Signed Numbers: A “Product” of Misconceptions

Abstract

One of the first encounters with a mathematical idea that jars a student's intuition takes place when he is asked to accept that the product of two negative numbers is positive. There are, of course, many different explanations at various levels of abstraction that have been used to persuade the neophyte of the reasonableness of the assertion. One can offer an argument based upon a model, and this is done in such programs as UICSM and Madison Project.1 One can offer a plausibility argument based upon the behavior of positive numbers and faith in some sort of continuity principle, as suggested by Dubisch.2 One can (quite ironically, given the context that “modern” mathematics is an attempt to replace rules by explanations and justifications) assert, as Courant and Robbins do, that, though appeal may be made to various channels for psychological reasons, ultimately the conclusion is a matter of definition.3 The prime psychological motivation would be the desire to preserve principles already existing in the set of positive reals, as is done by SMSG.4