The Uses of Spurious Proofs in Teaching Mathematics

Abstract

A spurious proof is a mathematical proof that seems to be logically cogent at every step, but reaches a conclusion that is clearly impossible. This occurs because a step of the proof has been cleverly written to conceal its own falsehood, and the hidden falsehood ultimately creates the incongruous conclusion. In the past, close reading of spurious proofs in order to discern which step is the false one has been a niche endeavor of recreational mathematics, and only occasionally used in the classroom as one among many types of long-form problems. However, spurious proofs have several distinctive—though sometimes neglected—values in the educational context. One of these sources of value is spurious proofs’ creation of “cognitive conflict” that stimulates critical thought in the contexts of both general problem solving and learning the proof process. Another underappreciated source of value is spurious proofs’ potential fitness for developing the “productive disposition” aspect of mathematical proficiency that sees math as a sensible and useful endeavor. This aspect of proficiency is widely recognized in the literature, but just as widely overlooked in the classroom, likely because it is usually something that must be taken or rejected wholesale. Spurious proofs, though, give teachers a rare means to reinforce this value in their students through an active problem-solving process. With a spurious proof, the coherency of the mathematical system is challenged when the proof offers a result that contradicts well-known mathematical knowledge, but still seems to be logically supported; the student must then actively validate the system’s coherency by searching out the proof’s logical flaw. In this way, a student is actively engaged in affirming the consistency of the mathematical system rather than simply accepting it, or not. After presenting some background information on the history and typology of spurious proofs, this thesis explains their value from cognitive conflict and coherency reinforcement in more detail, and then offers examples of how they may be put to use in the classroom beyond their generic role of simple long-form problems.