is not passive e it is fundamentally active and requires multiple decisions and actions about what to attend to, what to ignore, and requires interpretation of events which in turn inform and influence subsequent responses. van Es and Sherin (2002, p. 573) identify three aspects of noticing: (a) identifying what is important or noteworthy about a classroom situation;

(b) making connections between the specifics of classroom interactions and the broader principles of teaching and learning they represent; and (c) using what one knows about the context to reason about classroom events Erickson (2011) contends that a teachers' prior teaching experiences greatly influences their noticing patterns thus accounting for the differences found in the noticing patterns of novice and experienced teachers. Experienced teachers are able to attend to a broader range of relevant aspects in the classroom, can concomitantly monitor teacher and student actions and the effect of those teacher actions on student understandings. In contrast, novice teachers direct their attention towards more superficial aspects of the classroom environment such as classroom routines, have less systematic scanning patterns of students (Erickson, 2011), are more likely to connect instructional problems to management issues rather than broader instructional decisions (Gonzalez & Carter, 1996) and focus less on individual student learning and the relationship with broader learning goals (Erickson, 2011). Studies have shown that the provision of certain experiences and carefully designed supports, such as the incorporation of video technology (Sherin & Han, 2004; Sherin & van Es, 2005) and activities designed to improve observation skills (Star & Strickland, 2008), can bring about improvements in the ability of pre-service teachers to both ‘notice’ and reflect critically on salient aspects of the classroom environment. Gains have been made in the ability to articulate clear learning goals (Jansen, Bartell, & Berk, 2009), to attend to features of the classroom environment (Star, Lynch, & Perova, 2011; Star & Strickland, 2008), to use evidence to support claims relating to the effectiveness of teaching practices (Morris, 2006) and to focus on student thinking and learning (Star & Strickland, 2008; van Es & Sherin, 2002). 2.2. Lesson study as a tool to support reflection and noticing in ITE Lesson study (LS) originated in Japan as an approach to support the continuing professional development of teachers. Interest in LS has grown since its first published description in 1997 (Lewis & Tsuchida, 1998) and the subsequent reference to Lesson Study as a strategy for change and improvement in education in The Teaching Gap (Stigler & Hiebert, 1999). Lesson study is a professional development practice in which small groups of teachers collaborate to develop, discuss, teach, and systematically reflect on one lesson. The primary purpose is not the production of an exemplary or model lesson; rather, to provide an avenue and focus for discussion on effective practices that bring about improvements in learning outcomes for students. Over the past decade there have been an increasing number of international studies of LS in ITE (Murata & Pothen, 2011). A large part of this work is motivated by the belief in the importance of teachers developing an inquiry stance in teaching (Hiebert, Morris, Berke, & Jansen, 2007). LS can be used to support this inquiry stance in ITE by providing pre-service teachers with the skills to learn from teaching by looking at their own teaching (Nemser, 1983; van Es & Sherin, 2002). As part of their overview of LS in ITE contexts, Cajkler, Wood, Norton, and Pedder (2013) categorize studies as those falling within the traditional lesson study structure of ‘plan-teach-observe learning-evaluate’ (referred to as ‘formal’ LS by Myers (2012)) and variants on this traditional structure. The three studies that undertook the formal structure revealed positive outcomes ranging from improvements in participants' mathematical knowledge and pedagogical content knowledge (Leavy, 2010), the development of deeper reflective and critical observation practices (Chassels &

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Melville, 2009; Sims & Walsh, 2009) and increased understandings of curriculum and pedagogy (Leavy, 2010; Chassels & Melville, 2009). Cajkler et al. (2013) also identify LS research studies in ITE which did not engage in full LS cycles in schools and focused instead on ‘Microteaching LS’ where lessons were taught to peers. These studies also report positive outcomes for participating preservice teachers and include improvements in content knowledge (Cavey & Berenson, 2005), the development of collaborative practice and reflection (Marble, 2006) and a move from teacher-centred to more student-centred pedagogy (Fernandez, 2005). LS provides an opportunity within this study to support preservice teachers to focus on teaching and learning using the lesson as the ‘unit of analysis’. By focusing on the lesson as it is taught in a real classroom, we provide an authentic context in which to engage pre-service teachers in the exploration of and reflection upon teaching. The LS lessons were video recorded in an effort to generate a shareable visual record of the lessons which would serve as a focus for reflection (Masats & Dooly, 2011; Sherin & van Es, 2005).

2.3. Examining teaching In this study, we are interested in the types of knowledge and understandings developed by pre-service teachers as a result of engaging in LS. Recent research utilizes conceptualizations of teacher knowledge as guiding frameworks to identify and categorize the types of understandings developed by pre-service teachers while engaged in LS. We utilize the Ball, Thames, and Phelps (2008) practice-based theory of mathematical knowledge for teaching, a theory building on the work of Shulman (1986). This framework is particularly salient as the categories of teacher knowledge therein were generated from looking at teaching in action in classrooms (see Fig. 1). Subject Matter Knowledge (SMK), involving understanding the concepts and facts of the discipline, is a significant component of the knowledge needed to teach (see Fig. 1). A growing body of research has identified the development of SMK arising from specific interventions in ITE (Hannigan, Gill, & Leavy, 2013; Leavy, Hourigan, & Carroll, 2015) including engagement in lesson study (Leavy, 2010; Leavy, 2015). This paper, however, does not report on the development of SMK; rather, we turn our attention to Pedagogical Content Knowledge (PCK). We use the PCK dimension of teacher learning as an organizing framework to report on knowledge for teaching developed from the study. Pedagogical content knowledge (PCK) was suggested as a major component of teaching expertise, by Lee Shulman (1986; 1987). PCK is a type of knowledge unique to teachers and is positioned at the intersection of their content knowledge (what they know about what they teach) and pedagogical knowledge (what they know about teaching). We are interested in PCK as many researchers and

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professional organisations have emphasised the value of PCK in ITE and in continuing professional development for teachers. Due to its usefulness as a construct for teacher education, interest in PCK has permeated research in teacher education in general and in subject matter education more specifically. PCK, according to Shulman (1986) incorporates the most useful forms of representation of ideas, the most powerful analogies, illustrations, examples, explanations, and demonstrations - in a word, the ways of representing and formulating the subject that make it comprehensible to others . . . [It] also includes an understanding of what makes the learning of specific concepts easy or difficult: the conceptions and preconceptions that students of different ages and backgrounds bring with them to the learning (p. 9). Ball and her colleagues categorise PCK into three categories, namely knowledge of content and students (KCS), knowledge of content and teaching (KCT) and knowledge of content and curriculum (KCC) (Fig. 1). KCS “combines knowing about students and knowing about mathematics” (Ball et al. 2008) and includes knowledge of common student misconceptions, mathematics that is perceived as interesting or difficult, and common approaches used by children when presented with specific tasks. KCT, however, provides teachers with the understandings required to plan their teaching so that misconceptions are challenged. The knowledge required to address student misconceptions is evidenced in the use of examples that highlight misconceptions, in the sequencing of instruction to address misconceptions and in the presentation of tasks that build in complexity and at a speed that provides sufficient consolidation of understanding. There is a third knowledge subdomain related to PCK and that is knowledge of content and curriculum (KCC). This type of knowledge is of less interest to us in this study as explicit knowledge of curriculum within our national context is widely accessible through readily available reference materials. However, there is a possible component of this knowledge that interests us e this relates to familiarity with the materials that can be used to support student learning. We believe, however, that we can adequately capture this when focusing on KCT in particular on the models, representations and procedures that support student learning (see Table 3).

3. Methodology This research was conducted with 25 pre-service primary teachers enrolled in a 3 credit mathematics education module. All participants were in the final semester of their 3rd year of a 4-year Bachelor of Education degree programme in a university in South Western Ireland. Participants ranged from 20 to 24 years old; 20 were female and 5 were male. They had completed 5 consecutive

TEACHER KNOWLEDGE Subject-Matter Knowledge (SMK)

Common Content Knowledge (CCK)

Specialized Content Knowledge (SCK)

Knowledge at the mathematical horizon

Pedagogical Content Knowledge (PCK)

Knowledge of Content and Students (KCS)

Knowledge of Content and Teaching (KCT)

Fig. 1. Conceptualization of teacher knowledge.

