That is the main struggle for many math teachers, they try to do all these fun and interesting explanations, and the kids just ignore it and go directly for the formulas and forgets everything else. the problem seems easy to solve until you have experienced trying to apply it yourself to a real class of kids needing the material for real grades. It can be done, but a teacher who could do it could make way more money in entertainment etc, since that is what is required to get kids to pay attention.

I did not do well in either set of subjects in K-12 -- even to the point that my graduation from high school was threatened. In college I forced by way back through it all by sheer force of will and got my A's.

There's something fundamentally broken in math/science pedagogy as these subjects aren't really all that difficult. There's far too much time spent memorizing things that are trivial to look up and way too little time understanding how to use them.

An analogy might be learning to cook, and spending all of your time remembering precisely how many spoons, bowls, cups, and cloves of garlic or other ingredients you have. And doing that kind of thing for years, and maybe seeing a demo once of pouring water into a cup. And tests might contain problems like "a party of 5 is coming over for dinner, are you able to set places for all attendees for a 7 course meal?"

The real message being sent is this: "Sorry kids, actually cooking from recipes is only for academics, and to get your PhD and be allowed into the hallowed halls of these academic cooks you must come up with one original recipe (edibility will be determined by peer review)".

In college I retook everything from Algebra up and found the math pedagogy focused more on symbolic manipulation and getting used to how that works in each subject rather than drilling arithmetic in various guises. Tests that required various pre-derived formula were usually just an open book problem. And what mattered was how one went about solving the problem, not the rightness or wrongness of it. Calculators were absolutely expected so you didn't waste time fighting with trivial mistakes.

The sciences usually had a mandatory lab portion that forced application of math to the problem space. Because the labs typically had you collecting your own measurements, it forced you to work through the calculations yourself anyways since there was nowhere else to look up the answer. Again the methods and approaches were where the grade came from, not the slavery to memorization.

Still, while I think the approach I encountered in college was much better than grade school, it still wasn't as good as it could be.

This does a considerably better job of context/interest than the math example did.

Expression and personal power over reality are where I get the joy.

A painter can create world and share a feeling. An author can manifests a memory. A musician can transmit a human experience without language.

Human imaginings about magic are immemorial.

Math describes reality. Math also can describe an extrapolation further.

Programming can manifest from the descriptive language of math into the real. It can use it as a pigment for a new kind of picture. Programming helps with everything below, and it enables new things slightly above.

By "above" I mean the layers of abstraction. A programmer isn't an painter, but a programmer/painter has an additional axis of art.

Programming is our species apex of material transcendence. I don't believe it's the top of the pile, but I have no conception for what's above. Its capacity for encapsulation seems to grow as the dreams do.

Programming grows to predict, programming grows to create. Everything is just a new library, a new framework, a new environment. How long before its limits are found, so we can find the next epiphanies?

That might be true for higher level math, but anything at the graduate or undergraduate level has been already curated to be interesting.

A big problem for CS papers, particularly in PL (programming language research), seems to be heavy reliance on assumed knowledge of Greek letters and notation in the very field-specific way you like to use them. People would understand your paper if only they had read your previous three, which alluded to what you might have meant by these Greek letters, but only by figuring out the two citations they each have in common. If you are bad at pronouncing Greek letters and it’s a PDF so you can’t copy them, you can’t even google what you see. Even if you could, it wouldn’t help. Notation is ten times harder to search for.

(I have never, ever had this problem reading a law paper, not even slightly, not even once.)

There’s an interesting demo here from Will Crichton about how to prepare better documents for conveying understanding in PL. He has a thing to show you the “read as” on hover. https://twitter.com/wcrichton/status/1442891297333800966 https://willcrichton.net/nota

Oh, Lord; NO!

I would like see all Human Narratives/Unnecessary frivolities/Assume-reader-is-a-Idiot language banished from the Teaching of ALL Maths/Science.

What we we need is a focus on the direct teaching of Principles along with their Real World Applications.

I know you mean this as a joke, but many cooking books are like that, and I for one value knowing the history (and science, as in "The Food Lab", highly recommended) of what I'm cooking or eating

You've just described practically any modern recipe website.

Makes me wonder if that's the difference, do people not check math papers unless they are specifically interested in that very proof or such?

