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Comment by jcranberry | Hacker News Reader

>"There is a general feeling in the pure math community that popularizing mathematics is betraying mathematics,”

I don't think this is true at all. My experience with math professors is that they either would welcome more popularity in the field, or they simply don't care.

I DO think that they would be very dissatisfied with the kind of farcical representation of physics that sometimes goes on (looking at you Michio Kaku).

I also think it's very misleading to say mathematics has nothing to do with the real world. The obvious contradiction would be physics, since it's applied math, but I think the quoted person was talking about mathematical research, which is a much fairer point. While it's true that a lot, if not most, of mathematical research has little to offer in terms of applications, but a lot of times "useless" math can still describe real world phenomena, sometimes very beautifully.

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You and the author are conflating two different things. The pure math community probably DOES feel that popularizing only applied mathematics is betraying pure mathematics, and they're probably right. We don't really teach proofs outside of highschool geometric proofs and maybe a single class in university. However there's no good reason why we can't popularize both applied AND pure mathematics.

The real tragedy is that pure mathematics is hidden from students for too long and many opt-out of that path in favor of "Actuarial Science" and applied maths.

Real talk: pure maths was way more fun, interesting, and applicable to my life as a software engineer than all the applied math and modelling that I wrote/learned/etc. Pure mathematics expanded my mind unlike any other subject matter.

> The real tragedy is that pure mathematics is hidden from students

Physicists make a similar complaint. "Why would anyone want to study physics, when kids are not shown any interesting physics?"

Chemistry education research characterizes pre-college chemistry education content as incoherent, leaving both students and teachers confused and deeply steeped in misconceptions.

Biology education... starts by deemphasizing it's core organizational principle, and goes downhill from there.

> The real tragedy

Here's another. "It has been more than a decade since science education research surprised us with how badly science education was failing students. But even now, how often are students told? Instead, K-13, students have been left to think their confusion and cluelessness, are strictly their own fault, their own failing. Left to think science is boring. They are not told that these are known defects of the wretched content we're giving them. Fixing the content is hard. But telling them isn't. At what point does our failing to notify them become an ethical failure?"

> pure maths was way more fun, interesting, and applicable to my life [as a software engineer]

It can be fun to create ELI5-ish stories about math in everyday life. Math as recognizing patterns. So for instance, moving one way on a "circular" path, you come back around to the same position. As with boardgames, clocks, parking, etc. Missed doing something at lunch today... perhaps try lunch tomorrow. Missed a parking space and can't back up... perhaps drive around the block. Category theory for kids.

Could you give counterexamples of good instruction or grounding in these fields: physics, chemistry, biology?

Or maybe a clearer description of what constitutes the good and bad approaches?

What are the core organising principles, in your view, of biology (and physics and chemistry, if you can state them)?

It's an area of general interest to me.

The core organizing principle of biology is evolution and natural selection.

Why only natural selection instead of both natural selection and sexual selection?

Because sexual selection is a subcategory of natural selection.

Ah interesting. They were contrasted as different categories of selection in my school textbook. What do you call the other category/categories?

Any roles for metabolism and/or genetics in there?

Sure, but everything is a consequence of natural selection. If you don't teach that, you aren't teaching biology.

I'd argue that the set is intrinsically related.

Variation. Metabolic efficiency (to expressed effect). Selection. Inheritance (via genetics).

Take away any one part, you lose the whole.

No variation -- nothing on which to select.

No metabolic efficiency (and behavioural effect) -- no criterion for selection or efficiency.

No selection -- no bound on behaviour or constraint on metabolic input or expressed behaviour.

No inheritence -- any variation or selection lacks survivability to the next generation.

I'm flabbergasted. I just learned more about biology than I have in something like 20 years of state schooling. Genuinely, thank you so much for your comment.

Well gosh. Thanks :)

It's long but well worth watching if you have any interest in science education.

https://m.youtube.com/watch?v=WwslBPj8GgI

I'd very much appreciate if you could give some sense of what specific highlights or benefits I might expect to gain from an 80 minute time investment?

There's an 18 minute cut from the original lecture, I'm thinking of starting with that.

https://m.youtube.com/watch?feature=youtu.be&v=rvw68sLlfF8

Possibly related: the Scholastic Method -- origin of the lecture method of education.

https://en.m.wikipedia.org/wiki/Scholasticism#Scholastic_met...

Dates from a time when books were expensive but time was relatively cheap.

