Constructions and justifications of a generalization of Viviani's theorem.
This qualitative study actively engaged a group of eight pre-service mathematics teachers (PMTs) in an evolutionary process of generalizing and justifying. It was conducted in a developmental context and underpinned by a strong constructivist framework. Through using a set of task based activities embedded in a dynamic geometric context, this study firstly investigated how the PMTs experienced the reconstruction of Viviani’s theorem via the processes of experimentation, conjecturing, generalizing and justifying. Secondly, it was investigated how they generalized Viviani’s result for equilateral triangles, further across to a sequence of higher order equilateral (convex) polygons such as the rhombus, pentagon, and eventually to ‘any’ convex equi-sided polygon, with appropriate forms of justifications. This study also inquired how PMTs experienced counter-examples from a conceptual change perspective, and how they modified their conjecture generalizations and/or justifications, as a result of such experiences, particularly in instances where such modifications took place. Apart from constructivsm and conceptual change, the design of the activities and the analysis of students’ justifications was underpinned by the distinction of the so-called ‘explanatory’ and ‘discovery’ functions of proof. Analysis of data was grounded in an analytical–inductive method governed by an interpretive paradigm. Results of the study showed that in order for students to reconstruct Viviani’s generalization for equilateral triangles, the following was required for all students: *experimental exploration in a dynamic geometry context; *experiencing cognitive conflict to their initial conjecture; *further experimental exploration and a reformulation of their initial conjecture to finally achieve cognitive equilibrium. Although most students still required the aforementioned experiences again as they extended the Viviani generalization for equilateral triangles to equilateral convex polygons of 4 sides (rhombi) and five sides (pentagons), the need for experimental exploration gradually subsided. All PMTs expressed a need for an explanation as to why their equilateral triangle conjecture generalization was always true, and were only able to construct a logical explanation through scaffolded guidance with the means of a worksheet. The majority of the PMTs (i.e. six out of eight) extended the Viviani generalization to the rhombus on empirical grounds using Sketchpad while two did so on analogical grounds but superficially. However, as the PMTs progressed to the equilateral pentagon (convex) problem, the majority generalized on either inductive grounds or analogical grounds without the use of Sketchpad. Finally all of them generalized to any convex equi-sided polygon on logical grounds. In so doing it seems that all the PMTs finally cut off their ontological bonds with their earlier forms or processes of making generalizations. This conceptual growth pattern was also exhibited in the ways the PMTs justified each of their further generalizations, as they were progressively able to see the general proof through particular proofs, and hence justify their deductive generalization of Viviani’s theorem. This study has also shown that the phenomenon of looking back (folding back) at their prior explanations assisted the PMTs to extend their logical explanations to the general equi-sided polygon. This development of a logical explanation (proof) for the general case after looking back and carefully analysing the statements and reasons that make up the proof argument for the prior particular cases (i.e. specific equilateral convex polygons), namely pentagon, rhombus and equilateral triangle, emulates the ‘discovery’ function of proof. This suggests that the ‘explanatory’ function of proof compliments and feeds into the ‘discovery’ function of proof. Experimental exploration in a dynamic geometry context provided students with a heuristic counterexample to their initial conjectures that caused internal cognitive conflict and surprise to the extent that their cognitive equilibrium became disturbed. This paved the way for conceptual change to occur through the modification of their postulated conjecture generalizations. Furthermore, this study has shown that there exists a close link between generalization and justification. In particular, justifications in the form of logical explanations seemed to have helped the students to understand and make sense as to why their generalizations were always true, but through considering their justifications for their earlier generalizations (equilateral triangle, rhombus and pentagon) students were able to make their generalization to any convex equi-sided polygon on deductive grounds. Thus, with ‘deductive’ generalization as shown by the students, especially in the final stage, justification was woven into the generalization itself. In conclusion, from a practitioner perspective, this study has provided a descriptive analysis of a ‘guided approach’ to both the further constructions and justifications of generalizations via an evolutionary process, which mathematics teachers could use as models for their own attempts in their mathematics classrooms.