Commentary on Theories of Mathematics Education

390 S.J. Hegedus
should be intimately bound up with the scientists establishing the epistemological
identity. As the hypo-thesis is refuted through a suitable methodology being constructed
with respect to the epistemological deficiencies, thus a thesis is established
and the reflexive procedure gains effect. Through axiomatic bases, a methodology
proves suitable if algebraic interpretations of the qualitative data can be made with
respect to inferred observations. A reflexive procedure, which observes the constructing
process and the nature of the epistemologist in this procedure, is necessary.
If not, the process will be corrupted and the circular process of scientific discovery
will break down. As the discovery is made, the process will generate a syn-thesis.
For the procedure to be made scientific, rigorous and effective, the scientist must acknowledge
his epistemological identity. The following set of questions offer a nonexhaustive
account of generic lines of knowledge inquiry, which might be made:
• Can I realize my position in the process of scientificity?
• Can I react to my position in this process?
• Am I associating assimilation of knowledge and synthesis of a scientific procedure
as a reflection of socially-constructed reality?
• What is the nature of the knowledge I have established?
• Have I answered the questions I have deemed necessary to be established?
• How does my knowledge exist?
• What are the assumptions made about this knowledge?
• What are my questions?
• Are my questions convincing?
• Are my answers convincing? I.e., do they use a suitable methodology? To answer
this, have I assessed my method of data collection? Is it scientific with respect to
a reflexive practice?
• Have I assessed the limitations of the ‘new’ knowledge I am generating?
• How does my epistemological identity affect my ‘new’ knowledge?
• Have I assessed my ‘truth values’ with respect to the constructionist phenomena
I am involved in?
The Dynamic
In recognizing a 3-fold doctrine, one is not adhering to mechanistic attributes of the
process of scientific discovery. It is deemed necessary that a dynamic is inherent in
the process of scientific discovery by its very nature of it being circular and motional.
The circular nature of the process will be outlined and illustrated at the end
of this section but let me attend to the very principle of the dynamic. The dynamic
is ‘motion’; it is that part of the nature of ‘new’ knowledge which sets it apart from
‘old’ knowledge and is given force by nature of the epistemological identity. Underlying
the mechanism of discovery which allows a hypo-thesis to be refuted and
a thesis to be established within a methodological and epistemological construct;
it is the dynamic of knowledge which allows syn-thesis and transfers knowledge
into ‘advanced’ knowledge. When this process allows re-generation into ‘advanced’