math-pedagogy

[tight star]

this is an interesting question

i think0 one reply0 you could0 say is that this tells0 you the behaviour of f for a whole0 _range_0 of values near your point0

like you could0 ask a similar question about0 learning integration when we have numerical integration solvers

being0 able to symbolically integrate helps0 for things like $\int \frac{1}{x^2 + a^2} dx$, where0 you have an integral depending on a parameter

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[turbid zenith]

Fixed. Thank you.

Oh great, now I'm getting quoted in the main discussion. Ugh.

[tight star]

linear maps are also incredibly well understood, compared to general functions, so it's very useful to approximate a possibly difficult function with a linear one

[turbid zenith]

Assume this is a student who will never take multivariable calculus.

So "It defines the derivative in higher dimensions later" won't be a very compelling answer.

[tight star]

where0 did i mention this sorry0

none of what i said assumes multivariable calculus

[turbid zenith]

If you're calling them "linear maps" I usually assume someone's talking about higher dimensions

[tight star]

ok

[turbid zenith]

Not because you only use that in higher dimensions but because by the time you're celling them "maps" instead of "functions" that's how far along you are 😛

I don't know why I've noticed that lol

[tight star]

the notion of "tangent line" is a little hard to define in 1D

without using0 the linear approximation point0 of view

[tight star]

i mentioned this as well

[turbid zenith]

That does make sense yeah

But you can also understand the behavior of a whole range of points by plugging in a whole range of points

[tight star]

the linear approximation point0 of view also makes0 the various "rules0" of derivatives make a lot more sense0

[tight star]

that might0 take ages and ages

[turbid zenith]

Not anymore.

[tight star]

???

[turbid zenith]

I'm trying to take the point of view of a very skeptical student who's like "we've got computers that can do this in the blink of an eye"

[tight star]

it's not like every0 function is efficiently computable what

well that's just factually incorrect

[turbid zenith]

"Every function I'll ever run into is"

[tight star]

i mean now it feels0 like you're just artifically changing the scenario

[turbid zenith]

I'm doing this because I'm trying to find just the right way to motivate tangent line approximation, where I don't have to say "we don't ever actually have to USE the tangent line approximation, but here it is anyway"

[tight star]

wdym i use the tangent line approximation all the time as a physicist

[turbid zenith]

Okay so like the small angle approximation?

[tight star]

that's one example but like

"linear response" is a very important concept

used all the time in kinetic theory and statmech

the underlying idea is very simple

in general, if you poke a physical system it responds in a complicated way

[turbid zenith]

Neither of which I've studied, so definitely interested!

[tight star]

however, what is almost always the case is the following

[turbid zenith]

That sounds like it could be the start of a good motivation

[tight star]

the response of the system is approximately proportional to the nudge0 you give it

this is very important in stability analysis for example

think0 about0 turbulence, say

very very complicated to do computations for

even now

but it's an important everyday phenomenon that e.g. airlines have to deal with

sometimes you don't need the exact0 way the system behaves though

just the approximate proportionality factor

this can already be enough to tell you whether it's stable or unstable

[midnight scarab]

What if you don't know the function explicitly? What if all you know are some derivative measurements at some specific points?

[tight star]

do small0 perturbations tend to grow in size over time, or shrink0?

questions like these0 are what the derivative helps0 to answer

[midnight scarab]

The examples I have in mind lead perhaps more naturally to the idea of numerical integration, but these are related ideas... Like say you have some velocity or acceleration measurements at some discrete times

[turbid zenith]

What might be a situation where this is true?

[tight star]

whole0 field0 of dynamical systems

[midnight scarab]

Plenty of experimental set-ups. And also many theoretical cases, where you only have access to say a few derivatives at a point (because the problem is complicated!)

[turbid zenith]

!!! Okay THAT'S compelling, actually

Since one of the things I'm teaching in my Calc I course is Euler's method

[tight star]

and here even 1d dynamical systems use the derivative a lot

[turbid zenith]

And that's a great place where you use the tangent line to "step through" approximating a function

[tight star]

or often0 you can take a 1D "slice0" of a dynamical system

[turbid zenith]

That I can very very much get into

[tight star]

yay

yeah one big lesson from my dynamical systems course was

[turbid zenith]

Yeah I promise I wasn't trying to come at this being personally combatative myself by the way 😛 I'm just trying to find the juuuust the right "hook"

[tight star]

sometimes the exact0 solution isn't what you're after0?

often0 it's more helpful to think0 about0 phase0 space0 trajectories

[turbid zenith]

One that's not too abstract and seems immediately useful

[tight star]

and that's a lot more qualitative, and not as simple as just evaluating a bunch0 of points

this is where0 fixed0 point9 and stability analysis can be incredibly useful

[tardy ember]

this is possibly not the sort of idea you want, but i sometimes find that computers tell me that something is true but not why

[tight star]

quote0 from richard hamming

[turbid zenith]

Although I still have a hard time imagining an experimental setup where you know the exact derivative at each point ... more of what I tend to imagine is that I know the function values at each point and what I want to do is interpolate

But that's secant line approximation at that point

[tight star]

in this sense0 the derivative gives0 you an insight into the behaviour of the function near your point0 beyond what mere numerical values can do

[tardy ember]

as an extreme example (albeit not one related to derivatives), my analysis of 5-player nim here uses a computation that runs in a couple of seconds, combined with an inductive argument, and honestly even now i have no idea why the results i proved are actually true, just that it does work out if you look through enough data

[turbid zenith]

CGT MY BELOVED ❤️

[midnight scarab]

If you have an accelerometer, it tells you the instantaneous acceleration at a given time

[turbid zenith]

I'm definitely gonna have to read this lol

[turbid zenith]

Oh yeah that's a good point

[tardy ember]

simpler computations aren't just a way to squeeze more ability out of modern computers, they also make it easier to step through everything with your mind and acquire an intuitive understanding of what's "really" happening, instead of just a bare fact coming out of a mysterious box that you have no real ability to verify for yourself

[midnight scarab]

In a similar vein, the way a car knows how much you've travelled is by integrating the speed measurement (from the angular velocity of the wheels)

[turbid zenith]

I'll tell you the one application that always seemed mysterious to me was economics, with marginal revenue etc

Because you'll have the idea of "marginal revenue is the extra revenue you get by producing one more unit of good"

But then you have MR(x) = R'(x) rather than R(x+1) - R(x), and often x is a discrete variable in the first place

I guess you're trading off accuracy for ease of use

The derivative is slightly inaccurate but it's easier to compute

[tardy ember]

it's like how if you want to compute something continuous you can just make it discrete, compute it in tiny-but-not-zero increments (of time or whatever), so you can put it on a computer, and it will hopefully work well enough even though it's not perfect

except the opposite, we want to understand something that's discrete but in tiny increments, so we replace it with something continuous, because that's easier to deal with from a theoretical perspective

[turbid zenith]

Thank y'all for humoring me on this, by the way. Some of the things you're saying have been things I've also said to students but I've never been 100% comfortable with the answers.

Sometimes I feel like I don't do a good enough job convincing the skeptic.

And I'm always looking for that just-right unassailable motivation.

[tight star]

Idk if that even exists

[turbid zenith]

Of course not! But it doesn't keep me from looking for it :V

[tight star]

Yeah I get that

One must imagine sisyphus happyy

[turbid zenith]

Oh here's an interesting thought

I wonder if error propagation could somehow be used as a motivator here

Since differentials are often used to talk about tolerances

Though again what's stopping you from plugging in the endpoints yourself ... but then those endpoints might not actually capture the full range if the function isn't monotonic, but neither would the tangent line approximation necessarily

[midnight scarab]

I think it's a nice application, but even students comfortable with calc are always so confused about error propagation so maybe not the best motivation

[tight star]

I would0 say that integrals feel easier to motivate for me than derivatives

And you can kinda0 motivate derivatives from integrals via FTC

[turbid zenith]

In a funny way yeah 😛

I mean, integrals came first historically

But I can't imagine teaching integrals before derivatives like Apostol does

[midnight scarab]

Integrals are more natural in the world of geometry. Algebraically derivatives are much simpler (and probably more natural)

[turbid zenith]

Have you ever tried to define what “natural” means in that sense?

Just a thought I just had

[midnight scarab]

I meant it in a very pedestrian sense. If you want to compute an area, of course you're going to cut things up into known pieces. And then you just need the insight of taking the infinitely small if necessary

Approximating the area of a circle by inscribing polygons is a "natural" idea

[spiral elbow]

"why do I need to use the tangent line to approximate f(3.002) when I can compute it directly?"
hmm, I feel like this question is missing the point of derivatives. Like, it is a linear approximation, but the reason we're interested in derivatives isn't to be able to approximate functions, is it?

[turbid zenith]

But tangent line approximations are a standard part of the calculus curriculum

e.g. to estimate sqrt(26) or whatever

That’s pretty much what all the examples tend to be, which is why it seems so uncompelling

[cosmic ibex]

Right, I think we all agree on that -- the question here was how to best respond to that misunderstanding.

[spiral elbow]

I guess that's one application of calculus, but I feel like in that case the question is justified - why do we need all the machinery of calculus to compute something that a calculator can do?

[turbid zenith]

And to get only an approximation for that matter

[spiral elbow]

estimating sqrt(26) using calculus seems like a toy example, I honestly wouldn't try to convince a student that this is genuinely useful or the main purpose of calculus

[cosmic ibex]

"Linear approxmation" works better as an explanation of "what it is" than of "what it is good for".

[spiral elbow]

maybe I'm interpreting the question too literally, but I would just answer that approximating function values isn't the main purpose of calculus, and then you could explain the other applications that people have mentioned

[turbid zenith]

In Stewart the very first example they give of linear approximation is sqrt(3.98) and sqrt(4.05)

Then they give a brief note about small angle approximations in physics, and then go into differentials for error estimation

[cosmic ibex]

(Though I do recall a recent case where the linear approximation was the application that saved the day for me: I needed to do a map projection conversion for every pixel on the screen, and each of those conversions needed like a dozen trigonometric and hyperbolic operations. It would be been noticeably slow to do it from scratch pixel for pixel -- but doing it just once in the center of the viewport and keeping track of the partial derivatives turned out to yield acceptable error and a tremendous speedup).

[turbid zenith]

Radius of a sphere has such and such an error, what’s the error in the volume

[vestal tangle]

you can motivate it as a tool for visualizing the derivative

if you want to see the derivative in the context of the larger graphh, for example in desmos, you need to extend it out of the little infinitesimal domain it livess in

[turbid zenith]

What if they've already done the derivative? 😛

[vestal tangle]

how was it visualized? you can say well in retrospect here's how the visualizations we looked at before can be done on a computer

[turbid zenith]

Zoom in "infinitely close" to the curve until it looks straight, and the derivative is the slope of that straight line

[vestal tangle]

that's a good way to do it

but if you want to see that straight line in the context of the larger graph, then tangent line

it does actually seem useless aside from that

[cosmic ibex]

Perhaps something like harmonic oscillations could be a possible motivation? A mass vibrating around an equillibrium position. If the force depends on position in some known but complicated way, it can be very difficult to analyze the motion exactly. But if we pretend the force is instead a linear function -- and to find which linear function we match derivatives! -- it's much easier to solve the situation and find, for example, the frequency of the oscillations, and if the amplitude is small the results will be quite reliable.
In other words, the linear approximation is not for actually computing values, but to support further symbolic reasoning about them.

[halcyon glade]

Another possibility is to ask questions like "how does the calculator compute sqrt(26)?" It's not exactly linear approximations most of the time, but it is related

[turbid zenith]

It’s funny how many textbooks will justify stuff like Taylor series by going “hey have you ever wondered how calculators find sin x?”

Meanwhile they don’t actually use Taylor series but rather CORDIC

[surreal lily]

Didn’t they use taylor in the past? Or did my calc teacher lie to me?

[cloud zealot]

from the wikipedia article on CORDIC:

CORDIC was conceived in 1956[10][11] by Jack E. Volder at the aeroelectronics department of Convair out of necessity to replace the analog resolver in the B-58 bomber's navigation computer with a more accurate and faster real-time digital solution.

i thinkk that predates stewart (the current calculus textbook market leader) by a good margin

the firstt edition of stewart came out in 1987

[midnight scarab]

But was the previous algo (or some (by hand) algo in the past) use Taylor series?

