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In forming your answer you may treat the qualifier math or maths as optional, since part of the question is whether there is anything peculiar to the subject of mathematics that demands anything peculiar of the teacher.‡

The question is hardly new, but it came up most recently in the context of answering another question about good ways to respond to people who say,"I was never much good at maths at school", especially if one is motivated to give "answers that would actually educate the other person".

A moment of reflection tells me that there's another question in the back of my mind when I muse on this question in the present setting —

Bonus Question. Working on the assumption that people visit this site for the sake of learning and sharing mathematical knowledge, what do your answers to the question of teacherly virtues tell you about the qualities of social-technical system design that would best serve that purpose?

‡ "peculiar" distinctive, not "peculiar" ha ha …

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  • $\begingroup$ My answer is very different depending on whether you mean high school or college. Could you clarify? $\endgroup$
    – Qiaochu Yuan
    Commented Nov 14, 2009 at 21:19
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    $\begingroup$ @Qiaochu Yuan : It's respondent's choice, or else feel free to give a parametric answer (a.k.a. "2-parter"). $\endgroup$
    – Jon Awbrey
    Commented Nov 15, 2009 at 4:22
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    $\begingroup$ There is a commentary in the June-July issue of the Notices. Dealing primarily with teachers of elementary or middle school math. $\endgroup$
    – Gerald Edgar
    Commented Jul 22, 2010 at 20:13
  • $\begingroup$ Eric Mazur has a very good talk about peer instruction (around 1 hour and 20 minutes long): youtube.com/watch?v=WwslBPj8GgI $\endgroup$
    – JRN
    Commented Mar 19, 2011 at 3:00

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I have answered this question with what I hope are some qualities a teacher can learn in order to improve their class. This is as opposed to listing qualities of a truly exceptional math teacher. I wrote this thinking of middle and high school level math, but most of it could be applied at other levels as well.

1) A good math teacher should motivate the math and engage the students. Take the example of solving linear equation. One can start by telling students the formal rules for how to manipulate an equation, but I think students will find this very dry, and won't understand why they are doing what they are doing. It becomes and exercise in memorization. Instead, one can start with problems that can be solved with such equation. One can first get students to solve them with other techniques (e.g. guessing and checking or using some sort of graph). After a while one realizes there should be an easier way, which turns out to be solving a linear equation. This way the students understand why the formal math was developed, understand how to apply it, and see how it is related with other ideas (like graphs). Right now you might ask where one can find good problems to use in this way. I think I will ask that as a separate question...

2) A good math teacher makes their students do math. I think it is crucial that every student, in every math class, every day, solve some math problems. Some of these should be easy (i.e. just practice solving equations, once they have been introduced), and some should require more creativity. It is of course a mistake to drill students with boring problems until they hate the subject, but it is also a mistake to let them do "interesting" or "discovery based" math all the time, and not make them practice the techniques they discover.

3) A good math teacher should convey the beauty of the subject. One of the other answers said ``infectious enthusiasm" was needed. That would be great, but in reality not all math teachers can be that charismatic. Even without a great deal of charisma, I believe it is possible to show students the wonder of extracting a simple answer from a seemingly difficult question, and the beauty of the tools that help one do this. Often it is enough that students see that their teacher believes this. So in particular, I do not think it is a good idea to say things like "I hated math when I was your age too, but we'll get through this".

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Every great teacher has his or her own teaching style and philosophy but here are some thoughts on things that I think make for a pretty good teacher (some more math-specific and some less so).

I think good teachers respect their students. I think teachers can effectively engage students by treating teaching and learning as a collaborative process and by showing their students that their thoughts and opinions are valued. It can be really good motivation for students to feel that they're working with their teacher to develop their understanding.

Following on from that, I think good teachers can get their students talking. Having students ask and answer questions about the material being covered is a great way to get them really thinking about the ideas for themselves. Also, some teachers really encourage their students to talk to each other and I think that's a great way to show students how much they can learn from each other, independent of their teacher.

Along those lines, great teachers should challenge their students. They should encourage their students to get out of the mindset of "I only know as much as my teacher tells me". In math, I feel like many students end up expecting all the problems they encounter to be similar to some example they've been shown. An invaluable skill that great teachers pass on to their students is the ability to take their knowledge and skills and apply them in unfamiliar situations. It's important to show students that they can do things on their own!