I also find the concept of lectio, meditatio, quaestio (vetted!), and disputationes to be interesting, perhaps an ideal model for the article / discussion online format. The better for having everyone actually listen to, and think over TFA.

I teach mathematics at a community college. The lecture resonated with me and I found it captivating. Here is a synopsis based off of memory:

Masur discovered that there is something called the Force Concepts Quiz. He learned that students of physics across the nation did very poorly on the test.

The test asks basic questions. For instance, a large truck and a small car collide head on. Which vehicle has more force imparted on it? This type of question requires no computations. It's easy if you really understand the concepts and hard otherwise.

He teaches at Harvard and thought his students would do well. They didn't. They did terribly. What he found is that regardless of the quality of the teacher (note he is a very good lecturer), type of class, or size of class students universally did bad. It didn't matter the quality of the institution either.

Conclusion, we are teaching physics wrong. He found that students who could do well on the Force Concepts Quiz could do tedious calculations and mastered that part of the course. Students who could master the calculus computations may or may not be able to pass the Force Concepts Quiz. Hence he decided to concentrate more on concepts and less on calculations. He does not lecture in the classroom.

In my own experience I've found similar things hold in mathematics. For instance, I ask College Algebra II students to give me an example of an equation that has no solution. Most can't do it even though 1=2 is a simple example of such an equation.

The question is if "1=2" is really an equation?

Wikipedia says "an equation is a statement of an equality containing one or more variables", but your example lacks a variable.

How about "-1=x²"? If x can be a complex number a solution is "x=i", but what if we only use real numbers? Wikipedia says "There are two kinds of equations: identity equations and conditional equations. An identity equation is true for all values of the variable. A conditional equation is true for only particular values of the variables." Does that imply that without a solution "1=x²" is not actually an equation? The two parts are not equal after all.

If you prefer, 1 + 0 x = 2. We agree that is an equation. It has no solution. Typically in math we don't write 0x in expressions.

Suppose we start with 2x+5 = x - 4 + x. If we solve this by showing all work we'll end up at the end with

5 = -4

At this point students are told this means that there are no solutions. But what they miss is what an equation really means. Anyone who really understands the concept of an equation can easily give an example of an equation with no solution.

An equation is a statement that two expressions are equal. What one tries to find is the solution set. That is, the collection of tuples (if more than one variable is present) that makes the expressions equal when evaluated at the tuple. An equation is really a question. How can these two expressions be evaluated at the same input to produce the same output?

Lastly, We agree that f(x) = 1 and g(x) = 0 are valid functions. Here is the equation 1 = 0 stated in a more sophisticated way. f(x) = g(x) where f and g are defined above.

My personal definition of equation is just "an equation is a statement of equality", no variables required. But if you want to talk about solutions, you usually need variables. (The example "1 = 2" could be saved by saying it has no solutions, while "1 = 1" is solved by the empty assignment of variables.)

The distinction between identity equations and conditional equations is basically one of quantification. (Quantification should really be introduced as soon as you have variables, but is unfortunately often left implicit.)

An identity equation is essentially a statement of the form "for all valid assignments of the variables, ... = ...", where a valid assignment is constrained by conditions like "x is a real number". (Those are also often implicit when they really shouldn't be.) This means that "1=x²" is not an identity equation when x is taken to be real, but it is one when x is constrained to be a root of unity. (This is just another way of saying 1=x², but all identities are like that.)

A conditional equation amounts to "there is a valid assignment of the variables such that ... = ...". Then "1=x²" is a conditional equation that can be made true on the real numbers. The solutions are variable assignments that make it true, in this case 1 and -1. If you constrain x to be an imaginary number, then it has no solution; the equation and the constraint on the variable contradict each other.

If you are taking college level algebra, you should be familiar enough with equations to set up a contradiction like that.

I agree that "1=2" would maybe cause more confusion. How about "x + 1 = x + 2".

I suspect in context that's not a fair question, because students are probably be expecting a technical definition of no solution - maybe imaginary roots, maybe something else - instead of a much simpler inconsistent statement.

Those two answers live in different conceptual categories, and most math lessons and problems live exclusively in one category and not the other.

This is related to Force Concepts because physics-as-math is taught as a series of algebra problems with a rather distant relationship to lived experience rather than the other way around.

Teaching physics as experience is harder than it looks. When I learned physics the course really tried to emphasise experience, but it did it in a stylised and abstract way with canned "experiments" using special equipment. This actually made it feel more distant and uninvolving, not less.