I'd imagine there were faster convergent approximations

Just like you wouldn't compute log 2 by summing the alternate harmonic series, or pi from the Taylor series of arcsin

[cloud zealot]

just to add on a bit, the relevant thingg to consider is were taylor series ever used on earlyy computers to numerically approximate trig functions

[cosmic ibex]

I don't think using CORDIC is universal in calculators nowadays. The advantage of CORDIC is that there are no general multiplications in its inner loop -- but if the calculator is built with beefy enough hardware to have fast multiplication circuits anyway, that advantage begins to diminish; I think I've read that something like piecewise polynomial approximations can be just as attractive.

[cloud zealot]

not just throughout history

[midnight scarab]

Well, apparently Taylor expansion was (possibly) used at least at some point https://en.wikipedia.org/wiki/Madhava's_sine_table

[austere delta]

I think we've known about more efficient methods than Taylor series since long before any form of calculators.

Taylor series are more useful for theoretical usecases than computations

[cosmic ibex]

Also, Taylor series are only halfway convenient when your angles are in radians -- which, for practical calculations, they weren't before computers.

[tight star]

What would people say are examples of “mathematical maturity”?

In particular I’m thinking about fields like graph theory or category theory which in principle don’t have many explicit prerequisites but in practice need some “mathematical maturity” to understand properly

[cosmic ibex]

In most cases, I think "mathematical maturity" is code for "comfortable with proofs, and in particular at ease with the idea of needing to puzzle out your own proofs without following a particular recipe presented in class".

[tall bolt]

I think there’s also an element of being able to handle objects being multiple things at once, for example being able to work with something which is both a group, and a topological space, I’ve found people get tripped up as things start to have more relevant levels of structure at any given time (and I remember finding this a bit tough a couple of years ago too)

But yeah mostly what you said I think

[cosmic ibex]

That's a good point too.

[vestal tangle]

i think it's about having fully internalized all the typical jargon

for all, there exists, x implies y, etc

[turbid zenith]

Wow, there's actually an article on mathematical maturity

https://en.wikipedia.org/wiki/Mathematical_maturity

And here's an MAA article
http://sigmaa.maa.org/rume/crume2019/Papers/106.pdf

[tardy ember]

this arguably isn't that different from what you said but i think i'd more specifically identify it as being able to read through mathematical jargon, and see the underlying ideas separately from the words or notation used to communicate them

or rather, i think that's a part of it

familiarity with proofs is definitely also part of it, and i think also just being comfortable with abstraction in general

[tardy ember]

if i wanted to define the entire concept i think it's something like "every part of skill at mathematics that isn't specific to one field/theorem/type of object/etc"

[viral pike]

we should isolate the Description column

[cosmic ibex]

That looks pretty dedicated to presenting Tao's model ... which I don't disagree with as such, but it feels a bit much to elevate it to the Encyclopedic Truth about what maturity means.

[viral pike]

The list of mathematicians that the public knows has 1 element.

Living mathematicians.

[midnight scarab]

I was about to say (from looking at the embed) that this sounds like Tao's post

[turbid zenith]

It’s just based on what’s citable I guess

Anyone could add more if they want though

[copper root]

just what I thought

[swift hatch]

There is no bad formatting in ba sing se

[viral pike]

come again?

[swift hatch]

I have absolutely no idea what you were talking about (I've done it)

[viral pike]

I don't follow (what is "ba sing se"?). But anyway, looks better now!

[swift hatch]

it is a meme originally from avatar the last airbender; the point being that there absolutely was bad formatting but "oh I have no idea what you're talking about"

[viral pike]

exotic 😆

[viral pike]

is that a commonly used term - never heard

[swift hatch]

I have no idea, I did not change any of the content (other than the post-rigorous example to be an actual example)

[viral pike]

yeah it came with the big Tao update 2023

But "underclassman" doesn't appear in Tao's article.
But that's enough internet forensics for me today.

[pure light]

it's by analogy to the much more common "upperclassman"

[quasi maple]

In Avatar: The Last Airbender, the capital of the Earth Kingdom - Ba Sing Se - 's military is in a war, but forces its population, via hypnotic suggestion of needed, that there's no war within the city's walls

Hence the line "There is no war in Ba Sing Se"

[viral pike]

Seems to be canon

[copper root]

I can take a phrase that's rarely heard – flip it. Now it's a daily word

gotta integrate shit into your vocab, man

[quasi maple]

"gotta differentiate shit out of your vocab, woman"

-# (how tf does "woman" as a vocative sound far more rude than "man" as a vocative wtf English)

[copper root]

'tis all in your mind, man

[copper root]

maybe use "girl"

nevertheless, I'm neither

[quasi maple]

well for one it was a joke on "taking a phrase" and "flip[ping] it"

[copper root]

😱

[copper root]

yeah

[jaunty yew]

<@&268886789983436800>

[copper root]

couldn't find the role lol

[jaunty yew]

btw you can turn off pings when you reply

[copper root]

@jaunty yew I know 😏

[quasi maple]

You just choose not to tho

my view of this channel's somehow more orange

[turbid zenith]

(Although the Iverson bracket is superior!)
$$\int\frac 1x\dd{x} = \ln|x| + C_1[![x<0]!] + C_2[![x>0]!]$$

[burnt vessel]

Solid Angles

[halcyon glade]

wait until the student pulls out the "1/x is a continuous function"

[turbid zenith]

If I had a student who said that, why would they be in my class XD

[white fulcrum]

damn I'm the teacher in this scenario

[turbid zenith]

How so?

[white fulcrum]

until this was posted I didn't understand what it meant

[white fulcrum]

it's continuous on its domain (x/=0), right? but ofc not cont on the entire real line

[halcyon glade]

yeah, it's not clear what continuous on the entire real line would even mean in this context

perhaps one way to say it is that no matter what point you add at x=0, the function will be discontinuous

[slim path]

Almost everywhere?

[storm prairie]

$\hat{\RR}$ is an addition of a single point that makes 1/x continuous, is it not

[burnt vessel]

Coolempire93

[quasi maple]

... Would it not be better to use an indicator function on the Union of those sets?

Or am I misunderstanding the technical correction there

[tardy ember]

the point is that a function like this is an antiderivative of 1/x, because at every nonzero real number x its derivative is 1/x, but it isn't of the form ln |x| + C for any constant C

since the domain, R \ {0}, is disconnected, you can add a constant to just one component of it without affecting the derivative

[tardy ember]

that would just be the same as writing 1, because the union of those two sets is the entirety of the domain

[quasi maple]

Er... neither interval contains 0

[pure light]

ln|x| + 1 and ln|x| + 1_{ (-infty, 0) U (0,infty) } are the same function is the point

[tardy ember]

and neither does the domain of 1/x, or ln |x|

that's why i specifically said "the domain", and not "R"

[quasi maple]

oh fair enough

[dim arrow]

what does that notation mean ?

[little drum]

it's the indicator function on a set

so 1_E(x) = 1 if x is in E and 0 otherwise

[turbid zenith]

What I've always seen in introductory calculus textbooks is that a function $f$ is continuous at $x=a$ if $\lim\limits_{x\to a}f(x)=f(a)$

[burnt vessel]

Solid Angles

[turbid zenith]

Which means three things need to happen at x=a:

1. The function needs to be defined
2. The limit needs to exist
3. The function needs to equal the limit

By that definition f(x) = 1/x would be discontinuous at x = 0 because it fails 1 (and therefore 3)

[sleek field]

Can anyone explain me these questions

[turbid zenith]

So you could say that "continuous on R" would mean defined on all of R and continuous wherever it's defined

[turbid zenith]

This is a channel for educators for discussing how to teach math. If you have questions about calculus you should ask in one of the help channels. 🙂

#❓how-to-get-help

[sleek field]

Oh okay

[midnight scarab]

I can't resist commenting about the fact that amusingly in general topology, limits are usually defined in terms of neighbourhoods (without removing a), so 3 is actually part of 2

Also interestingly, in France this is also how it's defined with epsilon-delta

[turbid zenith]

Interesting!

I will say I've never been a big fan of how in some spaces online, I've seen people act like "continuous on its domain" is the only correct definition of continuity and that the introductory calculus definition of continuity is somehow wrong

e.g. "Of course 1/x is continuous, why would anyone ever say otherwise?"

[midnight scarab]

Yeah that's a bit of a smart-ass thing. Even if they're right formally, there's a strong implicit in such claims that we're talking about continuity on R

[turbid zenith]

Yeah it's the same energy as "Well ACKTCHYUALLY the area of a circle is 0, the area of a disk is πr²"

[pure light]

at the same time it feels a bit weird to say "sqrt(x) is discontinuous at -5"

[austere delta]

I mean, that's perfectly consistent with the introductory definition.

A point is continuous at x=a if it satisfies the limit thing, and it is continuous if it's continuous at every point of its domain.

I guess with the exception that you can't really define limits at isolated points

Are people taught that 1/x is discontinuous?

[turbid zenith]

They're taught it has a discontinuity at x = 0

[cosmic ibex]

I think that's not uncommon.

[turbid zenith]

An infinite discontinuity, specifically

[midnight scarab]

Perhaps a way to formalise this (and answer cloud's comment) is to talk about continuity (or rather the existence of a continuous extension to) limit points of the domain

[turbid zenith]

I think an acceptable phrasing might be "1/x is continuous on its domain but not continuous everywhere" and then you just have to choose which one the unqualified "continuous" means.

My guess is that unqualified "continuous" should mean on the domain

[turbid zenith]

But then what about f(x) = x/x?

Unless you're saying that the question of continuity should only apply to limit points of the domain

Which I think would make sense

[midnight scarab]

I'm not sure what the issue is?

[turbid zenith]

I'm saying you wouldn't say f(x) = x/x is continuous at x = 0

[midnight scarab]

I would say it admits a continuous extension to x = 0, yes

[turbid zenith]

But that makes a bit more sense as a question than asking whether f(x) = sqrt(x) is continuous at x = -1

[turbid zenith]

Okay, then that's in direct conflict with what's taught in introductory calculus

That would be called a "removable discontinuity"

[turbid zenith]

Unless I'm misinterpreting what you meant by this — I read it as "continuous = admits continuous extension" because of the "yes"

[midnight scarab]

Yeah the yes was ambiguous

[turbid zenith]

English really needs to go back to having two kinds of yes and no XD

[midnight scarab]

But I'd make a distinction between the two

[midnight scarab]

Yet another 🥖 win

[turbid zenith]

English used to have four!

Yes, no, yea, and nay

[midnight scarab]

But anyway I do think that if we're careful about continuity VS existence of a continuous extension, I do think "1/x does not admit a continuous extension to R" is strictly speaking more proper than "1/x is not continuous on R"

If we're using the general definition of continuity

Of course, if we use the calc def of continuity of left/right limits agreeing and agreeing with the value at the point then it'd be slightly different

[turbid zenith]

But I would say that "1/x has an infinite discontinuity on R" is even more descriptive as to why that happens

If a function doesn't admit a continuous extension there are multiple reasons that could happen

So I think the curriculum is trying to be as descriptive as possible and build students' intuition for the functions' behavior

[midnight scarab]

Yeah exactly, we're using the special structure of R to say more by looking at left and right-sided limits

[austere delta]

Not a removable singularity?

[midnight scarab]

That's in complex analysis

[austere delta]

I wasn't aware that people in real analysis called singularities discontinuities. But I didn't do high school in English so 🤷‍♀️

[halcyon glade]

Personally this feels like an abuse to me bc I would use that to mean that the function is defined at x=0 but with the "wrong" value

Idk man at this point let's just burn everything down and start from scratch with the terminology this is too confusing x_x

[turbid zenith]

Both usually count as removable!

Because you can make the function continuous there by defining or redefining just the one value

[austere delta]

Both are removable, one being a removable singularity and the other a removable discontinuity

[halcyon glade]

Part of me wonders what the point of being sticklers about "x/x is not the same as 1" is, it's difficult for me to think of a situation either practically or mathematically where it matters

[turbid zenith]

Okay this I haven’t heard!

[halcyon glade]

I can be happy with that

[turbid zenith]

Difference quotients

[midnight scarab]

This seems to be related to the idea that continuity is for points of the domain

[turbid zenith]

In fact every derivative as a limit has this

Thinking about it now

So I think the point is to get students thinking that you can only really do this in the safety of a limit

[turbid zenith]

So what would one call the behavior of x/|x| at x = 0?

A jump singularity?

