Inspired by a quote from the University of British Columbia’s Philip Loewen, David likes to envision a classroom as an imaginary triangle, where the vertices are the instructor, the students, and the course material. He believes that if a struggle were to exist – which is usually the case – the struggle should be between a student-instructor alliance and the material. It follows that teachers shouldn’t hold out on students when it comes to clearly explaining what is expected of them, even if that means formally recognizing that marks aren’t ‘just a number’ to the students. Says David, “There is inherent value to the student in the mark and in the degree, and it’s important to acknowledge that it’s okay for the student to ask ‘what’s going to be on the exam?’ even though – on the surface – that has nothing to do with the material of the course.” He is quick to clarify that this doesn’t mean to figuratively hand out copies of the exam, but to give students a fair idea of what they need to know how to do in order to succeed.
In 2008, David was a well-deserved winner of the Faculty of Mathematics Award for Distinction in Teaching (along with Robin Cohen). He has accumulated an astonishing amount of theoretical insight on pedagogy over the years. Among his doctrines is the principle that each specific branch of mathematics is a sequence of seven progressive courses; 1A calculus would be course number one, while a typical faculty member’s research is past the seventh course.
He also conceptualizes teaching math as a continuum of abstraction. “The top level is the biggest picture, which is the simplest, but most abstract, therefore most difficult to come to grips with in an intellectual way. Gradually moving down to the very concrete level, it is most complex, but easiest to understand: "You take the numbers and do this with them and you get the answer," David explains. According to him, it’s a matter of striking a balance between the two extremes so that the mathematics is both comprehensible and useful for students in solving problems. It’s quite baffling, but it obviously works for him, so let’s not question.
David likes to challenge his students by including some questions on his exams that are beyond the scope and the level of the course, but does not penalize them so long as they meet the core requirements of the syllabus. “You could look at old exams and ask yourself, ‘Are students performing much worse than they used to?’ and you also have to ask questions like, ‘Is it really true that they don’t really have a shot at succeeding in the next course?’” he says.
In his other role as Associate Dean of Undergraduate Studies, David is responsible for overseeing the assignment of teaching posts, ensuring that the appropriate resources are available for each course, polishing up the curriculum, and running the Mathematics Undergraduate Office, among other duties. The other major issue that he deals with is academic integrity and punishing related offences. “That’s a very unpleasant part of the job, but somebody’s got to do it. Lucky me,” David remarks.
Amidst all his duties as instructor and associate dean, he somehow manages to find time to do research. However, as much as he loves solving math problems on a blackboard, David has a difficult time motivating himself to compile a written summary of his findings. It’s also a rather boring task compared to the thrill of research, and not to mention frustrating. Often, while partway through his write-up, he would discover some small detail that causes his entire solution to unravel.
Fortunately, David is persistent. He is currently examining the properties of K3 surfaces. While they are generally assumed to have rational curves defined over infinitely large field extensions, nobody has found an example or proven the existence of an example that illustrates this. “You’re in a room full of swans and you assume that they’re all white, but it’s pitch dark so you can’t tell,” David analogizes.
David McKinnon’s ‘basic’ tips to good instructing:
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Preparation
“Walking into the room, you’re going to have to explain in detail exactly how this technique works, do a few examples of it, and be able to field random, off-the-wall questions from students who had never seen it before, who are misunderstanding it in ways you’ve never heard of before.” -
Teaching
‘voice’
“You can’t go to a superb instructor’s classroom, sit down, look at what they do, and try to copy it. Almost every time it comes out sounding forced and unnatural, and makes your teaching worse. You have to teach the way you want to teach.” -
Clear
handwriting
“It’s amazing how much of a difference that clear handwriting can make. If the students can read what you’re writing on the board, it’s so much easier to follow than if they can’t.” -
Clear
speech
“Lots of people can speak perfectly clearly one-on-one, but don’t know how to project to a room. If that’s your problem, get a microphone! Ask the students at the back, when it’s just you and them, whether they can hear you. In the middle of class, they wouldn’t answer you.” -
Eye
contact
“When you’re talking to someone, look at them; if you don’t, they won’t hear you.” -
Blackboard
etiquette
“When you’re writing on the blackboard, don’t stand in front of it. Write it, and get out of the way so they can see it. You’re not invisible!”