Introduction to Proofs, an Inquiry-Based approach

A Free text for a course on proofs

Jim Hefferon
Mathematics Department, Saint Michael's College
jhefferon at

Introduction to Proofs is an undergraduate text. It is Inquiry-Based, sometimes called the Discovery Method or the Moore Method. It is Freely available, so you can download it.


Getting Introduction to Proofs

Click to download the compact version of Introduction to Proofs. I copy this and hand it out on the first day (it prints onto seven sheets of paper). You can instead get a larger-sized version that includes a Preface and some additional material.

Also here are the first day's slides that introduce the class's style. In the first couple of weeks I show a few slides on logic each day, just covering enough to get students started, And, I require that by the end of the semester they do their hand-ins in LaTeX, so to help with that I distribute a LaTeX for Undergraduates, and a Cheat sheet for LaTeX math.

Finally, if you are into LaTeX then you can clone the repository.

What is "inquiry-based"?

This teaching style requires that students work directly with the mathematics. It is the core experience of the class. That is, this style shows students how to be, and in fact requires that they be, active learners. Consequently, it is a good fit for this course.

I'll illustrate with my version of a day's class. As background, students have pledged not to work together or to use any resources such as other books or the Internet. So each day they arrive having worked on each exercise, on their own.

Typical is four exercises. To start, I randomly pick four students to put their proposed answers on the board. (While they are writing their answers I collect hand-in problems from the rest of the students, or in the first few classes do beamer slides with some elementary logic, or perhaps answer questions about LaTeX. Or, sometimes we just talk about news in the math world or what courses are offered next semester.)

Then the work begins. Students talk through the proposed answers sentence by sentence and sometimes word by word. This discussion is filled with misconceptions to work through, ideas that folks eventually come to see are not fruitful, as well as good ideas that may initially have trouble getting heard.

During the discussion I speak as little as I can. I also try not to nod or frown, no matter how good, or bad, are the things being said.. But the discussion eventually comes to an end, and typically that end is correct. (Sometimes I do guide, as with speaking up after the class has decided that the proof is OK to say, "In the second paragraph, exhibiting the n=3 and n=5 cases is not enough to show that in all cases the square of an odd is odd.")

The advantage over a lecture is that, as well as having seen what is right, students now also understand why wrong things are wrong.

The four presented problems often fill the period. But sometimes we have time for another problem. Some problems traditionally give a lot of trouble, such as that there are infinitely many primes, so I try to arrange to do these in class.

Class ends with a new set of problems to take home, and soemtimes with the announcement that one of them to be handed in, and perhaps that it must be done in LaTeX.

Three comments. First, I find that students learn this material better this way, and that's the point, isn't it? Second, with this approach, no matter where a student falls on the bell curve, even at the end of the semester they are still understanding what is happening. In particular, I want that the students who will teach mathematics will have a solid understanding of arguments, including an understanding of when arguments are wrong. Finally, third, I must acknowledge my debt to the Inquiry-based learning community and particularly to Prof Jensen-Vallin's wonderful Proofs book.

Can you help with Introduction to Proofs?

If you are an instructor who adopts this book then I'd be glad to hear your comments, including just the comment that you find it useful. (I save bug reports. and periodically revise.)

If you have some material that you are able to share back then I'd be interested to see it. Of course, I reserve the ability to choose whether to include it. I gratefully acknowledge all the contributions that I include, or I can keep you anonymous if you prefer. My email is at the top of this page.


This text is Free. Use it under either the GNU Free Documentation License or the Creative Commons License Creative Commons Attribution-ShareAlike 2.5 License, at your discretion.

To bookstores: thank you for your concern about my rights. I give instructors permission to make copies of this material, either electronic or paper, and give or sell those copies to students. If you have further questions, don't hesitate to contact me at the email address on the top of this page.

To instructors: if you want to modify the text then feel free. However, I ask as a favor that you make clear which material is yours and which is mine. When I get questions or bug reports then having to work out what is happening gets frustrating all around unless the authorship is clear. It would be great if you could change the cover to include something like: "\fbox{The material in the appendix on logic is not from the main version of the text but has been added by Professor Jones of UBU. For this material contact \url{}.}"

You may also be interested in my Linear Algebra text, also Freely available. It is for a first US undergraduate course, covering the standard material while focusing on helping students develop mathematical maturity. It has been widely used for many years.

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This site Joshua is located in the Mathematics Department of Saint Michael's College in Colchester, Vermont USA.

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