[halcyon glade]

I like that

[austere delta]

Even just "jump" is fitting isn't it

[turbid zenith]

Pffff one syllable words aren’t rigorous mathematical terms 😛

[midnight scarab]

I'm not going to argue with the fact that jump discontinuity is the normal terminology 😄

I was just saying that's how I interpret the distinction jagr's terminology was making

[turbid zenith]

Not unless the word requires at least three levels of abstraction to see what it’s describing

e.g. I still have no clue what a sheaf is

[midnight scarab]

topoi (or toposes) have entered the chat

[turbid zenith]

And any time I’ve tried to learn it’s left my brain the next day

[midnight scarab]

An association of a ring to each open set, whose definition is local in the same sense as the notion of continuity or analyticity is

[tight star]

mfw local-global equivalence

[midnight scarab]

And scheme is a dialect of the LISP programming language

[turbid zenith]

I will forget that by tomorrow!

[midnight scarab]

I'm not sure I can give you a convincing reason not to 🙃

[tight star]

idk i think sheaves are a pretty cool concept

[midnight scarab]

I mean they're definitely cool, and natural, and useful for AT and AG

[turbid zenith]

I think my issue is I don't know what a sheaf is meant to capture

[tight star]

.

[turbid zenith]

The explanations I've seen are just "here's the formal definition"

[tight star]

https://en.wikipedia.org/wiki/Pasting_lemma

this is the best starting point imo

[midnight scarab]

It's just saying that if a function is continuous/analytic/any other local property on an open U, it's also open on a smaller open set V and other obvious things (that's a presheaf), and then to get a sheaf the key thing is a gluing property: i.e. if you have continuous/... functions on each U_i which agree on U_i cap U_j, then you have a function on their union which is continuous/...

So indeed the key thing is just this gluing property, which encodes the local character of the property

[turbid zenith]

Hmm I see

[vestal tangle]

from the free book "sheaf theory through examples"

it gives a cute high level description of the idea in terms of sensors looking at different parts of a piece of paper

https://direct.mit.edu/books/oa-monograph/5460/Sheaf-Theory-through-Examples

[halcyon glade]

Is sheaf theory actually useful for machine vision with many sensors? Does anyone know if ppl use it

(Not to hate on sheaf theory, just wondering about applications)

[vestal tangle]

every problem in every domain has been solved already by the neural network

[halcyon glade]

Untrue

[white fulcrum]

'the' network?

[rapid tusk]

the neural hivemind clearl

[torn apex]

um, yes. very very very much yes although it's highly under utilized

I made visual explainers for artists and category theorists awhile ago. You can skim it in 5 mins:

• https://ikrima.github.io/topos.noether/x/descent-part1-the-cover.html
• https://ikrima.github.io/topos.noether/x/photogrammetry-for-category-theorists.html

[turbid zenith]

So many students cheating this semester

[midnight scarab]

Disappointing

Is this a required class for non-scientists?

[turbid zenith]

Yup.

Is anyone else dealing with a rise in cheating?

[white fulcrum]

I'd like to find accessible proofs of gradeschool geometry facts, specifically volumes/surface areas of shapes like pyramids and cones and such

I'm thinking stuff like the onion and pizza proofs for the area of a circle

[torn apex]

That sucks. I'm hearing a lot of that from college professor friends and the extra burden of managing that. I also hear a decent amount of contradictory evidence where some of my friends have found the students to be way more engaged and asking deeper questions.
I wonder if the divide falls along undergrad/grad classes and they're seeing the effects of selection bias (e.g. entry grad students are presumably much more motivated than freshman)?

[turbid zenith]

Yeah I'm sure some of it has to do with whether it's a class related to the student's major

In my case it's the liberal arts math class that all majors have to take

[jaunty yew]

A sphere has 2/3 the volume of its circumscribing cylinder. Also has a historical element to it too.

[quasi musk]

My students homework is largely just copying the solutions manual

For upper division math

It makes me very sad

[turbid zenith]

What class?

[halcyon glade]

there was continually a large portion of students cheating at hw when I was a ta

[quasi musk]

optimization

It combines vector calculus, linear algebra, and some numerical methods

[torn apex]

I keep forgetting you are playing "Math Educator" on ultra-hard mode 😬
If it helps psychologically, the non-cheating students I've talked to are all livid about the cheating.
So you're probably still having a large impact even if it's drowned by all the cheating behavior.

[halcyon glade]

I feel like the only thing to do is to make it as explicit as possible that cheating is not okay and will be met with consequences, with some specific examples of what cheating means to rule out the "I don't know if this is cheating" defense

[quasi musk]

Ah, it's not even worth the effort

What am I gonna do? Report them for academic dishonesty, and it's going to become my word against theirs, and they'll say they just got it right. Then get dragged into several meetings about it. OR I can just say "Don't do this", and move on with my life and research

[turbid zenith]

Thankfully in my case the thing they're cheating on, it's a little more obvious they're cheating

They're having to do readings about math (the history, the culture, current issues) and pick quotes to discuss

So when you get ChatGPT to flat out make up a quote, it's REALLY easy to tell :V

Lots of "it's not just A, it's also B"

[midnight scarab]

Yeah in that case you can just give them a zero for making up a quote (which is dishonest either way, but I digress) and hopefully that'll at least teach them to do some minimal verification of llm output

[quasi musk]

That's nice

[turbid zenith]

But yeah somehow for math questions it's MUCH harder

Because how do you tell the difference between the student getting the right answer and the AI getting the right answer

(Solution: Back to in person testing)

[quasi musk]

This is exactly why I don't fuss over it too much, because we will have in person quizzes/tests

[turbid zenith]

I'm gonna have to start designing calculus tests again soon!

It's been a while 😅

[torn apex]

I can only imagine how livid someone would be if they didn't cheat but got accused of cheating.
I got so tired of getting accused of being an LLM on social media that I started putting my messages into Claude and asking it to help me make it sound less autistic.
What an odd time to be alive 🫢 🤷‍♂️

[little drum]

what service is this? I'm curious to see what my writing gets lol

[tawny slate]

i think you might be missing the point here. why would you use a tool that is as obviously flawed and useless as this

but also i am not kidding when i say that almost any online "check for AI text" service also comes with a "humanize my text with an AI" service

[little drum]

idle curiosity

sometimes I just see something and I'm curious to try it out even if I wouldn't actually put any stock in the result

[tawny slate]

also what makes a person sound autistic and sound like an LLM isnt the same thing

autism is difficulty understanding social context and clues. LLMs also struggle with this too but this kind of usage is only going to alter speech style to be more like an LLM without addressing the substance behind it, which makes things worse, not better

id rather talk to an autistic person's original words and thoughts because you can glean more social clues and context from their mental state and make a more accurate model of what to expect when interacting with them, and this is what I mean when I say that it is slop

[quasi musk]

This is a fun linear algebra problem. Given a vector $\vec{v}$, and T a transformation from one coordinates, say $B_1$, to another, say $B_2$, show that $$T([v]{B_1}) = [v]{B_2}$$

[burnt vessel]

MoonBears-C-

[tight star]

I have been thinking about this a lot recently

[quasi musk]

The answers I'm getting are often just something along the lines of it's true because that's what coordinate transformations do

[tight star]

I believe I finally understand change of basis satisfactorily

[quasi musk]

Yeah, I tried mapping out how to do it in discussion, and basically told them "Linear maps are determined by where they send basis vectors, so go open up your Linear Algebra book, and find a way to do this carefully"

The books answer is to basically just "invert" the matrix given by the coordinate vectors, and hit it by the vector

[tight star]

Right, that’s not quite how I think about it

I need to sleep soon but I can explain more tomorrow

[quasi musk]

But they're basically starting with (the notation in the problem uses x' and x for the different notation), x' = x therefore x' = Tx

Which is exactly how you solve the simpler problems like (1,2) = 1e_1 + 2e_2, but if v_1 = (1,0) and v_2 = (1,1), then (1,2) = a(1,0) + b(1,1) = (v_1, v_2) (a,b)^t

So you invert the matrix to solve for a and b

[torn apex]

This looks like just a change of basis problem, no? If you look at the transform of the basis vectors, it should characterize the transformation. I vaguely recall the Jacobian playing the same role but can't recall if it's necessary for Affine transforms or just Shear/Conformal transforms. Maybe I'm missing something?

[quasi musk]

It's a change of basis problem, but it's difficult in that "it's true because it's true"

Really it should be something like $$x = \sum_j x_j e_j$$ and $$e_j = \sum_k a_{jk} v_k$$

[burnt vessel]

MoonBears-C-

[quasi musk]

The $a_{jk}$ become the entries for T, and now you can write it down precisely

[burnt vessel]

MoonBears-C-

[torn apex]

If you google computer graphics change of basis, you'll find a lot of intuitive visualizations (not just of change of basis but all of undergrad linear algebra).
Change of basis bugs are the bane of every graphics/animation programmer as they often don't compose "nicely" the way artists think they should
Ex: it often inverts or introduces negative scales which break everything

[halcyon glade]

How are you defining T

Or is the question just how to find the entries for T

It feels fairly straightforward if you write out the definitions of everything, like if you have basis vectors e1, ..., en for B1 and f1, ..., fn for B2 and you know that
e1 = a_11 f1 + ... + a_1n fn
...
en = a_n1 f1 + ... + a_nn fn
then it just follows from definition that multiplying by the matrix (a_ij) will work right

And then you can solve for a_ij via row reduction as usual

[quasi musk]

yeah! Some students did this. Most students wrote x = x' therefore Tx = x' in some way

Handwriting reveal

I think the interesting thing here is that you can use what you said, but you end up invoking the uniqueness of coordinates to say that x_j' is this expression summed up

[torn apex]

side tangent: what software did you use to make this?

[quasi musk]

OneNote

I have a microsoft surface pro

I prefer it to an iPad since it has a full OS instead of an iOS

[quasi musk]

Yeah, this is an applied math class, so a lot of the students aren't comfortable with precise arguments. They are more comfortable with pushing formulas around and saying "that's a wrap"

Most of my education has been strictly in the pure math camp, but I'm dipping my ankles into applied math

[halcyon glade]

It is possible to say that there was no serious applied mathematics in France for forty years after Poincaré. There was even a snobbery for pure math. When one noticed a talented student, one would tell him 'You should do pure math.' On the other hand, one would advise a mediocre student to do applied math while thinking, "It's all that he can do!" ... The truth is actually the reverse. You can't do good work in applied math until you can do good work in pure math.
~Dieudonné, one of the founders of Bourbaki

Not sure that that last part is true but ah well

[quasi musk]

I'd say to do applied mathematics well, it helps to have a good framework for how to prove things in undergrad. Obviously there are counter-examples like Steven Strogatz, who did excellent work in applied math as an undergrad, but wasn't so good at pure math

But even then I'd say he was probably pretty decent at pure math, despite his lack-luster grades in pure math classes

[halcyon glade]

Yeah I think doing rigorous math helped me a lot to gain mathematical intuition to apply to biology

[quasi musk]

It's helped me a lot in Numerical PDEs, which is "applied" but doesn't really have anything applied about it, except for requiring some computer code to implement algorithms

[tall bolt]

Side note but why does your math mode align right lol

[quasi musk]

I used double dollars

For gradescope, you have to put everything in double dollars

But for discord, it's single dollars to get it to render

[midnight scarab]

I imagine it's just texit truncation

[pure light]

it's an issue with TeXit where math aligns to the center and text aligns to the left, leaving the right to get cropped out and throwing the whole thing off

[tight star]

As a first introduction, what approach to tensors have people found works best? I’ll list some examples but I’d also be curious if there are alternatives:

• Multidimensional array of numbers
• Transforms like a tensor
• Multilinear map
• Tensor product space, abstract def
• Tensor product space, universal property
• Tensor product space via space of bilinear forms on V* x W*
• Tensor-hom adjunction

[turbid zenith]

Multidimensional array of numbers has worked best for me.

Plus their mathematical properties of course, but it helps a lot to have something concrete to work with.

For another example, I remember trying to learn what the tensor product was, and any time I asked I got all this mumbo-jumbo about modules and bilinear maps, and I had no idea what was going on

[tight star]

Which ones do you use

[tight star]

Hm, so what’s your understanding of the tensor product now?