Great teachers provide positive encouragement and credit where it's due. When you're in the middle of learning something, it can be hard to take stock of how far you've come or whether you actually know more than you did weeks or months ago. So it's important for teachers to keep track of their students' progress and to let them know about it.

Especially in math, I think good teachers provide motivation and explanation for the material. It's so easy to get caught up in formulas and theorems and simply ignore where they came from but it's so important to make sure that students realise that math doesn't just come out of nowhere. Even in cases where rigorous explanations are a little beyond the students, good teachers can appeal to their students' intuition and give basic ideas about why things are true. Being able to show students that math is all about logic and reasoning and that it should make sense I think is the mark of a really incredible teacher.

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A good teacher should

  • be able to figure out when a student's understanding becomes better than his own to recommend some reading or to pass the student on to a more knowledgeable teacher
  • develop individual approaches for gifted students (say, emphasizing theoretical vs problem based depending on the student)
  • be realistic in estimating the impact of his teaching work on students, and not be lazy, e.g. repeat important things again and again
  • have enough "infectious enthusiasm"; without it any efforts will be pointless

(clearly there's much more, but that's something that comes to my mind right away)

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Initially, this was supposed to be a comment about point 3) in Peter Tingley's answer, about a math teacher being able to "convey the beauty of the subject", but it got too long so I turned it into an answer.

While I generally agree with it, I think it is also a tad too idealistic. Dr. Johnson, in one of his invectives against the Scots, barked that "much may be made of a Scot if he be caught young". Contrapositively, if a student arrives at university without any appreciation for Mathematics, there is only the slimmest of chances that he will gain it there. And teenagers? Mathematics has to compete in a teenager's mind with Lady Gaga and porn sites; with mtv rap and online gaming; with entertainments that range from the irrational to the inane. Call me an elitist cynic, but the plain matter of fact is that the majority of students will be completely unmoved by the beauty of mathematics, much as great literature can only be understood and enjoyed by a relative minority within each generation. I guess what I am saying is that a quality that a good teacher must have is a thick-skin and somehow, by whatever means, keep the flame alive inside his own heart so that when that special, receptive student does come along, he will be able to kindle the fire and infect him with the rapturous love for Mathematics. Or in the words of Samuel Beckett, "Try Again. Fail again. Fail better."

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    $\begingroup$ This is second time in my life I see a reference to something called "Lady Gaga", and both of them have been here on MO. I wonder if that means my inner flame is safe... $\endgroup$
    – Victor Protsak
    Commented Jul 22, 2010 at 23:52
  • $\begingroup$ I think that there is a wonderful point between the lines. Keeping in mind that you have to compare with Lady Gaga can be one of the most helpful things while teaching mathematics. Most (if not all) of us will agree that mathematics can be beautiful. However, we cannot hand it at a student and expect them to see the beauty. I believe that every teacher should also try to be entertaining and colorful. Of course, there is a limit to that (we are not stand up comedian), but I believe that really putting an effort into presenting it in a fun and entertaining way (like something on MTV) would help. $\endgroup$
    – Niemi
    Commented Jul 31, 2011 at 18:59
  • $\begingroup$ Excuse this off-topic comment, but this reminded me of another Dr. Johnson quote. This is the occasion when Boswell was first introduced to Johnson through a mutual acquaintance, Mr. Davies. Boswell writes, "Mr. Davies mentioned my name, and respectfully introduced me to him. I was much agitated; and recollecting his prejudice against the Scotch, of which I had heard much, I said to Davies, 'Don't tell where I come from.' 'From Scotland,' cried Davies, roguishly. 'Mr. Johnson, (said I) I do indeed come from Scotland, but I cannot help it.' I am willing to flatter myself that (cont.) $\endgroup$
    – Todd Trimble
    Commented Oct 17, 2011 at 16:28
  • $\begingroup$ (cont.) I meant this as a light pleasantry to soothe and conciliate him, and not as an humilating abasement at the expence of my country. But ... [he] retorted, 'That, Sir, I find, is what a very great many of your countrymen cannot help.' $\endgroup$
    – Todd Trimble
    Commented Oct 17, 2011 at 16:30
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The math teachers that I like the most are the ones that make the subject they teach look very easy. For instance, my abstract algebra teacher makes all the definitions justified and look very simple. He always finds the simplest arguments for every proof, and since he does everything in the right order it just seems to flow, and we almost could guess every proof that he's about to give us, because they all seem very natural.

