(Page 1 of 7) - Does proof prove?: Students' emerging beliefs about generality and proof in middle school authored by Bieda, Kristen., Holden, Christopher. and Knuth, Eric.

DOES PROOF PROVE?: STUDENTS’ EMERGING BELIEFS ABOUT

GENERALITY AND PROOF IN MIDDLE SCHOOL

Kristen Bieda

Christopher Holden

Eric Knuth

University of Wisconsin - University of Wisconsin - University of Wisconsin

Madison

## email not listed ##

This paper presents results from a multi-year research study

that examined the development

of middle school students’ competencies in mathematical reasoning and proof. Written
assessment data collected from 78 middle school students over a period of three years shows
that while improvement does occur from 6

to 8

grades, results indicate that competencies

needed to generate mathematical proof remain to be developed.

Proof has always been a central aspect of the discipline of mathematics and the practice

of mathematicians, but it is a relatively recent aspect of mathematics education for students at
all grade levels. Traditionally, students’ first encounters with proof occurred during high
school geometry courses, where the formal two-column proof was often the only method of
proving students encountered, and explorations of proving general mathematical statements
in other mathematics courses such as algebra were typically not supported. However,
researchers such as Schoenfeld (1994) and Wu (1996) assert that proof cannot be separated
from mathematics and it is an essential part of the process of doing and communicating
mathematics in all content areas. From its inclusion in the 2000 Principles and Standards for
School Mathematics (NCTM), reasoning and proof has gained increased attention as a central
part of mathematics education for students at all grade levels. The 2000 NCTM document
recommended that students be encouraged to view reasoning and proof as fundamental
aspects of mathematics, know how to make and test conjectures, and evaluate and select
various types of reasoning and methods of proof. Existing research, however, indicates that
students’ understandings of proof are weak in light of these recommendations (e.g.,
Balacheff, 1988; Bell, 1976; Healy & Hoyles, 2000; Porteous, 1990; Senk, 1985).
Understanding the notion that a proof treats the general case is critical for students’ success in
evaluating and generating mathematically correct proofs. A number of researchers have
documented evidence that students tend to view empirically-based arguments as sufficient
justification for demonstrating the truth of a mathematical argument (see Porteous, 1990;
Fischbein & Kedem, 1982; Balacheff, 1988, Healy & Hoyles, 2000). Existing research,
however, has not studied students’ competencies in proving over a cohesive grade band (e.g.,
middle school or high school). The purpose of this paper is to present results from a
longitudinal study of middle-school students’ conceptions of proof. We explore how
students’ understandings of proof change during their middle school education by exploring
the following questions related to notions of generality: Do students tend to generate
empirically-based arguments or proof-like arguments to justify a mathematical statement?
and To what extent do students recognize that a proof treats the general case?

PROOF FRAMEWORK

Researchers have hypothesized that the development of students’ proving competencies

might follow a developmental progression and, indeed, various frameworks have been
proposed that reflect such a developmental progression (e.g., Balacheff, 1988; Bell, 1976;
van Dormolen, 1977; Waring, 2000). Building upon these aforementioned frameworks, we
used four levels of proof concept development: Level 0 – students respond either “I don’t
know” or give information already presented in the problem (for this paper, Level 0 also
includes non-codable and no response/I don’t know); Level 1 – students consider checking a