Knowledge of content and curriculum (KCC)

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Table 1 Activities and data collection associated with the lesson study research. Primary activities

Data collection structure and method

 Introduce lesson study (LS) Stage 1  Explore key readings relating to LS Weeks 1e2 Stage 2 Step 1: Collaboratively Planning the  Identify the targeted number concepts Weeks Study Lesson  Develop a trajectory of instruction relating to targeted 3e8 number concepts (content mapping)  Identify the key foci for each lesson study group (LSG)  Each LSG designs their lesson

Step 2: ‘First teach’: Seeing the Study Each of the 5 groups: Lesson in Action  Teach the lesson  Observe the lesson and make notes

Step 3: Discussing and Revising the Lesson

Each of the 5 groups:  Reflect on the taught lesson  Revise the original lesson

Step 4: ‘Second teach’: Seeing the Study Lesson in Action

Each of the 5 groups:  Teach the revised lesson  Observe the lesson and make notes

Step 5: Discussing and Revising the Lesson

Each of the 5 groups:  Reflect on the lesson  Make final revisions to the second lesson plan  LSG presentations  Individual pre-service teacher reflections

Stage 3 Weeks 9e10

 Observations of group discussions relating to LS  Notes from researcher diary and researcher meetings (i.e. refined observations)  Photographs of content mapping  Observations of group discussions  Notes from researcher diary and researcher meetings (i.e. refined observations)  Record of resources used to research and design each lesson  All lesson plans and changes to lesson plans  Observation of lesson by researchers  Observation notes of LSG members  Notes from researcher diary and researcher meetings (i.e. refined observations)  Photographs (teacher activities, student work)  Student materials (copies of work, written records of responses)  Notes taken during group meeting of researchers and LS participants following the lesson  Notes from researcher diary and researcher meetings (i.e. refined observations)  Record of changes made to revised lesson and justification of those changes  Observation of lesson by researchers  Observation notes of LSG members  Notes from researcher diary and researcher meetings (i.e. refined observations)  Video records of the lesson  Student materials (copies of work, written records of responses)  Notes taken during group meeting of researchers and LSGs following the lesson  Notes from researcher diary and researcher meetings (i.e. refined observations)  Record of changes made to revised lesson and justification of those changes  Observation of presentations by researchers  Video records of LSG presentations  Copies of presentations  Notes from researcher diary and researcher meetings (i.e. refined observations)  Individual pre-service teacher reflections

Table 2 Structure and content of the instructional sequence of early number lessons. Lesson

Day

LS group

Main concept focus

1 2 3 4 5

Monday Tuesday Wednesday Thursday Friday

LSG LSG LSG LSG LSG

Rote counting, Rational counting, Subitising, Conservation, Cardinality Rational counting, Creating sets, Equivalence (and non-equivalence) Comparing and Ordering Recognising the numeral, Matching numeral to set/array, Creating sets, Numeral construction Part-part-whole relationships, Partitioning

1 2 3 4 5

mathematics education modules focusing on mathematics content and pedagogy relating to number, algebra, geometry, measures, data handling and probability. All participants had completed a variety of school placements in grades 1e6 and in multiclass settings constituting 10 weeks of classroom observation and teaching experience. Participation in the research was voluntary and ethical approval was obtained for the study from the College Research Ethics Committee. The researchers were the mathematics education lecturers who taught the course within the ITE institution. They assumed the role of ‘knowledgeable others’ (Lewis & Tsuchida, 1998) throughout the semester providing information about the subject matter content and providing guidance when necessary. While the research study focused on the impact of Lesson Study participation on pre-service

teachers' understandings generally, this paper explores the research question: What types of mathematics PCK are developed by pre-service primary teachers as a result of engaging in Lesson Study? 3.1. Procedure The LS focused on the teaching of number in an early years classroom e more specifically, developing understandings of the number 5. The LS cycle took place over a 10-week semester with meetings twice weekly for a total of 3 hours per week. Individual LS groups (LSG) also met outside of class time. The semester was divided into three stages (see Table 1). Stage 1 (weeks 1e2): The first stage focused on introducing lesson study. Participants were provided with key LS readings (cf.

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Table 3 Test definitions for two sub domains of Pedagogical Content Knowledge (developed from Ball et al. 2008). Knowledge domains

Knowledge sub domains

Pedagogical content knowledge

Knowledge of content and students Knowledge of content and students supports teachers in carrying out the following activities: (KCS)  Selecting exemplars that motivate and interest students [KCS 1]  Selecting questions and tasks that seek out the presence of misconceptions [KCS 2]  Anticipating student misconceptions and errors when presented with a mathematical task [KCS 3]  Ascertaining the relative difficulty and ease with which students can solve specific tasks [KCS 4]  Interpreting the mathematical meaning associated with student responses [KCS 5] Knowledge of content and teaching Knowledge of content and teaching supports teachers when carrying out the following activities: (KCT)  Designing the sequencing of the content of instruction [KCT 1]  Identifying strategic examples to demonstrate the meaning of concepts [KCT 2]  Evaluating the strengths and weaknesses of specific representations and procedures in supporting understanding [KCT 3]  Selecting models, representations and procedures that support the development of mathematical understandings [KCT 4]  Selecting appropriate mathematical language, analogies and metaphors [KCT 5]

Test definitions

Fernandez & Yoshida, 2004; Lewis & Tsuchida, 1998; Stigler & Hiebert, 1999) and contributed to structured class discussions on the readings. Stage 2 (weeks 3e8): The second stage, the lesson study cycle, lasted 6 weeks. The multiple steps and key activities associated with each step are presented in Table 1. The first three weeks were spent planning and designing each lesson (see step 1 of stage 2, Table 1). During the first week, participants read research papers, recommendations from professional organizations and reviewed international curricula in an effort to develop a sense of content, trajectories and approaches to teaching early number concepts. The following week participants shared and generated a list of key mathematical concepts that might be addressed when teaching early number (23 concepts were identified which were further refined and reduced to a list of 7). With the guidance of the researchers, they sequenced these concepts to reflect the order in which they might be addressed as part of classroom instruction. Once the sequence was established, a five lesson teaching sequence was determined and mathematical concepts were assigned to specific lessons (see Table 2). Participants were then randomly assigned into 5 groups with 5 members in each group. Each group was responsible for designing and teaching one lesson, this involved identifying instructional activities and tasks that developed understandings of the required mathematics concepts (Table 2). The lesson format adhered to guidelines put forward by Ertle, Chokshi, and Fernandez (2001) and incorporated specific reference to: steps of the lesson (learning activities and key questions), student activities, expected student responses, teacher response to student activity/response, and goals and methods of evaluation. The two researchers worked closely with each LSG, contributed ideas to the lesson design where appropriate and were available during planning sessions for consultation. The lesson plan went through a series of revisions until a final version was completed. The integrated nature of the number concepts required careful coordination and planning between groups to ensure a coherent sequence of instruction. This was facilitated through regular weekly reporting from groups to all class members regarding their instructional design. Groups were told of the importance of participation from all members and two arrangements were put in place to encourage participation. Firstly, each group was required to keep a log of each meeting and identify decisions made and the assignment of tasks within the group. Each group was also required at the end of the semester to submit a document detailing the roles, responsibilities and tasks completed by each group member. All group members

were required to sign this document signalling their agreement with the reported distribution of the workload. During week 6 of the semester, participants taught their sequence of lessons in school 1 in a Junior Infants classroom (children aged 4e5 years old) across 5 consecutive days (see step 2 of stage 2, Table 1). Within each group, one person was selected to teach the lesson. In some groups, the teacher was selected by ‘pulling straws’, however, in other groups individuals volunteered to be the teacher. During the teaching of the first lesson (called the ‘first teach’), the researchers, classroom teacher and all LSG members of the particular group assumed the role of observers. Following the lesson, the LSG and researchers met immediately to discuss their observations and reflections and determine what changes were required to refine the lesson (see step 3 of stage 2, Table 1) which was taught two weeks later (week 8) in school 2 to a different junior infants class. This ‘second teach’ (see step 4 of stage 2, Table 1) was video recorded using a professional video crew. Again, the researchers and members of the LSG observed this lesson and participated in the subsequent reflection. This final lesson was further refined based on these reflections (see step 5 of stage 2, Table 1). Stage 3 (weeks 9e10): The final stage was dedicated to reflecting and reporting on the LS process. Each group provided a 30-min in-class presentation reflecting on and critiquing their lesson. All LSG presentations were video recorded. Participants also submitted an individual reflective assignment three weeks after the presentation. Guidelines and a rubric were provided to support participants when designing their presentation and writing their assignments. Guidelines and rubrics promoted a focus on observations and reflections relating to the following: their espoused theories of teaching and learning mathematics, the development of children's understandings of number, participants own learning in relation to content and pedagogical aspects of early number and participants experience of the LS process. However participants were not limited to these foci and were invited to address other observations and aspects of the LS of their choosing. 3.2. Data collection This was a collective case study (Stake, 1995) of five cases of teaching as it was enacted within a real-life context (Creswell, 1997; Yin, 2009). The cases were considered real life contexts in that the teaching occurred in live classrooms within local schools. This is in contrast to teaching situations often experienced in ITE that take place in the absence of children or outside of schools (for example,

