Assorted links with tips, tricks, and bits of philosophy.
Suppose you want to teach the “cat” concept to a very young child. Do you explain that a cat is a relatively small, primarily carnivorous mammal with retractile claws, a distinctive sonic output, etc.? I’ll bet not. You probably show the kid a lot of different cats, saying “kitty” each time, until it gets the idea.
Can we make mathematics intelligible?, by Ralph Philip Boas, Jr.
Organizers of colloquium talks everywhere exhort speakers to explain things in elementary terms. Nonetheless, most of the audience at an average colloquium talk gets little of value from it. Perhaps they are lost within the first 5 minutes, yet sit silently through the remaining 55 minutes. Or perhaps they quickly lose interest because the speaker plunges into technical details without presenting any reason to investigate them. At the end of the talk, the few mathematicians who are close to the ﬁeld of the speaker ask a question or two to avoid embarrassment. This pattern is similar to what often holds in classrooms, where we go through the motions of saying for the record what we think the students “ought” to learn, while the students are trying to grapple with the more fundamental issues of learning our language and guessing at our mental models.
On proof and progress in mathematics, by William P. Thurston.
Sometime in my second year in university I suddenly discovered that most of the things I studied really mean something; that proofs are not just long chains of logical deductions discovered by evil math geniuses, that behind every worthwhile argument there’s a beautiful picture that fits together with everything else. Except that my profs tended not to talk about these pictures, and certainly not about everything else. They didn’t even tell me that these pictures were there.
Teaching philosophy, by Dror Bar-Natan.
This post is about a very simple idea that can dramatically improve the readability of just about anything, though I shall restrict my discussion to the question of how to write clearly about mathematics. The idea is more or less there in the title: present examples before you discuss general concepts. Before I go any further, I want to make very clear what the point is here. It is not the extremely obvious point that it is good to illustrate what you are saying with examples. Rather, it is to do with where those examples should appear in the exposition. So the emphasis is on the word “first” rather than on the word “examples”. If this too seems pretty obvious, I invite you to consider how common it is to do the opposite.
My favourite pedagogical principle: examples first!, by Tim Gowers.
Common sense, cognitive psychology, and my personal experience agree that, no matter how interesting a lecture, students’ attention is much keener in minutes 1-10 than in minutes 40-50. I’ve found it makes a big difference to break every lecture up with some kind of active or reflective activity. For example: I write an assertion on the board, like “If a matrix is diagonalizable, it has distinct eigenvalues.” I ask for a show of hands on whether the assertion is true or false. Then I ask each student to turn to the person sitting next to them and try to convince that person of the truth or falsity of the assertion. After one or two minutes, I bring the class back together for another show of hands, followed by a brief discussion. I think the class finds it gratifying that the process tends to converge on the right answer; it seems to me to emphasize the useful lesson that something is mathematically correct because it can be argued, not because I say so. (I first heard about this technique from Eric Mazur at Harvard; I’m not sure whether it’s original to him.)
Notes and links on teaching, by Jordan Ellenberg.
- Mathematical writing, by Donald E. Knuth, Tracy Larrabee, and Paul M. Roberts.
- Mathematical Education, by William Thurston.
- Real-life module statistics: A happy Harvard experiment, by Kari Lock and Xiao-Li Meng.
- (To read) Teaching labs the Compass way: Engaging students in authentic research practices and regular self-reflection, by Punit R. Gandhi, Jesse Livezey, Anna M. Zaniewski, Daniel L. Reinholz, Dimitri R. Dounas-Frazer.
Michael Atiyah has a section devoted to the power of good examples in his Advice to a Young Mathematician. Here is his final paragraph (the whole thing is very much worth reading):
But most of all a good example is a thing of beauty. It shines and convinces. It gives insight and understanding. It provides the bedrock of belief.
I am going to translate a passage of good English into modern English of the worst sort. Here is a well-known verse from ECCLESIASTES:
I returned, and saw under the sun, that the race is not to the swift, nor the battle to the strong, neither yet bread to the wise, nor yet riches to men of understanding, nor yet favor to men of skill; but time and chance happeneth.
Here it is in modern English:
Objective consideration of contemporary phenomena compels the conclusion that success or failure in competitive activities exhibits no tendency to be commensurate with innate capacity, but that a considerable element of the unpredictable must invariably be taken into account.
This is a parody, but not a very gross one . . . The first sentence contains six vivid images, and only one phrase (“time and chance”) that could be called vague. The second contains not a single fresh, arresting phrase, and in spite of its 90 syllables it gives only a shortened version of the meaning contained in the first.