Commentary on Theories of Mathematics Education

<strong>Commentary</strong> on DNR-Based Instruction
in Mathematics as a Conceptual Framework
Bharath Sriraman, Hillary VanSpronsen,
and Nick Haverhals
In the domain of proofs in mathematics education, the DNR 1 theory created by
Guershon Harel attempts to bridge the epistemologies of what constitutes proofs in
professional mathematics and mathematics education. The DNR theory is one that
has been gradually developed. In one of the “early” papers that proposed this theory,
Harel (2006a) wrote:
Pedagogically, the most critical question is how to achieve such a vital goal as helping
students construct desirable ways of understanding and ways of thinking. DNR has been
developed to achieve this very goal. As such, it is rooted in a perspective that positions
the mathematical integrity of the content taught and the intellectual need of the student at
the center of the instructional effort. The mathematical integrity of a curricular content is
determined by the ways of understanding and ways of thinking that have evolved in many
centuries of mathematical practice and continue to be the ground for scientific advances. To
address the need of the student as a learner, a subjective approach to knowledge is necessary.
For example, the definitions of the process of “proving” and “proof scheme” are deliberately
student-centered. It is so because the construction of new knowledge does not take place in
a vacuum but is shaped by one’s current knowledge. (p. 23, pre-print)
Harel’s views in a sense echo the recommendations of William Thurston, the 1982
Fields medal winner, whose article On Proof and Progress (see Hersh 2006) gets
widely cited and reprinted in both the mathematics and the mathematics education
communities. Thurston outlines for the lay person:
(1) what mathematicians do
(2) how (different) people understand mathematics
(3) how this understanding is communicated
(4) what is a proof
1 DNR = duality, necessity,andrepeated-reasoning.
B. Sriraman () · N. Haverhals
Department of Mathematical Sciences, The University of Montana, Missoula, USA
e-mail: sriramanb@mso.umt.edu
N. Haverhals
e-mail: nicolas.haverhals@umontana.edu
H. VanSpronsen
Department of Mathematical Sciences, Michigan Technological University, Houghton, USA
e-mail: hdvanspr@mtu.edu
B. Sriraman, L. English (eds.), Theories of Mathematics Education,
Advances in Mathematics Education,
DOI 10.1007/978-3-642-00742-2_35, © Springer-Verlag Berlin Heidelberg 2010
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