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peer teaching or micro teaching contexts). Each LSG constituted a case. Participant observation was the principal data collection technique consisting of the researchers observing the three stages of the LS process (Table 1). Observational data were collected during planning meetings (stages 1 and 2), classroom teaching (stage 2), LSG planning meetings, group reflections following the first and second teach of each lesson (stage 2) and group presentations (stage 3). The second data source was the analysis of documents produced by the participants e group lesson plans (stage 2), group and individual reflections on teaching following the lesson (stage 2), final individual reflections (stage 3), researcher reflections (stages 1e3), group presentations (stage 3) and concept assignments. Other data collected were photographs taken during the first teach (stage 2), copies of children's work (stage 2), and video records of the five lessons (stage 2) and of the group presentations (stage 3). Alignment between the data sources and the lesson study process is presented in column 3 of Table 1. The issues of reliability and validity are particularly important when carrying out qualitative methodologies as much of the data is in the form of opinions, attitudes and beliefs which may well contain a certain degree of bias. A number of measures were taken to combat this. Firstly, the authors engaged with the study participants over a prolonged period and collected data at every stage e.g. the documenting of developments using notes, observations of lessons, recording of lessons, presentations, documentation (See Table 1) (Creswell, 2009). Transcripts reflected verbatim accounts of classroom practice as well as presentations (McMillan & Schumacher, 2001). While all data collection methods have limitations, the overall use of a ‘multi-method’ approach in this study helps overcome the weaknesses of the individual methods making the findings more reliable (Creswell, 2009). For example, the researchers were acutely aware of the limitations of self-report data, given that there may be a mismatch between pre-service teachers' perceptions and the reality. Given these limitations, the researchers strove to gather data from a number of angles and perspectives thus facilitating triangulation or ‘convergent validity’ (Burke Johnson & Onwuegbuzie, 2004; Cohen, Mannion, & Morrison, 2000; McNiff & Whitehead, 2002). The findings reported in this paper focus on the development of PCK and draw from data collected both inside (see Table 1) and outside of the LS process. Data collected outside the LS process derived from a concept assignment relating to teaching early number. The assignment involved each participant selecting a concept or approach central to early number and designing a brief digital presentation (similar to a ‘video podcast’) which would be helpful to other pre-service teachers. This data source provided insights into how knowledge developed within the context of LS was applied to tasks following the completion of LS. 3.3. Data analysis Data analysis took the form of naturalistic inquiry (Lincoln & Guba, 1985). The focus of the data analysis was to provide a detailed description of the pedagogical content knowledge (PCK) understandings developed during the LS process. Data collection and analysis were ongoing throughout the Lesson Study process (Table 1). Categorizations of knowledge were made based on the descriptions and exemplars outlined by Ball et al. (2008). A list of test definitions were used which were had previously been developed by Leavy (2015) arising from the subdomains identified by Ball et al. (2008). These test definitions were used as guides to support the

identification of the development of participant knowledge as they planned, taught and reflected upon mathematics lessons. As the focus of this study was on two of the three PCK subdomains (KCS and KCT), test definitions were used for these two knowledge domains (see Table 3). However, as outlined by Leavy (2015) ‘these test definitions are not definitions of the knowledge subdomains identified by Ball et al. (2008); rather, they are criteria that must be met in order to be classified as belonging to this subdomain’. 3.3.1. Observational data Observational data (see section 3.2 and Table 1) collected by the researchers were interrogated immediately following the observed events. Researchers met and shared observations, and supporting data, regarding the development of and challenges around pedagogical content knowledge. 3.3.2. Document analysis Data arising from documents (see section 3.2 and Table 1) were collated. Analysis of the lesson plans produced by each group was an ongoing process from weeks 3e8 (Stage 2, Steps 1, 3 and 5, Table 1). Each LSG drafted their first lesson plan and sent it to the researchers. The researchers separately reviewed each lesson plan and sent annotated feedback on the lesson plans to the respective LSG groups (Stage 2, Step 1, Table 1). Feedback was then discussed with the researchers in class sessions or in meeting times outside of class. Each LSG sent a revised lesson plan to the researchers and the feedback and meeting cycle continued until agreement was reached that the lesson plan was ‘fit for purpose’. This process occurred again prior to ‘teach 2’ (Stage 2, Step 3, Table 1). As part of the analysis, the researcher annotations were collated and the data were organised and treated as five cases with each case consisting of a series of revised lesson plans. Hence, lesson plans for each group were examined across time, and the challenges experienced and changes made to the lesson plan were identified in the annotated feedback, observations of meetings and in lesson revisions. 3.3.3. Concept assignment The 25 concept assignments were collated into similar group topics resulting in 8 main categories (rational counting, partitioning, subitising, one-to-one correspondence, comparing, recognising and creating numerals, problem solving, and representations and resources). Within each category, data were analysed in an effort to identify evidence of PCK relating to early number and several primary themes of relevance emerged. In terms of data analysis, the findings are compared to uncover ‘…convergence, difference or some combination’ (Creswell, 2009: 213). Cohen et al. (2000) suggest that when different methods of data collection produce the same findings, one can be confident that the results are valid. All the above data were collated, digitized and then analysed to identify dominant themes. Each theme related to some aspect of pedagogical content knowledge that was developed during the LS cycle. Examples of themes were: importance of teacher modelling, the role of subitising in developing number understandings, the critical role played by context, the use of multisensory approaches, the role of questioning to reveal mathematical understandings. The primary task involved mapping the identified theme onto the Ball et al. theoretical framework. For each theme, this involved using the test definitions (see Table 3) to guide its' categorization as falling within either the KCS or KCT knowledge sub domain (see Fig. 1). Accuracy of the categorisations across the three data sources was ensured by the researchers working together to agree on the categorization of themes and the strength of evidence for these claims. There was

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an initial list of 18 themes and in most cases, several themes were categorised within the same knowledge subdomain resulting in a merging of the 18 themes into broader categories of PCK subdomains. For example, one of the PCK subdomains which targets the selection of ‘models, representations and procedures that support the development of mathematical understandings’ (See KCT4, Table 3) encompassed three of the themes: the use of ICT, animation and video to illustrate concepts (theme 5), teacher modelling of activities and procedures (theme 6) and the importance of multisensory and play based approaches including song and rhymes (theme 7). In an effort to reflect the consistency of themes within and across the data sources, these categories were scrutinized and validated (or not) using triangulation (see Table 4). Triangulation was aided by juxtaposing the multiple perspectives gained from analysis of the observation data, documentary data and the concept assignments in addition to the video records of the lessons. Table 4 illustrates where the particular PCK understandings as described in the test definitions appeared within and across the data sources. For some data sources, in this case the LS Stage 3 documentary data, the categorisation of the qualitative data into the knowledge sub domains is represented using descriptive frequencies. We found this useful due to the large quantities of data falling within this data source. Across the remainder of the data sources, we simply identify whether the PCK as described in the test definitions appeared

