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 a Free undergraduate text. It is Inquiry-Based, sometimes called the Discovery Method or the Moore Method.


Here is Introduction to Proofs

Click to download the full Introduction to Proofs. You can instead get a compact version that copies onto six or seven sheets and so is perfect for handing out on the first day of class; this is what I do.

There are beamer slides covering enough logic to get students started. You can also clone the repository, if you are into LaTeX.

What is "inquiry-based"?

This teaching style has students work directly with the mathematics.

I'll illustrate by discussing my version of this class. The students have four grades: one each from a mid-term and a final, a grade from the written work on their hand-in problems, and one reflecting their contributions to the in-class discussion. Thus the in-class work is the major part of the course — exercises are the core experience.

Here is a typical day. As background, students have pledged not to work together or to use any resources such as other books, the net, students who have taken the course before, etc. I tell them that working with others is a great thing but that a variety of experiences is also great. Consequently, they arrive each day having worked on each exercise, on their own. Typical is four exercises. I shuffle index cards for who will go to the board — the first person picked gets their choice of the four, the next person chooses from the remaining three, etc. (I leave out people who were picked last time. In addition, twice in the semester each student can take a pass.)

While those students are at the board there are hand-in problems to collect, or in the first few classes beamer slides that with some aspects of logic, or perhaps questions on LaTeX (I distribute a LaTeX for Undergraduates). Or, sometimes we just talk about things, such as the Clay prizes, or news in the math world, or interesting blogs, or what math courses are offered next semester.

With the proposed answers in sight the work begins. Students talk them through sentence by sentence and sometimes word by word. This discussion is typically filled with misconceptions to work out, bad ideas that take a while to go nowhere, and good ideas that have initial trouble getting a hearing. As an instructor I struggle with letting it play out. I speak as little as I can stand, try not to nod or shake my head, and if the room allows it then I sit where people can not easily look to me. (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.")

But the discussion does eventually come to an end and usually it is the right end, and with the bonus that students now understand why wrong things are wrong as well as understanding what is right.

The presented problems can take the entire period. If there is time then we will do another. (Some exercises traditionally give people trouble, such as that there are infinitely many primes or that for a function being a correspondence is equivalent to being invertible. I try to arrange to do these in class.) Class ends with a new set of problems, perhaps once a week with one of them to be handed in, and if so perhaps it must be done in LaTeX (at the start I give extra points for LaTeX but eventually it is required).

A note about the hand-ins: I grade them out of 10 and they are due the next class. A student who does not see how to do it can choose to hand it in the class after that, having heard the discussion, for a maximum grade of 5/10.

Three final points. First, I find this style of teaching harder than lecturing because I have given up the flow of control, But students learn better than with other approaches and that's the point, isn't it? The second point is related: with this approach no matter where students fall on the bell curve, even at the end of the semester they are still involved with the discussion and are still understanding the material. In particular, I want that the folks in this class who will go on to teach mathematics will understand proofs, and understand when argument are wrong, and this approach gives me that. Finally, I must acknowledge my debt on this project 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 jhefferon at


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 — that is one of my motivations for starting the project. However, I ask as a favor that you make clear which material is yours and which is from the main version of the text. When I get questions or bug reports then having to work out what is happening gets frustrating all around unless authorship is clear. In particular, it would be great if you could change the cover to include something like this: " \fbox{The material in the second 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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