The most effective thing that I have been able to do over the years for students learning to do proofs has been to make things more explicit. The ultimate end of a process of making things explicit is, of course, a formal system—which this text contains. Some people have thought that it makes proof too easy. My own view is that it does not trivialize anything important; it merely exposes the truly trivial for what it is. It shrinks it to its proper size, rather than allowing it to be an insurmountable hurdle for the average student.

The system in this text is based on a number of formal inference rules that model what a mathematician would do naturally to prove certain sorts of statements. Although the rules resemble those of formal logic, they were developed solely to help students struggling with proof—without any input from formal logic. The rules make explicit the logic used implicitly by mathematicians. After experience is gained, the explicit use of the formal rules is replaced by implicit reference. Thus, in our bottom-up approach, the explicit precedes the implicit. The initial, formal step-by-step format (which allows for the explicit reference to the rules) is replaced by a narrative format—where only critical things need to be mentioned. Thus the student is lead up to the sort of narrative proofs traditionally found in textbooks. At every stage in the process, the student is always aware of what is and what is not a proof—and has specific guidance in the form of a “step discovery procedure” that leads to a proof outline.

The inference rules, and the general method, have been used in two other texts—intended for students different from the intended readers of this text:

(1)* Outlining Proofs in Calculus* has been used as a supplement in a third-semester calculus course, to take the mystery out of proofs that a student will have seen in the calculus sequence.

(2)* Deductive Mathematics—An Introduction to Proof and Discovery for Mathematics Education* has been used in courses for elementary education majors and mathematics specialists.

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