A Teacher’s Conception of Definition and Use of Examples When Doing and Teaching Mathematics

Abstract

To contribute to an understanding of the nature of teachers’ mathematical knowledge and its role in teaching, the case study reported in this article investigated a teacher’s conception of a metamathematical concept, definition, and her use of examples in doing and teaching mathematics. Using an enactivist perspective on mathematical knowledge, the authors give an account of the case of Lily, a prospective, then beginning, teacher who conceived of mathematical definition as an object with particular form and function and engaged in purposeful, specialized use of examples when doing and teaching mathematics. Lily’s case illustrates how a teacher’s interpretation of examples (as exemplifications or single instances) and conception of the form and function of definitions can influence her doing and teaching mathematics. An implication is that teacher preparation should foster teachers’ abilities to use examples purposefully to provide students with rich opportunities to engage in mathematical processes such as defining.

Notes

¹ Doing mathematics includes a teacher’s use of mathematical processes (e.g., defining) and work with mathematical products (e.g., definition). A teacher could engage in doing mathematics in a variety of settings.

² Lily is a pseudonym.

³ For a definition to be minimal, it “should not contain parts which can be mathematically inferred from other parts” (Vinner, 1991, p. 65) of the definition. As an example, Vinner explained that defining rectangle as a quadrilateral with three right angles would be preferred to a definition mentioning four right angles, because a fourth right angle is a logical consequence of three right angles.

⁴ Although Lily had been working with the polyhedra in sets Alpha and Gamma previously during the interview, she neither recognized them as being prismatoids nor noticed that the vertices of the polyhedra in both sets lie on one of two parallel planes.

⁵ Although Lily used “iff,” her instruction did not directly address the biconditional nature of iff.

⁶ Although the defining example in which markings were used to indicate the congruence of opposite sides (see Figure 8) did not correspond with a listed characteristic, in the lesson’s postinterview Lily stated that, “both pairs of opposites sides are congruent” should have been included as a characteristic.

⁷ Assuming a definition to be arbitrary could be likened to conceiving of the function of a definition from a definition-as-stipulated perspective.