Introduction to Proofs: an Inquiry-based Approach by Jim Hefferon is for a Mathematics major course that is inquiry-based, sometimes known as “Moore method.” It is Free.
Highlights
- Inquiry-based The text provides the material's outline, a sequence of statements along with a few definitions and remarks.
- Covers needed material Number Theory up to the Fundamental Theorem of Arithmetic, sets to DeMorgan's Laws, functions to two-sided inverses, and relations to equivalences and partitions.
- No prerequisite We enroll sophomore Mathematics majors, who typically have taken Linear Algebra, just to ensure that they have mathematical aptitude.
- Free You are free to use the text and free to share it. With the LaTeX source, you can modify it to suit your class.
- Flexibility The text comes in two presentations, one of which is compact enough — seven sheets of paper — to just hand out on the first day. The second presentation adds a preface and a section on infinity.
- Extras My first day's slides introduce students to Inquiry-Based learning. The slides on logic organize the discussions that arise naturally. I go through them a little per day in the first two weeks.
What is Inquiry-Based?
The class works through the material together, so the instructor is more of a guide or mentor than a lecturer. Through peer discussion, as well as learning what is right, students also come to understand why the wrong things are wrong. This is the best way to develop mathematical maturity. The New England Community for Inquiry-Based Learning in Mathematics is a great place to get started looking for more.
What I do
We run a semester course, meeting three times a week. I cover the first three chapters.
At the first meeting, I explain that in this course we develop each person's ability, as a future professional, to work independently. Students pledge that they will not work together outside of class and will not use any resources such as other books or the Internet. Consequently, each person arrives at each class having thought carefully about each exercise, on their own.
Usually there are four exercises. I randomly pick students to put proposed solutions on the board. (I shuffle index cards. The picked students negotiate among themselves over who will do which one, a student picked in the prior class will not be picked today, and each student is allowed twice in the semester to take a pass.)
With that, the work starts. The class as a whole discusses the proposals, sentence by sentence and often word by word. There are misconceptions to get past and ideas that the class comes to see are not fruitful, as well as good ideas that may initially have trouble getting heard. But the discussions eventually come to an end, and usually that end is correct.
I do my best not to speak, and to not even nod or frown. Sometimes I must guide, for instance saying after a class has decided that the proof is OK, “Exhibiting the n = 3 and n = 5 cases is not enough to prove the result for all odd numbers.”
The discussion often takes the entire period but occasionally we have a little extra time. By experience I know that some problems will be a challenge and I try arrange the assignments so that there is time to give students a boost with these. For instance, I might give an example suggesting how to show that corresponding finite sets have the same cardinality.
With that, class ends with a new set of problems to work on at home, and once a week with the designation of one of them to be handed in at the start of the next class, to be graded.
I will close with a few comments. One is that in the first two weeks, I start class with fifteen minutes of working through slides that cover elementary logic. These provide a vocabulary and sharpen the discussions, particularly about fine points such as vacuous implication.
The second comment is on who is in the class. We enroll sophomores who have had Calculus III and Linear Algebra, so we can expect that they have an aptitude, as well as some basic scaffolding of mathematical reasoning on which to base this class's development. We limit to twenty students because in a too-large class people can hope to hide, both from being picked to propose solutions and from full participation in the discussion. Besides, with more than that, the marking burden discourages an instructor from giving complete attention to each student's work.
Finally, about grading. I count four things: in-class contributions, the weekly hand-in problems, an in-class midterm, and an in-class final. In all four, I give credit both for mathematical correctness and for mathematical writing. I weigh the four equally, but students agree that the in-class discussion is the core experience. Success flows from that.