An analogy is a sort of similarity between two things that are compared, and correspondingly analogical reasoning is a process of transfer of structural information from one system, the base (called the source) to another system (called the target) through mapping relational correspondences between the two systems. Analogical reasoning as a vehicle to develop generalizations and proofs is often either overlooked or not given adequate attention within the existing South African school/teacher education mathematics curriculum. This quality case study explored the extent to which pre-service mathematics teachers can develop their abilities to conjecture and justify a geometric generalization through analogical reasoning within a dynamic geometry environment. Data was collected from 8 pre-service mathematics teachers from a Faculty of Education at a university in Cape Town South Africa. Data gathering instruments included observing learners working on a scaffolded task-based activity, document analysis, and interviews. The task-based activity first required the PMTs to solve an equilateral triangle problem through conjecturing, experimentation, generalizing, and constructing a logical proof. The second activity required the PMTs to solve a related problem linked to an equilateral quadrilateral either via experimentation using Sketchpad, analogical transfer, or logical proof. For the equilateral quadrilateral problem, six PMTs, who developed their conjecture generalization by means of empirical induction from dynamic cases in a GSP context, and the two PMTs who did so by analogy-making, expressed a desire and a need for an explanation as to why their conjecture generalization is always true. In seeking to construct an explanation for their equilateral quadrilateral conjecture generalization, PMTs discovered a strategy via analogical reasoning either immediately or after some scaffolded guidance. This discovery in itself is an example of the discovery function of proof. The findings emphasize that mathematics teachers and teacher educators should be prepared to initially receive generalizations made on empirical or analogical grounds when students face a problem for the first time. Further to this, they may expect their students through differentiated scaffolded guidance to prove their conjecture generalizations via analogical transfer from the source to the target.

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Conjecturing and Proving a Geometric Generalization Through Analogical Reasoning in Geometer’s Sketchpad Context

  • Rajendran Govender

摘要

An analogy is a sort of similarity between two things that are compared, and correspondingly analogical reasoning is a process of transfer of structural information from one system, the base (called the source) to another system (called the target) through mapping relational correspondences between the two systems. Analogical reasoning as a vehicle to develop generalizations and proofs is often either overlooked or not given adequate attention within the existing South African school/teacher education mathematics curriculum. This quality case study explored the extent to which pre-service mathematics teachers can develop their abilities to conjecture and justify a geometric generalization through analogical reasoning within a dynamic geometry environment. Data was collected from 8 pre-service mathematics teachers from a Faculty of Education at a university in Cape Town South Africa. Data gathering instruments included observing learners working on a scaffolded task-based activity, document analysis, and interviews. The task-based activity first required the PMTs to solve an equilateral triangle problem through conjecturing, experimentation, generalizing, and constructing a logical proof. The second activity required the PMTs to solve a related problem linked to an equilateral quadrilateral either via experimentation using Sketchpad, analogical transfer, or logical proof. For the equilateral quadrilateral problem, six PMTs, who developed their conjecture generalization by means of empirical induction from dynamic cases in a GSP context, and the two PMTs who did so by analogy-making, expressed a desire and a need for an explanation as to why their conjecture generalization is always true. In seeking to construct an explanation for their equilateral quadrilateral conjecture generalization, PMTs discovered a strategy via analogical reasoning either immediately or after some scaffolded guidance. This discovery in itself is an example of the discovery function of proof. The findings emphasize that mathematics teachers and teacher educators should be prepared to initially receive generalizations made on empirical or analogical grounds when students face a problem for the first time. Further to this, they may expect their students through differentiated scaffolded guidance to prove their conjecture generalizations via analogical transfer from the source to the target.