Developing the professional development model

In the first instance the professional development model drew on findings from the literature chapters. The initial focus was placed on investigating teacher perceptions of early algebra and then to widen their understanding of this area by the introduction of an overview of early algebra concepts. The subsequent and on-going re-design of the model for professional development drew on researcher observations from the classrooms. For example, it was observed that the teachers needed professional development in facilitating students to generate and explore conjectures. In response, a task was designed to enable the teachers to explore possible conjectures which students would make and how these could be justified. The study group meetings, teacher interviews, and discussions also provided further information regarding the need for professional development activities. For example, both during study group meetings and interviews the teachers requested support in how to facilitate students to ask questions. Consequently in the following study group meetings research articles were used as the basis for

Algebraic concepts Mathematical practices Classroom practices Tasks and enactment Developing early algebra in primary classrooms

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a discussion on facilitating student questions. The focus for professional learning within the model comprised four separate but related components:

 Understanding of early algebraic concepts.

 Task development, modification, and enactment.

 Classroom practices.

 Mathematical practices

These foci of teacher learning/inquiry when combined with phases reflecting the possible trajectory of foci formed the conjectured framework of professional development as shown in Table 1. This was termed a conjectured framework because the focus of the study did not analyse whether each of the professional development activities was effective in regards to teacher development of algebra ears and eyes.

Table 1

Conjectured Framework of Professional Development to Develop Teachers’ Algebra Ears and Eyes Understanding of early algebraic concepts Task development, modification, and enactment

Classroom practices Mathematical practices

PHAS

E

ONE

Concept map of early algebra.

Teacher analysis of student reasoning (task based interview results). Teacher examination of - Equals sign. - Relational reasoning - Commutative property. Overview of early algebra concepts. Examination of MEP curriculum material for links to early algebra concepts.

 Classroom development of - - Collaborative discourse. - - Mathematical explanations.

Used framework to reflect on current classroom practice and set goals.

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PHAS

E

T

WO

Used framework and classroom video-recorded lessons as reflective tools with a focus on - Developing collaborative group work. - Developing whole class discussions.

- Strategies to integrate early algebra.

 Developed new concept map of early algebra. Compared and contrasted previous understanding.  Examined a case of student reasoning to build understanding of relational reasoning. Examination of common errors when using variables.

 Used task to predict and plan for student responses.

 Identified and

adapted MEP tasks to develop algebraic reasoning.  Wrote number sentences using different numbers or properties which drew on relationships.

 Orchestrating a productive whole class discussion.

Classroom development of making conjectures and justification - Generate possible student conjectures.

- Predict student justification of a conjecture.

- Justify a conjecture in three ways (using representational material, a verbal explanation, and symbolic form).

PHAS E T HRE E Collaborative lesson planning.

Developed overarching aim for the learning community.

 Lesson study post-lesson observation reflective meeting. Focus on

- Pedagogical strategies. - Student responses. - Task design.

 Collaborative lesson planning.

Identified and

adapted MEP tasks to develop algebraic reasoning.

 Classroom development of generalisation. Used framework to reflect on current classroom practice and set goals.

 Reflective discussion. Focus on

- Understanding of early algebra. - Task design.

- Pedagogical strategies to support early algebra. - Student reasoning and participation.

 Lesson study post-lesson observation reflective meeting. Focus on

- Pedagogical strategies. - Student responses. - Task design.

Initial instructional activities included in the professional development were selected to explore and challenge teachers’ existing understanding and beliefs about early algebra. For example, an opening activity involved teachers drawing a concept map of their understanding of early algebra. This activity supported both the teachers and the researcher to reflect on their current understandings of early algebra. In subsequent meetings, it provided further opportunities for the

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teachers and researcher to continue reflecting on their developing understanding of early algebra.

Research articles were used as multi-purpose tools in the professional development. A range of articles and excerpts from research texts1 were shared with the teachers during the study group meetings. These were used to extend teacher understanding of early algebra, to provide models of classrooms which would support early algebraic reasoning, and to promote reflection on current practice. Discussions held after reading each article required the teachers to respond to questions such as “what did you find interesting?” or “are there any ideas that you could bring to your classroom after reading that?” The articles also served the purpose of developing links between research and classroom practice as advocated by Watson (2009).

