Commentary on Theories of Mathematics Education

252 J. Cai
For example, Lesh and Zawojewski (2007) called for a models-and-modeling perspective
as an alternative to existing views on problem solving.
The chapter by Goldin in this volume can be viewed as an attempt to suggest
one future direction of problem solving in school mathematics. The major theme in
Goldin’s chapter is that the domain of discrete mathematics has a great potential to
increase students’ interest in exploring mathematics and develop problem-solving
heuristics. In this commentary, I would like to situate my discussion of his chapter
in the context of future directions of mathematical problem-solving research. I will
start with a discussion of Goldin’s idea about discrete mathematics as a content domain
to solve problems, then I will specifically discuss instructional objectives and
problem-solving heuristics in the discrete mathematical domain. I end this commentary
by pointing out a future direction for problem-solving research.
Discrete Mathematical Domain and Problem Solving
In his chapter, Goldin used a dichotomy of “traditional mathematical topics” and
discrete mathematics to make his arguments that the domain of discrete mathematics
provides better opportunities for mathematical discovery and interesting nonroutine
problem solving than does traditional mathematical topics. According to Goldin,
students are “turned off” by traditional school mathematics, but discrete mathematics
comes to the rescue because experiences in discrete mathematics may provide
novel opportunities for developing powerful heuristic processes and powerful affects.
Goldin further provided several sequential reasons why experiences in discrete
mathematics may provide novel opportunities for developing powerful heuristic
processes and powerful affects. For example, discrete mathematics involves fewer
particular formulas and techniques. Because of involving fewer particular formulas
and techniques, there are more opportunities to create interesting and nonroutine
problem solving activities for students to explore. Because of the nonroutine nature
of problems in discrete mathematics, these problems not only invite students
to engage in the problem-solving activities, but also the success of solving these
problems in discrete mathematics “can reward the engaged problem solver with a
feeling of having made a discovery in unfamiliar mathematical territory.”
Since Goldin’s chapter was originally published in a ZDM special issue related
to discrete mathematics (Volume 36, Number 2, 2004), it seems to be understandable
that he would use the dichotomy of “traditional mathematical topics” and discrete
mathematics to make his arguments. On the other hand, it is questionable and
unproductive to use a dichotomy of “traditional mathematical topics” and discrete
mathematics to make his arguments. However, the essential message is clear that
the kinds of problems used in classrooms matter when engaging students in problem
solving and learning about mathematics. Tasks with different cognitive demands are
likely to induce different kinds of learning (Doyle 1988). Only worthwhile problems
give students the chance to both solidify and extend what they know and to stimulate
their learning. According to the National Council of Teachers of Mathematics