— Xxxxx Selection Test. Imagine you have a four cards on a table and every card has a letter on one side and a number on the other. With the cards placed on a table, you see D K Turn over the fewest cards to establish the truth of the following statement “Every card which has a D on one side has a 7 on the other.” Examples 2.1.3 and 2.1.4 are two problems whose statements are precise and unam- biguous; nevertheless, students struggled with the correct interpretation. Another possible situation is a problem whose statement is open to interpretation, where the terms may not be properly defined or the logic is suspect. In this case the student’s task could be to fix the statement so that its meaning is clear and its conclusion is true. A celebrated discussion of this situation is given in Xxxx Xxxxxxx’x Proofs and Refutations (1976), a socratic dialogue in which a teacher and their students attempt to prove or disprove the statement “For all polyhedra, V — E + F = 2”. In this case the students begin with differing and unclear interpretations of polyhedron, vertex, xxxx, and face. The extract below (Xxxxxxx et al., 1976, p. 14) demonstrates one of the issues the students have to resolve: how to define the set of objects called ‘polyhedra’ to ensure that the conclusion is true? They must first decide whether to alter their interpretation of the terms in the conjecture, or the conjecture itself: “them has to give way. But why should the theorem give way, when it has been DELTA: But why accept the counterexample? We proved our conjecture—now it is a theorem. I admit that it clashes with this so-called ‘counterexample’. One of proved? It is the ‘criticism’ that should retreat. It is fake criticism. This pair of nested cubes is not a polyhedron at all. It is a monster, a pathological case, not a counterexample. GAMMA: Why not? A polyhedron is a solid whose surface consists of polygonal faces. And my counterexample is a solid bounded by polygonal faces. TEACHER: Let us call this definition Def. 1. Through a sequence of proposed proofs and counterexamples, the class eventually reaches both a proof of the statement and rigorous definitions for the terms used in it. Traditional direct instruction avoids situations such as this, precisely because fruitful definitions (e.g. continuity, convergence) are difficult to formulate.
