Commentary on Theories of Mathematics Education

DNR-Based Instruction in Mathematics as a Conceptual Framework 349
Fig. 3 Another trial with
integer dimensions
Fig. 4 Trial with rational
dimensions
Fig. 5 Trial with irrational
dimensions
Fig. 6 Dimensions
represented by algebraic
expressions involving a
variable
11. The teacher indicated this fact to the students, which prompted them to offer
fractional values and later irrational values. Figures 4 and 5 depict these exchanges
(with the value 2/3 and √ 2, respectively).
12. After these repeated attempts, some students expressed doubt as to whether
dimensions for the four regions that preserve the given configuration can be
found. The teacher responded by recapitulating the solution process carried out
thus far, and concluded with the following question, which he wrote on the
board:
Can a figure representing the problem conditions be constructed for A =
100?
The students were asked to work on this question in their small working
groups.
13. A few minutes later, one of the groups suggested searching for the length of A
by substituting a variable t for it. The teacher followed up on this suggestion.
An outline of the exchange that ensued follows (Fig. 6).