I suspect to do a better job you have to get out the classroom. E.g. introduce force, momentum, and acceleration to kids on bikes and skateboards, not with wooden trolleys on ramps.

It's a fair question in that a student's inability to give an example of an equation that has no solution indicates that we aren't really teaching students what an equation is/means. If one can't instantly give an example of an equation with no solution then they don't really understand what an equation is. Which means something is wrong with our teaching.

The numberphile channel on YouTube is almost nothing but pure math and number theory and is pretty popular.

I think you're right that pure math is hidden too long, and with too many "applied" detours along the way. When I was getting my BS in math, I started hearing concepts my junior/senior year that would have done a lot to clear up the motivation and context for lower level classes, even if wouldn't understand them fully until later. Particularly in regards to linear algebra. I did go the Actuarial Science/applied math route in the end, so your comment rings true, but I'm not sure I would characterize it as a bad thing. I wasn't going to be a world-class researcher and the marginal return on a graduate degree in pure math wasn't there.

That said, there was a small feeling of "And now the time for fun and games is over" when I stopped studying pure math. I enjoy my job, but math provides a certain satisfaction that business problems do not.

As someone who ended up majoring in pure mathematics, I always thought it was a shame that theres this wall of arithmetic and calculus between you and the first pure mathematics class.

I suck at those, but I do well with abstractions and proofs. Makes me wonder how many pure mathematicians the field misses out on.

My university kept the pure math courses for "honors" students; I'm still outraged by this to this day. It's very misleading and as someone who didn't do great in the "regular" (i.e. rote repetition/memorization calculus and linear algebra courses), it kept me out until a prof in one of these classes introduced me to real math. At that point it was a much steeper learning curve to switch to the "honors" route but I'm glad I did it.

It does seem strange that you have to typically "pay your dues" with 4-ish semesters of calculus (calc i, ii, ii, diffy-q, and maybe pre-calc) before you take anything else. It's very strange as a gate-keeper to the subject.

There's this joke in applied math departments that the best way to spook a pure mathematician is to whisper in his ear "We've found an use for your theorem..."

I found an application for a theorem that was originally proved by the authors because for its interesting geometrical meaning. One of the authors who I talked to really liked my application.

So in my impression pure mathematician might not be motivated by applications, but if other people find cool applications, they typically love it.

There's another traditional joke among pure mathematicians: as hard as we resist, every theorem will eventually find its way to practical application.

More of an aphorism than a joke, but we'd all laugh.

Is there a link to a paper? :)

"We've found an use for your theorem..."

I'd think that would be more likely to spook the English department.

Rhetoric.

So ... this is a thing.

https://en.m.wikipedia.org/wiki/Rhetoric

I got a BA from NYU and took a fair amount of grad classes and spent most of my time in CIMS. Firstly I'd probably want to hear what you think of as applied mathematics, because if you're talking about the field of applied mathematics that lies outside of engineering and physics, it gets about as much "popularization" as pure mathematics.

Putting that aside, of the professors who had pure focuses, I'd say more often than not they tried to do their responsibilities as educators and put forth examples to try and make the material more accessible—but that kind of thing seemed to be how they thought on the matter. Actually, the applied mathematicians were very similar. I felt that most of their thoughts were constructive and there wasn't really any sense of comparison with the traitorous physicists or anything.

In the first place, before any popularization of mathematics can occur, it needs to be introduced in a way that doesn't leave all our secondary school students traumatized for their entire education, which goes back to the education system.

Speaking of the lack of education in proofs and popularizing math, do you (or anyone else here) know of good resources to practice and learn more about proofs? I've asked several professors and TAs in my CS program and they say "practice," but they couldn't point to any good tools for self-teaching and I don't really feel confident evaluating the material I've found on my own.

Velleman's "How to Prove It" is a nice textbook on how to develop mathematical proofs. You can apply the techniques to any other field you are interested in, and that is probably the best way to practice: by trying to prove things you already have some intuitive sense about.

Picked up a copy, thank you! And thanks for the suggestion on proving things in area I already have an intuition for, maybe I'll do that in conjunction with reading the book.

We can't popularize pure mathematics because, by its definition, pure mathematics is not related even tenuously to every day life, so popularization requires that the audience do years of idiosyncratic study.

> popularizing only applied mathematics is betraying pure mathematics

I'd say it's betraying mathematics.

> The obvious contradiction would be physics, since it's applied math

XKCD : Purity

https://xkcd.com/435/

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