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 21 st
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.