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within the particular data source. As can be seen (see Table 4), some data sources provided evidence of the development of PCK understandings across all test definitions (the group lesson plans, in this case) whereas other data sources did not reveal knowledge development across as many test definitions (observational data). The tracking of the appearance of the PCK across the sources (i.e. triangulation), as presented in Table 4, facilitated the presentation of findings which were well-supported. We set the criterion that evidence for appearance of the PCK as described in the test definitions needed to be present across five of the six data sources. This process resulted in the identification of knowledge development within some, but not all, PCK subdomains. As it is not possible to report fully on all of these subdomains, this paper will focus on a selection of these in the findings that follow. 4. Findings The findings arise from the analysis of data gathered within and outside the Lesson Study process. Within LS we report on data across all three stages. These multiple data sources were used to identify and categorize development of participants’ pedagogical content knowledge (PCK) as a result of engaging in LS. The support for our claim of PCK development arises from the use of baseline data gathered from observations in stage 1 (when participants shared initial thoughts about teaching number concepts) and in the

Table 4 Appearance of PCK sub domains (KCS and KCT) test definitions across data sources. Data sources within lesson study

Data sources outside LS

Documentary data

Selecting exemplars that motivate and interest students [KCS 1] Selecting questions and tasks that seek out the presence of misconceptions [KCS 2] Anticipating student misconceptions and errors when presented with a mathematical task [KCS 3] Ascertaining the relative difficulty and ease with which students can solve specific tasks [KCS 4] Interpreting the mathematical meaning associated with student responses [KCS 5] Knowledge Designing the sequencing of the of content content of instruction [KCT 1] Identifying strategic examples to and teaching demonstrate the meaning of concepts [KCT 2] (KCT) Evaluating the strengths and weaknesses of specific representations and procedures in supporting understanding [KCT 3] Selecting models, representations and procedures that support the development of mathematical understandings [KCT 4] Selecting appropriate mathematical language, analogies and metaphors [KCT 5] Knowledge of content and students (KCS)

Observational data

Concept assignment

Group lesson plans (stage 2)

Group and individual reflections on teaching (stage 2)

Group Final individual presentations reflections (stage 3) (stage 3)

Planning meetings, classroom teaching, researcher reflections (refined observations), group reflections following teaching and presentations (stages 1, 2, 3)

Individual concept assignments

Yes

Yes

52

5 groups

Yes

Yes

Yes

No

17

3 groups

Yes

Yes

Yes

No

4

1 group

No

No

Yes

Yes

5

0 groups

No

No

Yes

Yes

15

5 groups

Yes

Yes

Yes

Yes

21

5 groups

Yes

Yes

Yes

Yes

4

3 groups

Yes

No

Yes

Yes

4

1 group

Yes

Yes

Yes

Yes

58

5 groups

Yes

Yes

Yes

Yes

19

3 groups

Yes

Yes

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initial step of stage 2 (in their initial lesson plans) that provide insights into initial PCK around teaching early number. These baseline data were compared with observations of PCK over the duration of stages 2 and 3 (see Tables 1 and 3) as well as beyond the LS process by means of the concept assignment. The findings are presented using the knowledge sub domain categories (Table 3) as an organizing framework. In choosing the data to present, we select areas for which we found evidence of knowledge development across five of the six data sources (see Table 4). We report on three aspects of knowledge development arising within KCS: Selecting exemplars that motivate and interest students (KCS 1), Selecting questions and tasks that that seek out the presence of misconceptions (KCS 2) and Interpreting the mathematical meaning associated with student responses (KCS 5). Within the KCT subdomain we also present data relating to three areas: Designing the sequencing of the content of instruction (KCT 1), Evaluating the strengths and weaknesses of specific representations and procedures in supporting understanding (KCT 3) and Selecting models, representations and procedures that support the development of mathematical understandings (KCT 4). The section concludes with the presentation of a short case study to illustrate the multiple aspects of KCS and KCT that participants drew from when reflecting on children's responses to a single task. 4.1. Knowledge of content and students (KCS) KCS is knowledge relating to how students learn mathematics. It combines teachers’ knowledge of mathematics with their knowledge of students and how they might best learn a particular topic. 4.1.1. Selecting exemplars that motivate and interest students [KCS 1] One component of KCS is the ability to select exemplars that motivate and interest students. With respect to the teaching of mathematics in the early years, we argue that an awareness of the important role played by context in supporting and motivating students constitutes KCS. Without exception, all 25 participants expressed appreciation of the critical role played by context in early years mathematics. It is evident from Teacher #13 that it was not until participants engaged in lesson study and observed the impact of the use of contexts in the classroom that they were convinced of its importance. ‘To be honest, in our first plan we only put in the context ’cause the lecturers kept saying how important context was. We related all our activities to the cookie monster and helping him get ready for Suzie's party. It was only after teaching the lesson, and seeing the videos of the other lessons, that I can begin to imagine how difficult it would be to carry out similar activities without the use of a context. I am so convinced! The context enriches the learning environment and makes it accessible to all regardless of their intellectual ability, social skills or language skills' Teacher #13, individual reflection (Stage 3) All LSGs also reported the benefits of using a context to promote learning When teaching rational counting and counting the number of items in a set, we wanted to link with the context of Suzie's birthday. So we said Suzie was making her party bags and she wanted to put 5 toys in each bag but she did it the night before and she got really tired and she isn't sure if there is 5 in every set.

So we set it up as a problem for them to solve and we felt that would be really motivational for them. Group 1, LSG presentations (Stage 3)

4.1.2. Selecting questions and tasks that seek out the presence of misconceptions [KCS 2] Teacher questioning is a critical skill in terms of gaining access to student mathematical thinking. During the post ‘teach 1’ meeting, the LSGs and researchers discussed this issue and the following researcher note comments on the evolving awareness among participants of the importance of questioning. It was necessary with many groups to address the impact of the question type on the student response, engagement and learning as well as the effect on the teachers' ability to assess. The respective groups committed to using more open-ended questions in Teach 2 Observations from refined researcher notes (Stage 2, Step 3) A heightened awareness of the relationship between assessment and questioning is evident in teacher #23 individual reflection. ‘I knew that method of questioning is always used as an assessment tool. But I have learned a lot from looking at the videos. I now see that just because you ask the children questions it doesn't mean that you are assessing what they have learned’. Teacher #23, individual reflection (Stage 3) Teacher #18 demonstrates coordinated thinking around a number of instructional decisions relating to the nature of teacher questions, the role of language in assessment of understanding and listening to student responses. ‘Working with my group and thinking about what children said during the lesson has made me realize the importance of children talking about maths. Through talking about their experience of the activity, they grow in knowledge while the teacher can listen out for any misconceptions they may have. This will mean in my planning I will prepare good open-ended questions (how do you know that, tell me why do you think that, tell me more about that, what if I did this). I will ensure to scaffold, prompt and probe the children to articulate their responses’ Teacher #18, individual reflection (Stage 3)

4.1.3. Interpreting the mathematical meaning associated with student responses [KCS 5] Developing KCS understandings is also evident in participants' ability to interpret the mathematical meaning associated with student responses. In the meeting with the LSG following ‘teach 2’ (Stage 2, Step 4), these types of understandings were evident from the dialogue. The groups reflected on what they had noticed about children's strategies. One group, LSG #2 (see Table 2), analysed responses to an activity requiring children to sort cubes into cups by colour, count them, and match the correct dot card to the correct amount of cubes. It is interesting that LS participants noticed and considered both verbal and non-verbal responses of children when generating insights into their mathematical understanding.

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Teacher educator: Eabha: Jim: Eabha:

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What did you notice about the children's mathematical understandings? For the dot card and cube activity, there was an issue with cardinality. I asked one of the kids, how many in this tower and they went ‘1, 2, 3, 4, 5’ and I asked her again how many and she said ‘1, 2, 3, 4, 5’. The kids also started placing the cubes on the dots so it was nice to see the 1:1 correspondence. We didn't actually plan this. And as well I noticed that the children started ordering the dots cards. I asked one of the children why have you put the dot cards like this and they said ‘That's the way you count’.

