Commentary on Theories of Mathematics Education

Problem Solving for the 21 st Century 269
section, vary across nations with insufficient recognition of, or communication between,
the various research hubs addressing this important form of problem solving.
The long-standing work on modeling in some European countries (e.g., Kaiser
and Sriraman 2006; Kaiser and Maass 2007) and the substantial research on interdisciplinary
model-eliciting activities in the USA and Australia (e.g., Lesh 2008;
English 2009a) remain in many ways isolated from one another. While different
hubs of research on models and modelling are making substantial advancements,
such as improving engineering education (e.g., Zawojewski et al. 2008), one wonders
what further achievements could be made if the knowledge across hubs were
more accumulative. Nevertheless, the research on models and modelling is becoming
more interdisciplinary and is providing new opportunities for improving classroom
problem solving.
Problem-solving research has also failed to accumulate adequately with respect
to theory advancement and subsequent implications for the classroom (Lesh 2008).
While we do not advocate the production of a “grand theory” of problem solving,
we suggest that mathematics education researchers work more collaboratively in
building a cohesive knowledge bank—one that can help us design more appropriate
21 st century problems and one that can provide tools that enable us to more reliably
observe, document, and assess important mathematical developments in our
students.
Advancing the Fields of Problem-Solving Research
and Curriculum Development
The Nature of Problem Solving in Today’s World
Although we have highlighted some of the issues that have plagued problem-solving
research in past decades, there are emerging signs that the situation is starting
to improve. We believe the pendulum of change is beginning to swing back towards
problem solving on an international level, providing impetus for new perspectives
on the nature of problem solving and its role in school mathematics
(Lester and Kehle 2003). For example, a number of Asian countries have recognized
the importance of a prosperous knowledge economy and have been moving
their curricular focus toward mathematical problem solving, critical thinking,
creativity and innovation, and technological advances (e.g., Maclean 2001;
Tan 2002). In refocusing our attention on problem solving and how it might become
an integral component of the curriculum rather than a separate, often neglected,
topic we explore further the following issues:
• What is the nature of problem solving in various arenas of today’s world?
• What future-oriented perspectives are needed on the teaching and learning of
problem solving including a focus on mathematical content development through
problem solving?