Developing early algebraic reasoning in a community of mathematical inquiry
Jodie Hunter
University of Plymouth
Background of the study
Significant changes have been proposed for mathematics classrooms of the 21st century in order to meet the needs of a “knowledge society” (Kaput, 2008). Developing inquiry learning communities where all students have opportunities to engage in mathematical practices which underlie algebraic reasoning has been an increasing focus in both national and international research and curricula reforms (Blanton & Kaput, 2002; National Council of Teachers of Mathematics (NCTM), 2000). This focus in part is due to growing acknowledgement of the insufficient algebraic understandings many students develop during schooling and the way in which this denies them access to potential educational and employment prospects. To address this problem, one response has been to restructure curriculum and develop changes in classroom practices to integrate early algebraic reasoning into everyday mathematics lessons (Kaput). This approach develops a wider definition of algebraic reasoning which includes both content and process, therefore within this frame algebra is both an evolving self standing body of knowledge (cultural artefact) and a human activity – a way of reasoning, talking, doing and thinking (Kaput; Watanabe, 2008). However, while primary teachers take a critical role in reforming classroom practice, many teachers themselves have had little experience with the rich, connected types of algebra which support young students development of early algebraic reasoning. Therefore the overall aim of this study is to investigate how teachers can develop early algebraic reasoning in communities of mathematical inquiry. The following key questions will be addressed by the study:
· How can curriculum material be used as a tool to develop early algebraic reasoning?
· What pedagogical strategies support engagement in early algebraic reasoning?
· What classroom practices facilitate engagement in early algebraic reasoning?
· What shifts occur in the way children engage in classroom activity when a culture of early algebraic reasoning has been developed?
Context of the study
During the 2009/2010 school year, a year-long classroom intervention study was undertaken at two schools involving six teachers. Included in the study were a Year Two class, three Year Three classes and a Year Five class. One school is located in an outlying suburb of London (Hillview) and the other on the Channel Islands (Beaumont). The teachers volunteered to be involved in the study. The schools use the MEP curriculum (based on Hungarian style teaching methods) which was developed by CIMT, University of Plymouth. Previously the teachers were involved in an observational research project investigating how the curriculum was enacted within different classrooms and schools.
Data collection
The following work with the teachers and their classes and accompanying data has been collected during the course of the study:
· Three half-day meetings and an after-school meeting at the individual participating schools. At these meetings we read relevant research articles and discussed these, collaboratively worked to adapt the curriculum material and lesson plans to incorporate early algebraic reasoning, watched classroom episodes and reflected on these. All of the meetings were video recorded.
· All children involved in the project classrooms were interviewed at the beginning of the study to investigate their attitudes and understanding of learning mathematics. They were also asked some task-based interview questions to assess their reasoning of some key early algebraic concepts such as the equal sign and the commutative property. The results of the task based interview questions were shared with their teachers at the initial meeting. The children were interviewed again at the end of the study.
· Between twelve to sixteen classroom observations were undertaken in each classroom. During these lessons, I was a participant observer. One large video camera was used to record the lesson as a whole and the whole class discussions. Two smaller cameras were used to capture pairs or small groups of students working together on the tasks. Still photographs were also taken during the lesson focusing in particular on student explanations of their reasoning to the whole class. Following the lesson, four students were interviewed individually using photo-stimulated recall. The teacher was also interviewed about the lesson, including their reflections on student reasoning, the developing norms in the classroom and focus of the lesson and future lessons.
· Both groups of teachers were also involved in a lesson study cycle which involved collaborative planning of a lesson, the teaching of the planned lesson while the other teachers observed and a reflective discussion following the taught lesson. The lesson study cycle was undertaken twice at Hillview school and three times at Beaumont school.
· A final interview was undertaken at the end of the data collection with each teacher.
Findings
The findings indicate the need to develop a shared rich understanding of what constitutes early algebra including both content and process. For the two groups of teachers involved in this project developing this shared view of early algebra was achieved through reading and discussing relevant research articles, solving and representing algebraic tasks, collaboratively planning for the classroom and using video recorded classroom episodes to reflect on their practice.
The teachers both modified curriculum material to facilitate opportunities for early algebraic reasoning and also drew upon spontaneous opportunities during classroom activity. To successfully plan for algebraic reasoning, in-depth consideration of the possible pathways of classroom tasks was undertaken. This included predicting potential student responses and misconceptions and then carefully listening to student responses as they worked collaboratively on the tasks and structuring the following whole class discussion to facilitate reasoning.
Changes in the ways the tasks were presented to students also supported a shift in student focus from finding ‘the answer’ to noticing patterns in the tasks and using these to solve the problems or make conjectures. For example, students were asked to solve one problem and then use the solution to solve the next problem or to read a problem and discuss with a partner the patterns which could be used to solve it.
Also of importance was a focus on key content areas which are highlighted in research literature as common areas of difficulty. For example, the initial interview data suggested that the students predominantly viewed the equal sign as an indication to write an answer or perform an arithmetical operation. Therefore the teachers modified the curriculum material to address student understanding of the equal sign and facilitate understanding of the equal sign as relational equivalence. Activities which focused on the equal sign were integrated into the classroom activity throughout the year.
It was recognised by the teachers and researcher that for early algebraic reasoning to be successfully integrated into the everyday classroom activity, it was necessary to develop an appropriate classroom context where student thinking was a focus and discussion was used as a means to facilitate reasoning. An initial focus in the classroom was on developing classroom practice where students worked collaboratively and engaged with each others’ reasoning. Specific pedagogical actions were used by the teachers to facilitate students to engage in collaborative interaction.
Also essential to the development of algebraic reasoning was engaging students in processes such as making conjectures, testing these and justifying their ideas. In the classrooms, there was an expectation that mathematical reasoning would be justified in multiple ways including verbally, visually and numerically. Although these classroom practices are not unique to early algebraic reasoning, they were fundamental in developing a culture where early algebraic reasoning prospered.
Developing early algebraic reasoning in a community of mathematical inquiry led to shifts in the ways that the students participated and engaged with the activity in the classroom. This was evident both through the video recorded lessons, teacher interviews and photo-stimulated interviews with the students. There were changes in the ways in which the students approached tasks. For example, they became competent at using patterns and relationships to solve problems rather than calculating step by step. Another change which was evident in the photo-stimulated recall interviews following the lesson was increased engagement with others’ reasoning during whole class discussions. In the latter stages of the project during the interviews students were able to recall the mathematical reasoning of their peers and describe in-depth how these ideas were shared, investigated and developed.
Summary
The grant provided by ESCalate has supported me to undertake the data collection for this project. It has also supported me to disseminate my research findings through presentations at BSRLM and BERA. I anticipate future conference presentations at the PME conference, ICME conference, CERME conference and MERGA conference which will further disseminate the findings of this project. Additionally the project will form the basis for journal articles for both educational research publications for academics and teachers. This project has also been successful in unanticipated ways as the teachers who were involved have now been invited to speak to teacher trainees at a University close to their school and are also pursuing further education opportunities themselves in mathematics education.
I wish to take this opportunity to thank ESCalate for the invaluable support which I have been provided with.