20: The Transition to Advanced Mathematical Thinking: Functions. Limits, Infinity, and Proof

Advanced mathematical thinking—as evidenced by publications in research journals—is characterized by two important components: precise mathematical definitions (including the statement of axioms in axiomatic theories) and logical deductions of theorems based upon them. However, the printed word is but the tip of the iceberg, the record of the final “precising phase” that is quite distinct from the creative phases of mathematical thinking in which inspirations and false turns play their part.

A major focus in mathematical education at the higher levels is not only to initiate the learner into the complete world of the professional mathematician in terms of the rigor required, but also to provide the experience on which the concepts are founded. Traditionally this has been done through a gentle introduction to the mathematical concepts and the process of mathematical proof in school before progressing to present mathematics in a more formally organized and logical framework at college and university.