Mis-Out and Mis-In Examples: The Case of Rational Numbers

This paper focuses on the definitions and the mis-out and mis-in examples of rational numbers that four prospective elementary teachers presented while working on rational number assignments. The participants were first asked to respond, individually, to an Individual Rational Number Assignment, consisting of items aiming at detecting their personal concept definitions of rational numbers and identifying the entities that they regarded as rational numbers. Then, to share their work with another prospective teacher, to identify similarities and differences in their responses, and to list issues that were raised during the individual or pair work, that they would like to discuss in class. The data exposed a tendency to provide one definition of rational numbers, to identify the term “rational” with “natural”, not to include a clarification that a rational number is a number, and a controversy regarding including (or not including) a statement that b ≠ 0 in the definition. Other observations related to a tendency not to categorize negative numbers (and perhaps also zero) as rational numbers and an inconsistency between their responses to the question “what is a rational number?” and their classification of examples of rational numbers. Recommendations for topics for discussion with prospective teachers, in light of the responses to the assignments, are suggested and methodological issues for considerations are proposed.