| The researcher characterized the pragmatic reasoning of students' mathematics learning using certain technology. A "Natural Mathematics computer Interface" designation, NMI, was introduced and predicated on its virtual use of things like compass-rule, or pencil-paper traditional mathematical inscriptions. The NMI provided capacities for manipulative geometric constructions and transformations, or symbolic interfacing to a Computer Algebra System. Two separate case studies facilitated empirically-based characterization and reflection concerning students' explorations, experimentations, and deductions in this NMI use setting. Over the course of a semester, one student studied Geometry proof (an elementary education major), and one Markov Chains (a lower division mathematics major). Four distinctive types of perceived mathematical embodiments were observed to be used by the students. These abstract embodiments, and related reasoning acts were described in the context of C. S. Peirce's Pragmatic Reasoning theory. NMI interactivity, and the means of a mathematical semantics level organization (via interface lay-out), were seen to be important contributors to the students' pragmatic reasoning. The abstract types of mathematical embodiments revealed were named: (i) Interface-procedural, (ii) Natural Mathematics Computational, (iii) Applications , and (iv) Generic. These mean, respectively, (i) interpreted merely as memorized interface procedures, (ii) resultant from interactive computation, interpreted as mathematical in a (sometimes) surface sense, as the student may not understand the underlying mathematics directing the computation, (iii) resultant from a student's interest in a real-world application used to analogously consider a mathematical model and its interpretation, and (iv) clearly abstracted and generalized, internal or mentalesque mathematical explanations or systematizations. |
