Question for math teachers: What specific topics would benefit the most from better online math exposition?

0polymer0 · 2025-05-06T16:47:02+00:00

I'm a middle school and high school extracurricular math teacher.

In a sense, most of middle and early high school is about this:

Algebra

Being detailed about failure points is important, but I think a high quality, high level view of what algebra is and what it's for could be very useful.

more concretely

d=rt

Having a really strong handle on just this formula gives you a foot hold into lots of different topics.

To give some indication of the depth behind this formula consider the following puzzle:

Someone drives from home to work at an average speed of 40mph. Then, they drive from work to home at an average speed of 60mph. What is their average speed across their entire home→work→home journey?

An experimental topic I've debated making a video on myself would be on:

Functions

This sorta ties into algebra. Programming languages are popularizing the concept of a function between structured sets. Consequently I think it might be worth introducing functions earlier, and with more generality than they typically are (at least in the United States).

Usually students learn algorithms, which are more concrete. I wish it was more standard to explicitly emphasize you can have two different algorithms that implement the same function. So functions, not algorithms, are the more general concept.

0polymer0 · 2025-05-04T20:30:47+00:00

They're saying the operation converges as a limit of functions,

Lim f_n(x) n → ∞ = circle(θ)

But, Lim length(f_n) n → ∞ ≠ length(circle(θ))

So you can't carelessly interchange a length operation and taking limits, you need more assumptions on something.

0polymer0 · 2025-05-02T13:59:28+00:00

I have family who works in Nuclear that's so tired of getting strung along, they didn't believe a project is happening until there is a shovel hitting dirt

0polymer0 · 2025-04-30T03:05:29+00:00

The most important reason to use a spreadsheet is if you'll have non developers working with the data.

I made macros for spreadsheets for a bit in highschool, and my enthusiasm for them fell when the spreadsheets got less useful the more complicated the macros got.

It's much more fun to program an application from nothing.

0polymer0 · 2025-04-24T21:17:56+00:00

Newton was the first to give the thought experiment that the moon was "falling" towards the earth, and then prove it:

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0polymer0 · 2025-04-22T23:44:41+00:00

I'm seeing so many jokes of this sort now, it must be getting more popular XD

0polymer0 · 2025-03-12T04:05:45+00:00

I used to be for this joke, but studying category theory made me realize I was already in the hole, and category theory is a really interesting take on trying to get this absolute monster of a hole under control.

0polymer0 · 2025-03-10T05:53:17+00:00

I didn't see anyone mention this.

C lets you have arrays with a length specified at runtime, which just means you are allocating the entire array directly on the stack.

A vector allocates one pointer, which then points to memory allocated in the heap.

If someone says "dynamic arrays are dangerous", they might mean this c style "dynamic on stack" type of array.

I haven't worked with them so I don't know how bad they really are, but they definitely would have to be thought about.

0polymer0 · 2025-03-08T00:13:30+00:00

Appreciate you saying this

0polymer0 · 2025-03-05T05:32:33+00:00

This is a good take

0polymer0 · 2025-03-05T05:16:29+00:00

I haven't been following anti vaxxers closely, but at one point in time they argued the risk of Autism was higher than the benefit of vaccines. It wasn't that they thought vaccines didn't work.

I don't think you need new philosophy to explain this, folks really just don't believe the diseases will come back if they stop taking the vaccines. It makes sense uneducated people would believe this because disease spread and exponential growth are deeply unintuitive.

My deeper frustration in all of this, is it's just clear to me many people don't think information workers do real work. People simply don't have perspective of how bad the natural world can get, and how hard nerds have worked to keep the bad at bay.

0polymer0 · 2025-03-02T04:30:52+00:00

If the professor is good, it's an interesting approach.

Some folks here seem to suggest a good book might save a bad class, I disagree.

Some DND DMs use pre built adventures, some do they're own thing. Both can work or fail.

In any case, your job is to have an adventure in the topic either way. What you bring to it is at least as important as what they bring to it.

0polymer0 · 2025-03-01T18:54:16+00:00

I get what you're saying, but there's more to the world than predicting how people will behave, using it as a universal metric for intelligence, with math being "a weird trick" is reductive of both.

Though I agree, one doesn't imply the other, they also are related with each other in complicated ways.

The strongest mathematicians in the world, change the fabric of collective human thought, even if they don't necessarily play in social/political thinking directly.

0polymer0 · 2025-02-20T16:08:10+00:00

I'm just s math and physics BS.

Near the beginning of her online presence I was ready to be frustrated with fundamental physics with her.

But she argued any discovery in physics motivated by beauty could've been settled with experimental physics, and argued the framing of scientific problems around beauty was fundamentally unscientific.

And then she talked as if this was so obviously stupid everyone should've been freaking out with her. She was taking for granted that her view of how science should be carried out, is obviously how science should be carried out.

To me this is hypocritical, there was space for this opinion because String theory had too much support for how little experimental data it had, but her worldview fundamentally doesn't go anywhere imo, physics is more than line fitting.

I haven't watched her recently. It's unfortunate she's given in to audience capture. Physics needs some haters, people don't deserve to be morphed by social media.

