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University Students Uses of Mathematical

UNIVERSITY STUDENTS’ USES OF MATHEMATICAL DEFINITIONS THAT ARE NEW TO THEM

  Valeria Aguirre Holguin Annie Selden New Mexico State University

  University students often do not know there is a difference between dictionary definitions and mathematical definitions (Edwards & Ward, 2004), yet in order to succeed in their university mathematics courses, they must often construct original (to them) mathematical proofs. Only a little research has been conducted to discover how university students handle definitions new to them (e.g., Dahlberg & Housman, 1997). Our specific research question was: How do university students use definitions to evaluate and justify examples and non-examples, in proving, and to evaluate and justify true/false statements? Data were collected through individual task-based interviews with volunteers from a transition-to-proof course. Each student was provided with a particular definition and asked to consider examples and non-examples, construct a proof, and consider true/false statements, in that order. Altogether there were five definitions: function, continuity, ideal, isomorphism, and group, but each student was asked to consider only one of the five. We used content analysis and grounded theory for the analysis. We concentrate here on the definition of function, which was the Bourbaki ordered pair definition. Results suggest significant interference of students’ previous knowledge, which was not always appropriate, in students’ understandings of this definition. We observed a tendency to ignore the given definition and use only their concept images (Tall & Vinner, 1981). One of our conjectures with regard to the definition of function is that such interference occurred due to the students having worked with functions in several previous courses. Is there a way to prevent students from using their inappropriate previous knowledge? We expect less interference as we analyse the later interviews, when the definitions were not only new to the interviewed students but also more abstract.

  References

  Dahlberg, R. P., & Housman, D. L. (1997). Facilitating learning events through example generation. Educational Studies in Mathematics, 33(3), 283-299. Edwards, B. S., & Ward, M. B. (2004). Surprises from mathematics education research: Student (mis) use of mathematical definitions. American Mathematical

  Monthly, 111, 411-424.

  Tall, D., & Vinner, S. (1981). Concept image and concept definition in mathematics with particular reference to limits and continuity. Educational Studies in Mathematics,

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