Commentary on Theories of Mathematics Education

<strong>Commentary</strong> on DNR-Based Instruction in Mathematics as a Conceptual Framework 377
literature that arose in the 1980s, only to learn later that implementing change in the
teaching and learning of problem solving was a much greater task than originally
anticipated (c.f. Schoenfeld 1992).
There is an additional issue in implementation, even if we disregard the actual
training of teachers to successfully achieve the goals set out by DNR, of the apparent
extra time necessary to properly allow students to work through the process of
self-motivation, discovery, and attending to the intellectual and instructional needs
and principles described by Harel. In the ever-shrinking time allotted in our classes,
both at the secondary and post-secondary levels, with increasing demands on the
content to be addressed, is this time demand feasible? Can we achieve these seemingly
important goals of duality, necessity, and repeated reasoning, and also the content
specific goals in our areas? Until further research gives us an indication of what
a typical classroom using DNR would look like, it is impossible to say whether the
time needed for this inquiry-based, discovery-based learning is worth the reward.
There are other issues that will not be answered until instructional methodologies
are designed, implemented, and studied, such as how much repetition is necessary
to achieve our goals, how we move our students through this process, and whether
these methods address multiple learning styles. These are not flaws in the framework,
necessarily, but rather things to consider while moving forward within the
framework.
All this being said, both positive and negative views in focus, we believe that
DNR-based instruction does present a new conceptual framework of value in mathematics
education research, particularly in the domain of proof writing. There are
larger issues to consider that are not yet resolved, however the ability of this theory
to bring together the major ideas related to instruction in mathematics proof writing
give it the potential to be a strong backing for future research and a reasonable
theory upon which to construct new teaching tools and research designs.
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