Doctoral Degrees (Mathematics and Computer Science Education)

Permanent URI for this collectionhttps://hdl.handle.net/10413/7140

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Constructions and justifications of a generalization of Viviani's theorem.
(2013) Govender, Rajendran.; De Villiers, Michael David.
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.
The role and use of sketchpad as a modeling tool in secondary schools.
(2004) Mudaly, Vimolan.; De Villiers, Michael David.
Over the last decade or two, there has been a discernible move to include modeling in the mathematics curricula in schools. This has come as the result of the demand that society is making on educational institutions to provide workers that are capable of relating theoretical knowledge to that of the real world. Successful industries are those that are able to effectively overcome the complexities of real world problems they encounter on a daily basis. This research study focused, to some extent, on the different definitions of modeling and some of the processes involved. Various examples are given to illustrate some of the methods employed in the process of modeling. More importantly, this work attempted to build on existing research and tested some of these ideas in a teaching environment. This was done in order to investigate the feasibility of introducing mathematical concepts within the context of dynamic geometry. Learners, who had not been introduced to specific concepts, such as concurrency, equidistant, and so on, were interviewed using Sketchpad and their responses were analyzed. The research focused on a few aspects. It attempted to determine whether learners were able to use modeling to solve a given real world problem. It also attempted to establish whether learners developed a better understanding when using Sketchpad. Several useful implications have evolved from this work that may influence both the teaching and learning of geometry in school. Initially these learners showed that, to a large extent, they could not relate mathematics to the real world and vice versa. But a pertinent finding of this research showed that, with guidance, these learners could apply themselves creatively. Furthermore it reaffirmed the idea that learners can be taught from the general to the more specific, enabling them to develop a better understanding of concepts being taught. Perhaps the findings and suggestions may be useful to pre-service and in-service educators, as well as curriculum developers.

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Browsing Doctoral Degrees (Mathematics and Computer Science Education) by Author "De Villiers, Michael David."