LSG #2 conversation following teach 1 (Stage 2, Step 3)

In this same LSGs presentation at the end of semester, participants shared the piece of video of children engaging in the activity above. The following dialogue occurred between the LSG #2, the class members and one of the teacher educators.

isín: Ro Teacher educator: isín: Ro Anna: isín: Ro John:

It was really interesting watching the children do this activity. I worked with two children and they did it in different ways. What sort of strategies did they use? Some children count the cubes and then count the dots to find a card that matched the same amount of sweets. They were definitely using their one-toone correspondence skills when counting. Most children I saw counted the number of cubes and then used their subitising skills to recognise the dot card that matched it. Others count the cubes and then matched the cubes to the correct dot card by placing each cube on a dot using their one-to-one correspondence skills. I just thought it was really interesting that all of the children came to the correct answer in different ways using skills they had developed.

LSG #2 presentation (Stage 3)

Participants also referred to the importance of analyzing closely what children say particularly when their response is mathematically incorrect. This analysis of student responses revealed the complexity of children's mathematical thinking and lead one participant to remark that ‘the capacity of very young children to learn and understand difficult concepts is the biggest thing I have learned’ (teacher #16, individual reflection, Stage 3). In the quote below we see how the opportunity to engage in group reflection around a particular teaching moment resulted in the development of KCS. In this case the fact that we had captured the teaching on video supported participants in revisiting the lesson (Stage 2, Step 4) and seeking the source of the student errors (see Fig. 2). ‘I have spent so much time in the past looking for the right answers that I was ignoring where the children may have been coming from when they got an answer wrong. In our lesson we represented the number 5 in different ways on a die. The children often subitised one of the fives wrong and when our group looked back and discussed it we realized it [the five dots] was laid out like the way a pattern of number six was on a die and this is the reason they were getting confused. This allowed me to see that children usually have reasoning behind their thinking.’ Teacher #3, individual reflection (Stage 3)

4.2. Knowledge of content and teaching (KCT) This type of knowledge provides teachers with the understandings required to plan their teaching to support learning and challenge misconceptions. Development of KCT was

Fig. 2. Representation of the number 5 that led to student errors.

manifested across the data in three areas.

4.2.1. Designing the sequencing of the content instruction [KCT 1] In the early stages of lesson study, conversations during stage 1 and the initial step of stage 2, participants viewed the focus on early number concepts as a less demanding topic than mathematical concepts studied later in the primary school. Over the course of the study, however, the researchers agreed that participants had developed greater understandings: ‘Overall there is a heightened awareness among the various groups of the interrelated nature of number concepts and an appreciation for the complexity of early number concepts’ (Observations from refined researcher notes (Stage 2, Step 5)). This is reflected in Stage 3 documentation also, where participants acknowledged this and reported on the impact when selecting activities to develop a particular number concept: ‘I now realize that number concepts are linked. They are all integrated and one cannot be taught without touching another. You need to reinforce previously taught concepts and broaden children's knowledge and use of those concepts by applying them to other areas in number’ Teacher #6, individual reflection (Stage 3) Furthermore, participants realized the impact of these interrelated early number concepts on the pedagogical decisions teachers needed to make, in particular the importance of outlining a sequence of instruction when teaching number. Both during LS stage 3 and ‘outside LS’, references were made to the need for a developmental trajectory of concepts that must be addressed in the teaching of number. The quote from Teacher #9 illustrates how group analysis of the video revealed the source of a child's problem (when attempting to make a set of 5 objects). ‘Following the lesson study there are many things I would do differently when planning an early years number lesson. Firstly, I will identify the number concept I want to teach and then I will identify the prerequisite number concepts that are necessary. It

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is important to assess if children possess these concepts otherwise it will be difficult to teach. In one lesson, one of the children we viewed in the video had made a set of 6 objects when asked to make a set of 5. When we looked at the video together we realized that he did not seem to have a grasp of one-to-one correspondence. This made it impossible for him to make sets of 5 objects!’ Teacher #9, individual reflection (Stage 3) Analysis of the data reveals that the experience of working together to design the series of 5 lessons highlighted the importance of sequencing as reflected in the researcher notes: ‘There was an improvement in the respective groups’ attention to sequencing between and within lessons…’ (Observations from refined researcher notes (Stage 2, Step 5)). The concept assignment (data outside LS) also provided evidence to support development of KCT

1. Katie, who was a member of the LSG #3 (see Table 2), made specific reference to sequencing in her concept assignment focusing on conservation. When addressing the order in which to provide experiences in comparing, her co-ordination of perspectives relating to the child's experiences and their familiarity and use of language were notable: When comparing sets it is important to begin with the concept of ‘more’ as it is easier for children to grasp than ‘is the same as’. The concept of ‘more’ should be firmly established before moving to the concept of ‘less’ because they don't use the term as much. We need to start with where they are at.

content of instruction starting with the concrete stage where children are given the opportunity to work with counters, manipulatives or real life objects to construct their understanding of the concept. She linked this to the lesson study (LSG #4, see Table 2) where ‘the children are developing their one-to-one correspondence and rational counting and constructing an understanding of the number 5 by working with the real life materials-the ingredients for making buns’ (see image 1b). She then outlined the pictorial/semi-concrete or representational stage, where students use 2D-objects such as pictures, to develop their understanding of the concept. She used the Lesson Study activities again to illustrate this stage where ‘two different types of pictures were used - Picture cards and Dot cards. In this way the children were given the opportunity to gradually move to more abstract thinking as the activity increased in complexity’ (image 1c). Finally, she described the Abstract stage which culminates in children being challenged to use abstract thinking. She recommended the use of modelling to support the development of abstract thinking and to develop their justification and reasoning skills.

There was also evidence of developing awareness at a more macro level of the importance of coordinating the children's learning trajectory alongside the carefully staged introduction of number concepts necessary to build robust understandings. ‘We began by taking the concept of one-to-one correspondence and built on this. We were able to incorporate other mathematical concepts when their need became apparent in terms of building children's understanding of the number concept i.e. seriation, sorting, subitising. In this way, we took a concept and found the children's level of competence with the concept. We then built on this to move them from the level they were at, while slowly introducing new concepts.’

Katie, concept assignment (slides 9e11) Teacher #8, individual reflection (Stage 3) Analysis of the concept assignments submitted following the completion of LS provided further evidence of greater appreciation for the critical role of moving from concrete to abstract when designing a sequence of instruction. In fact, 14 of the 25 assignments specifically recommended this approach within the context of number recognition and early number work. One of the participants, Ella, chose Concrete to Semi-Concrete to Abstract Approach when representing numbers (Slide 1, concept assignment, image 1a) as the theme of her concept presentation. She made reference to Jerome Bruner's Theory of Representation and summarised the essence of her argument that ‘in order for any child to learn anything that is abstract they need to start with the concrete then the pictorial and finally the abstract approach’ (Slide 7, concept assignment). She provided examples of how to sequence the

The researchers believed that ‘Participants are demonstrating greater ability to coordinate multiple pedagogical considerations in relation to the design of an early number lesson’ (Observations from refined researcher notes (Stage 2, Step 5)). Their growing awareness of the complexity of early number is also evident in the quote from Teacher #4 below in relation to the pacing of instruction. ‘In developing a good sense of number for young children it is important for it not to be a rushed process. Number is a very tricky area. It is vital to develop a sound number sense from a young age. It is important to pace the lessons and gradually develop the various number concepts.’

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Teacher #4, individual reflection

4.2.2. Evaluating the strengths and weaknesses of specific representations and procedures in supporting understanding [KCT 3] Participants became focused on the importance of conceptual understanding and demonstrated a criticality around the selection of tasks and activities that promoted deep understandings as opposed to a focus on procedures: ‘Even though teaching how to write the numeral is a low level activity in terms of learning demands, and hence could justifiably be taught at the beginning of the week, participants decided to delay its introduction so as to focus first on the development of conceptual understanding’ Observations from refined researcher notes (Stage 2, Step 1) In the quote below, we can see that the participants suggest that instruction relating to the numeral ‘5’ occur after the development of conceptual understanding of ‘fiveness’ of five (or a strong number sense for any number). ‘My thinking on early concepts has changed from a somewhat stereotypical view of the order of teaching being: teach the children the numeral and then do some activities and then move to a new numeral. I now have a more discovery learning view on number. The children need to discover the meaning of the numbers themselves and the writing of the digit should not be the main goal of early years instruction. Teacher #15, individual reflection Participants’ ability to identify weaknesses associated with specific procedures and make modifications to address these weaknesses was evident in the response posed by a classmate following the LSG #1 presentation (Stage 3).

