To improve the chances of sharing my experiences, I decided to record at least a brief, one-sentence "nugget of wisdom" from every talk I attended, be it research- or teaching-oriented. Oftentimes, I jotted down other things, too, like references to look up later, or more extensive ideas. But for (pretty much) every talk I went to and paid genuine attention to (which was about 95% of the ones I attended), I wrote down something memorable to take away from that presentation. Gathered below is a curated list of some of those "nuggets". (I've fleshed out some ideas that were chickenscratched and brief, and I've tossed out some that were too similar to others.)
- Create a "moment" or "story" for every class you teach; in years to come, students might forget the material, but they'll remember a significant moment.
- We mathematicians have trained our brains to switch from "proof mode" to "example mode"; we need to help students do the same, or even think to do so.
- In an intro to proofs course (or any course, really), a good activity is to have students convince their classmates that their work is valid and clear.
- It sometimes seems like students read a proof like a story, so they can get a good overall sense for it, but they will miss little statements that are either incorrect or subtle or otherwise difficult to follow; can/should we teach them to fixate on particular lines of a proof?
- If you want to write a "fun" exam problem (or even have a theme throughout the exam), be sure to know your audience and make sure it won't be disruptive.
- There are many comic strips that have mathy content; an activity where students have to explain the joke to someone else can build their conceptual understanding.
- Beginning class with a joke or short story, even if it takes up a few minutes of time, can get the class more engaged at the beginning and yield more participation/attention throughout the session.
- First year students may view math more cohesively (as opposed to procedurally) than we give them credit for, and our reaction to that shapes their future learning.
- Students apparently misunderstand what a function is; but what can we do about that?
- Baseball is known for being a sport full of statistics, but folks should be aware that the PGA Tour's ShotLink system will willingly share with you, for free, more data than you know what to do with!
- Have students take conceptual quizzes before class (perhaps online), and shape discussion at the beginning of class based on those results.
- Mentoring an undergrad research project is both time-consuming and rewarding, in large amounts (of both).
- Why is it so hard to come up with combinatorial proofs for identities that are trivially true algebraically?!?!
- When training future teachers, be sure to teach them how to give good criticism; it's not all about praise.
- Adjacency matrices and their properties can be great topics to use "easy/intro" graph theory to motivate linear algebra.
- A great speaker can make a topic that I really don't know anything about seem completely fascinating and understandable.
- If you want to motivate epsilon-delta definitions, use in-class numerical explorations/"lab work" to make the notion of "closer and closer" feel like cheating; then, when the epsilon-delta definitions are introduced, it will feel rewarding.
- Using primary literature (i.e. journal articles) in undergrad classes improves understanding, but it's very difficult to choose appropriate articles. (We need more people writing such articles!)
- One way to introduce students to reading scholarly articles is to choose a particularly readable article and have students turned in an annotated copy after reading, making notes and highlighting parts that are relevant to your course.
- We need to ensure that high school students understand there is so much more to math than just calculus, especially talented students who might be jaded by math by the time they enter college.
- (re: last point) It would be great to provide activities to AP Calc high school teachers to use in school after the AP exam, but we need to keep the needs and struggles of those teachers in mind (e.g. provide them a guide to the math involved, to save them time/effort).
- As much as possible, be playful with math; show students that one can unwind with math, that it's not always "stressful".
- There are many academic journals that embody exposition instead of creative research, but I wonder why these don't have the same level of "prestige" attached to them as other journals.
- Seeing "3/10" on a proof is demoralizing to a student; seeing something like "progressing" (e.g. on a descriptive scale) is better, but still, students want and need a sense of where they stand. Is there a happy medium?
- Historically, divergent series were once a contentious and important topic; nowadays, not so much. What happened, when, and why?
- When we plan courses, we should ask ourselves, "What are the skills we want students to have 10 years from now?" We can contribute to that positively, and we really should.
- Encourage false starts in class, and be willing to follow through to a dead end; if students complain about their notes, just say, "Put a box around this part and, underneath, explain WHY it's wrong, then move on."
- When giving online quizzes, make the last question, "Ask one question about this section (or earlier material)" and give credit for this. Making the students formulate a question pays dividends for them and us.
- I appreciate the teaching talks from large universities where their class sizes are 200+ so they can have lots of data to analyze, but my classes are so dissimilar that it's hard to compare. I also appreciate the teaching talks from small colleges where their class sizes are more like mine (10-20), but their results are almost entirely anecdotal. Is there any way to have large-scale data about small classes?
- If you're making students present in class (be they homework solutions or final projects or anything), tell them what skills they're supposed to gain from doing so (beyond just sharing with the class).
- All of those "series convergence tests" we teach in Calc II are fundamentally comparisons (direct comparison, comparison to a geometric series, etc.) and presenting them as such gives that section a narrative students can hang on to.
- The mainstream does an undeniably terrible job of presenting mathematicians and their work accurately and meaningfully; we need to do better self-promotion by writing articles ourselves.
- Might it be worthwhile to encourage (or at least support) "guess and check" methods of integration? Integration by parts and u-substitution have an inherent "guess" step anyway...
- If you have a class of size 6n+3, you can split the class into triplets for 3n+1 sessions such that each student works with every other student exactly once. (Such arrangements are fairly easy to find by hand for small n but the difficulty increases greatly...) [This is related to the "Kirkman Schoolgirl Problem"]
- Paul Seeburger has a collection of awesome applets for Calc class demonstrations.
- Consider adding a "How confident are you?" button to online quizzes (or even exams). Gathering such data can be enlightening in the long term, and can guide class discussions in the short term.
- Consider letting students choose the weighting scheme for your syllabus (within limits you define). Letting them take ownership pays off for everyone.
- Take an equilateral triangular board, whose "squares" are hexagons, with n hexagons on each side. There exists a Knight's tour on such a board if and only if n is at least 8. One proof of this is a neat "non-standard" induction: show a bunch of base cases and then show KT(n) and KT(n+1) imply KT(2n) and KT(2n+1).
- Letting students write and turn in exam corrections for credit can increase their engagement and show that we actually care.
- Fractals can be a great, engaging motivation for studying infinite series.