The interplay of empirical and deductive reasoning in proving “if” and “only if” statements in a Dynamic Geometry environment

This paper is situated within the ongoing enterprise to understand the interplay of students' empirical and deductive reasoning while using Dynamic Geometry (DG) software. Our focus is on the relationships between students' reasoning and their ways of constructing DG drawings in connection to directionality (i.e., “if” and “only if” directions) of geometry statements. We present a case study of a middle-school student engaged in discovering and justifying “if” and “only if” statements in the context of quadrilaterals. The activity took place in an online asynchronous forum supported by GeoGebra. We found that student's reasoning was associated with the logical structure of the statement. Particularly, the student deductively proved the “if” claims, but stayed on empirical grounds when exploring the “only if” claims. We explain, in terms of a hierarchy of dependencies and DG invariants, how the construction of DG drawings supported the exploration and deductive proof of the “if” claims but not of the “only if” claims.