[turbid zenith]

Then I saw this and it clicked immediately

And the rest of the abstraction fell into place after that because I had something to build it on

[turbid zenith]

I think of it as a "free product" of sorts where you're not placing any additional constraints other than wanting it to play nicely with addition and distribution

Where if you place additional structure on it you can get things like the dot product

I'm sure I don't have a full understanding yet but that gave me the first actual foothold into it

[tight star]

Hm, ok

[turbid zenith]

Here's the article that came from
https://www.math3ma.com/blog/the-tensor-product-demystified

I will forever advocate for starting with explicit computational and/or visual examples first instead of formal definitions 😛

[empty gull]

if I'm learning them abstractly from an abstract algebra position the motivation I always liked was to construct the tensor product of two modules explicitly and say that this is a very natural way through which we can change the base ring over which we consider these modules

[turbid zenith]

See I have absolutely no concept of modules past "vector space but over a ring"

[empty gull]

well yes this is exactly what they are

but even in like vector spaces this is a thing right

[turbid zenith]

Yeah so I wouldn't be able to answer a question about "so what"

[halcyon glade]

I like the "array of numbers that transforms like a tensor" definition and the "universal property" definition, just my personal taste though

[turbid zenith]

Except maybe some stuff about how now you can have different minimal generating sets with different sizes

e.g. you can generate Z with {1} but also {2,3}

[halcyon glade]

It works the same for vector spaces, e.g. you complexify a real vector space by taking a tensor product with C and that turns it into a complex vector space (so no need to think about modules in general here)

[tight star]

so, what would be your explicit computational/visual example of a tensor?

[empty gull]

well I think the motivating example should be turning a vector space over R into a vector space over C

[turbid zenith]

I gave it above

At least right now

Eventually I need to learn the physical applications in which case that would give a better initial motivation

[tight star]

so... $\begin{bmatrix} 1 \ 2 \ 3 \end{bmatrix} \otimes \begin{bmatrix} 4 \ 5 \end{bmatrix} = \begin{bmatrix} 4 \ 5 \ 8 \ 10 \ 12 \ 15 \end{bmatrix}$?

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[turbid zenith]

But I'm not at that point yet. But based on everything I know about what helps me learn mathematics, the ideal entry point is (1) a motivating application followed by (2) an explicit computation that comes in "just in time".

[empty gull]

yes this is also called the kronecker product

[turbid zenith]

And then stepping back and saying "okay what happened here? What's the phenomenon in general?"

[empty gull]

I guess this is a more physics oriented way to think about it

because to me the natural way to consider things would be something like

take a real vector space and say you wanna study the diagonalization properties of some matrix

but it has complex eigenvalues so what can you do

so a natural idea would be well maybe we can turn our vector space into a complex vector space

and in comes the tensor product to save the day

[halcyon glade]

The natural thing to me is "you have two vector spaces and you want to make a new vector space made out of choosing a vector from each independently"

[empty gull]

well I guess

why would you wanna construct such a vector space to being with

[halcyon glade]

it's natural if your vector space represents the state of some physical system, like if you know how particle 1 behaves and how particle 2 behaves and they're independent, I should be able to make the state space of how particle 1 and 2 together behave

it's sort of a "null" assumption, against which you could look for e.g. effects of interactions between the particles

[empty gull]

I guess the fundamental issue with motivating these kinds of more "abstract" mathematical definitions is what perspective do you want to take

do you want to take the "real world" application and motivate it through that

[halcyon glade]

or complexification feels the same to me tbh, you independently combine a complex number with an existing vector space

[empty gull]

or through some problem that's more of mathematical interest inherentlly

and I think that in some sense neither will capture everyone's attention

[turbid zenith]

It's probably obvious that I teach very few math majors 😛

[empty gull]

haha well that's fair

[turbid zenith]

I'm teaching the full Calculus sequence again starting next semester, and it's mostly physics/engineering majors

[halcyon glade]

I feel like combining two independent spaces is the core idea, regardless of whether you do a more practical or mathematical motivation

[tight star]

hm so do tensors come up in what you teach?

[turbid zenith]

They don't! I'm extrapolating from what's helped me learn them and what I know about my students

So I'm basing this on "if I had to explain what a tensor is to the kinds of students I teach, what would most likely work"

[tight star]

i see i see

as a physicist myself, one of the big things about tensors is the transformation law

[turbid zenith]

Are there any places in the traditional calculus sequence that make you think "this is all just tensors"?

[tight star]

as far as i understand, this is not really relevant for the algebraic applications

[tight star]

i think they start becoming relevant in multivariable calc

[halcyon glade]

(recently I've been thinking about optimal transport, where you want to move piles of sand into holes efficiently—the tensor product is the strategy where the starting location of the sand is independent from the ending location, an obviously inefficient strategy in most cases because usually you'd want to move the sand pile to the closest hole)

[tight star]

in ordinary calc, it's hard to see their application

[halcyon glade]

the two most obvious cases where tensors show up to me are both in multivar calc: differential forms and determinants (but even then, it's only really helpful to understand alternating tensors, not tensors in general)

[tight star]

hm in quite a few physics applications i've used symmetric tensors

[turbid zenith]

Okay that makes a lot of sense

I'm actually avoiding determinants until we get to coordinate transformations

[tight star]

determinants are... a whole battle on their own

i'm not actually sure starting with a computational example is the best idea there

[halcyon glade]

algebraists: "it's the eigenvalue of the matrix when applied to the one-dimensional space of top alternating forms"
geometers: "it's just the volume, with possibly a minus sign"

[halcyon glade]

yea I think it's better to start with the concept and then show how to compute them with row reduction

[tight star]

-# category theorists - it's a natural transformation

[tight star]

row reduction is underrated honestly

[rapid tusk]

the best feeling is when you get one row to be almost all zeros lmao

[tight star]

honestly it's something my course didn't really cover

[halcyon glade]

row reduction is crazy for both computations and proofs

[turbid zenith]

Geometer all the way XD

[tight star]

i should learn more about it tbh

[midnight scarab]

I think the most natural and visual place where tensors show up is continuum mechanics.
For example, if you ask how does fluid velocity change between neighbouring points, or the deformation of a solid.
And then you naturally ask how the matrix you get transforms when you pick a different basis. And then the physical, basis-independent object, whose matrix representations in different bases are related via "transforms as a tensor", is a tensor

[turbid zenith]

If I'm introducing determinants, I want to start with "how does the area/volume change when I do this transformation"

[halcyon glade]

I think "I give you all the coordinates of the corners of the shape, how would you find the volume?" is also a fun start

maybe a little more concrete

and then afterwards you can say "well now let's look at the shape you get when you apply the matrix to a unit cube... now that we know how to calculate the volume of the image, we know how stretchy the transformation is"

[midnight scarab]

Yes, I believe one should start by defining the determinant of n vectors as the volume of the parallelepiped. And once they're comfortable with computing determinants and its multilinearity property, you can naturally introduce the determinant of a matrix as the volume of the parallelepiped defined by the images of the canonical basis, whence the determinant of a linear transformation as the transformation of the volume

[rapid tusk]

I first learned that interpretation when I was doing contest math in middle school lmao

there’s the “shoelace formula” for areas of polygons in R^2 which is just the determinant dressed up differently

[white fulcrum]

this uses the volume of the cone, any proofs for that?

[white fulcrum]

it's times like these I wish i were 3b1b

[tawny slate]

fun fact: if you take the following problem:

3 random numbers x, y, z between 0 and 1 are chosen at random uniformly. what's the probability that x is the smallest?

and then use geometric probability to graph the 3d plot, you get this shape

so if anyone has difficulty visualizing why 3 copies of that pyramid fit the way they do into a cube, this demonstrates by symmetry you only need to construct one of these by this probability problem

which isn't all that difficult, because you only need to plot the boundary conditions (x=y, x=z, x=y=z) and the shape draws itself

this method also has the nice property in that you can extend it to arbitrary numbers of dimensions

[white fulcrum]

the triple pyramid proof is, dare I say, trivial. the next step is to establish that the volume doesn't change when you slant the pyramid, which is apparently cavalieri's principle. how do we establish this? In the above picture I'm not sure what the justification is for comparing the cone volume to the pyramid volume

to clarify - if i understand correcly - taking the tip of the cone and moving it around horizontally within the bounds of the top of the cylinder doesn't change the volume of the cone. is that an instance of the principle?

[vestal tangle]

multilinear map

a function that eats some number of arrows and some number of stacks and spits out a real number seems like the easiest way to think about them

it's also very visual

the array of numbers that obeys a transformation rule doesn't seem very enlightening to me. that's probably the worst

and you can visualize not only what the multilinear map eats but you can think of the map itself as a bundle of arrows and stacks

and that gives you an easy way to connect it to the tensor product

the tensor product being the thing that lets you bundle the arrows and stacks together into the maps

[rapid tusk]

yes i believe so, imagine you project all the cross sections onto the base; if the apex is directly above the center its a buncha concentric circles, if its say above a point on the base then you get a buncha internally tangent circles, but notably these are the same circles as the first case (by similarity), so then cavalieri’s principle applies

you can also think of it like “two identical integrable functions have the same integral on some interval”

[tight star]

I think it can be ok if you derive it properly

[tight star]

Well, a formal sum of such bundles, right?

[tight star]

on a related note, how do people tend to explain change of basis? i keep coming back to it and wondering whether i understand it correctly or not

[cosmic ibex]

The argument seems to be most useful if you've already developed a concept of the volume as a (more or less handwavy) integral over cross-sectional areas, but don't have enough calculus available to evaluate it symbolically with the FTC. Then it gives you a fairly concrete reason why the integral of x² from 0 to 1 has to be 1/3, and you can then get cones and other pyramids just by linearity properties that are comparatively easy to handwave.

[turbid zenith]

I think you can also make it a bit easier to digest if you do the pyramid FIRST

And once students are comfortable with that, then do the cone by showing how it reduces to the pyramid

[turbid zenith]

That way students will be less likely to get lost partway through a six step process

[halcyon glade]

One could start with this comic haha

I remember in high school physics I had to watch this absolute banger of a video https://www.youtube.com/watch?v=bJMYoj4hHqU

[tight star]

<@&268886789983436800> spam

[turbid zenith]

So I've been reading a lot about elasticity of a function, which is defined as $$\dv{(\ln y)}{(\ln x)}=\dv{y}{x}\cdot\frac{x}{y}\text.$$ Apparently, in the Stata statistical software, the command for this is $\texttt{eyex}$, where it seems like they're using $\texttt{ey}$ to mean $\dd{y}/y$ and $\texttt{ex}$ to mean $\dd{x}/x$. It also has $\texttt{dydx}$ for the derivative as well as $\texttt{dyex}$ and $\texttt{eydx}$ for the two kinds of semi-elasticity. How terrible of an idea would it be to define a notation like $\frac{\mathrm ey}{\mathrm ex}$ or $\frac{\epsilon y}{\epsilon x}$ for elasticity? 😛 Is it even worth creating a notation for it?

[burnt vessel]

Solid Angles

[turbid zenith]

(When creating course materials do y'all ever find yourselves asking yourself what the best notation would be to use for things?)

[zinc dove]

Yes

Initially, I used mostly what's in one book, now I mix from manu sources, staying concistent through the subject

[halcyon glade]

I think as long as you give disclaimers about what's custom and what's standard, whatever's clear and consistent is good

[zinc dove]

Yes, I clarify that different authors use different notation

Although there are some standard notation widely used

[cloud zealot]

https://www.rxddit.com/r/math/comments/1shf2w5/im_thinking_of_making_videos_on_mathematical/

[white fulcrum]

I got now how to show them that the cavalieri's principle is valid. last thing I want is to show that it's true for any base shape (even a 'strange' one), not just the square pyramid. I only know the integral way to do it. I think I can get across that the area of a shape scales with the square of its 'size' or 'radius' easily

[cosmic ibex]

Do you mean "I don't know how to show them ..."?

[white fulcrum]

no, cav's principle is easy to show I think. i'll take a stack of coins or dice and stack them neatly, then slant the stack and note that the 'volume' (number of dice, amount of metal in coins) doesn't change

[cosmic ibex]

Ah, so it wasn't really a question, just a plan. Gotcha.

[tawny slate]

question

regarding cav's principle

how would you guys explain to students why this is not valid, but cav's principle is valid?

as usual, pedagogy because I personally understand but im not sure the best, simplest way to explain to someone else would be, is there a clean super elegant explanation somewhere?

like for this one, i know I can just take the diagonal of the square to disprove the validity of this method, but you cant apply this method to show cav's principle is valid rather than invalid

[rapid tusk]

when youre dealing with limiting cases

the error actually has to go to zero

here the error is always 4-pi

[cosmic ibex]

I think the real take-home point of the troll limit is that we need to be careful about defining what we want arc length to mean before we start applying limiting arguments to them willy-nilly; otherwise we have no way to argue whether they are or aren't valid.