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the children were engaged for this length of time because the maths was fun; it was as if while the children were playing they were learning about partitioning unbeknownst to themselves.’ Teacher #19, individual reflection (Stage 3) Observation of group discussions revealed a sense of criticality around the selection of multisensory and play based activities. During the Question and Answer session at the completion of the LSG #1 presentation (Stage 3, Week 10) we can see participants’ appreciation for these approaches and their acknowledgement of the mathematical underpinnings of the activities.

Classmate: Maria: LSG#1 group: Maria Darren:

What activity went best? Suzie knocking on the door Yeah They loved it. In both lessons that's the activity that went best. They were all really engaged. As well it was based on maths. With the knocking at the door, they had to use one-to-one correspondence between the knocks and the number names. They counted on their fingers or in their heads. While they enjoyed it, there was a lot of maths focus and it was more abstract than counting physical items.

LSG #1 presentation (Stage 3)

The importance of the teacher modelling of concepts and procedures was represented in the reflections of the majority of participants: ‘The analysis of reflections revealed their ability to make links between teacher modelling and other instructional decisions such as representations and language’ (Observations from refined researcher notes (Stage 3)). The quote from teacher #3 emphasises the need for similarity and coherence between both teacher and pupil materials used during the modelling whereas teacher #18 refers to the potential of modelling to overcome language barriers and the importance of engaging in multiple repetitions when modelling.

Classmate: What was the biggest challenge you faced in your lesson? Lucy: We were looking for other activities for rational counting so we decided to use the counting jar. We just had a glass jar at the top of the classroom and we had marbles. All the children had to put their heads down on the table so they couldn't see and they had to listen to how many times Suzie dropped a marble into the jar. Maria: Yeah, so in the first teach, the children shouted out the answer and it was difficult to know if all the children were using 1:1 correspondence or understood cardinality. Lucy So for the second teach, we ensured that we changed it so all the children had to hold up the correct number of fingers to represent the number of marbles they heard. So we could gain insight into all the children's rational counting not just one child who shouted out the answer.

4.2.3. Selecting models, representations and procedures that support the development of mathematical understandings [KCT 4] All data sources revealed a greater appreciation for the importance of pedagogical decisions around the multiple ways to represent mathematical concepts. The majority of participants emphasized the importance of multisensory approaches combined with play-based approaches. Participants stated that they had underestimated the power of these approaches in supporting the development of mathematical understandings of young learners. ‘This lesson study opened my eyes in regard to play based approaches to teaching. During the first teach our lesson ran way over time; these children did maths for an hour as opposed to the standard twenty minute lesson in the infant years. However,

‘The lesson study has taught me the importance of modelling for young children particularly with concepts such as rational counting and making sure they carry out one-to-one correspondence correctly. In my teaching I will now ensure even more modelling of activities and also ensure that the resources I use to model on the board match the materials I give out to children’ Teacher #3 individual reflection (Stage 3) ‘The importance of modelling is one of the considerations I have realised. Modelling is vital in a classroom where there is no adult support for each group and when children must understand how to complete an activity and what is expected from them. It

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can also be highly beneficial where there are children who don't understand the language being spoken by the teacher. In these situations, there is great value in having the teacher model the activity and then having a student some to the top of the classroom to demonstrate’ Teacher #18, individual reflection (Stage 3)

(continued )

Laura Student teacher Laura Student teacher Laura

Emily starts to move off task 2 He has [pointing to the additional blocks] 1, 2 3 3 more. So how many more has Elmo than the Cookie Monster? I have 3 here [pointing to the 3 blank spaces]

Transcript from video of classroom teaching (LSG #3)

4.3. Illustration from a data excerpt: the case of comparing groups (LSG#3) The development of PCK understandings is presented using the KCS and KCT subdomains as an organizing framework. This presentation of the data masks the simultaneous and interrelated nature of the development of understandings across the study. At most stages of LS participants were drawing from knowledge categorised as falling within several PCK subdomains. This reality is illustrated in LSG#3 ‘Teach 2’ (Stage 2, Step 4) (see transcript below), where the children are working with the student teacher to compare the number of chocolate chips on Elmo and Cookie Monsters cookies (see Fig. 3):

Following viewing the video excerpt in the LSG presentation, the statement from Katie (the student teacher who taught the lesson) indicates insights into the relative difficulty and ease with which students can solve specific tasks (KCS 4). They seem to be able to compare at a basic level. But it is the last question ‘how many more chocolate chips does one character have?’ that was the toughest question for them. They can identify that Elmo had more, but when asked how many more does Elmo have, she said ‘Elmo has 5 more’, so demonstrating an inability to compare using number. While I tried to scaffold and elicit from her, she was struggling with this aspect of the concept’ Katie, LSG#3 Presentation (Stage 3) The knowledge demands relating to selecting appropriate mathematical language, analogies and metaphors [KCT 5] is also apparent in relation to this episode. Analysis of the researcher reflective notes in response to the LSG #3 post-lesson reflection meeting highlights this awareness.

Fig. 3. Representing the number of chocolate chips owned by Elmo and Cookie Monster.

Student teacher Laura Student teacher Laura Student teacher Student teacher

Student teacher Laura and Emily Student teacher Emily Student teacher

Who has more chocolate chips? Elmo has more. How do you know Elmo has more? Because the Cookie Monster has only 2. If you were going to pick a cookie, which one would you prefer - Elmo's or the Cookie Monster's cookie? silence Do you like chocolate chips?

What was really evident to me when participating in the reflection session following today's lesson is that the language difficulty and the challenge for children in articulating their mathematical thinking has really become apparent to the student teachers. They all mentioned the language difficulties they observed with the groups they worked with during the lesson. To be honest, I think the teacher, Katie, is being a little too hard on herself in terms of her shortcomings teaching the lesson. There was a lot going on in that session. The concept of comparison is complex even if the children were a couple of years older and all first language English speakers. That was very very tricky for them. And it is a big challenge even for an experienced teacher. Language keeps coming out all the time in all the group reflections and just the complexity of language is much more apparent to them now. I hope the student teachers can see how it's building in complexity over the week. Observations from refined researcher notes (Stage 2, Step 5)

children nod So if you wanted a cookie would you have the cookie with the 5 chocolate chips or the cookie with 2? Laura and Emily point to Elmo 5 because the 5 has … More Now do you think you'd be able to tell me how many MORE chocolate chips Elmo has than Cookie monster (pointing to the grid which compares 5 blocks and 2) 5 So Cookie Monster has 2 and Elmo has 5, how many extra does the Elmo have? silence Let's have a count. So the Cookie Monster has 2 and Elmo has 5 how many more does Elmo have?

In the concept assignment (outside LS), Katie also refers to this excerpt in her presentation on ‘Conservation of Number’. She states that ‘errors made in early comparing number quantities activities may be due in part to difficulties with one-to-one correspondence’. She suggests presenting children with a well-known conservation task (image 5) to assess the presence or absence of one-to-one conservation. Furthermore she states that ‘many children's inability to conserve number quantities at the junior infants stage will prevent them from being able to compare number quantities that are arranged in different ways. For example, a child may think that the second set of circles has more than the first set.’ This task (image 2) reveals knowledge relating to Selecting questions and tasks

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that that seek out the presence of misconceptions which falls within the KCS 2 domain. Furthermore, Katie's developing KCT 4 is then evident when she demonstrates her understanding of the role played by the selection of models to support developing children's understandings. She recommended the use of compare grids as a method of organising and rearranging sets to support children in comparison. She recommended these representations as they support the development of one-to-one correspondence. She states when using grids ‘one object is placed in each box which supports the even distribution of objects and makes it easier for children to compare sets’.

