Homotopy Theory

Aaron Mazel-Gee

3:58 AM

what do i do about the fact that my calc students are really bad at math

Qiaochu Yuan

4:48 AM

nothing

Arnav Tripathy

6:33 AM

what's the status of equivariant bott periodicity ?

what kind of proofs do we have for this guy now

Sean Tilson

10:22 AM

@ArnavTripathy What kind of groups do you care about?

@AaronMazel-Gee What could you possibly do? Oh how I wish you could know my pain. As bad as your calc students are I have had worse. I once heard a grown women ask what counterclockwise was. When someone told her she didn't believe them because "one hand goes one way and the other hand goes the other." When someone pointed at a clock and said no they don't, she asked for a second opinion. This person was over 40.

Nat Stapleton

12:08 PM

@SeanTilson I love that story.

Jon Beardsley

12:23 PM

@SeanTilson @AaronMazel-Gee i once had a student ask me if the fact that $a<b$ implies $b<a$ is some kind of rule that they need to memorize.

@AaronMazel-Gee in my experience, the best thing you can do is maintain a positive attitude and stay healthily detached, seriously. it is not our job to ask why nor to fix higher education, at least, not for the moment.

Zhen Lin

Sometimes I wonder if perhaps everyone would be happier if they didn't make calculus mandatory

Jon Beardsley

I don't know. Somehow I'm not sure that's the solution. Our students can't get something so we just eliminate it from the curriculum?

Saul Glasman

at a lot of universities, even in the US, calculus isn't mandatory for everyone, right?

Jon Beardsley

12:39 PM

Correct.

I'm not saying it should be mandatory for everyone but, I don't know, it's a pretty important development in human thought.

I don't know. At JHU we give it to pre-med students, and they don't seem to really wanna take it, for the most part, and I'm not sure that they should have to either.

Saul Glasman

maybe calculus isn't important for actually seeing patients, but another part of the job of a doctor is keeping up with the literature

and surely a lot of life sciences papers use some calculus

Jon Beardsley

i agree

i think the last time i thought about this in depth i came to a similar conclusion.

i've basically decided to stop thinking too hard about it, since it's clearly something i can do absolutely nothing about

infinitesimal

1:01 PM

you can lead a horse to water...

... but asking if the fact that a<b implies b<a is some kind of rule that they need to memorize clearly shows a lack of a thirst for reasoning.

Denis Nardin

2:03 PM

Do you know the story of the doctor who reinvented (and published!) integrals?

Jon Beardsley

Yeah I've heard that. I think Jack told me about it. That's great.

Denis Nardin

Or the fact that a large percentage of doctors cannot reliably interpret statistics about whether a medicine is reliable or not?

It's scary...

Jon Beardsley

That's my biggest concern, is that they need to understand how to read statistics a little more than integrals.

Aaron Royer

4:06 PM

@ArnavTripathy As far as I know, the only proof for general compact Lie groups in the literature is Atiyah's index theory proof. I'd be happy to be wrong about this, though.

Sean Tilson

5:53 PM

Don't even get me started. I don't know what would happen if calculus became mandatory.

I used to think that if someone wanted to go to university that they ought to know how to add fractions. This seemed pretty basic. I was disabused of this notion after teaching a particular course multiple times. I know longer believe that is what is at the bottom.

I think the bottom is thinking it is not reasonable to be expected to learn how to add fractions. Imagine that. It isn't so uncommon as you might think.

I think there is a point though about what we require pre-med and other students to learn.

I don't think doctors necessarily need to know calculus (how could I actually know as I am not a practitioner of medicine) but they ought to be able to learn it. It has to be easier than surgery.

@SaulGlasman I don't see why an internist or general practitioner should be keeping up with the literature. It isn't the job of a high school teacher or lecturer (in the US) to do the same in mathematics.

Espen Nielsen

@SeanTilson A high school graduate who doesn't know how to add fractions, but somehow got accepted into university? That's pretty extraordinary.

In Norway, in order to be accepted for a STEM degree at university, certain minimum grades in mathematics and science are required in high school. Is there no such requirement in the USA?

Saul Glasman

6:11 PM

@SeanTilson I'm led to believe that even GPs are supposed to keep up with the medical literature to a certain extent (and go to conferences, etc) so that they know what the effective new treatments are. I'm not claiming that they uniformly do a good job of it

Basic research in math obviously isn't going to have an impact on high school teaching the way basic research in medicine might have an impact on clinical practice

Sean Tilson

@EspenNielsen There are many ways one can attend university. Some of them are through programs that would not grant a degree. Keep in mind that the state pays for very little of the education by comparison. "If someone wants to pay... who should stop them?" is one attitude.