A priori we might get into similar trouble if we use too handwavy arguments for Cavalieri's principle .

On the other hand, a too rigorous argument may end up being too full of pedantry for the audience to follow at all, so it's not a given that's better either.

[tawny slate]

this isnt particularly satisfying I think because you are first assuming that the real circumstance is length pi. i think troposphere kind of hits at the heart of what makes this difficult

[midnight scarab]

The crux of the issue is that there are order 1/epsilon triangles of side length order epsilon, and for each of these triangles the error is order epsilon, so the total error is order 1

Whereas for Cavalieri you have 1/epsilon times errors of order epsilon^2, so total error epsilon -> 0

[tight star]

Yes this is also how I’d explain it

The idea of calculus is roughly the following

To solve a big problem, you do the following stages:

• Split it up into a bunch of small problems
• Solve the small problems
• Combine the small solutions to a big solution

When you do this, there are two competing factors

Say you split into N small problems

Ideally, each individual problem will be easier to solve, since it’s smaller

But there are also more problems to solve

Calculus works when the rate at which smaller problems get easier is faster than the rate at which the number of problems increases

[tight star]

So here, when you split up the square into N segments

Each “small problem” is basically a small right-angled triangle

But the issue is that you’re estimating the hypotenuse of this triangle by summing the lengths of the other sides

That gives an error of O(1/N), which is not fast enough to balance out the linear growth rate of the number of problems you’re making

However if you use the true length of the hypotenuse, then compared to the actual length of the arc you only have an error of o(1/N), essentially because of differentiability

And that is fast enough to overcome the growth rate of the number of problems

So taking N -> infinity is sensible

[tight star]

This “divide and conquer” strategy isn’t unique to calculus; the main unique thing is that calculus deals with when you take the number of small problems to infinity

[turbid zenith]

So I’m actually curious where the orders come from here

And also I feel like there’s still the issue of what we’ve chosen as our definition… in order to compute error we need a “correct” value in the first place

So I think it needs to enter somewhere that the definition we’re choosing is supposed to agree with “what if I laid a string along the curve?”

[midnight scarab]

Indeed, that's why approximating curve lengths is more subtle than approximating areas, for which you have an obvious monotony property

The analogue for curve lengths is more involved, see

[tight star]

sure, i think this is where differentiability is crucial

[midnight scarab]

^

The TL;RD is that if you approximate a curve by a broken line (a polygonal chain), adding more points on the curve to the chain will increase (or keep constant) the length of the chain

If the upper bound is finite, the curve is called rectifiable and this defines the length of the curve

Indeed if the curve is continuously differentiable, then it is rectifiable and this definition of its length agrees with the definition by integration of the norm of the tangent vector

[turbid zenith]

Okay I can certainly see why that would be monotonic

Where do we get order 1/n vs 1/n^2 from?

[midnight scarab]

Yeah if you draw the original segment and the two new segments, they form a triangle. So it's just an instance of the triangle inequality AKA the straight line is the shortest path

[midnight scarab]

Well, the generic length of a figure of side epsilon is epsilon, and the generic area is epsilon^2

The only way to get a decent approximation is to make your approximation and the correct answer to agree to linear order

So for example taking the two sides of a right-angled triangle instead of the hypotenuse is "egregious", and you get an error of the same order as the side length

Whereas approximating a small arc by the chord is correct to order 1 in the arc length

[turbid zenith]

That’s after already defining the hypotenuse method to be correct I guess

I’m not entirely sure how would actually calculate the order of the error in taking the two sides though

I get the vague sense of the order but that’s sort of just on vibes

[midnight scarab]

There's two things:
First of all, if you have a right triangle with a and b order epsilon, then approximating sqrt(a^2 + b^2) by a + b is wrong by something of order epsilon

Then the other thing is that you can approximate the arc length by the hypotenuse of the triangle, i.e. by the chord

In the sense that they agree to order epsilon (and so disagree to order epsilon with a+b)

[turbid zenith]

Oh so you’d literally compute sqrt(a^2 + b^2) - a - b where a and b are multiples of ε

I’m just trying to see how to get an ε to pop out

[midnight scarab]

a^2 + b^2 - (a+b)^2 = 2 a b ~ epsilon^2

[turbid zenith]

This gives me enough that I think I could justify why the troll limit doesn’t work

[midnight scarab]

Yeah the only part that might need convincing is that the chord is the correct approximation to the arc length, to linear order. Though I think it's intuitively quite believable

[turbid zenith]

I think that can be done from the point of view of what real world idea we’re trying to capture

[midnight scarab]

Essentially all you need to see is that if you take an extra point on the arc, the height of the triangle is order epsilon**^2**, so the improved approximation agrees to order epsilon with the chord

[turbid zenith]

Because otherwise a priori how would you know to do that even for the hypotenuse of a right triangle?

You could make the same argument either way

[midnight scarab]

epsilon^2, typo

[midnight scarab]

To do what?

[turbid zenith]

To use the correct method

At some point you need to agree on a correct definition and you can’t prove that… uh… by definition

I’m having a hard time wording this

[tight star]

it's interesting that the way pythagoras is proven is using areas, not lengths directly

[turbid zenith]

But when you decide on the correct definition of arc length you can’t really prove it’s correct by showing it matches up with a known arc length because that requires you already have a correct definition of arc length 😛

[midnight scarab]

That's true, but I think you can't really do more natural than approximating by a broken line with increasingly small segments

Ok actually maybe you can also formalise the "length of rope" idea

[turbid zenith]

Right, that’s no longer a mathematical decision though

At some point you’re having to go off of some kind of intuition of what you’re trying to capture

[midnight scarab]

By defining a curve of length l to be a continuous map $\gamma$ from [0,l] to the plane such that $\abs{\gamma(b)-\gamma(a)} = \abs{b-a}$

[burnt vessel]

afqt

[turbid zenith]

So the argument it seems needs to be (1) here’s why we chose the hypotenuse method as the correct definition, and (2) here’s why the “remove corners” method doesn’t converge to the hypotenuse method

[midnight scarab]

I mean, it's not really about the hypotenuse

It's just about chords

[turbid zenith]

Okay that’s what I meant

Diagonal line

[halcyon glade]

You can choose the remove corners option, if you want to talk about non-Euclidean geometry and taxicabs 😛

[midnight scarab]

So yeah, for (1) the argument is this

[turbid zenith]

Putting it that way it seems the hypotenuse really is the right way to think about it

Because your way of measuring the length of a chord is based on the Euclidean metric

My students never believe me when I tell them they have no idea how important the Pythagorean theorem is 😂

[pure light]

a key part of archimedes' estimate of pi is that he bounded the circumference of the circle both below and above, using an inscribed and circumscribed polygon respectively

each of those bounds required adding an axiom, being:

• the shortest distance between two points is a straight line
• given two convex paths between a pair of points, the inside one is shorter
  See here for some discussion of those axioms, although the focus is on his area of a circle

[midnight scarab]

Do we really need the 2nd point as an axiom?

[pure light]

the second one applies to curves, not just broken lines. so that theorem is a good motivation for the axiom but not a replacement

[midnight scarab]

Ah right. So you just take this as an axiom, which I guess is quite similar to defining curve length as the sup of lengths of broken lines with points on the curve

[turbid zenith]

https://bill-j-shillito.github.io/calculus-scbi/online/sec-sensitivity.html
I've been reworking my first lesson from my calculus book a good bit based on some of the feedback I've gotten before I move on to any future sections, so I have an idea of what kind of "feel" would be good.

A summary of what's been done:

• I'm focusing more tightly on the "sensitivity" concept, going into both sensitivity between two points (Δy/Δx) and sensitivity at one point (dy/dx).
• Still drawing attention to the fact that "infinitely small" changes are a vague concept and we'll make it more precise later.
• I'm actually delaying the word "derivative" until the second lesson so I can use it in the sense of "derived function", i.e. given a function f, the derivative is a new function that tells you the slope of f.
• Added some stuff about approximation of changes — not full-on linearization of a function, that'll come later, but just enough to show that it does a decent job of approximating small changes.
• The first few exercises are designed both to give students practice with computing slopes and to notice patterns in those slopes for linear, constant, and quadratic functions, to get them ready for the next lesson when we actually do the derivative.

I did end up getting rid of the "estimating the area of a circle" activity for now because that's really more akin to integration than anything. So I'll introduce that later instead.

Any feedback or questions anyone has about the explanations or exercises would be welcome.

Also wondering if the prose (everything before the exercises) is something that could be read before the first day of class.

[midnight scarab]

Minor nitpick but the Objectives still use the word "derivative"

[turbid zenith]

. . . oh yeah I still need to fix those 😛

[midnight scarab]

It might be helpful to draw the secant in the figure below definition 1, since it's mentioned in the the sentence below

[turbid zenith]

Also I know some graphs are missing

[turbid zenith]

What I actually might want to do is add a “zoom box” to show the graph more clearly, and also call it the slope of the segment

Since in some sense you don’t really need the whole line

[midnight scarab]

Idk why I read the aside as "We will figure out how to make it better" instead of a note to yourself...

[turbid zenith]

I’ll be getting into the tangent line later I think, when we get into linearization

[vestal tangle]

nit: "in tandem with each other" seems redundant

[white fulcrum]

hey how do blind ppl learn math? I have absolutely no idea where I'd even start with that

[austere delta]

I was involved in teaching a blind person for a little while, but it got cut short for reasons unrelated to blindness.

Anyway, they had their own system for writing down equations in a way the screen reader could read back to them. Other than that they learned linear algebra similarly to anyone else I guess.

[lethal hornet]

could they do geometry? what was their intuition like?

[austere delta]

Didn't really get to that, but in general blind people understand shapes and volumes etc. So I don't see why not

[tight star]

<@&268886789983436800> spam

[elfin sleet]

I think it mostly could, or at least the first little bit. Here are some things I think would cause problems for students reading it on their own

• you mention a "simple empirical model"; I don't think students at this level necessarily know what that is. You can probably just reword it to avoid tripping them up
• my gut feel is that students are not going to follow the paragraphs that are right under "sensitivity at a point". I think it's mostly that you haven't stated that the sensitivity at a point corresponds to the slope at that point, which maybe you're delaying saying but it's kind of necessary to know in order to understand those paragraphs. Otherwise, it's not clear what's the relevance of the graph looking straighter as we zoom in, nor exactly what you mean by "mixing sensitivities"
• for the sign of sensitivity section, I think it would be helpful to give an example of a negative sensitivity and what it means before jumping to the general case. I think for most students it won't have occured to them that it can be negative, much less what that would mean, so I think they'd be thrown by a sentence starting "if dy/dx < 0..."

[turbid zenith]

Would "experimental" model be more accessible than "empirical"?

But past that, these definitely make a LOT of sense. I would need to use a different model but I can certainly come up with one.

Or maybe I can better emphasize negative sensitivity/slope in the part where it's talking about the sign of the derivative.

[midnight scarab]

Idk if it's cheating, and whether perhaps this is enlightening cheating, but to get a negative derivative you can just "multiply the function (or the variable) by minus one". So the likelihood of survivability of someone in the car (or of a pedestrian in front of the car) decreases with speed

Or, for another example, the life expectancy of rabbits decreases with the increase in the number of predators

[elfin sleet]

Yeah I think so. I might also avoid the word model and say something like "through experiments, we can establish the following formula that relates L to v". (fyi you'd have to replace at least one later use of the word model too if you went with that)

[midnight scarab]

I do like the emphasis on the fact that it's a model

Or if you want to avoid the word "model" at least something along the lines of "is well approximated by the formula"

[elfin sleet]

Yeah I agree conceptually I'm just not convinced that these students will actually know what a model is. I don't think I heard it used in that way in high school. Maybe I'm wrong though

[halcyon glade]

I think it's probably good to spend a little time to explicitly teach them if you were actually teaching this to someone

high schoolers can definitely understand what a model vs observations is by the time they're learning calculus

[tight star]

I wanted to ask - have you tried this out yet? How did it go?

[tame tulip]

isnt sqrt(24.999) computed by linear approximation

i assume sqrtx is solved by newton raphson if not for precomputed tables

[turbid zenith]

Tried what out exactly?