4.4. The challenges of lesson study While participants were very positive about the perceived benefits of engaging in Lesson study, they also acknowledge the challenges associated with the approach: The first challenge that we came along was time. I'm not sure how the other groups got on but we thought it was a very time consuming process and I don't mean that in a bad way. When we are in college we have lots of other things going on and we have other pressures and deadlines and the time it took to do lesson study was extensive. Damien, LSG #1 Presentation, Stage 3 Other groups addressed the challenges associated with communication between groups to ensure coherence between lessons: In the first two weeks, we didn't really know where to start. Because we were the Thursday group, we didn't really know what the Wednesday group had planned on covering nor what the Friday group hoped to cover. So definitely at the early stages we were in a hard place and we didn't know what to cover exactly Moll, LSG#4 Presentation, Stage 3 While all LSGs were aware of the benefit of having a group of 5 people working on a lesson, they were aware of the related need to modify activities into the future when they were teaching alone:

Katie:

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5. Discussion and conclusions This study is framed by research on teacher noticing and reflection and utilises Japanese lesson study in its design. The design represents a strong ‘formal’ approach (Myers, 2012) to lesson study incorporating two ‘plan-teach-observe learningevaluate’ cycles undertaken in primary classrooms. Our aim was to explore the growth in pre-service teachers' pedagogical content knowledge arising from engagement in and reflection upon teaching primary mathematics within a lesson study structure. We use the Ball et al. (2008) practice-based theory of content knowledge for teaching as an organizing framework to identify areas in which development of mathematical knowledge for teaching occurred. We focused specifically on the pedagogical content knowledge of the Ball et al. framework as this focus provides an opportunity to investigate the application of knowledge of this construct into the classroom context. The observation of 5 lesson study groups as they planned, taught and revised mathematics lessons in local schools provided unique insights into the types of knowledge that pre-service teachers draw on when teaching early number concepts. Improvements were identified in the pedagogical content knowledge sub domains of knowledge of content and students (KCS) and knowledge of content and teaching (KCT). The use of the case illustration was selected to highlight the interrelationships between the knowledge subdomains and reveals that the nature of teaching entails considerations to multiple dimensions of teaching and learning. In this study, the activity of exploring knowledge in practice provided an authentic context within which pre-service teachers attended to the multiple different components of PCK simultaneously (evident in the quote that introduces this paper). This, we argue, resulted in highly integrated and robust pedagogical understandings that transferred outside of the study context (i.e. lesson study) and were evidenced in other aspects of participants teaching lives (i.e. concept assignments). It can be argued that the learning that took place is indicative of the understandings that may be expected from teachers reflecting on practice. However, it is also interesting to note that the observed growth in PCK related to content and pedagogical practices that had already been the focus of numerous pedagogical courses earlier in the ITE programme. However, the collection of baseline data (in stages 1 and initial lesson drafts) indicated that this knowledge had remained in some ways inert. It appears that lesson study was successful in triggering the development of PCK, a finding of other studies with pre-service primary teachers (Leavy, 2010). In fact, as was evident in a number of quotes presented earlier in the paper, some of the participants themselves remarked that the experiences provided by lesson study ‘convinced’ them of the benefit of the practices they now advocated. While acknowledging the challenges associated with lesson study, the pre-service teachers themselves also recognised and reported on the nature of their learning as a result of engaging in Lesson Study. This led the authors to consider the particular features of the lesson study process which supported the development of PCK subdomains. Firstly we consider that the structure of the lesson plan utilised in this study (cf. Ertle et al. 2001) required LSGs

... all of the lessons are very good for learning and we will be able to use this experience on our school placement but realistically we would have to make changes for an individual teacher, you couldn't do that in one lesson, not with the resources. We would split it up over a number of days... What else [looking at other members of the group] Sharon: Materials. We did have a lot going on and if it was our own infant classroom- realistically … for example, for the washing line activity, there aren't going to be 2 teachers to hold either end of it. But you could use the interactive white board or hang the line in the classroom … LSG #3 Presentation (Stage 3)

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to address the relationship between planning, teaching and assessment. This level of detail helps parse the complex interrelationships that often remain hidden in teaching and required pre-service teachers to justify the pedagogical decisions that they made. Cajkler et al. (2013) contend that LS opens up these links between planning, teaching and assessment to observation and scrutiny on the part of the pre-service teacher. They refer to the complex situation of classroom teaching as a ‘pedagogic black-box, which can remain either partially or wholly shut up in individually-oriented teacher placements’ (2013, p. 550). The opening of the ‘pedagogic black box’ was facilitated, we contend by the focus on individual and group reflection in lesson study. For example, during the planning phase the need to draw on research evidence to provide justification for pedagogical approaches and specific activities was emphasized. This resulted in a ‘justification micro-culture’ where pre-service teachers realised that they should be able to defend or explain their decision-making processes around planned pedagogical decisions. Secondly, Lesson Study engaged pre-service teachers in all aspects of teacher noticing. Within the context of this study, while lesson study provided a structure that supported pre-service teachers in attending to noteworthy events (i.e. the first aspect of the van Es and Sherin (2002) conceptualization of what it means to notice), the combination of mentor supported reflection and group dialogue provided the impetus and support for preservice teachers to begin to reason about noteworthy events and, with these supports, make informed teaching decisions based on the analysis of these observations from the first teach. These informed teaching decisions are articulated in revisions made to the lesson plans and subsequently in modified pedagogical practices in the second teach. A third hypothesis to account for gains in pre-service teacher understanding is that Lesson Study provides a focus on children's mathematical thinking by shifting the focus of attention from what the teacher is doing and provides an impetus to focus on what students are saying. The structure of the LS plan required participants to attend to possible student responses and gives recognition to the role of knowledge of children's thinking in informing instruction. Reflection following lessons focused participants on observations of student learning, the impact of pedagogical decisions on this learning and any changes in pedagogical actions required to bring about desired outcomes in understanding. Finally, the focus on observation combined with the opportunity to work with small groups of students during individual, pair and group work further facilitated learning. This learning was enhanced further by the opportunity to analyse video from the second teach of the lesson; video footage which captured both the teacher and student behaviour during the lessons. Even though participants had been present for the lesson, examining the recording provided a shared common experience of the lesson, promoted selfawareness of their practices, and facilitated the re-examination of critical teaching incidents as they occurred. It also provided the opportunity for participants to focus attention on student actions, a critical area that inexperienced teachers tend not to attend to (Carter, Cushing, Sabers, Stein, & Berliner, 1988). The dual role assumed by the authors as both lecturers and researchers is a complex undertaking. While the authors endeavoured to lead pre-service teachers through the various stages of Lesson Study which involved assuming the role of ‘knowledgeable others’, in parallel they strove to research the impact of the process on participants' PCK understandings. Given the authors' involvement in the process, the potential for bias must be acknowledged and addressed. While it may be conceived that researcher objectivity can only be achieved where the researcher remains emotionally detached from the study participants, in this scenario it was neither desirable nor possible to have data collected by ‘external’ researchers (Burke Johnson & Onwuegbuzie, 2004;

