טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentBuchbinder Orly
SubjectCounter Examples in Mathematics: Generation Processes and
Modes of their Use
DepartmentDepartment of Education in Science and Technology
Supervisor Professor Orit Zaslavsky


Abstract

The present study examines the learning processes and student understanding related to the concept of counterexamples: the ways in which students understand and use the concept of counterexamples. In particular, the study concentrates on development of understanding of the role counterexamples play in refuting false conjectures, and the tendency to use counterexamples in checking and evaluating conjectures.

Despite the seeming simplicity of counterexamples, empirical studies showed that students sometimes posses wrong conceptions associated with counterexamples, their generation and usage.

There is a consensus among mathematics educators and researches that students’ understanding of a concept is influenced by their overall experiences with the concept. This implies to the concept of counterexample as well. The purpose of this study was to present students with a kind of learning environment and experiences which facilitate the generation and use of counterexamples in order to reflect on learning processes the students go through when they engage in activities that prescribe the use of counterexamples.

For the purposes of this study a teaching unit that addresses the students’ difficulties with counterexamples was especially designed (in two parallel versions adjusted for both high and low level students). The unit was implemented in two classes: high and low level students in a high school.

The findings suggest that engaging in different kinds of activities that emphasize various aspects of counterexamples, helped students to improve their understanding of a concept of counterexample and its use. The same effect was observed in both research groups, regardless of the mathematical level or age of the students.

The analysis of students’ responses revealed that students:

  • Came to recognize counterexamples as a legitimate kind of proof (disproof) that can stand by its own.
  • Became aware of the domain of validity of mathematical universal statements.
  • Increased caution in overgeneralization of conjectures.

In addition, students in both research groups improved their content knowledge, reasoning and communication skills.

The main contribution of this study is the development of a learning environment that focus on counterexample use and generation; its implementation in two classes, at two different levels of mathematics learning; providing a detailed description of learning processes (of students who participated in a study) associated with counterexamples.

The research findings show that there are ways to incorporate in, school context, powerful activities that have a potential to create learning situations in witch students can develop their understanding of mathematical concepts and improve their reasoning skills through dealing with counterexamples.