Commentary on Theories of Mathematics Education

212 S.J. Hegedus
The authors then build the idea that the concept of semiotic mediation is based
upon the semiotic capacity of thinking. First, the use of formal notations (as per
abstraction) for formal mathematical reasoning, and secondly, manipulation of signs
that are decontextualized from the origins of meaning. This construct helps us see
the relevance of the “seeing-through” metaphor and the role of semiotic mediation as
a theoretical and methodological construct. Through analysis, we need to establish
where people establish meaning, what they see and whether they see through to a
meaningful interpretation via their media. So what does the media do to enhance or
constrain semiotic mediation?
The paper provides an excellent overview of relevant literature from the evolution
of sign systems and tools which helps introduce the theory of semiosis. There is also
a strong theoretical synthesis of the works of Rabardel, Vygotsky and Wertsch.
Finally, the authors offer extremely rich and new theoretical constructs that I
believe will aid students and researchers for many years. First, the construct of “Domains
of Abstractions.” This focuses on an environmental appreciation of the cognizing
individual operating within the proximal media. As the authors eloquently
propose, “it is the environment where a general idea finds a dress that makes it
visible at the eyes of the students.” Second, the construct of “executable representations.”
Here the authors not only describe the physicality of certain new digital
forms of expression, but also the experience that a user (student) might have in
reflecting-upon, and internalizing for future use, the crystalline or organic structure
that a dynamic form has in a mathematical environment. This is then extended to
some basic educational ideas. We need to enable our students to see-through mathematical
structures and by offering them dynamic, digital environments we extend
the semiotic space and offer them much more opportunity to think, see, and accommodate
meaning.
In conclusion, I believe this paper has much to offer in terms of offering researchers
and students the right kinds of theoretical lenses to observe significant
aspects of learning, especially with technology, but I also think that there is an untapped
academic space. Here we need to focus on pre-service, or early education
programs, where teachers of tomorrow understand and implement such theories into
their preliminary practice.