Commentary on Theories of Mathematics Education

<strong>Commentary</strong> 2 on Problem Solving for the 21 st Century 299
solving heuristics/strategies to be applied to non-routine problems (a problemsolving
perspective).
Teaching via mathematical modeling is the alternative approach promoted by
the chapter authors. Our everyday world is a commoditization of complex social,
political and commercial systems. Sophisticated technologies and ever-increasing,
multifaceted real-world situations demand a type of problem-solving ability beyond
the traditional classroom approach.
A mathematical model is defined as a system of elementary operations, relationships,
and standards used to describe, explain, or predict the behavior of another familiar
system (Doerr and English 2003). Mathematical modeling is a process beginning
with a realistic complex situation to accomplish some goal, usually generating
Byzantine artifacts or conceptual tools (Lesh and Zawojewski 2007). Traditionally,
mathematical modeling only is taught at the post-secondary level.
The chapter authors present five advantages of students doing mathematical modeling
at the elementary levels. First, mathematical modeling allows students to experience
quantities (e.g., probabilities) and operations (e.g., weighing data sets) in
realistic situations. Second, mathematical modeling offers students richer learning
experiences, allowing students to create their own mathematical constructs. Third,
mathematical modeling explicitly uses multi-disciplinary situations. Fourth, mathematical
modeling encourages students to create generalized models, applicable to a
range of related circumstances. Last, the authors argue that mathematical modeling
at the elementary level is designed for small-group work.
English and Sriraman present, in considerable detail, two examples of mathematical
modeling at the elementary level. The first example is the scenario The
First Fleet, a fifth-grade Australian recreation of locating the first settlement for
the British fleet in 1788. This four-session unit provides students with background
information, data listings of environmental suitability elements and inventories of
equipment, livestock and plants/seeds. Students are to make and present a mathematical
model that best selects the optimal site for the settlement from five possible
sites.
The chapter further describes a study using First Fleet in fifth-grade classrooms
in Brisbane, Australia. Illustrated is one group of students’ cyclic development of:
prioritizing problem elements, ranking elements across sites, proposing conditions
for settlement, weighing elements, reviewing the models, and finalizing site selection.
In the group, students identified and prioritized key problem elements, explored
relationships between elements, quantified qualitative data, ranked and aggregated
data, and created weighted scores—all before being introduced to these processes.
The second example of mathematical modeling at the elementary level is an application
of statistical reasoning. Data modeling engages elementary students in
extended situations where they generate, test, revise and apply models to solving
real-world problems. An example is exploring the growth of a flower bulb. Data
modeling allows for problem posing, generating attributes, measuring attributes, organizing
data, representing data, and drawing inferences.
The chapter concludes that the research in mathematical problem solving is stagnant.
English and Sriraman argue the time has come to consider other options for ad-