Commentary on Theories of Mathematics Education

<strong>Commentary</strong> on Problem Solving Heuristics, Affect, and Discrete Mathematics 253
(2000), worthwhile problems should be intriguing, with a level of challenge that invites
exploration, speculation, and hard work. Mathematical problems that are truly
problematic and involve significant mathematics have the potential to provide the
intellectual contexts for students’ mathematical development.
Instructional Objectives
Goldin distinguished two different types of instructional objectives. The first is
Domain-specific, formal objectives—the desired competencies with conventionally
accepted mathematical definitions, notations, and interpretations. The second is
Imagistic, heuristic, and affective objectives—the desired capabilities for mathematical
reasoning that enable flexible and insightful problem solving, but they are
not tied directly to particular notational skills. Then he used his representational
model for mathematical learning and problem solving based on five kinds of internal
representational systems to discuss how the two types of instructional objectives
are related to the five internal representations. He emphasized the importance of
achieving the second type of instructional goals and strongly argued that discrete
mathematics offers the best possibilities to achieve both types of goals.
The question is not which type of instructional objectives is more important because
both types of objectives are obviously important. Perhaps, the real question is
how these two types of instructional objectives when interwoven can be achieved.
Even though Goldin suggests that discrete mathematics offers the best possibilities
to achieve both types of goals, he offers few specifics to actually implement these
goals together in classroom.
These two types of instructional objectives can be translated into the development
of basic mathematical skills and development of high-order thinking skills.
Obviously, both basic skills and high-order thinking skills are important in mathematics,
but having basic skills does not imply having higher-order thinking skills
or vice versa (e.g., Cai 2000). Traditional ways of teaching—involving memorizing
and reciting facts, rules, and procedures, with an emphasis on the application of
well-rehearsed procedures to solve routine problems—are clearly not adequate to
develop basic mathematical skills. Is it the case that students learn algorithms and
master basic skills as they engage in explorations of worthwhile, mathematically
rich, real-world problems? Or in general, how can the development of basic mathematical
skills intertwined with and support the development of higher-order thinking
skills? These questions are critical for future research in mathematics education, in
general, and problem solving research, in particular.
In discussing these two goals, in addition, we should not ignore the testing effect.
In the current testing practice world wide, the main focus has been on the first
instructional objectives. Because of various practical reasons, this testing practice
has such a huge effect that it becomes a driving force and directs classroom instruction.
That is, if the testing does not include the assessment of the second instructional
objectives, it is unlikely that classroom instruction will address the second
set of objectives, even in the discrete mathematical domain. In other words, even