Commentary on Theories of Mathematics Education

298 A. Zollman
Even though all national and international documents and reports cite problem
solving as the critical objective, English and Sriraman mention “the 800-pound gorilla
in the room” that limits research. This gorilla, of course, is high-stakes testing.
Studies now discuss the mathematics scores on international and state tests, rather
than core concept development and transfer of problem-solving learning. Teachingfor-the-test
has supplanted teaching-for-problem solving.
The second limiting factor (Begle 1979) is that traditionally mathematics first
is developed then applied to problems. English and Sriraman want more research
on problem-driven conceptual development—that is, concept development through
problem solving. They want to see research on generation of important mathematics,
not just the application of previously taught mathematical procedures and concepts.
A further limit on problem-solving research, according to English and Sriraman,
is our limited knowledge of students’ problem solving beyond the classroom. They
want, again, much more research on problem solving, out of the mathematics subject
area context and into other subjects and into the real world. How does problem
solving transfer to new contexts?
Richard Shumway (1980) adapted Bernard Forscher’s essay “Chaos in the Brickyard”
(1963) to mathematics education research. Research makes a lot of very nice
bricks (research studies), but there is not a design plan of what kind of bricks to
make and where each individual brick should be placed in relation to other bricks to
build something substantial and useful. So it still is today, lament English and Sriraman,
as their last limiting factor. Although they do not advocate a grand theory of
problem solving, they refer to the isolated nature of research studies as a hindering
factor in building a cohesive knowledge depository.
The remaining section of the chapter discusses advancing the fields of problemsolving
research and curriculum development. First is a brief into the nature of problem
solving in the 21 st century. English and Sriraman are optimistic a new perspective
on problem solving is emerging—one that is interdisciplinary and applicable to
real-world work situations.
English and Sriraman suggest a future-oriented perspective on problem solving
that transcends current school curricula and national standards. To this end, they
suggest adopting the Lesh and Zawojewski (2007) definition of a problem, namely:
A problem occurs when a problem solver needs to develop a more productive way of
thinking about a given situation. In this definition the key word is develop, an action.
As English and Sriraman suggest, a more productive way of thinking involves an
iterative cycle of describing, testing, and revising mathematical interpretations while
identifying, integrating, modifying and refining mathematical concepts.
Again, English and Sriraman suggest clarifying the connections between the development
of mathematical concepts and the development of problem-solving abilities.
This would allow a powerful, alternative approach of using problem solving to
develop substantive mathematical concepts. This alternative approach differs from
both: (a) the traditional approach requiring concepts and procedures to be taught
first, then practiced through solving story problems (a content-driven perspective),
and (b) the traditional approach presenting students with a repertoire of problem-