Investigating students' opportunities to develop proficiency in reasoning and proving: A curricular perspective
Stylianides, A.J.
PhD Thesis, University of Michigan, Ann Arbor, 2005
Despite widespread agreement that all students should have opportunities to
acquire proficiency in reasoning and proving in school mathematics, too little
is known about how to design instruction to accomplish this goal. Curriculum
materials undoubtedly have an important role to play in providing tasks for
students' and teachers' classroom work, and in providing guidance for teachers
in developing students' capacity in reasoning and proving. At this time, however,
we lack detailed knowledge of how reasoning and proving is treated in contemporary
curriculum materials. This study takes step toward addressing this limitation,
with particular attention to one reform middle school curriculum and two research
questions:
(1) What opportunities are designed in the curriculum materials for students
to engage in reasoning and proving?
(2) In what ways do the curriculum materials appear to be designed for helping
students develop a capacity for both inductive and deductive modes of reasoning?
This study includes both conceptual and empirical analyses. I begin by defining
reasoning and proving so that it is both honest to the discipline of mathematics
and useful in a curriculum analysis. In brief, I define reasoning and proving
to encompass the breadth of the activity associated with identifying patterns,
making conjectures, providing proofs, and providing non-proof arguments. Using
my definition of reasoning and proving and the findings of pertinent educational/psychological
research, an analytic framework is proposed to investigate the opportunities
that are designed in the mathematics curriculum for students to engage in reasoning
and proving. The analytic framework is used to analyze the algebra, geometry,
and number theory units of Connected Mathematics. Also examined is the support
the curriculum materials provide for teachers to enact the proof opportunities
designed.
Some major findings from my curriculum analysis are the following: (1) about
40% of the tasks are designed to engage students in reasoning and proving; (2)
many more opportunities are designed for students to use inductive than deductive
modes of reasoning; (3) reasoning and proving opportunities are distributed
unevenly across grade levels and content areas; and (4) very limited guidance
is provided to teachers to support their use of proof tasks.
The framework and the empirical findings contained herein have potential implications
for curriculum design, and they provide pointers to possible links between students'
difficulties in reasoning and proving and the treatment of this practice in
curriculum materials. Also, the framework and associated methodology can provide
a platform for conducting detailed analyses on the interplay of curriculum materials
and teaching/learning in reasoning and proving.