Not saying the status quo is optimal of course just why it is the way it is now and probably won't change soon

Oh my goodness! So there are actual people who prefer that? I had to chew my way a fair share of books like that and I can’t stand them. Clearly the person who thought of these definitons, propositions and proofs had some reason to think of them. Sometimes they were trying to solve a problem, sometimes they were combining ideas, sometimes they were looking for structures with certain aesthetic properties. There was always a why behind why they thought about this and not something else. Sometimes there are multiple possible such reasons, that is fine. In that case the author can select whichever they fancy the most. Every time i had the misfortune to read a book like what you describe it felt like I was eating powdered milk without reconstituting it. There was a clear chain of thought between ideas and they choose to just hide it.

I understand that math requires work. One needs to get a paper and a pencil and work out examples, check proofs, play with definitions. But why wouldn’t the author write down what made them care about the next item?

> it is helpful to keep this as an aim.

Why?

Until mid-20th century, mathematics had never been communicated in this austere manner.

When you are a working mathematician, you never start with a definition. You start with a context in which your exploration begins.

It might be a question which someone else asked that interests you (and there is a story as to why). It might be that you don't even have a question, but the objects of your study are not studied enough, so you hope you stumble into one. You can easily tell why this is an interesting thing to look at.

You do some calculations, take a look at a few examples, see if you can make a stab at the chaos in front of your eyes and find a pattern, which we formally call a conjecture.

Then you see if this pattern holds in other cases, and why. This shapes the backbone of the proof.

Once you have a basic idea of a proof, you can formulate a theorem. In the formulation, you list all the conditions to which your proof applies. The pattern might be much more general, but your proof might work e.g. "only in cases when the order of the group is invertible in the base field", or some other.

After all the work is effectively done, you decide that a concept that appears repeatedly in your reasoning deserves to have a name. Something convenient to call it by, so you don't have to repeat yourself. So you make a definition.

You then decided to share your joy with this world, and write a paper.

You listen to the "beautiful advice", and throw out anything that makes your paper interesting.

Context goes out of the window, along with any hope for the reader to have any idea why your paper is worth looking at. At best, you'll advertise this in talks, or explain over beers. Side note: you will have to drink a lot of beers to make it in math.

Then, you follow the advice again, and lay things out in the order exactly opposed to the one you were thinking in: definition - theorem - proof - conjectures - examples - context.

Wait, you already scrapped context, and the examples you started with aren't illustrating the results you ended up proving.

So you tidy up your paper, come up with more specific examples, and remove anything that wasn't on the direct path to your result.

Having climbed to a place where you can see better, you pull the ladder up. Good luck to anyone outside the group of five people who are actively working in this niche!

And finally, you write an abstract to your paper, where you mention the things you defined. The abstract doesn't make any self by itself, one needs to be in-the-know to get half of it, and read your paper to understand it.

In practice, it acts as a "No Trespassing" sign for the outsiders (i.e. anyone not in direct contact with the five people you have beers with at the Annual Niche Field Conference).

Satisfied, you lean back and post it to arXiv.

It's been a beautiful day, you think. This practice is difficult to enforce, but you kept it as an aim, and got pretty close to perfection (as exemplified by a Bourbaki text, or anything by Serge Lang, but I repeat myself).

Somewhere not too far away, a student in the class you're teaching cries.

----------------

I said "you", but as someone who's written a couple of math papers, that's really me too. We are all taught in a horrendously backwards (literally!) manner.

This perversion of the beautiful art isn't a new observation. I can't write better about it than Vladimir Arnold[1] (a titan whose name is, I hope, familiar to you).

It's worth a read to anyone who has ever studied mathematics:

[1] https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html

In other competitive fields such as banking, basketball and football you have the higher ups caring about the pyramid below them, if only as a place to recruit new talents.

Among the math higher ups, only Jim Simons cares about the "math pyramid" so to speak.

One has to be pragmatic, the goal of getting the population interested in math is GDP and median quality of life.

I know those things are very mundane for mathematicians who are absorbed in their world trying to be the ones cracking the Rienmann hypothesis, but even as that individual you have slightly better odds at making it if your surroundings look like Zurich or Cambridge vs. Baltimore or Mobile.