McNiff, 1988). The authors already had a working relationship with the pre-service teachers in question, therefore making it possible to enter naturally into the educational context under study, without causing opposition to the idea or altering the situation (Cohen et al., 2000). In order to combat the potential impact of the authors' personal biases regarding the impact of Lesson Study on participants, both authors used a reflective diary to make personal beliefs explicit throughout the study (Creswell, 2009). While the exclusive use of qualitative data may also be perceived as a limitation, in light of the nature of the study and the small number of participants, it was considered that the data collection methods were ‘fit for purpose’ and provided a richer more detailed picture. The findings of this study have important implications for the design of teacher education programmes. As well as providing a vehicle for the development of appropriate pedagogical content knowledge, the experience of engaging in LS helped move preservice teachers towards taking a more critically reflective stance on their teaching and developing a clear understanding of their students’ thinking and the pedagogical content reasoning that influenced that thinking. This is captured well by Beth (LSG #5) in her comment: I think the looking at the child's thinking is the biggest learning for me. I wouldn't have considered that a whole lot before. You know when you are on teaching practice you are teaching to the lesson, thinking about what you are going to say and do whereas here you are more focused on how you expect the children to respond … thinking more about how the children are thinking and learning. Acknowledgements The authors would like to acknowledge the support of faculty seed funding from our institution Mary Immaculate College, Faculty Seed Funding. References Ball, D. L., Thames, M. H., & Phelps, G. (2008). Content knowledge for teaching: What makes it special? Journal of Teacher Education, 59(5), 389e407. Burke Johnson, R., & Onwuegbuzie, J. (2004). Mixed methods research: A research paradigm whose time has come. Educational Researcher, 33(7), 14e26. Cajkler, W., Wood, P., Norton, J., & Pedder, D. (2013). Lesson study: Towards a collaborative approach to learning in Initial Teacher Education? Cambridge Journal of Education, 43(4), 537e554. Carter, K., Cushing, K., Sabers, D., Stein, P., & Berliner, D. (1988). Expert-novice differences in perceiving and processing visual classroom information. Journal of Teacher Education, 39(3), 25e31. http://dx.doi.org/10.1177/ 002248718803900306. Cavey, L. O., & Berenson, S. B. (2005). Learning to teach high school mathematics: Patterns of growth in understanding right triangle trigonometry during lessonplan study. The Journal of Mathematical Behavior, 24, 171e190. Chassels, C., & Melville, W. (2009). Collaborative, reflective, and iterative Japanese lesson study in an initial teacher education program: Benefits and challenges. Canadian Journal of Education, 32(4), 734e763. Cohen, L., Mannion, L., & Morrison, K. (2000). Research methods in education (5th ed.). London and New York: Routledge Falmer Academic Publishers. Creswell, J. W. (1997). Qualitative inquiry and research designs: Choosing among five traditions. Thousand Oaks, CA, USA: Sage. Creswell, J. W. (2009). Research design- qualitative, quantitative and mixed methods approaches (3rd ed.). USA: SAGE publications. Darling-Hammond, L. (2000). How teacher education matters. Journal of Teacher Education, 51, 166e173. Darling-Hammond, L., & Bransford, J. (2005). Preparing teachers for a changing world: What teachers should know and be able to do. San Francisco, CA: JosseyBass. Davis, E. (2006). Characterizing productive reflection among preservice elementary teachers: Seeing what matters. Teaching and Teacher Education, 22(3), 281e301. Erickson, F. (2011). On noticing teacher noticing. In M. Sherin, V. Jacobs, & R. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers' eyes (pp. 17e34). New York, NY: Routledge. Ertle, B., Chokshi, S., & Fernandez, C. (2001). Lesson planning tool. Retrieved June 08, 2009, from http://www.tc.columbia.edu/lessonstudy/doc/Lesson_Planning_

A.M. Leavy, M. Hourigan / Teaching and Teacher Education 57 (2016) 161e175 Tool.pdf. van Es, E. A., & Sherin, M. G. (2002). Learning to notice: Scaffolding new teachers' interpretations of classroom interactions. Journal of Technology and Teacher Education, 10, 571e576. Fernandez, M. L. (2005). Learning through microteaching lesson study in teacher preparation. Action in Teacher Education, 26(4), 37e47. Fernandez, C., & Yoshida, M. (2004). Lesson study: A Japanese approach to improving mathematics teaching and learning. Mahwah, NJ: Lawrence Erlbaum Associates. Gonzalez, L. E., & Carter, K. (1996). Correspondence in cooperating teachers' and student teachers' interpretations of classroom events. Teacher & Teacher Education, 12, 39e47. Hammerness, K., Darling-Hammond, L., & Bransford, J. (2005). How teachers learn and develop. In L. Darling-Hammond, & J. Bransford (Eds.), Preparing teachers for a changing world: What teachers should know and be able to do (pp. 358e389). San Francisco, CA: Jossey-Bass. Hannigan, A., Gill, O., & Leavy, A. M. (2013). An investigation of prospective secondary mathematics teachers’ conceptual knowledge of and attitudes towards statistics. Journal of Mathematics Teacher Education, 16(6), 427e449. http:// dx.doi.org/10.1007/s10857-013-9246-3. Hiebert, J., Morris, A. K., Berke, D., & Jansen, A. (2007). Preparing teachers to learn from teaching. Journal of Teacher Education, 58(1), 47e61. Jansen, A., Bartell, T., & Berk, D. (2009). The role of learning goals in building a knowledge base for elementary mathematics teacher education. The Elementary School Journal, 109(5), 525e536. Leavy, A. M. (2010). The challenge of preparing preservice teachers to teach informal inferential reasoning. Statistics Education Research Journal, 9(1), 46e67. Leavy, A. M. (2015). Looking at practice: Revealing the knowledge demands of teaching data handling in the primary classroom. Mathematics Education Research Journal, 27(3), 283e309. http://dx.doi.org/10.1007/s13394-014-0138-3. Leavy, A. M., Hourigan, M., & Carroll, C. (2015). Exploring the impact of reform mathematics on entry-level pre-service primary teachers attitudes towards mathematics. International Journal of Science and Mathematics Education. http:// dx.doi.org/10.1007/s10763-015-9699-1. Lewis, C., & Tsuchida, I. (1998). A lesson is like a swiftly flowing river: How research lessons improve Japanese education. American Educator, 22(4), 12e17, 50e52. Lincoln, Y. S., & Guba, G. E. (1985). Naturalistic inquiry. Newbury Park, CA: Sage Publications. Marble, S. T. (2006). Learning to teach through lesson study. Action in Teacher Education, 28(3), 86e96. Masats, D., & Dooly, M. (2011). Rethinking the use of video in teacher education: A holistic approach. Teaching and Teacher Education, 27(7), 1151e1162. McMillan, J. H., & Schumacher, S. (2001). Research in education: A conceptual introduction (5th ed.). New York: Longman.

175

McNiff, J. (1988). Action Research: Principles and practice. London: Routledge. McNiff, J., & Whitehead, J. (2002). Action Research: Principles and practices (2nd ed.). London and New York: Routledge/Falmer. Morris, A. K. (2006). Assessing pre-service teachers' skills for analyzing teaching. Journal of Mathematics Teacher Education, 9(5), 471e505. Murata, A., & Pothen, E. B. (2011). Lesson study in preservice elementary mathematics courses: Connecting emerging practice and understanding. In L. Hart, A. Alston, & A. Murata (Eds.), Lesson study research and practice in mathematics education: Learning together (pp. 103e116). New York: Springer. Myers, J. (2012). Lesson study as a means for facilitating preservice teacher reflectivity. International Journal for the Scholarship of Teaching and Learning, 6(1), 1e21. Nemser, S. F. (1983). Learning to teach. In L. Shulman, & G. Sykes (Eds.), Handbook of teaching and policy (pp. 150e170). New York: Longman. Sherin, M. G., & Han, S. Y. (2004). Teacher learning in the context of a video club. Teaching and Teacher Education, 20, 163e183. Sherin, M. G., & van Es, E. A. (2005). Using video to support teachers' ability to notice classroom interactions. Journal of Technology and Teacher Education, 13(3), 475e491. Shulman, L. S. (1986). Those who understand: Knowledge growth in teaching. Educational Researcher, 15(2), 4e14. Shulman, L. S. (1987). Knowledge and teaching: Foundations of the new reform. Harvard Educational Review, 57, 1e22. Sims, L., & Walsh, D. (2009). Lesson study with preservice teachers: Lessons from lessons. Teaching and Teacher Education, 25(5), 724e733. Stake, R. E. (1995). The art of case study research. Thousand Oaks, CA: Sage Publications. Star, J., Lynch, K., & Perova, N. (2011). Using video to improve preservice mathematics teachers' abilities to attend to classroom features. In M. Sherin, V. Jacobs, & R. Philipp (Eds.), Mathematics teacher noticing: Seeing through teachers' eyes (pp. 117e133). New York, NY: Routledge. Star, J., & Strickland, S. (2008). Learning to observe: Using video to improve preserves mathematics teachers' ability to notice. Journal of Mathematics Teacher Education, 11(2), 107e125. Stigler, J. W., & Hiebert, J. (1999). The teaching Gap. Chapter 7. New York: Free Press. Tabachnick, B. R., & Zeichner, K. M. (1999). Idea and action: Action research and the development of conceptual change teaching of science. Science Education, 83(3), 309e322. Yin, R. K. (2009). Case study research: Design and methods (4th ed.). Newbury Park, CA: Sage Publications. Zeichner, K., & Liston, D. P. (2006). Teaching students to reflect. In D. Hartley, & M. Whitehead (Eds.), Teacher Education: Major themes in education (pp. 5e34). London: Routledge.