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Vol.2-430 PME-NA 2006 Proceedings
STUDENTS STRATEGIES FOR CONSTRUCTING MATHEMATICAL PROOFS IN
A
PROBLEM-BASED UNDERGRADUATE COURSE
Jennifer Christian Smith
University of Texas at Austin
[email protected]
This paper reports the results of an exploratory study of
students proof strategies in the context
of a problem-based undergraduate number theory course. The
students strategies for
constructing proofs varied depending on context, but our
analysis demonstrated that they were
primarily engaged in making sense of the mathematics, rather
than attempting to reproduce
particular proof types or strategies.
Mathematics educators and education researchers have reported
students difficulties with
mathematical proof and point out the conflict between the nature
of this essential mathematicalactivity and current approaches to
teaching it (Hanna, 1991; Harel & Sowder, 1998; Moore,
1994; Raman, 2003; Selden & Selden, 1995; Usiskin, 1980;
Weber, 2001). This recent interest
has led to an increased effort to teach proof in innovative
ways. One instructional approach thatemphasizes student-centered
learning processes is inquiry-based or problem-based teaching,
in which the central activity of the course is to engage
students in mathematical inquiry or
problem solving, rather than to present them formal mathematics
in the form of a lecture.Our study focused on students proving
processes and strategies for constructing proof in
inquiry-based undergraduate number theory course at a large
state university. In this particular
course, students worked outside of class to solve problems and
prove theorems. The students
then presented their solutions during class meetings and the
instructor led whole-groupdiscussions of their work. The role of
the instructor in the course was that of facilitator and
advisor; he did not lecture or present himself as the arbiter of
mathematical truth. The students
in the course were expected to determine the correctness of the
presented solutions through their
discussion. The course served as a transition to proof course at
the university, so for most ofthe students, this was their first
course in which formal mathematical proof was the primaryfocus. The
instructor did not tell them what constituted a mathematical proof;
rather, he expected
the students to construct an understanding of proof by
participating in the course.
In order to closely examine how students in this non-traditional
context learned to constructmathematical proofs, we employed an
exploratory case study design guided by two research
questions: What processes do undergraduate students employ when
proving mathematical
statements? What strategies do students use to construct
mathematical proofs in an inquiry-basedundergraduate course? Six
students were selected from those enrolled in the
above-described
course. During a series of four task-based interviews conducted
over the span of the semester of
study, the students were asked to construct proofs of various
number theory statements.
The interview transcripts were analyzed using open coding
techniques, and a framework ofinterconnected paths for proof
emerged. A visual representation of these paths is shown in
Figure 1. Students proving processes consisted of four phases:
use of initial strategies,
construction of informal arguments, construction of a formal
proof, and validating or reflectingon the final proof/argument. Our
results demonstrated that students proving processes were not
necessarily hierarchical in nature, but shifted fluidly and
frequently between these phases. The
students were primarily engaged in making sense of mathematics
and often used concrete
_____________________________
Alatorre, S., Cortina, J.L., Siz, M., and Mndez, A.(Eds) (2006).
Proceedings of the 28th
annual meeting of the
North American Chapter of the International Group for the
Psychology of Mathematics Education. Mrida, Mxico:
Universidad Pedaggica Nacional.
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Reasoning and Proof Vol.2-431
examples to understand the statement, clarify the strategy or
gain further insight into theproblem.
Reflection was an important component in students progress in
constructing proofs and
appeared to be the mechanism by which students shifted from one
phase of the proving processto another. We found that the students
did not tend to reflect on the final proof or argument
spontaneously, so the influence of our intervention was included
in the analysis.Overall, our results also showed that individual
students strategies for constructing proofs
varied greatly, in contrast to the more static tendencies for
proof frequently seen discussed in the
literature. The participants in our study did not seem to prefer
the same sequence of phases foreach proof attempted, nor did they
appear to use the same strategies each time. We hypothesize
that the problem-based structure of the courses facilitated the
development of their relatively
flexible and sophisticated strategies for proof.
References.
Hanna, G. (1991). Mathematical proof. In D. Tall (Ed.), Advanced
Mathematical Thinking (54-61). Dordrecht, The Netherlands: Kluwer
Academic Publishers.
Harel, G. & Sowder, L. (1998). Students' proof schemes:
Results from exploratory studies.
CBMS Issues in Mathematics Education, 7, 234-283.Moore, R. C.
(1994). Making the transition to formal proof.Educational Studies
in Mathematics.
27, 249-266.Raman, M. (2003). Key ideas: What are they and how
can they help us understand how people
view proof?Educational Studies in Mathematics, 52, 319-325.
Selden, J & Selden, A. (1995) Unpacking the logic of
mathematical statements. Educational
Studies in Mathematics, 29, 123-151.
Usiskin, Z. (1980) What should not be in the algebra and
geometry curricula of average college-bound students?Mathematics
Teacher, Spring 1980, 413-424.
Coordination possiblevia reflection (R)
or intervention
Choose
strategy
Startover
(b)
Initial Strategies1
Construction
of Informal
Argument2
Constructionof Formal
Proof3
Validate or Reflect onFinal Proof/Argument4
Deeperinsights
(a)
Interviewer
Intervention(I1, I2, I3, IR)
Figure 1: Students Proof Paths