Matter of fact you have better odds if your country can extend the areas looking like Zurich and Cambridge and reduce the areas looking like Baltimore or Mobile.

Among all those symbols only ">" and "<" are somewhat intuitive, all the others you have to learn what they mean.

Even "=" is derivative of "<" and ">" because by reasoning you can understand that you get to it by rotating the 2 lines about 30 degrees after realizing that you are dealing with 2 numbers which are in fact the same and not one being bigger than the other

"Tedious"? Yup! Can use Dedekind cuts or maybe something called the normal completion, and especially the first is darned tedious!

There is a good math writer G. F. Simmons, of Introduction to Topology and Modern Analysis, who stated that the two pillars of analysis were linearity and continuity -- nice remark. He also stated that really to understand, have to chew on all the arguments, etc. or some such.

Then I decided to study the proofs really carefully, so to "chew", and in hopes of finding techniques I could use elsewhere. When I mentioned that study technique, objective to my department Chair his remark was "There is no time." -- he also had a point.

Commonly there is an intuitive explanation of what is going on and some views that can provide motivation to study the stuff at all.

There are a lot of books and papers. As a student, I saw a lot of the books, got copies of some of them, put them on my TODO reading list, etc. Eventually, after falling far enough behind on the list, I wondered just where all those books were coming from? It dawned on me, profs need to publish so they do. They are also supposed to have grad students and do. Then the grad students take the advanced course by their major prof and end up with a big pile of notes. Then the grad student, as an assistant prof, wants to publish so cleans up the pile of notes and contacts the usual publishers to publish a book. The top university libraries are essentially required to buy the books, so they get published and bought. And, then, often, there the books sit, gathering dust. I won't say that writing those books was a total waste, and I won't say that students should spend more time reading those books. Or, the books are there on the shelves. They are not really difficult to find. The books have work that was done. Maybe the work is useful now; maybe someday it will be useful; whatever, the work is done, the results found, and there in case they do become useful.

In the meanwhile, back to the mainline of math education, research, applications, usually there can be some helpful intuitive explanations and motivating example applications!

Apparently some authors just give up and assume that their books will mostly just gather dust. But once I wrote Paul Halmos, likely my favorite author, and got back a nice letter from him with "It warms the heart of an author actually to be read, and clearly understood, by ordinary humans." -- at the time I had no academic affiliation and was just reading his book on my own. So, Halmos was surprised that an ordinary human would be reading and understanding his book.

Ah, in what I wrote, I left out that also in linear algebra in n dimensional space, the Pythagorean theorem still holds, that is, an n dimensional version holds!

=============

From the comment on that video:

>It is more fun to proof that the set of a 2 by 2 matrices with everywhere the same value x with x not equal to 0 is a group. The determinant of such matrices is 0 but it is still a group.

This is only surprising if you don't understand 2x2 matrices as operators on a plane, in which case the exercise is a cruel perversion (why on Earth would anyone want to consider such matrices, or check that they form a group, with identity that's not the identity matrix?! And how would one come up with this to begin with?!).

"Even though the determinant is zero" is a symptom of a conceptual gap. Of course the determinant being zero has nothing to do with these matrices forming a group! You can embed GL(2) into GL(3) by filling the rest of the entries with zero, and of course this will still be a group: because it acts on the XY plane in just the same way as before, and matrix multiplication still gives their composition because we defined it to work this way.

And of course matrices of the form [x x; x x] form a group. A better question would be, why wouldn't they?

Take the following kindergarten-accessible definition of a group: actions that you can undo, repeat, and combine.

Let's hand-wave the above exercise with this definition. What does [t t; t t] do? Let's take t=1. [1 1; 1 1] takes a point in [a, b] in a plane, and sends it to a point [a+b, a+b].

Doesn't seem like you can undo that, because you don't know whether [3, 3] came from [1, 2] or [2, 1]. Bummer.

Well no surprise, the operation [1 1; 1 1] smashes the whole plane into a single line spanned by [1; 1], aka y = x (the image is spanned by column vectors). We might as well ignore anything off this line, 'cause we can't tell points away from the line apart after applying [1 1; 1 1].

What does [1 1; 1 1] do on its image, the line y = x? It sends [a; a] to [2a, 2a] on the same line. So [1 1; 1 1] acts like multiplying by 2. That's certainly something you can undo.

