(Page 1 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.

FROM PRIMITIVE KNOWING TO FORMALISING:

THE ROLE OF STUDENT-TO-STUDENT QUESTIONING IN THE

DEVELOPMENT OF MATHEMATICAL UNDERSTANDING

Lisa Warner

RobertaY. Schorr

## email not listed ##

Rutgers University-Newark

In this paper, we examine the development of inner city middle school students’ ideas and the
student-to-student interactions and questions that contribute to this development within the
context of the Pirie/Kieren model. We analyze data collected from

an inquiry oriented,

problem based mathematics class in which students were repeatedly challenged to explain
their thinking to each other, and defend and justify all solutions. In this instance, we
document how one student was able to move from primitive knowing to formalising. Further,
we note that this student (and her classmates) were able to use this knowledge several months
later when solving a structurally similar problem.

OBJECTIVES/PURPOSES

Prompting students to talk about mathematics is an important goal of education (NCTM

2000; Sfard, 2000; Dorfler, 2000; Cobb, Boufi, McClain, and Whiteneck, 1997). Cobb,
(2000) notes that student exchanges with others can constitute a significant mechanism by
which they modify their mathematical meanings. Carpenter and Lehrer, (1999) state that “the
ability to communicate or articulate one’s ideas is an important goal of education, and it also
is a benchmark of understanding.” (p. 22) Researchers such as those cited above (and others,
see for example, Schorr, 2003; Maher, 2002; Shafer and Romberg, 1999) maintain that it is
important to provide students with opportunities to discuss their ideas with each other, defend
and justify their thinking both orally and in writing and reflect upon the mathematical
thinking of others. One important component of this involves students’ questioning the
mathematical thinking of their peers. This report focuses on the impact of student questioning
on the development of mathematical thinking. We do this within the context of the
Pirie/Kieren theory for the growth of mathematical understanding (Pirie and Kieren, 1994).

Our central premise is that when students have the opportunity to question each other

about their mathematical ideas, both the questioner and the questioned have the opportunity
to move beyond their initial or intermediate conceptualizations about the mathematical ideas
involved. As students reflect on their own thinking in response to questions that are posed by
their peers they have the opportunity to revise, refine, and extend their ways of thinking about
the mathematics. As they do this, their earlier conceptualizations and representations become
increasingly refined and linked. We stress the role of representations in this dynamic since
“the ways in which mathematical ideas are represented is fundamental to how people can
understand and use those ideas.” (NCTM, p.67) In this paper, we will trace the development
of ideas (using the Pirie/Kieren model) and the student-to-student interactions and questions
that contribute to this development.

THEORETICAL FRAMEWORK

In 1988, Pirie discussed the idea of using categories in characterizing the growth of

understanding, observing understanding as a whole dynamic process and not as a single or
multi-valued acquisition, nor as a linear combination of knowledge categories.

In 1994, Pirie

& Kieren described eight potential layers or distinct modes within the growth of
understanding for a specific person, on any specific topic. The inner-most layer, called
primitive knowing, is what a person can do initially and is the starting place for the growth of
any particular mathematical understanding. When a person is doing something to get the idea