Jon Bannon
• Member for 7 years, 10 months
• Last seen this week
• Albany, NY

Maryam Mirzakhani, who just won the Fields Medal, and also was the first Iranian student to win a gold medal in the IMO in 1995 with a perfect score. My colleague Mohammad Javaheri was on Iran's IMO ...

Imagine you had to look up every word you wanted to use, because you had a poor vocabulary. This would get old, very quickly. The trouble is, many people don't have a genuine need to internalize ...

I would discuss with these students the rather healthy point of view of Terry Tao on the subject. Summary One can roughly divide mathematical education into three stages: The "pre-rigorous" ...

As you told the student, the easiest way is to regard the Lebesgue integral as beginning with a partition of the range, rather than the domain. Perhaps a more refined way to view this is that the ...

This is a bit tongue-in-cheek, but it is good to lecture after students think they have mastered something via an active learning experience, provided the lecture is the result of long effort to find ...

To supplement the "games not problems" answer above (I love that answer): Certain games provide interesting opportunities for discovery...even if you don't know anything about the rules! An example ...

This is an excellent question. Some good advice on this can be found in the writing of Bill Thurston, some of which I have posted in an answer to this question on Math Overflow. The opening of the ...

If I were faced with this situation, I'd have a discussion with the student about what a number is. We believe that 1,2,3,4,… are numbers. It is a stretch to say that 0 is a number…which is why it ...

This answer/comment might be glib, but my motto is: If it's not in your head then you can't think with it. Another image (I paraphrase) that is helpful in this regard is due to Alain Connes: If I fly ...

My favorite little tidbit on this topic is due to Jean Pierre Serre: One should aim to be "precise, yet informal". Here, "informal" means not using symbols and notation, i.e. not using "formalism". ...

I think this question, with editing, has a lot to do with mathematics education, in that it points to the problem of learning to read and interpret mathematical writing. I would instead ask something ...

This is a great question. Here are some thoughts on it. A theorem statement is a sign of an idea that tends to be useful in the pattern of mathematical inquiry in a given subdomain. A good theorem ...

Check out http://jiblm.org. There are lots of scripts here, some better than others. A nice book in this style is "Distilling Ideas" by Brain Katz and Michael Starbird. I also recommend the ...

Below are some great and inspiring books by an excellent mathematician. (In the Really Big Numbers book, on the page where counting by tens is discussed there is an inspiring error (?)…Big Bird is ...

I've been tempted to try to use spaced repetition in the past, as there is certainly some kind of linguistic component to learning mathematics. This always meets the paradox that many mathematicians ...

I'd like to enthusiastically reinforce celeriko's comment above. It seems that students often don't pick up on the fact that really "juicing" something basic is what allows for fluent thought about ...

As a counterpoint to Polya: The psychology of invention in the mathematical field by Hadamard. As much as we all love Polya, Hadamard's study indicates that mathematicians don't often think in ...

This is a rather good collection. It is an essay by Reuben Hersh entitled Math Lingo vs. Plain English: Double Entendre.

The Calculus textbook of Hughes-Hallett, Gleason and McCallum et. al. includes ConcepTests that are rather good for this purpose. Sorry this is not a free online resource, but I hope it helps.

Another thought occurred to me regarding this, after having read Reuben Hersh's collection of essays. There is a quote of Bill Thurston, which I paraphrase as "thinking is the same as seeing". In a ...

The relation with squares is pretty nice: $\sum_{i=1}^n {F_i}^2 = F_{n} F_{n+1}$ The video link shows how suitable this concept might be for consumption by a general audience.

This book of Art Benjamin is fantastic. There is a new version of it, but I cannot remember the title. I should mention that although the book contains many specific "tricks for calculation" as ...

I think this question is important. I'd love to see an actual answer to it and cannot upvote it enough. I don't have an answer, but would like to share some intuitions/speculation. I think the ...

I do not believe there are any good such strategies, especially if you are in a Ph.D. program. I think your question is an important one because the comprehensive exam system at some places conveys ...

There is evidence that both a computational and conceptual approach are needed: https://www.jstor.org/stable/3482237 The paper of Sfard linked to does seem to agree that the scale must tip first ...

Since there have been a number of questions along these lines, the following observation may be useful: As a mathematician, I've often read summary background material in a very high powered ...

How about giving them the five room puzzle on the plane and then on the torus? This is a rather visceral way to appreciate why "holes matter", and may segue nicely into topics in topology.

For many courses, note taking is concerned with the organization of data for the purposes of recall. Let's call such organized data "information". In mathematics, the purpose of writing is more, ...