The same works for other matrices; [x x; x x] acts like multiplying by 2x on that line. You can undo that as long as x is not 0.

And you can repeat/combine these operations because who's gonna stop you?

There is nothing left to prove.

This "hand-wavy" argument, of course, is something that gives much more understanding than a "formal" proof from the "definition". That proof is "fun" because it is surprising - and it is surprising because it doesn't make sense.

And I would argue that it's much easier to make a mistake there - and conclude that it's not a group because such-and-such axiom doesn't hold.

The hand-wavy argument, though, ultimately comes from (or would lead to) an understanding that matrices act on their eigenspaces - and little more needs to be said (given that all of those matrices share an eigenspace).

Furthermore, this gives an example of a group representation for a group of nonzero real numbers with operation defined by xy = 2xy.

Of course, such a definition is utterly confusing; why would anyone come up with such a thing, other than to torture people? Why would* one want to redefine the product of real numbers to be something else?!

Seeing people work it out on Math StackExchange [3] is painful.

The people giving the answers are confident that a * b := 2(a+b) is both is and isn't a group! This alone should tell you that at some point, rigor becomes a hindrance. This is that point.

I say, the correct answer is that if someone gives you a group without a thing that it acts on, ask for your money back.

Of course that thing isn't a group, but what breaks if we just say it is? Since nobody is giving me a refund on this, let's see how it would act on itself. An element a would act by sending b to (2a) + 2b, so we have translation and scaling. Can we undo this? Sure, we just need to shift back by (-2a) and scale down by 1/2. But scaling down isn't an option here, so tough luck.

It's not the only problem, of course; but the student is left utterly confused (again, see comments in [3]!) by this exercise, whose point seems to be that "arbitrarily messing with definitions sometimes works and sometimes doesn't".

But it feels* like this should be a group. Let's fix it. The rule "a * x -> 2(a+x)" is kosher; we can take it to be the action of a on the real line.

What does the composition look like?

well, a * b * x = a * (2b + 2x) = 2a + 4b + 4x

That "4x" there tells us that the group generated by these actions is larger than just the generating set. Nothing in our generating set can multiply by 4 (again, that would be a way to see that the rule doesn't define a group). The exploration can then go on further to examining which subgroup of the affine transformations of the real line this generates. It's interesting!

I trust you that rigor being forced on you could have improved your mathematical reasoning. But in that case, you are exceptional - or there was more to it than "do it this way just because". The most common case, in my experience, is represented in the [1][2][3] (particularly in the comments): it makes people confused, wrong, and lost.

I'd rather have them never seen a definition of a group than go through that kind of brain damage.

>I agree that you cannot teach math in a purely bourbakist style. I prefer a "visual" style like that inspired by the books of Arnold, Strang, Needham, and I am the sole teacher in my lab that seriously uses the word "amplitwist" to refer to the complex derivative :)

In that case, they might be crying tears of joy or grief over all the years they were taught otherwise :)

[1] https://www.youtube.com/watch?v=q_JqHQPbmUk

[2] https://math.stackexchange.com/questions/919040/proving-a-gr...

[3] https://math.stackexchange.com/questions/1108349/prove-that-...

Yes! A long time ago I read a wonderful little book on just this bit of history:

https://www.amazon.com/Zero-Biography-Dangerous-Charles-Seif...

The square root of a number takes us from an area to the length of the side of the square corresponding to that area. The cube root is the same for a cube. What is the 10th root of x?

It's a number that, when multiplied by itself ten times equals x.

OK. How do you compute this number?

The best I can offer at this point is, for simplicity, a brute force search or, for faster results, a bisection search algorithm.

In other words, the "and then a miracle occurs" moment is right there. The fact that I can key these numbers into a calculator and get the answer isn't the kind of explanation I want to use for my kid. I don't want to say "once you get here you pick-up your calculator", because the legitimate question then might be "If it's magic, why don't I just pick it up at the start of the problem?"

To be clear, I don't mean "miracle" as anything other than "this shit is hard-to-impossible to explain or calculate by hand". That said, you could probably run through a quick bisection search by hand and likely converge on a low error answer in 2 to 5 cycles.

The meaning of of the e root of b explained with exponentiation and the exponentiation is explained with the root.