(Page 2 of 8) - From Primitive Knowing to Formalising: The Role of Student-to-Student Questioning in the Development of Mathematical Understanding authored by Warner, Lisa. and Schorr, Roberta.

of what the concept is, he/she is working in the image making layer. A person working in
this layer is tied to the action or tied to the doing. Working in the image having layer is
when one reaches a “don’t need boundary”, where he/she is no longer tied to the action or the
doing. When a person “…can manipulate or combine aspects of ones images to construct
context specific, relevant properties” and ask themselves how these images are connected,
one is property noticing (Pirie & Kieren, 1994, p.66). When one no longer needs to talk
specific and can make a general statement, he/she is formalising. When formalising, “the
person abstracts a method or common quality from the previous image dependent know how
which characterizes his/her noticed properties” (Pirie & Kieren, 1994, p. 66). This theory is a
way to explain understanding and is a useful tool for understanding how understanding
grows. The structure of the theory is non-linear, repeating itself with many layers wrapped
around.

.In 2003, Warner, Alcock, Coppolo & Davis linked this theory to specific behaviors that

indicate mathematical flexible thought. Briefly stated, a person exhibiting mathematical
flexibility may be characterized as one who displays some or all of the following behaviors:
interpretation of their own or someone else’s idea (e.g. through questioning it and thus
showing it to be valid or invalid; through using, reorganizing or building on it); use of the
same idea in different contexts; sensible raising of hypothetical problem situations based on
an existing problem: creating “What if…?” scenarios; use of multiple representations for the
same idea; connecting representations (Warner, Coppolo & Davis, 2002). In this study, we
will illustrate movement through the first six layers (described above) as we focus on how the
transitions from one layer to the next occurred in association with student-to-student
interactions and questioning. We will also highlight how these student-to student interactions
and questions move students to new representations and the linking of these representations.

METHODS

The study took place over the course of 8 months, which involved two visits a week (50

minutes each session), for the first two months, and 3 to 6 visits a month for the remaining 6
months, in a diverse eighth grade inner city classroom with approximately 30 students. The
visits were part of a professional development project in which the teacher/researcher (first
author), who is a mathematics education researcher at a local university, routinely met with
local teachers, planned classroom implementations, and then modeled or co-taught lessons
with the teacher. After each lesson, the teacher/researcher would “debrief” with the
classroom teacher and a University mathematics education professor (the second author) to
discuss key ideas relating to classroom implementation, the development of mathematical
ideas, and other relevant issues. During the course of the eight months, several different tasks
were explored. The teacher/researcher, along with the classroom teacher encouraged the
students to exchange, talk about, and represent ideas; conjecture, question, justify and defend
solutions; discuss disagreements and differences; revisit ideas over time; and, generalize and
extend their ideas. Generally, the students worked in groups of 3-5, and each group discussed,
argued, and ultimately presented its solutions.

During each class session, two cameras captured different views of the group work, class

presentations and associated audience interaction. In addition, careful field notes were taken
after each session. This study focuses on 8 of the 62 videotapes generated in this manner, as
the students explore variations of a task. The problem task was as follows: John is having a
Halloween party. Every person shakes hands with each person at the party once. Twenty-
eight handshakes take place. How many people are at the party? Convince us.

This particular problem entails a context that may suggest a structure that ultimately leads

to a solution that is generalizable to a larger class of problems. In this case, such a solution is
[n(n-1)]/2.

Episodes were transcribed and coded to identify critical events, which in this case were