Commentary on Theories of Mathematics Education

<strong>Commentary</strong> 2 on Problem Solving
for the 21 st Century
Alan Zollman
We are attempting to educate and prepare students today so that
they are ready to solve future problems, not yet identified, using
technologies not yet invented, based on scientific knowledge not
yet discovered.
Professor Joseph Lagowski, University of Texas at Austin
Discussing problem solving for this century is a daunting task—a real “problem.”
English and Sriraman present a look at the past research on mathematical problem
solving and its impact on the curriculum. They identify several limiting factors affecting
problem-solving research and propose ideas to advance this research.
The first section of English and Sriraman’s chapter is a brief yet precise chronological
view of the relative short history of mathematical problem-solving research.
Polya (1945) in his precedent-setting book How to Solve It is credited as the key
figure who began the investigations for assisting students to problem solve in mathematics.
The teaching of strategies to problem solve in mathematics are rooted in
Polya’s heuristics of understand the problem, devise a plan, carry out the plan, and
look back. Based upon Polya’s writings, suggested problem-solving strategies of
work backwards, draw a picture, look for a related problem, guess and test, make an
organized table, are in every elementary textbook in the United States.
In books and reports on problem-solving research, from Begle (1979) to Silver
(1985) to Schoenfeld (1992) to Lester and Kehle (2003) to Lesh and Zawojewski
(2007), English and Sriraman point out all have a consistent, disconcerting finding:
the classroom teaching of problem-solving strategies and heuristics does little to improve
students’ problem-solving abilities. English and Sriraman argue the research
community needs to go beyond the questions of “Can we teach heuristics?” and “Do
these have an impact on students’ abilities?” They argue for more prescriptive, than
descriptive, questions such as: “What does it mean to understand?”, “How do these
understandings develop?”, “How can we measure these developments?” and “How
can we effectively integrate core concept development with problem solving?”
English and Sriraman also identify a decline in the recent literature on problem
solving and concept development via problem solving. Their next section of the
chapter deals with four limiting factors affecting problem-solving research.
A. Zollman ()
Department of Mathematical Sciences, Northern Illinois University, DeKalb, USA
e-mail: zollman@math.niu.edu
B. Sriraman, L. English (eds.), Theories of Mathematics Education,
Advances in Mathematics Education,
DOI 10.1007/978-3-642-00742-2_29, © Springer-Verlag Berlin Heidelberg 2010
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