טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
Ph.D Thesis
Ph.D StudentBuchbinder Orly
SubjectThe role of Examples in Establishing the Validity of
Universal and Existential Mathematical Statements
DepartmentDepartment of Education in Science and Technology
Supervisors Professor Emeritus Uri Leron
Professor Orit Zaslavsky
Full Thesis text - in Hebrew Full thesis text - Hebrew Version


Abstract

The purpose of the study was to examine students’ understanding of the interplay between examples and mathematical statements as well as the roles of examples and counterexamples in proving. This kind of understanding entails making a distinction between universal and existential statements, and realizing that although examples are not sufficient for proving a universal statement, a single example is sufficient for proving an existential statement. Moreover, a single counterexample is sufficient for refuting a universal statement but it is not applicable for refuting an existential statement. This critical kind of understanding is not explicitly addressed in the school curriculum and is left to students to develop indirectly mostly on their own. The nature of this understanding has not been fully conceptualized, nor has it been studied systematically prior to this study.

For the purpose of the study, a conceptual framework describing the interplay between examples and proving was developed. This framework provided the basis for designing special types of mathematical tasks that assess students' understanding and at the same time facilitate its development. Two parallel versions of these tasks (in algebra and in geometry) were implemented with six pairs of top-level high-school students. Each pair participated in a series of six task-based interviews.

Data collection included video recordings of the interviews, students’ written work and field notes. The data were analyzed using qualitative research methodology, which resonated with the research goals.

The findings provide a complex account of students’ understanding of the roles of examples in proving and revealed various inconsistencies in their understandings. Some inconsistences were manifested as discrepancies between the views students explicitly expressed and their actual performance. Others were manifested as discrepancies between students' performances on similar types of tasks. In addition, students exhibited fallacious reasoning that was based on sophisticated mathematical ideas that were incorrectly or inaccurately applied. 

The findings point to several elements that may explain students’ above obstacles. These include students' difficulties in determining the logical structure of a statement and their misinterpretations of the language used to describe the status of examples in proving.

In addition to the theoretical contribution to the field, the study has educational implications. The tasks, which were designed for the study, elicited students' understanding of the logical connections between examples, statements and proving. These tasks can be used or adapted to different mathematical topics and for various populations of students.