[tight star]

Like, motivating linear approximation to your students

[turbid zenith]

The last time I taught linear approximation (in a video for my Calc I class), the example I gave was calculating lighting in a video game using the inverse square root, pointing out that you need to do this multiple times per second, and then using linear approximation to estimate 1/sqrt(4.1)

And then at the end of the video I pointed out that while this isn't exactly what Quake III did for the "fast inverse square root", what they did do (a Newton-Raphson hack) still used linear approximation

So the overall message I used was "sometimes you need to trade accuracy for speed because you're doing lots of calculations"

[tame tulip]

i do agree linear approx should be motivated/emphasized better in calc 1

i didnt know it was important until i learned linalg and read something about calculus mixed with it

[tight star]

How would you motivate it?

[tame tulip]

no clue!

i couldnt stay in the realm of calc 1 only (id need multiple variables) to properly motivate it i'd say

though partial derivatives could be a completely calc 1 affair if you visualize it as a derivative with one "direction" fixed

[white fulcrum]

linear approx seems easy to motivate, linear functions are so easy to work with compared to just about anything else

right?

[tame tulip]

its not really that cool that it approximates the function, moreso that it forms the base for differential calc at a high level

but the latter thing would be hard to motivate in calc 1

especially considering that calc 1 is usually a general institute requirement and so no one cares

[devout horizon]

Yeah I think this is why the “college board” (to some degree) emphasizes linear approximation stuff on the AP Calc exams

[rapid tusk]

eh they dont do it very well imo

[mystic jolt]

Would something like this be usefull for people trying to learn advanaced mathematics?

[tawny slate]

helpful in what sense? i dont see how detailed sectioned proofs could possibly be not helpful

[turbid zenith]

Yeah I feel like it’s difficult to properly motivate it in Calc I

Compared to where linear approximation is essential to make problems tractable

So you necessarily need a kind of “toy” problem that’s still sorta believable

I added something recently to my intro to sensitivity about how using the sensitivity means you don’t have to repeatedly recalculate the function value

[icy timber]

you want to find the slope of a line that changes, so you try doing a bunch of linear approximations, but its really difficult

you figure out that derivatives can give you that in an easier way

counteractively you connected it to linear approximations

this is how my teacher motivated it, and whenever I think of derivatives I kneejerk linear approximation

somehow i lost that lol i should be more conscious of it

all it takes is one prof to make you feel scared and confused and dumb, and then have that branded on your academic record

i got a B in calc III lmao

I figured it out later

[icy timber]

(in fact, you do linear approximations by looking at two points and closing the distance between them to try and get the linear approximation "at that point")

this made differential changes seem simple, and i never even had to think about the coordinates all that much

pinning two points on a curvy line and trying to make it tangent is a very coordinate-free concept

like you might not even need a coordinate system

i bet you could make a physical model out of some metal wire to demonstrate the idea

[icy timber]

i did this in blender lol and my teacher literally had me present to the class

i literally zoomed in and showed that like,

_ -- is not the same as /

literally by pythagorean theorem

you cant do like, 1/3 / 1000 * 1000 to get the right number

a million errors does not make it right

i said literally three times

[icy timber]

I always wished that a prof would put up a bunch of papers on the wall like

the integral symbol, and under it: "a bunch of little things added up"

df(x)/dx: "slope of a function at x"

yk it might spoil it but you could like

unveil it as you go or something lol

like a big surprise / dramatic action

cause someones gotta make the calculator!! also, calculators cant do everything

youre the only one that can be creative

[icy timber]

this was a really logical approach to me as a 10th grader

the teacher actually asked us how we would figure out the slope somewhere and had us figure it out

then he suggested like, try making the points closer yk?

once we literally HAD the definition of a derivative he wrote "lim -> 0"

and then wrote what we call it

there was no instruction it was just "how would you guys do this?"

[limpid oar]

hi any high school teachers here?

[mystic jolt]

Is in having automatic summry of proofs, different level of hand-holding, issues,pull requests attributions and soo n

[mystic jolt]

It is essential: this is where the whole idea of calculus comes from. that is, the definition of a derivative

[midnight scarab]

But linear approximation is not how a calculator computes square roots. Actually even mentally, if I want to compute sqrt(4.1), I'd do the Babylonian algo starting with 2: 1/2(2+4.1/2) = 2 + 0.1/4 = 2.025
Rather than having to do the gymnastics of sqrt(4+x) = 2 sqrt(1+x/4) = 2(1+x/8) = 1 + x/4 and then setting x = 0.1 (in fact when doing it mentally I started off by forgetting to distribute 2 over x/8)

[icy timber]

Had no idea about this

[tame tulip]

Why not?

I thought calculators used Newton raphson to compute. Sqtt

[midnight scarab]

That's what I'm saying - they don't use linear approximation. Of course that's not to say the algorithms used don't involve derivatives.

As a matter of fact, Newton's method gives you precisely the recursive formula I used

It's called the Babylonian method, or Heron's method

[turbid zenith]

This isn’t far off from the lie that calculators today use Taylor series for transcendental functions!

[cosmic ibex]

Calculators (and computers) use many different methods to compute square roots, depending on tradeoffs between speed, hardware complexity, and program size, as well as which other operations the machine needs to be (efficiently) capable of. Many of the possibilities do not correspond neatly to any named classroom method.

[tame tulip]

Is newton raphson not a form of linear approx

Maybe I'm extremely wrong but I thought Newton raphson constructed tangent lines and used Linear approx recursively

[cosmic ibex]

The difference Afqt is trying to express is that even though it uses linear approximations internally, the output of approximating a square root with Newton-Raphson is not itself a linear approximation to the square root function.

[turbid zenith]

You know, I've never actually taught Newton-Raphson whenever I've taught calculus

I haven't found a good way to make it fit into the "story"

It always seemed kind of out of nowhere and not in the "I could have come up with this myself" set of topics

[tall bolt]

Isnt the geometric picture quite sensible? I feel like I remember watching a good video on it back in highschool when we were learning it

[turbid zenith]

For some reason the first geometric picture that comes to mind is always the one where it fails on the cube root function or whatever

I feel like when I learned it we spent so much time looking at when it doesn't work that I didn't internalize why it does work

[midnight scarab]

Iirc to approximately solve f(x)=0, you replace f by its linear approximation near your initial guess and solve that. If your initial guess was close to the actual zero, then the zero of the linear equation should be close (in fact, closer) to the zero of f

More intuitively, if you're at x_0 and f(x_0) > 0, f'(x_0) > 0, then you want to pick x_1 smaller than x_0, and f(x_0)/f'(x_0) is a natural estimate of "by how much", by dimensional analysis

[tight star]

yeah i thought newton-raphson was quite tied to the whole linear approximation angle

[turbid zenith]

I want to include it when I do my linear approximation lesson this next time

I just never got a good feel for it, and I'd need to find a way to make it feel natural rather than just something tacked on

[cosmic ibex]

I'd agree that Newton-Raphson feels a bit oversold. It's a neat idea to be aware of for the special cases where it's an improvement, but for general root finding you can get just as good iterative approximations by linear interpolation between known function values (plus some extra footwork to ensure quick convergence).

[turbid zenith]

I might include it as an extension problem

[midnight scarab]

Imo the most natural root finding algo is just dichotomy/biscetion, which gives a natural proof of IVT.
And then if you want to do better by taking into account the behaviour of the function, then you arrive at something like the secant method
Newton is a variant of this where you replace the secant between two points by a tangent at a single point, which ig has some advantages, but is mainly useful if you know the derivative analytically

So for example if I wanted a simple algo to compute cube roots, with a nice recursive formula, I'd just look at what Newton tells me

[rugged lark]

Don't they?

I thought it was between Taylor Series or CORDIC

At least for trig functions

[boreal agate]

i think its more of an "only cordic" situation

trig taylor series is just awful from a numerical perspective

[cosmic ibex]

There are definitely other methods.

Since floating point trig just needs approximation to a known, fixed precision for each input, a piecewise polynomial approximation with precomputed coefficients can be feasible and competitive in some regions of the tradeoff landscape.

(The core trouble with CORDIC is that it needs a trip round the loop for each bit of precision, so if you have a fast multiplier available, polynomials can become pretty attractive compared to that).

[turbid zenith]

I eventually need to actually learn how CORDIC works :V

[clever nacelle]

Does anyone know the group "Mojza"?

A student-led pakistani group with notes on certain subjects?

[halcyon glade]

This is probably not the right channel for this, maybe try #advanced-lounge

[clever nacelle]

I do understand, but i wanted to focus on their pedagogy practices

[fiery rain]

No that comment is on my deleted message

[midnight scarab]

Congrats, you've proved P and not P

[vagrant meadow]

how do yall deal with teaching/tutoring scenarios where a student is like
"what did i do wrong"
and you start to help them, but then it becomes painfully clear the student is missing about 60% of the prereq knowledge they should have, but become increasingly insistent on the "just help me with this problem" point?

last week i had a student ask what they did wrong on an integration by parts problem only to find out they didn't even know how to do the product rule.

[halcyon glade]

I think you just have to force them to take a step back and think about fundamentals

Something that might make it feel better (and more engaging for the student) is alternating between time talking through problem solutions and then long periods of time going through basics so that they can see how the pieces fit together

But yeah in my limited experience, it's gonna feel sucky no matter what so you just need to be there to manage the student's expectations and motivate them

[vagrant meadow]

given they had chatgpt out in front of them, and the attitude i got, i dunno if that would have been particularly appealing. there was definitely a huge emphasis on wanting to be given this particular fish, and not wanting to learn how to catch one.

i got the sense they just weren't interested in help from me so i let another tutor handle it. if i was the professor and we were in office hours i think this might work better, but ig in the tutoring center scenario i don't think it's worth wasting both our time.

[sick bough]

I'd usually ask the student to explain what they were trying to do -- if they describe a sound strategy to solve the problem they're doing, then it's most likely that the mistake is a small technical mistake, eg an arithmetic error, an incorrect algebraic manipulation, etc. In that case you just go through their working carefully and spot the error, and maybe give them some tricks for checking their work as they go to prevent such things in the future.

Ofc if what they describe is not a sound strategy, then you've already found their mistake, which is that they don't understand what they're doing. In that case, you need to figure out what the gap in understanding is and correct it.

[vagrant meadow]

suppose you ask for their strategy and they say "i don't know" lol
and when you try to explain they say "can you just tell me what i did wrong"

because yeah i definitely tried to help them be able to check their answer in the future by verifying the integral with a derivative, but that's when i realized they didn't know how to take a derivative and it was like "oh no"

[sick bough]

I think in that case you just have to kind of stand your ground a bit and make clear that the student needs to do a lot of work to get where they need to go and that it is their responsibility to do that work. It's your job to teach, but it's their job to actually learn.

[spiral elbow]

I would just be honest and say something like "do you want me to just give you the solution, or do you want to learn how to solve these yourself? If it's the latter then we need to take a step back and go over the basics"

[cosmic ibex]

But if they say it's the former?

[spiral elbow]

If it's a student that haven't paid attention at all, then they might genuinely think that differentiation has nothing to do with integration, and it will seem irrelevant to them. So I think it's useful to explain exactly why you need to go over the basics

[halcyon glade]

Just say it's not your job

[spiral elbow]

honestly, I would just cut my losses and say that's not what I'm there for

[vagrant meadow]

yeah I mean is it an unreasonable boundary to only help students that actually want tutoring. if they just want someone to give them the answers well they have chatgpt open just do that and not waste my time.

[halcyon glade]

"I can't effectively do my job unless we step back for a second and figure out the conceptual gaps that you have. That way, you can answer not just this question but others as well."

"I know it might be frustrating right now to slow down so much, but it'll improve your understanding and your ability to solve these problems in the long run."

[tawny slate]

i dont know if its appropriate, but i would also advise the student to not be using chatgpt, its a tool only for those who are already highly competent in the area they are asking about

[lime shore]

I feel like there needs to be richer language to better ask questions like "what is [blank]?" There are so many different roles that concepts fill simultaneously. Take the question "what is an equation?" Well, an equation is a logical statement declaring that distinct objects are the same. It's also a specific notation we use to represent this logical statement. They also often times represent other objects, like shapes or whatnot. They also have certain properties, like being able to algebraically manipulate them. How could you meaningfully ask about these different perspectives?

I don't think I have the right words rn to properly get across what I'm trying to say, but I'll try to elaborate.