I'm not sure I know how to summarize this quality exactly, but I'd say that every math teacher should be able to make the subject he's teaching look simple and easy by finding the simplest proofs and doing all the necessary intermediate steps that lead to a more difficult theorem.

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    $\begingroup$ From your description it sounds like your abstract algebra teacher is indeed a very good teacher. However, I don't think that "making it look easy" is what makes him so good. What you are describing instead sounds to me like he is very well organized! I often dislike teachers who try too hard to makes things look easy, because often this simply results in sweeping the fiddly bits under the carpet. $\endgroup$
    – Emily Peters
    Commented Nov 14, 2009 at 21:00
  • $\begingroup$ yes, I guess, I didn't really mean that he was "trying" to make his class easy, more that the way he explains makes it look easy, without even trying, because of his organization... That said, there are some fields that I doubt any teacher would be able to make look as simple... For example, my analysis classes have never seemed as easy, mostly because of the harder proofs... $\endgroup$
    – Jean-Philippe Burelle
    Commented Nov 14, 2009 at 21:10
  • $\begingroup$ I think the word 'easy' may be not quite right. Perhaps the right word is 'natural'. One of my favorite professors (actually the guy currently in the office next to Emily) had the miraculous ability to be clear while doing math on his feet. I think this is a consequence of having thought very carefully about things and worked them out for himself. Somehow it was possible to benefit from his effort/natural point of view to pick up on how to approach the subject. I feel that his material was so well-worn and honest that it was inspiring. What he presented looked simple, but you knew it wasn't! $\endgroup$
    – Jon Bannon
    Commented Nov 18, 2010 at 17:13
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First and foremost an excellent teacher has all the qualities of an excellent student. She is articulate, open minded and friendly. She is excited about the subject and is able to communicate her amazement and understanding to her students by articulating her own intuitions and expertise in a way that is accessible to everyone. She also provides a bird's eye view of the subject by interjecting various tidbits from the history of the subject and pointing to stumbling blocks. You'd be surprised to see how many teachers act like answering a simple question is beneath them, speak in monotone, purposely ask trick questions to make the students feel stupid, dismiss questions as not worthy of an answer and just plain act rude and unfriendly.

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    $\begingroup$ What makes this behavior even more despicable at the university level is that many of its worst practitioners are some of the field's top researchers.They're POed they actually have to come down from the mountaintop and teach,which is the most degrading of actions for someone of thier self-percieved character.I can't help but think of that scene from "A Beautiful Mind",where Russell Crowe's John Nash is really upset he has to teach at MIT and delibrately gives the students a homework exercise he knew they couldn't solve without a trick to both keep them out of his hair and demoralize them. $\endgroup$
    – The Mathemagician
    Commented Jul 22, 2010 at 17:29
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    $\begingroup$ While some people are better teachers than others, very few of the top researchers that I know fit Andrew L's description. $\endgroup$
    – Andy Putman
    Commented Jul 22, 2010 at 18:12
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    $\begingroup$ Andrew L, perhaps you could try to distinguish more between "things that are generally true," "things that are true in your experience" and "things that you have heard are true"? Whenever you begin discussing "top researchers," you deliver a lot of pronouncements that sound like they're in the first of these three categories, but I suspect that they belong on the last. The fact that you defend your perception by reference to a fictionalization of the biography of an exceedingly unusual individual really reinforces my perception. $\endgroup$
    – JBL
    Commented Jul 22, 2010 at 19:52
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    $\begingroup$ @Andrew L: So, going back to the three categories mentioned in my comment, you assert that you make pronouncements of "general truth" based on personal experience, rather than hearsay. Yet every time you make such a comment, someone has responded to note that your experiences differ noticably from their own. How do you suppose this can be reconciled? $\endgroup$
    – JBL
    Commented Jul 23, 2010 at 1:50
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    $\begingroup$ @Andrew L: It really troubles me that "Perhaps my views don't really reflect universal truth" isn't on your list. How hard have you considered this possibility? (Incidentally, I've never found it difficult to "be accepted as a mathematician by my peers," nor have I ever been subject of any sort of negative attention for my expressed love of teaching. And while I have certainly been subjected to my share of bad teaching in undergraduate and graduate level math classes (and in other subject areas!), none of it even remotely reminds me of your experiences.) $\endgroup$
    – JBL
    Commented Jul 23, 2010 at 12:36
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A little late - A good math teacher is someone who has a passion for mathematics and can share and instill that passion upon their students.