It is a complicated issue and I don't actually think there should be so many things keeping people from attending. The US has more than 300 million people, with so many different school systems. Some things should be basic but... we don't fund education in the US.

@SaulGlasman Just from being a patient in the US this is an impossible thing to accomplish. They get all of their information from pharmaceutical reps.

Denis Nardin

@SeanTilson My mother is a primary school teacher and I can tell you that they are expected to know pedagogical research literature, at least at the level of having a general idea of what the current ideas are

I don't see why doctors can't be expected the same (NB My mother does not live in the US though)

Sean Tilson

@DenisNardin Exactly Denis, she doesn't teach in the US.

Jeremy Hahn

@EspenNielsen There are students at Harvard that can't consistently add fractions with variables in the denominators, and there were students at my high school who can't consistently add ordinary fractions yet went on to do good work at good universities. They all had mediocre but not failing grades in math classes.

We are in a strange situation where a lot of people take and even understand swaths of calculus and trigonometry without really knowing fractions or algebra well. It's a necessary aspect of 'keeping up with the Joneses' and getting into a good college for a non-math career.

Sean Tilson

I am not talking about doctors or teachers in nice wealthy cities or suburbs. (I am sounding super combative and defensive and I don't mean to)

@JeremyHahn Just to make sure, I wasn't talking about people adding fractions with variables anywhere.

Espen Nielsen

6:25 PM

@SeanTilson Right, I see how that could work.

Sean Tilson

It certainly isn't ideal. There are some amazing brilliant people that I profoundly respect, admire, and adore that see no point to basic algebra (the use of solving linear equations in one variable).

But this is somehow irrelevant. Could they learn it? Yeah.

(meaning irrelevant from their daily lives not the conversation)

Espen Nielsen

@JeremyHahn Of course achievements in high school and achievements in university don't neccesarily correlate, and there is no single system that takes everyone's individuality into account.

It's probably not an issue that will be resolved in the near future.

@SeanTilson Same here. I know a lot of people who are much much smarter than me, and the fact that I happen to know more math than they do doesn't change that.

But in general it might be difficult to separate factual knowledge from smartness, and that is perhaps where the problem manifests.

Sean Tilson

I totally agree. But I only meant in terms of calculus and the what not being mandatory. These things are more about people being literate in some kind of logical reasoning that is implicit in algebra but that we somehow don't seem to teach.

Espen Nielsen

Yeah, I agree. I think most of my time TA'ing elementary number theory last semester was spent teaching students how to reason about mathematics, rather than carrying out calculations.

Adeel

6:49 PM

@JeremyHahn hey, nice to see you here!

Jeremy Hahn

@Adeel You too Adeel :). I've been lurking here for a while but never felt like I had much to contribute mathematically.

Adeel

7:13 PM

Ah, yeah there are a lot of people lurking here, I'd bet.

Jon Beardsley

7:49 PM

There are MANY people lurking. =P

Sometimes they send me e-mails.

Arnav Tripathy

11:16 PM

@AaronRoyer in the literature meaning there are other folk proofs floating around?

user105491

@SeanTilson @AaronMazel-Gee @SeanTilson @EspenNielsen One thing I can say for sure, being a high school student myself, is that people are becoming more competitive. I can say with some confidence that the level of those students entering universities in the future will know more mathematics than normally expected. I mean, people are scared that they're not competitive enough, after the hear about these extraordinary kids doing all these great things, and are raising the bar for themselves.

Also, I personally believe that calculus should be made mandatory because it's like a psychological …

Christopher Galias

Making things mandatory is a really easy way of discouraging interest.

user105491

11:45 PM

@ChristopherGalias Agreed. But some things should just be known; a common saying at our high school is why people who aspire to study biology should study mathematics. Of course, math is used in biology, but we don't know what might be used and when, so it's good to have a sound basis and then specialize to whatever field.

Espen Nielsen

It just feels like the algorithmic approach to calculus taught in high school exists in a vaccuum of sorts. Most people who graduate will probably never use it. Instead, I think teaching a module on classical Euclidean geometry would make much more sense, as far as developing skills like abstract reasoning and problem solving.

user105491

@Espen Isn't that (Euclidean geometry in two and three dimensions) the first math course a student encounters in high school? (And yes, I do agree that the method of teaching calc can be improved upon; it's just so ... dull.)

Espen Nielsen

It's there, but I think it should be approached like in Euclid's Elements. At least at my high school, we had a short module on classical geometry, but we quickly transitioned to vector geometry where algorithmic calculations (and some proofs) were emphasized.

user105491

@Espen I'm not really qualified to say much about that, because I was moved out of geometry and into calculus after the first half of the course, so I only have a vague ieda of the concepts covered. :-P

Espen Nielsen

I'm thinking of the kind of problems found on Japanese temple geometry tablets ("Sangaku").

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