[lime shore]

What something 'is' means so many different things. There's the specific properties of the thing, the purpose of the thing, where the thing exists relative to other things. The thing is usually cared about or conceptualized in a certain way for some reason. I think all of these aspects tie into what the thing 'is', and it's really difficult to specify what you're asking about. An example of this for me was the idea of a function. It took me an embarrassingly long time in highschool before I had a decent intuitive understanding of what a function is. I was told it was something that took some input and gave an output. That the same input always gives the same output. Sometimes I was shown a graph, and told that that's what a function was. Sometimes a function looked a lot like an equation. All of these things that a function 'was' felt so disparate and disconnected, that I had no clue how to actually conceptualize them. it wasn't until I read that a function is a "mapping between sets" that it finally clicked for me. I'm not sure why this specific framing did the trick for me, and obviously others will find that explanation entirely opaque; but it gave me the right outline for the concept that I could nicely fit all the specific instances I'd seen into.

How can we better ask what something is? And how can we better answer that question when someone who doesn't have much conceptual framework for math asks?

[white fulcrum]

math is arguably about all the different ways of expressing the same concept

[tawny slate]

i think with formal objects, they dont have a "true" identity of what they are, because they are idealized and completely abstract. in many ways, it matters more what they do rather what they "are"

what is the number 1? honestly just describing what 1 is does very little, only through actually using it do you get a better feel for it. sometimes, i think this is the mentality to have, to get used to some rudimentary definition to start, and play with it some, to get a better picture of what this thing really does, and therefore is

this should at least loosen the expectations a bit so the questions are less difficult to answer

[tall bolt]

Yeah, this is the concept of a schema in the research, people have thought about it quite a lot (I believe Kemp introduced it, at least in the context of maths education?)

[turbid zenith]

I do think sometimes there’s value in asking what something is without qualification as a diagnostic tool

For example I can imagine asking students “what is a derivative”

And I’d be fine with “slope of a curve”, “instantaneous rate of change”, “a measure of sensitivity”, etc

But I’d be worried if the answer was “that’s when you bring the power down and subtract 1”

[astral zinc]

I agree with everything except that a function is a mapping between sets. The concept of a function should not be tied to the concept of a set. A mapping from one collection (whatever that means) to another sounds better at least to me. Sets cannot be too big.

[turbid zenith]

But before you know that there can be objects that are larger than sets, “mapping between sets” is entirely reasonable.

[tall bolt]

Isn’t a function defined to be a mapping between sets? At the very least I don’t see it being used in any other context. A mapping between objects need not be a function, and in such cases we do tend to call them maps or morphisms

[cosmic ibex]

In set theory, things like the successor function on ordinals, or the power set operator, or Hartogs's function behave mostly just like functions, except being defined for too many inputs to be represented as a set.

[turbid zenith]

Yeah so you could have a function between proper classes

[astral zinc]

yea this is good as a starting intuition

[astral zinc]

If you want to formally define a function in ZF then yes, but the concept of a function is not something that depends on the concept of a set. This is what ZF thinks functions are, an implementation of the concept of a function, but not an actual function. A function is not a set of ordered pairs which are pairs of pairs, it is a black box that takes an input of some kind and returns an output of some kind. In a similar manner the concept of one does not depend on the symbol 1 or the definition of 1 in ZF, namely {{}}.

At least that is my philosophy.

[tall bolt]

I believe this is similar to the point Leinster makes in rethinking set theory and proposing ETCS tbf

But also like meh, these conversations don’t really interest me all that much, I don’t have strong feelings on the matter

[halcyon glade]

Yes, although sometimes people specifically say "set functions"—this is an odd tangent for this channel though

[cosmic ibex]

"Set functions" just as often seems to be a way to emphasize that the functions in question aren't necessarily continuous/measurable/nice.

[lime shore]

I think part of the reason why what something 'is' is so difficult for students to grasp, is because they don't know what they don't know. giving them a few examples could actually be kind of detrimental to their learning because it could lull them into a false sense of understanding. once they're confronted with the thing behaving outside of the framework they've developed, they're back to not knowing what the thing is.

at least in the schools that I've been to in the US, we aren't taught how to conceptualize abstract concepts, or even that abstract concepts should be thought of differently than everything tangible we think about

I think early math education should be focused more towards the mindset of math than the symbolic manipulation rules that kids are taught. Obviously teaching how to calculate things is essential and a large part of building understanding, but learning about how to intuit things is glossed over so much

I also believe answers to "what is [blank]" should be more focused on contextualizing the thing rather than giving concrete properties of the thing. When kids don't understand something, flooding them with more information doesn't clear stuff up. I find it better to provide the scaffolding. Then when they learn the properties, they're better equipped to put them together in a way that makes sense

[turbid zenith]

Can you give an example?

Especially of where examples could be detrimental

[lime shore]

examples aren't inherently bad whatsoever, but if you use them too much without first establishing an outline of the concept, people can easily get lost. It's difficult for people to extrapolate the important bits of an idea with just a handful of examples. Especially if they don't have the mathematical maturity to know how much they should be extrapolating

this all mostly comes from my personal experience with earlier math education. there were so many times where I was shown a couple trees, and it wasn't instilled in me that there was a forest beyond them

[turbid zenith]

Still do you have any examples of this?

That would help me get an idea what you mean

[halcyon glade]

If you need to flood with more information when you're explaining something, you didn't explain in the right order

Motivate with concrete examples first, then abstract

[lime shore]

my whole point is that starting with too many examples can lead to confusion. when I was taught what a function was in middle school and early highschool, I was told to think of it like f(x)=y. Or that I should think of it as a graph that doesn't cross the same vertical line twice. Or that a function was something that took an input and gave an output determined by what you gave it. this overload of different ways to think about a function without the abstract notion beforehand kept me from understanding it for a long while

[lime shore]

you never want to flood with too much information. when I'm learning something new, I want the outline beforehand, then some concrete examples to see where they fit in the outline.

getting the general idea first, then seeing how examples execute that idea works so so much better for me than the converse. even better if I see some things which don't quite count, but still behave in ways reminiscent of the general idea

[tardy ember]

i think i would also say that... instead of just saying "a function is this, and also this, and also this", you would ideally want to explain what exactly the connections between the perspectives are, so that they really do feel like multiple ways to view the same thing instead of multiple distinct concepts that confusingly have the same name

[halcyon glade]

All of the definitions you gave are abstract definitions of what a function is

I imagine if you had been given an even more abstract definition to start with, it would have been even more confusing

[turbid zenith]

And my whole thing is I can't imagine what "too many examples leading to confusion" would look like. That doesn't sound like examples to me.

"A function is something that takes an input and gives an output" isn't an example of a function. It's an attempted description of what a function "is".

f(x) = x² is an example of a function.

"The graph doesn't cross the same vertical line twice" isn't an example, and it's also not a definition. It's a property of a function, and a way you can tell whether a graph represents the graph of a function.

Also, f(x) = y isn't an example or a definition. it's a notation for a function.

I know all of this is obvious to people who are comfortable with functions, but the reason I'm bringing all this up is, I don't think any of this is an indication of "too many examples."

Personally, I would say an effective way to introduce functions in a middle school classroom would be:

1. Give a motivating example of a function, preferably based on some real-world phenomenon the students are familiar with
2. Give a (developmentally appropriate) definition of a function as something that turns inputs into outputs, where each input leads to exactly one output
3. Give more examples to help students lock in that intuition, but perhaps using different representations (this is where "The Rule of Four" comes in handy — words, data, graphs, and formulas)
4. As you do this, point out properties of functions, but tie them all back to the definition. The Vertical Line Test can show why a graph doesn't represent y as a function of x, but that's because it violates the "each input leads to exactly one output" part of the definition.

[white fulcrum]

oh no

[midnight scarab]

<@&268886789983436800> the usual MrBeast

[quasi maple]

ew don't use that as a label

that's unfair to the scammers

[quasi maple]

It's surprising how often I have to... well, not explain, but correct that understanding

It often doesn't happen in the (A Level) chapter that introduces functions, but in the calculus chapters, whereby a function is explicitly given the name f(x) = ... and a student will go "it's dx'ing time" and start dy/dx'ing all over the place their workbook

[turbid zenith]

Then you call it g(x) and they go blank?

[quasi maple]

Holy shet the amount of confusion that can stem from shit like that happening yh

[turbid zenith]

It’s interesting how certain variable names end up being sacred like that

[jaunty yew]

<@&268886789983436800>

[white fulcrum]

my students love ramanujan's magic square, the one with his bday

I highly recommend it

[placid knot]

why do we learn/teach pascals triangle in conjunction with arithemtic and geometric seq/series how can i motivate it (sorry high school levle pedagogy)

[lime shore]

hmm, I've only encountered it in relation to binomial coefficients. it was introduced as a combinatorial tool to count paths through a grid

[placid knot]

im specifically thinking about it in the context if the gaussian if that helps? (although thats more for me than for them)

[tawny slate]

what gaussian? related to seq and series? in high school?

[cloud zealot]

I found several papers on the pedagogy of real analysis cited in the bibliography of this open access real analysis textbook. Here are a couple of citations that particularly stood out to me along with their abstracts:

1. J. Barnes, Teaching Real Analysis in the Land of Make Believe, PRIMUS, vol. 14, Taylor and Francis, UK, 2007, pp. 366–372.. Abstract: This paper is essentially a story that can be used to help students understand some of the definitions found in a standard introductory real analysis course.

2. S. Seager, Analysis Boot Camp: An Alternative Path to Epsilon-Delta Proofs in Real Analysis, PRIMUS, vol. 30 (1), 2020.. Abstract: For many of my students, Real Analysis I is the first, and only, analysis course they will ever take, and these students tend to be overwhelmed by epsilon-delta proofs. To help them I reordered Real Analysis I to start with an “Analysis Boot Camp” in the first 2 weeks of class, which focuses on working with inequalities, absolute value, and multiple quantifiers. This helps with all the topics which follow, and when we finally reach limits of real-valued functions, the weaker students find epsilon-delta proofs much easier to handle, and are more likely to pass the final exam.

[cloud zealot]

also, the book itself seems pretty good, so i'd recommend checking it out

[turbid zenith]

No reason to apologize for any level of pedagogy!

But I always see it primarily taught when discussing the binomial theorem, just raising (x + y) to a power

I’ve never seen it taught with sequences

[halcyon glade]

Yeah it's nice to show why expanding (x+y)^n leads to a counting problem where the coefficients are nCk and then talk about how you would recursively calculate nCk and then Pascal's triangle motivates itself

[arctic coral]

my students perceived the homework exercises as easy despite not scoring too well in the exam, and the exam based mostly on those series
i feel so weird and so perplexed by this, and i think they are comparing their homework with another class under a different teacher

what do you think i should do, i hate adding auxiliary difficulty that has nothing to do with the subject, as it does stray from the objective and a waste of effort and time

[placid knot]

its all 1 unit here, in grade 11 theyre tested on pascals triangle in conjunction with seq/series. I guess I can think of the sum in the brackets as the difference and the exponent as the ratio, so maybe pascals triangle can be viewed as a composite of arithemetic+ geometric? idk if that's the right framing or valid.

[sick bough]

you're probably writing questions that are easy as homework but difficult under exam conditions -- this is actually quite easy to do, since students are much worse at mathematics in the exam hall. This is especially true now where many hw questions can be solved with chatgpt

Instead of adding artificial difficulty to the hws (which i agree is mostly pointless), you'd probably benefit from making the exam questions more structured. Students will find a given question a lot easier if they are guided to the key ideas through multiple parts instead of having to produce the key ideas from their knowledge (even if it's something they've done before). Alternatively, just allocate more points in the exam to basic bookwork like recalling definitions and proving standard basic lemmas (at least if it's a closed book exam -- open book is a lot harder to do this with ofc).

[sick bough]

Another facet of this is that a lot of hw questions are pretty impossible to solve only partially (at least up to minutia like forgetting special cases like when things are zero) -- the same question in the exam will have people scoring either 0 or near full points and all the people who get 0 will feel the question was very difficult since they couldn't make any progress and can't see a way they could have made any progress.

[white fulcrum]

I"m thinking of getting a math tattoo, and I want it to be something easy to explain to my students and strangers. I was thinking of a proof of pythag but I'm not sure which one would be best. another candidate is ramanujan's magic square (the one with his bday). suggestions welcome

[jaunty yew]

https://en.wikipedia.org/wiki/Sierpiński_triangle

[midnight scarab]

It might take a long time to tattoo

[tawny slate]

not quite sure why this is in pedagogy but shrug lol

personally, im less of a fan of printed symbols and actual images as tattoos, because they have so much room for interpretation and baggage for so many people, so if I had to pick something, id pick something abstract: https://en.wikipedia.org/wiki/Einstein_problem

the downside is that a single tile looks contextless and ugly, to make it look nice you basically have to texture a large patch of skin

[shadow radish]

I'm not sure if this falls here, but here goes.