Good math teachers/professors should be the ones who encourage their students to go beyond the classroom with their learning. They should encourage and help their students get involved in original research and advise them on what steps to take. The teachers have the experience and the students do not; thus, the teachers should share this experience with their students.

A good math teacher is one that teaches students what the true joy of mathematics is and how to function in a world outside school utilizing mathematics.

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Just knowing that it took Whitehead and Russell almost 370 pages of the first edition of Principia Mathematica to be able to state 1 + 1 = 2, and nearly 800 pages of P.M. to q.e.d. 1 + 1 = 2, I think that most of us will be satisfied that it CAN be done that painstakingly, if need be, while we meanwhile get on with our shortcuts to go on to more interesting results.

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    $\begingroup$ Seems like this should be a comment on Jose Brox's answer rather than an answer all on its own. The OP never said anything about painstakingly justifying everything from the axioms $\endgroup$
    – David White
    Commented Oct 17, 2011 at 16:35
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A good teacher should know his/her audience : what do they know? what do they expect? what is the goal?

A same lecture will be one's joy and another's torture.

You definitely won't treat the same subject in the same way in front of would-be mathematicians, would-be engineers, would-be biologists, would-be economists, etc! [and that's just mentioning students, not pupils...]

There are audiences in front of which you won't give any definition, any lemma, any theorem. You'll just take examples, first very down to earth to show how to compute/visualize something, then a little more general, but still not abstract, to show off the magic.

And even in front of mathematicians, you should beware : are you talking to a pack of specialists of the subject or in some sort of colloquium? On that matter, I can link you to William Thurston's "On proof and progress in mathematics" (which I recently read because someone pointed it out in an unrelated question).

90% of teaching isn't about the core of the subject, but being receptive to the people you're dealing with.

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I think it is crucial for the math teacher to give a detailed description of anything he does, justifying any step he takes from definitions and axioms, from the start. Of course, before/later he should add a lot of intuitive explanations and examples about how and why the definitions are chosen and the results are going to be true; but the teacher should keep well in mind that, for every step he spares, he will get a bunch of students who will reach to the conclusion that maths is not a logical subject, but merely a collection of boring and inscrutable pieces of symbolic relations that they should memorize, even more if they are immersed in a program that teaches them more algorithms than theoretical results (as happens in the highschools of many countries, for example here in Spain, where I live).

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    $\begingroup$ I disagree with this. Carefully justifying every step very much sends the message that math is a collection of boring symbolic manipulations. When I do mathematics, I don't justify every step, and I don't think kids should either. Unless it's the crux of the argument, I would never say to a kid "hold it right there; how do you know the sum of two even numbers is even?" The place for careful arguments is when something interesting happens, when your intuition (or something you "proved") says something you know is wrong. Kids should learn the value of being meticulous, not have it forced on them $\endgroup$
    – Anton Geraschenko
    Commented Nov 14, 2009 at 15:30
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    $\begingroup$ For an argument longer than 600 characters, see Paul Lockhart's "A Mathematician's Lament" (maa.org/devlin/LockhartsLament.pdf), which I first came across in this question: mathoverflow.net/questions/5074 $\endgroup$
    – Anton Geraschenko
    Commented Nov 14, 2009 at 15:34
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    $\begingroup$ The level of detail that suits the teachable moment at hand is of course a judgment call, but the rule of thumb that reason must be given where reason is due — that, I think, is basically sound. Our teaching seminars in grad school expressly cautioned against impressing students too deeply with the "Bag of Tricks" theory of mathematical prestidigitation. Sure, we all love those tricks, but leading students down the path of thinking that math is "nothing but tricks" is going a bridge too far. $\endgroup$
    – Jon Awbrey
    Commented Nov 14, 2009 at 16:06
  • $\begingroup$ @Anton : Thanks for the frabjous essay! I'll have to print it out and give it another couple of readings. $\endgroup$
    – Jon Awbrey
    Commented Nov 14, 2009 at 16:42
  • $\begingroup$ I agree (to a point) with Anton. The goal should be for students to reason precisely. More fine, detailed steps should be a means to that end. Demanding justification can add or subtract from this goal. You want justification where being ambiguous or unclear affects the reasoning. But demanding every minute justification may undermine the cause -- students might view such steps as symbol pushing disconnected from the "real" reasoning. $\endgroup$
    – Richard Dore
    Commented Jul 25, 2010 at 21:14

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