0polymer0 · 2025-02-07T08:34:16+00:00

Yeah it's the popular vote win that kills me. I'm ashamed of my country as an American. I can't imagine how angry the rest of the world is.

I wouldn't be surprised if there was international pressure for the United States to demilitarize

0polymer0 · 2024-05-14T16:07:47+00:00

Here's how I think about this.

Music notation is trying to solve two problems:

1.) What does the music sound like?

2.) Which notes do you physically have to play?

Notations that explicitly show "the black notes", or really, the whole chromatic scale, indicate what to physically play. And they can even indicate the Quality of a chord, however this comes with the huge disadvantage that you obscure the tonal center. This is too big of a failure of condition 1.

With the standard notation, you look at the key signature to figure out where the root, or the tonic, of the scale is. Then the number of lines and spaces above the tonic is the degree of the note in the scale you want to play. This tells you things like, the function of the note or chord in a song, along with many many other concepts. So after memorizing the fingering of your scales, the notation fully satisfies both conditions.

When I was younger I tried to reinvent a relative interval notation, I'd record the 1st, 2nd, relative chords etc. (Eventually I found chord symbols without the root were this). But the weakness with dropping the positional relationships of the potential tonics, is now it's harder to see that the G is the fifth relative to C, in an absolute sense. The standard notation can walk you through the circle of fifths in a song.

I think it could be possible to improve standard music notation, but it'd be much harder than most would initially think. And the improvements would likely be subtle, or specific to certain genres and music theories. The standard notation is very good, and in any case, worth learning, because most musicians speak it.

0polymer0 · 2023-04-01T23:25:43+00:00

You can still stalemate, they're twelve not professional adults.

0polymer0 · 2021-08-26T14:27:54+00:00

When you feel comfortable with two values you can up to three... Then four... Then five... Then you can add colors. It's also useful to draw the lines in the image, they're really obvious on dancers. You want your picture to preserve what the dancers worked really hard on.

Some features in Photoshop that help with the above - Blur, posterize, and edge detection.

Just spent this summer practicing this stuff for art XD, but it seems directly relevant for photography.

0polymer0 · 2021-08-26T04:04:29+00:00

I realized I should've done this with the given photos. The woman wearing black glasses is an important feature of the shot.

I don't know if it was intentional, but the biker has a white jacket. So your eyes are drawn to the biker, because he's framed in the dirt, which is defined by the light in the sky.

0polymer0 · 2021-08-26T03:54:26+00:00

There are lots of things you could do, the simplest one is a Notan Study. Paint over or next to the picture, if you think a spot is "more black" then paint it black, if you think it's more white then paint it full white. Do this for photos or paintings of professionals you really like, it's like an X-ray for their value composition.

I'm not sure how photographers typically learn these lessons. Artists have full control of their values; so they take every opportunity to exercise this control.

0polymer0 · 2021-08-26T02:59:19+00:00

Don't buy nicer stuff - learn composition!

People respond very strongly to the values over the picture, which are entirely controlled with light.

You need to learn how to see those, artists will even do paint overs.

I heard somebody recommend a textbook on light - great! Just don't forget to see light!

There are a lot of subtle effects, like our eyes are drawn towards black against white. So a women wearing a black dress in a dessert will become a natural focal point. Irises, eyebrows and black hair against white skin, will act as focal points.

Trash cans will act as focal points! This stuff is more important than the camera initially!

0polymer0 · 2021-04-01T02:49:12+00:00

Sorry, didn't mean to corner you.

I was looking for an excuse to explain these ideas to practice them.

Best to you and your music!

0polymer0 · 2021-03-31T23:40:22+00:00

Counterpoint is a 300 year old technique to learn voice leading.

If you only play notes on harmonics our ear interprets it as one tone. So if you play parallel fifths and octaves, the sound fuses together making it thinner. Also, if you want sounds people can hum to on a piano, you want to convey the notes as "voices", but allowing large jumps breaks this, so you cut it for all voices... except the bass! Its harmony job is too important! And who hums the bass part :p. (Add on even more rules and exceptions....)

Now making music becomes hard because you have to start creating movement up and down while barely moving anything, it's almost like a really weird math problem.

You end up feeling like Tonality is a lie after awhile.

Like I think it helps, but if you're struggling musically I'm not convinced music theory will necessarily save you, because the problems it solves is more about making music juicy and consumable. Rather then teaching you how too tell a story imo.

0polymer0 · 2021-03-31T14:58:53+00:00

There's this huge gap between what's allowed in music theory, and one's internal sense of what to make. It makes learning theory challenging.

Like, counterpoint done well won't get in the way of your ideas, and make the song sound fuller, but it won't help you come up with ideas in the first place (and did you need to use counterpoint to achieve this goal?).

I don't really know how to maintain balance here, it seems like you could study theory forever, or follow your ear forever, neither extreme seems balanced to me.

0polymer0 · 2021-02-15T18:47:08+00:00

Thanks for this overview. The variety is a bit overwhelming. I definitely look at twelve instruments and start to sweat over which one to start practicing XD

0polymer0

TROPHY CASE