Aspects of a Neoteric Approach to Advance Students' Ability to Conjecture, Prove, or Disprove

The author of this paper suggests several neoteric, unconventional, idiosyncratic, or unique approaches to beginning Set Theory that he found seems to work well in building students' introductory understanding of the Foundations of Mathematics. This paper offers some ideas on how the author uses certain "unconventional" definitions and "standards" to get students to understand the essentials of basic set theory. These approaches continue through the canon and are employed in subsequent courses such as Linear Algebra, Probability and Statistics, Real Analysis, Point-Set Topology, etc. The author of this paper submits that a mathematics student needs to learn how to conjecture, to hypothesise, "to make mistakes," and to prove or disprove said ideas; so, the paper's thesis is learning requires "doing" and the point of any mathematics course is to get students to do proofs, produce examples, offer counterarguments, and create counterexamples. We propose a quintessentially inquiry-based learning (IBL) pedagogical approach to mathematics education that centres on exploration, discovery, conjecture, hypothesis, thesis, and synthesis which yields positive results--students doing proofs, counterexamples, examples, and counter-arguments. Moreover, these methods seem to assist in getting students to be willing to make mistakes for, we argue, that we learn from making mistakes not from always being correct! We use a modified Moore method (MMM, or M[superscript 3]) to teach students how to do, critique, or analyse proofs, counterexamples, examples, or counter-arguments. We have found that the neoteric definitions and frame-works described herein seem to encourage students to try, aid students' transition to advanced work, assists in forging long-term undergraduate research, and inspirits students to do rather than witness mathematics. We submit evidence to suggest that such teaching methodology produces authentically more adept students, more confident students, and students who are better at adapting to new ideas. (Contains 36 footnotes.)