Commentary on Theories of Mathematics Education

254 J. Cai
though problems in discrete mathematics show promise in providing “the pathways
and structures whereby previously unsuccessful students come to feel, ‘I am really
somebody when I do mathematics like this,”’ the promises can hardly become reality
if the imagistic, heuristic, and affective objectives are not valued in instruction
and testing.
Problem-Solving Heuristics
Goldin argued that a discrete mathematical domain provides a good opportunity to
develop the heuristic processes, such as “trial and error,” “draw a diagram,” “think
of a simpler problem,” and so forth. On one hand, because of the nature of the tasks
in discrete domain, I agree with Goldin’s argument. Students do have more opportunities
to use heuristic processes in a discrete mathematical domain. On the other
hand, will the opportunities actually improve students’ problem solving skills and
learning related mathematical concepts? Goldin did not develop explicit arguments
to address this question. As we know, research has indicated that teaching students
to use general problem-solving strategies and heuristics has little effect on students’
being better problem solvers (e.g., Begle 1973; Charles and Silver 1988; Lesh and
Zawojewski 2007;Lester1980; Schoenfeld 1979, 1985, 1992; Silver 1985). In fact,
the evidence has mounted over the past 40 years that such an approach does not improve
students’ problem solving and learning of mathematics to the point that today
no research is being conducted with this approach as an instructional intervention.
If the teaching of general problem-solving heuristics has little effect on improving
students’ problem-solving skills, what is the benefit of teaching these heuristics?
A more general question is: what shall we teach to help students becoming better
problem solvers?
Teaching Mathematics Through Problem Solving: A Future
Direction of Problem Solving Research
Even though it has been called to have problem solving be integrated throughout
the curriculum, in reality problem solving has been taught as a separate topic in
the mathematics curriculum. Teaching for problem solving has been separated from
teaching for learning mathematical concepts and procedures. Lesh and Zawojewski
(2007) challenged the oft-held assumption that a teacher should proceed by:
First teaching the concepts and procedures, then assigning one-step “story” problems that
are designed to provide practice on the content learned, then teaching problem solving as a
collection of strategies such as “draw a picture” or “guess and check,” and finally, if time,
providing students with applied problems that will require the mathematics learned in the
first step. (p. 765)
While there is mounting evidence that such an approach does not improve students’
problem-solving skills, there is mounting evidence to support thinking of