### No one way to teach mathematics

There are many teaching methods for instructors of mathematics courses at university. From clickers to flipped classrooms, these and other approaches are useful depending on the topic and size of the lecture. I present two approaches—one well known and the other lesser known—which I have used with good results.

### The Moore method

I’ve used this method in my graduate courses for years now in various forms and am a big fan of the approach. In the Moore method, students discover the material in the course while guided throughout by the instructor. Here is how it works. Typically, no textbook is used.

- The instructor states relevant facts and definitions at the beginning of the lecture.
- The instructor presents the class with a theorem stated without proof.
- Students work on the proof.
- The students present their proofs and the instructor corrects them when necessary.

When executed properly, the method has potentially rich benefits for students. Students taught by this method learn how to problem solve and think critically. They learn what research is about, as opposed to being spoon fed material by their professors.

On the down side, you cover much less material. It requires patience on the side of instructor as students work through proofs, and the method also requires patience from students who are used to the more traditional note-taking form of university lectures.

I’ve only ever attempted this method at the graduate level, and with twenty or fewer students. It could be challenging to do it effectively with a larger group.

Students need mathematical maturity for the Moore method to work. When it does work, however, it is brilliant. I had one Masters student discover the Fano plane essentially on her own in the lecture! Rarely would a student come up with a new proof that I hadn’t considered. Often, students get stumped, but that is what happens in research. You try everything you know and either something clicks, or it doesn’t work at all and you come up with something new. My role is to be supportive, give hints when needed, and give suggestions on their presentations.

Theorems or examples presented have to be bite-sized. If the results are too deep or unwieldy, there is no reasonable way to expect students to come up with original ideas about them in a short span of time. This is one reason why I am not a fan of exams in graduate-level courses in mathematics. Students should have taken an ample number of course-based exams in their undergraduate degree.

The method I am using to teach my graduate course on *Topics in Discrete Mathematics* this semester is not full-on Moore, but hybrid Moore: I lecture traditionally, then occasionally switch to what I call “Experiential components.” Here, students work together in small groups on problems that I give them, which they then present for participation marks. It works well and helps foster a sense of team building.

There is nothing more rewarding as a professor than to watch a group of students first struggle with a concept, only to have a breakthrough and help each other in their understanding. Mathematics is learned incrementally and through many hours of hard work. It is also a social activity, and as the Beatles say “We get by with a little help from our friends.”

### The Nowakowski Method

What about large lectures, you ask? Or a method more suitable for elementary undergraduate courses with students fresh to advanced mathematics?

My co-author Richard Nowakowski from Dalhousie University shared with me years ago his method for teaching large lectures, which is especially useful for first year courses such as Calculus or Linear Algebra. In such courses, usually there is far less emphasis on proofs, and more focus on applications of theory and problem solving. The method is simple and I have found it incredibly effective.

Here is how it works.

- The instructor states definitions and notation.
- The instructor states the relevant theorems, algorithms or other theoretical pieces they want to communicate.
- The class breaks for five to ten minutes.
- The instructor completes examples for the remainder of the class.

The idea is that you get the theory out-of-the-way fast, which is the backbone of what you want them to learn but more viewed as tools. I would often have slides ready for that theory part, and post those on-line after class.

The break could come as soon as ten or twenty minutes into the lecture. One lecture I finished the theory in five minutes and the class laughed and loved it! The real work comes in the second part. I always, repeat ALWAYS write out my examples real-time, either on the white/blackboard, or on a document camera. I start with a simple example that illustrates the ideas from the theory. Then I ramp up the complexity. By the end, we are tackling tougher problems that challenge most of the class (these are the minority of the problems, but it is important to challenge the best students so they are not bored). The solutions I write out are not posted on-line to keep attendance up.

*The main thing students learn from with this method is watching you mess up*. They love it when I make mistakes. Forget a negative sign? Do an improper cancellation? Twenty hands go up! Students are engaged and that is so important.

I try to structure exams and tests so that they contain the examples I did in lecture. So there is an element of karma here: attend lectures and watch an expert solve questions from the the upcoming exam. Skip lectures (which is a real problem in first year courses) and you risk missing the solutions to problems that can get you a great mark on a test.

Incidentally, I was nominated for a provincial-level teaching award offered by TVO when I used the Nowakowski method to teach Calculus to Engineers years ago. I am eager to try this method again if I ever teach a large lecture and maybe I would even win the award this time.

Please let me know your teaching methods. I am eager to try new things with my mathematics lectures.

Anthony Bonato

I think it’s important to note whenever Moore is mentioned that his racist views were actively harmful to many black mathematicians. While he was hugely influential and supervised a lot of doctoral theses, there were many students who just didn’t have access to his classes because he refused to teach black students or explicitly told them they would get lower grades simply on account of being black. Raymond Johnson, one of the first black students at my grad school alma mater, Rice, writes about that here: http://math.rice.edu/People/Homepages/rlj/info/RJ.html

The math community’s continued veneration of Moore is harmful now because it tells black students that we would excuse someone hurting them if we liked other things they did. That’s not OK.

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Hi Evelyn, Great comment! It is definitely not OK, and you’ve made me and the readers here aware of Moore’s racists views. Johnson’s post is a powerful witness to those events and times.

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