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 Moore method or the discovery 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, which are extensively discussed in class, and a grade reflecting their contributions to the in-class discussion. Thus the in-class work is a major part of the course \mdash; 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 a great thing and this class proceeds in this way.

Consequently, on each class day students arrive having thought about and worked on each exercise, on their own. Typical is four exercises. I shuffle index cards to pick who will go to the board today — the first person picked gets their choice of the four, the next person chooses from the remaining three, etc. (I leave out the 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 is usually business to do. In perhaps one class out of three there are hand-in problems to collect. In the first few weeks of the semester I have beamer slides that discuss some aspects of logic such as the definition of implication. After the first few weeks there may be LaTeX questions, since some hand-ins must be in LaTeX. (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.) And sometimes we just talk about math things, such as the Clay prizes, or news in the math world, or interesting blogs, or good writing style for mathematics, or what math courses are offered next semester.

With the exercises are on the board the work begins. Students talk through the proposed solutions, 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 take its course (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 correct end, with the bonus that the process has allowed students to have come to understand why wrong things are wrong and 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 discussion can take the entire period for just the four problems. If there is time, as there will be in perhaps a half or a third of the classes, then I will propose another exercise for general discussion. (There are some exercises that 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 the homework so that we do these exercises in class.)

Finally, I assign a new set of problems, perhaps saying that one of them is to be handed in and 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 on the problem, for a maximum grade of 5/10. On the first day of the semester I comment that typically every person in the class uses this option at some point.

To close, there are two things I must say. First, I find this teaching harder than lecturing because I have given up the flow of control but it is also rewarding. Students learn better than with other approaches and that's the point, isn't it? Second, 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.

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 Associated 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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