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 Introduction to Proofs. You can also get a compact version that copies onto six or seven sheets and so is perfect for handing out on the first day of class.

There are beamer slides covering enough logic to get students started. I find the answers useful in thinking ahead for class (and they kept me honest in sequencing the material). 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.

The best way to see what this means is by seeing 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, students arrive each day having worked on each exercise, on their own. Typical is four exercises. I shuffle index cards to pick 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 LaTeX questions. (At the start of the course I distribute a LaTeX for Undergraduates tutorial; note that some PDF readers have trouble with this document but it displays correctly in Acrobat Reader.) 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 on the board 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, so as an instructor I can 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 in the back so people do not easily look to me). But the discussion does eventually come to an end and usually it is the right end, with the bonus that the process has allowed students to understand why wrong things are wrong as well as to see what is right. (Sometimes I guide a bit, 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 four or five problems can take the entire period. If there is time, as there will be in a half or a third of the classes, then we will do another exercise. (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.)

Finally, I assign a new set of problems, perhaps once a week saying that one of them is to be handed in, and if so perhaps that it must be done in LaTeX. That ends the day.

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. Students are not keen on what they think of as a 50 but on the first day I comment that typically every person in the class uses this option at some point.

To close, there are three things that I must say. First, I find this style of teaching harder than lecturing because I have given up the flow of control, but I also find it rewarding. Students learn better than with other approaches and that's the point, isn't it? The second is that even students on the back half of the bell curve are involved with the discussion and understanding the material, even at the end of the semester. In particular, I want that the folks in this class who will go on to teach high school mathematics understand what makes a good proof, and can spot a bad one. Finally, I must acknowledge my debt on this project to the Inquiry-based learning community and particularly to Prof Jacqueline Jensen-Vallin's wonderful Proofs book.

Can you help with Introduction to Proofs?

If you are an instructor who adopts this book, I'd greatly appreciate an email. I'd be glad to hear your comments but I'd also just be glad to hear about people using it. (I save any comments, especially bug reports. and periodically revise.)

If you have some material that you are able to share back then I'd be delighted 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

Various lengths

Introduction to Proofs's LaTeX code allows it to comes in three lengths. I teach a four-credit course and so the longest version has the right number of questions for a fifteen week semester, with three classes per week, and four questions per class. The shortest version omits some questions to end with the right number for three questions per class. The intermediate version is ... well, I think you know. (By "right number" I mean approximately the number of meetings times four questions per meeting times 1.05, for wiggle room.)

If you are not sure then take the longest one. It is the canonical version of the text and is the one linked-to above so you probably already did the download. It has all the questions that the other two have, and in class omitting is much easier than adding.

Book Compact book Answers
Maximum length download download download
Medium length download download download
Minimum length download download download


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. (Instructors may like to make an extra copy and prorate the price of student copies so that their copy is paid for.) If you have further questions, don't hesitate to contact me at the email address on the top of this page.

If you want to modify the text: feel free — that is one of my motivations for starting the project. If you are able to share back your modifications then I'd be glad to see them. If not that is fine. 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 having to work out what is happening gets frustrating all around unless authorship is clear. In particular, changing the cover to include a statement about your modifications would help. Something like this would be great: " \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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