It's that time of the week again, and a new "Ask an SBer" question is out. The question is: "What makes a good science teacher?"
As usual, since I'm the only math blogger around here, I'm going to shift the subject of the question a bit, to "What makes a good math teacher?". The answer is similar, but not quite the same.
In my experience, what makes for a good math teacher is a few things:
- The ability to teach. This should go without saying, but alas, it doesn't. There are an appalling number of folks out there who are brilliant mathematicians and genuinely nice people who have all of the other skills I'm going to mention, but have absolutely no concept of just how to get in front of a group of people and teach in a reasonable coherent way.
- Enthusiasm. Most people have an unfortunate sense that math is miserable drudgery. Teaching something mathematical, one of the most important things you can do is to just be genuinely enthusiastic - to make it clear that you love what you're talking about, and that it's something fun and exciting.
- Balance. The power of math comes from the way that it breaks things that you're studying into simpler abstractions. Abstraction is the key to the value of mathematics. But it's very easy to get caught up in the abstraction, and forget why you're doing it. Good math teaching is a subtle act of balance: you're studying abstractions, but you need to keep the applications of those abstractions in sight in a way that lets your students understand why they should care.
There are two teachers that come to mind when I'm talking about this, both in mathematical specialties of computer science.
One is Eric Allender, a professor at Rutgers University, who taught my first course on the theory of computation. ToC is a field that can get incredibly difficult, and can often push abstractions so far away from reality that it's hard to see what the point of it is. Eric had everything that I said above nailed down perfectly: he had the ability to stand in front of a classroom full of people and explain difficult concepts in a way that made them comprehensible; and he caught us up in his enthusiasm for the subject, so that we caught on to why these difficult abstract things were interesting; and he always kept things grounded in a way where it was clear to us why we should care about it.
The other is Errol Lloyd, a professor at the University of Delaware, why I did my PhD, and a member of my dissertation committee. Errol is a professor who studies algorithms - not quite as abstractly mathematical as ToC, but a subject that many computer science students dread. I certainly wasn't looking forward to it coming into the class: my undergrad experience in the topic was awful. (The main thing I remember about it was the professor who seemed to only own one shirt, which he never washed. It was a running joke among the students, because every time we saw him, the shirt was dirtier. Same stupid blue turtleneck, which was almost more grey than blue by the end of the semester.) In contrast to the dreadful undergrad experience, Errol's class was one of my favorite classes ever. Errol has the most astonishing teaching method I've ever seen. He doesn't directly tell you anything: he gets up in front of the class, and starts asking questions. But the questions guide you through the process of discovering the subject that he's teaching. And as he does it, he's excited and happy and very, very kinetic, bouncing around the classroom, peppering different students with his questions. So as a student, you're involved, and you're caught up in his enthusiasm. (For those who understand what I'm saying: imagine a professor who can lead you through the process of inventing LR parsing from scratch, without ever telling you how to do it - just asking the right questions to force you to work through the problems that led to the invention of the LR parsing algorithms.)
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This is a recognized formal pedagogy in mathematics. It goes by the name Discovery Learning these days, but graybeards call it The Moore Method, because it was formalized in this country by R. L. Moore (it fell out of favor because of personal issues with Moore, not his method). It is extremely popular at Texas universities.
It is also quite controversial when done at the undergraduate level. Many departments are often worried about "coverage", and don't like this method because the class generally runs slower (of course, how much students retain in a high coverage course is another issue). I have long used this technique in my "bridge courses" (intro to proof), where it is the skill, not the concepts that are important. But political considerations keep it from being used as a general technique.
Or more generally the Socratic Method, no? There's a good article about using it to teach maths (to younger children) here - http://www.garlikov.com/Soc_Meth.html
I am currently teaching first-year basic maths and statistics to Education students in remote parts of Manitoba (Mathematics is not my main area but I have a degree in Math/Comp Sci - I would not want to teach anything more advanced than first year). Many of the student are weak in maths and dread the prospect of having to do the courses.
The first thing I do is to agree that many find it a stressful area. This is probably because often people are very aware that they do not have the correct answer, but may not realize that what they did was 90% correct. On the other hand, in history or biology they could write an essay and be completely oblivious to the fact that they had forgotten three-quarters of the material. I also point out that many of the concepts, for example subtracting a negative number or that zero is a number, took a thousand years or more for people to come up with, so they shouldn't feel bad if it takes them a couple of days to get it.
As prospective teachers, they tend to be interested in people and appreciate anecdotes I give them about those who developed the techniques, and why they were done (why the heck did Viete want to solve quadratic equations in the 1500s?).
Many people who have difficulty with maths think that those who find it easy can just look at a problem and know how to do it - I point out that often competent mathematicians will have several false attempts, although they have probably learnt which methods are likely to be productive.
Above all, students need encouragement. So much of maths is about confidence.
Breadth of knowledge.
Point out associations of ideas across different fields.
I find this causes the "Aha!" effect of insight in students.
Simple example for primary school : taking square roots using compass and straightedge only.
Amen! I still remember (thirty-five years ago) that the grad student scheduled to teach my freshman calculus course at college was hit by a car (no! BEFORE the class started) and we got a full professor instead. He may have been a brilliant mathmematician, and he may have been great with post-graduate courses, but every single one of us failed the course. We never had any idea what he was talking about ... the university expunged that course from our records and offered us the chance to take it again, but he'd sufficiently scared me away from the field (I was 18, after all).
I love my field (languages) but I've often wistfully wondered what it would be like to know a bit more about math beyond basic algebra... One of the reasons I read this blog every day.
It's been a while, but I experienced the Moore Method in 1976 at Georgia Tech. One of the strangest and most wonderful experiences of my life. I was a freshman. My 1st quarter, there must have been 20 sections, each with ~80 students, of generic Engineering Calculus I. Yeah, 1600 seems about right. Quarters 2 & 3 there was 1 section, with ~20 students, of Honors Calculus taught by the Moore Method, Dr. James V. Herod in charge. Changed my life.
I could say more.
(funny -- come to think of it, "I could say more" is, in fact a Herod-ism I picked up 30 years ago!)
Hmmm. I posted a couple of innocent-seeming non-spam comments here a couple of hours ago, but I don't see 'em yet.
If it's just a slow moderation day, then please excuse my apparent impatience.
OTOH, will I ever see *this* comment? :-)
Amen to the Socratic method. The best classes I've ever been in have been taught this way. I've never had a math class taught like this, but science and *several* sociology classes have used this with me to great effect.
On the other hand, my best math classes have all linked the abstract to the concrete, like Mark's first teacher did.
Mark-
Hey! You're not the only math blogger around here! Growl. Snarl.
Jason:
No insult intended. I know you're a math guy, but I thought you don't blog about math, you mostly blog evolutionary bio. I haven't seen any other blogs focusing on math here at SB.
One detail that was probably too trivial for you to even consider, but that is vitally important: understands the material. I never had that particular problem at the graduate or undergraduate level, but I had some experiences in high school and earlier that just floored me.
Of course, if they fail on that point they likely fail the "enthusiasm" point anyway.
One thing I vividly remember from the Moore Method class I took was that you have to be verrrrrrry careful as to what you're using to designate entities. Using O sub K when you tend to finish each sentence with "okay?" really doesn't work. (Other than that, I adored that particular fellow student....)