The selection, design, and enactment of tasks were a central focus for professional development. The current study built on previous studies (e.g., Blanton & Kaput, 2008; Franke et al., 2008; Jacobs et al., 2007; Schifter et al., 2008; Stephens et al., 2004) that successfully used mathematical tasks to engage teachers in reconceptualising their understanding of algebra. In these studies the use of algebra tasks provided teachers with multiple opportunities to reflect on their own understanding of algebraic concepts and the mathematical practices which support students’ learning of early algebra. For example, in the current study the teachers were asked to

1

Monaghan, F. (2005). Don’t think it in your head, think aloud: ICT and exploratory talk in the primary mathematics classroom. Research in Mathematics Education, 7, 83-100.

Kazemi, E. (1998). Discourse that promotes conceptual understanding. Teaching Children Mathematics,

4(7), 410-414.

Carpenter, T. P., Levi, L., & Farnsworth, V. (2000). Building a foundation for learning algebra in the elementary grades. In Brief, 1(2), 1-8.

Carpenter, T., Franke, M., & Levi L. (2003). Thinking mathematically: Integrating arithmetic and

algebra in elementary school. Portsmouth: Heinemann.

Smith, M. S., Hughes, E. K., Engle, R. A., & Stein, M. K. (2009). Orchestrating discussions.

Mathematics Teaching in the Middle School, 14(9), 548 – 556.

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solve number sentences involving variables, develop their own number sentences, and at another time asked to develop different forms of justification for a conjecture. Specific tasks from the MEP curriculum material were used to provide ways for the teachers to identify opportunities for algebraic reasoning and also to investigate ways of modifying and further developing existing tasks.

Other research studies (e.g., Blanton & Kaput, 2008; Franke et al., 2008; Herbel-Eisenmann & Phillips, 2008; Jacobs et al., 2007; Koellner et al., 2011; Stephens et al., 2004) that highlight the usefulness of a focus on student thinking and reasoning to facilitate changes in practice also influenced the design of the professional development. In addition to designing and critiquing tasks, teachers in this study were encouraged to predict responses that students would give to algebraic tasks. They also engaged in activities which investigated student responses from task- based interviews conducted by the researcher and undertaken at the beginning and end of the study.

Prompting reflection on practice was a key element of the professional development. This required that the teachers were able to develop tools and skills for noticing relevant aspects of their practice (Franke et al., 2008; Ghousseini & Sleep, 2011). To support the teachers in this study to reflect on their own practice, they were provided with an adapted framework (see Appendix A) from Hunter (2009) which detailed classroom and mathematical practices linked to the development of algebraic reasoning. The framework provided the teachers with an objective lens to use when viewing video records from their own classrooms. It was also a useful tool to support them to reflect on their practices and set future goals.

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As the study progressed, teachers had opportunities to engage in a lesson study cycle which drew different elements of the professional development together. Lesson study is based on a Japanese model of teacher development which emphasises student learning and reflection on practice (Lewis, 1995; Stigler & Hiebert, 1999). It aims to increase teachers’ knowledge about mathematics, ways of teaching mathematics, and ways in which learners engage with and make sense of mathematics (Fernandez & Yoshida, 2004). Although a superficial view of lesson study is that it is focused on developing the ‘perfect’ lesson, the deeper intention of the lesson study cycle is to support teachers to engage with the processes of teaching and learning. Participating in a lesson study cycle can prompt teachers to reflect on their own approaches to the processes of teaching and learning and develop practices in ways which are meaningful within their working contexts (Burghes & Robinson, 2010; Stepanek & Appel, 2007).

In the lesson study process used in this study, each group of teachers worked as a community within their own school. The initial step involved the establishment of an over-arching aim which was relevant to each school. This collaboratively agreed goal established that the teachers wanted to develop creative and flexible problem-solvers. Following this, the teachers planned an area of focus for the study lessons. The foci corresponded to mathematical concepts their students had difficulties with or those which the teachers felt less confident about teaching. For example, at Hillview school the teachers wanted to address how their students over-generalised the commutative property to include subtraction and division. A lesson study cycle was devised which included lessons designed to facilitate student understanding and justification of the commutative property with a focus on the use of representations to model conjectures and justify reasoning. At Beaumont school the study lesson cycles aimed to develop student skill at solving multi-step word problems and part of the focus was placed on the equal sign. Through collaborative activity the ‘study lessons’ were planned and taught by one of the teachers and

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observed by the others. Teacher reflection was prompted from observation during the study lessons and the post discussion prompted by key questions (see Appendix B). The lesson was then re-planned and further developed on the basis of the student responses and consequently re- taught to a different group of students in another classroom while being observed by the other teachers and discussed again in a meeting following the observation.