I was asked by someone who works for a scientific vulgarization TV show to help explain a "parabolic calculator" to a general target audience, including a lot of kids.

Here is the subject. Consider the parabola $f(x) = x^2$. Choose two non negative numbers $a$ and $b$ which you want to multiply. If you trace the line between the points $(-a,a^2)$ and $(b,b^2)$, it turns out the $y$ intersect is $ab$.

The mathematical proof is quite simple. The slope has to be $\frac{b^2-a^2}{b-(-a)} = b-a$. The line is the one going through $(-a,a^2)$ with slope $b-a$, so it is given by
[
L(x) = a^2 + (b-a)(x-(-a)) = a^2 + (b-a)(x+a).
]
In particular, $L(0) = a^2+(b-a)(a) = a^2 +ab - a^2 = ab$.

Most of the algebra I can explain geometrically (using areas of rectangles) or using intuitive arguments like relating slope to salary. However, this ends up being quite a long explanation. I was wondering if some of you had any ideas as to how this could be explained more directly, while still keeping it light on the math. I reckon a geometric explanation might be good, but I struggle to find one that would explain most of the solution in itself.

[burnt vessel]

Sam

[arctic coral]

thank you so much, that helped me find little bit of peace of mind, as well as the direction i should put my thoughts toward it

[turbid zenith]

If you’re talking about something where you’ll have a screen and narration over it, you can always give a broad overview by voice and just have the algebraic nitty gritty on screen.

That’s what I sometimes do with videos when I know lots of messy algebra will be involved.

[shadow radish]

That's what I was probably ending up suggesting. I at least have a way to explain it, but it would be neat to have a prettier answer.

[white fulcrum]

it's here bc I wanna show it to my students and therefore it has to be somewhat accessible

also posts get lost in any of the general chats

[tawny slate]

makes sense then

[cosmic ibex]

<@&268886789983436800> Mrbeast

[white fulcrum]

again? jeez

[cosmic ibex]

Happens several times a day. They must have evolved beyond what the automod patterns catch.

[tame tulip]

mister beast

[rapid tusk]

miter bret

[cloud stream]

my calc teacher had us do the homework together peer-to-peer in class, so when we didnt know something we held ourselves accountable to learn it

[arctic coral]

good approach

[cloud stream]

something else thats been helping me is like, well, i found this book called mathematics for human flourishing and its been a really bit motivator to make me care about all the interesting parts

if i dont get why something works i just really want to know

this can be a hinderance in some classes

your profile description makes me think youd like the book as well

[arctic coral]

thank you so much for the suggestion i get similar question and something the answer isn't ready or surrounded by doubt that will me at least to lead them to that reference

hmm interesting really giving simple examples behind many math subdomains wich does also explain and put the lecture into a good pov

[scarlet pebble]

You might have a class of bad test takers but chances are they were have "outside" help while doing the homework or studying the material, like referring to notes while doing it and not making the connections they need to actually complete a problem without that reference. In my experience the thing that helped me the most throughout my education so far has been struggling with problems and repetition. Also if you are asking students just outright if they thought the homework was easy or how they did they will probably respond with that they did fine, ive seen many people in my classes who are basically failing but still reaffirm that they understand material. Most students will search for help outside of class so they probably wont act like they are struggling in class. Im not a teacher but I feel like the only thing you can do is inform them more about the effort that classes will take and that real effort and struggle will ultimately improve their skills. You of course could try to alter your teaching strategy but time is always limited and things need to be completed within certain time frames so it may not be realistic. About the comparing to other classes, people are always going to do that, i did and still do it. It really doesnt mean anything, students can complain or state that a class is easier, assigns less homework, has a better teacher, etc. but in reality most learning is truly done outside the classroom. I believe a teachers job is to introduce material and answer questions that occur when students further study that material themselves and encourage further development of their education and not to hand hold and teach every specific thing that is needed for a concept. A bit off topic but i feel like ive noticed more people than i would expect who dont spend enough time outside of class learning material, and kind of just use the class as their study time and thats it. I liked grays input, my multivariable calc teacher had group quizzes pretty often and encouraged group work often as well. Group quizzes though i wouldnt recommend (usually leads to one person doing most of the work) but having people hold themselves accountable and in a group of 4 theres most likely someone who would be able to explain it well enough to help out other students.

(this is also heavily biased because i enjoy studying math and find it much more interesting than the average person)

[arctic coral]

yeah thank you, that is true, probably didn't really test doing homework without external help

[turbid zenith]

Someone in my liberal-arts-for-mathematics class today, for feedback/suggestions for next time I teach it, said "can you do like ... less games, and more instructing?"

(I've made the deliberate decision to make games a common vehicle of introducing new material, and our last third of the class is focused on both classical and combinatorial game theory)

[cosmic ibex]

Out of curiosity, what s "liberal arts for mathematics"?

[torn apex]

What do you make of that feedback? Like what do they not like in the games and what do you think they meant by "more instructing"?

[midnight scarab]

By "games", do they mean playing a game in class (idk if you did that) or game theory?

[turbid zenith]

I assume both

[turbid zenith]

I think this student just wanted a more “traditional” math class that they could student their way through. They were basically never engaged and were always on their phone though.

For example when we did taxicab geometry they refused to accept that a taxicab circle is a “circle” because it’s not round

[turbid zenith]

Oh I should have said it the other way around, mathematics for liberal arts

[cosmic ibex]

That makes more sense :-)

[turbid zenith]

Sorry I’m not all here today

Got rear ended on my way to work so it’s been a day

[cosmic ibex]

Possibly "more instructing" means, "just tell us what the steps are already".

[turbid zenith]

Yeah. They’re probably used to “step 1 do this, step 2 do that, step 3 circle the answer”

[tawny slate]

i could understand that as a preference if they actually paid attention and had troubles, but if they were on their phones then i think thats more difficult to accommodate

[cosmic ibex]

"Yeah, doesn't hurt to check Discord until he begins teaching math instead of ranting about games".

[quasi maple]

Jeez am I glad I'm a tutor and not a teacher

[quasi maple]

I've actually tried to explain that in D&D, because the default setting in Roll20 is to use the taxicab metric

...Then again, I was explaining this to other players, and not maths students lol

[tawny slate]

i would start by being more general, talking about general movement in a grid

so things like sokoban also count

[cosmic ibex]

I would expect the pushback is more about the term "circle" than about "this is what the solution set for d(x,x0)=c looks like". But perhaps I'm too optimistic.

[turbid zenith]

Well the unit this comes from is our geometry unit, but the big meta themes are about generalization and axiomatization

So this was their introduction to “what if we changed how we measure distance? What else would change along with it?”

The way I introduced it was with a “treasure hunt” Battleship style game… you and your opponent pick a location for your treasure and then you take turns guessing where each other’s treasure is, and you have to give the “number of paces away” your treasure is from each guess

And by the end of it we looked at how the new “taxicab circles” can help you narrow down the location more quickly

[cosmic ibex]

I see.

[white fulcrum]

I just went down a rabbit hole of teachers losing their cool and yelling at students on youtube

[lethal hornet]

peak pedagogy

[torn apex]

@turbid zenith sooo I switched to using "vibe physics" (or "sportsball" for normies :P) as a pedagogical scaffold to teaching 8-10 year old kids about Noether's theorems and conservation of momentum and energy as time translation invariance. It started out as a joke but now it's turning into a real research paper on applied category theory, kinesiology, and the mathematics of learning.

Most shocking result is symmetry transfer learning where it only took 3 months to reach dexterity parity with non-dominant foot. I obviously can't show any of the kids so at the risk of getting roasted myself, here's a raw video snippet of the results.

I don't know how novel or cross-applicative it is to college classroom environments but at the very least, you (and this discord channel) might get a shoutout/special thanks in the research paper as a lot of this is the culmination of all the math edu content you sent my way

[white fulcrum]

can you elaborate on this? what do the handles have to do with noether?

[torn apex]

sure but what are handles?

[lethal hornet]

dribbling skills

applies to sports like basketball, soccer, hockey, etc

[torn apex]

haha, oh man I'm so not a sports person

The noether/symmetry part comes in two parts:

• The pedagogical part is teaching from Jakob's book, Physics From Symmetry, but curtailed to 10 year olds
• The "fun part" applied to sports is through learning about static friction, conservation of momentum on how to "trap the ball", how to apply learning drills on right foot to the left foot

in a very meta-way, I also built a coaching app powered by a neural net tailored for each kids progress that are built around this paper Neural Mechanics: Symmetry and Broken Conservation Laws in Deep Learning Dynamics.

The last part is where the research paper actually focuses on, which is showing the symmetry between learning dynamics across digital + biological substrates (if you can train a neural net, the neural net can train you!)

[white fulcrum]

Trying to explain to students how we know which rate goes on numerator or denominator. Im struggling to explain it. In my head all I think is "the other way gets you a nonsense answer" but obv that's not a good explanation

[austere delta]

Isn't it just if you want to count holes per hour you do holes/hours, if you want to count hours per hole you do hours/holes?

I don't think there's much more to be said...

[tawny slate]

i think the way i would explain it is

• firstly, notice that when you add rates like this, you can think of it as adding fractions, which means you can only validly add the numerators

• secondly, pretend, for instance, Daniel and Rob are cleaning houses together. because they work at the same time, and stop at the same time, it is time that is constant, but it is their work that is combined

• thus, the thing being combined, the work, must be the numerator, and the thing held constant, the time, must be denominator

[elfin sleet]

This is my instinct too, but then how do you justify that in this case you're wanting holes per hour?

[midnight scarab]

Well, if you want holes in 15 hours, then you want holes in 1 hour

[turbid zenith]

I think asking them to make sense of it first and foremost is EXACTLY the way to go.

Replace “per” with “for every”.

In any situation where your rate is a/b, you could come up with a situation where the rate should instead be b/a

[elfin sleet]

(assuming this is a response to me, sorry if I've misunderstood) if you mean that you would say it's because you're solving for holes then that doesn't work because in the first problem you're solving for hours but it's still houses/hour because that's what allows you to add them

[sick bough]

I think it's important to emphasise that doing it the other way isn't nonsense, it's just describing a frequency instead of a rate, and that's perfectly meaningful (see for example how runners will describe their paces as minutes per mile or km instead of in units of speed). The only time you get nonsense is when you try to add a speed to a frequency or something like that -- and that's for the same reason that adding a time to a distance is meaningless

[midnight scarab]

Ah, I see, I didn't understand originally that the question was which ones are additive.

[midnight scarab]

Not an explanation, but a good sanity check is that for question 1 you expect the final time to be less than the individual times, while in 2 you expect the combined number of holes to be greater than the individual one

[tawny slate]

to give an example the other way:

sam can build a home in 40 work hours. however, he has to take a break of 5 hrs for every home he builds. how many hours per home can sam build at?

here the number of homes is constant rather than the number of hours

(response directed at OP)

[quasi maple]

An aside - that feels pretty fast, mad props to Sam

[turbid zenith]

Out of curiosity does anyone have the new TI calculator? The TI-84 EVO?

[pure light]

seems to have only come out a few days ago, so i would expect the only people to have it are day 1 orders

[cloud zealot]

wack that this line hasn't been superseded by the nspire only because the nspire is banned from several standardized exam environments

[turbid zenith]

I just see videos of educators having it apparently

And literally my only question is did they find a better way to get to the hyperbolic functions 😛

If not, it's just sad

[rapid tusk]

how much has ti calculator monopoly been displaced by online tools like desmos/geogebra i wonder

[turbid zenith]

not enough

[pure light]

i've definitely seen a lot of students do all their homework calculations in desmos

can't avoid physical calculators for pencil-and-paper exams though

[turbid zenith]

They do have Desmos for the AP Calculus exam now apparently!

[rapid tusk]

that’s so cursed

[turbid zenith]

I love it

[rapid tusk]

i graduated high school right before they started integrating desmos into all those exams

idk maybe it’s bc I grew up in a relatively more analog era

[pure light]

it does make sense if you're planning to computerize anyway

although i'm not personally huge on computer exams

[turbid zenith]

One of my colleagues allows students to use Desmos on laptops for paper exams if they turn off their Wi-Fi

It's something I want to try next time I teach calculus (in the fall)

[pure light]

sounds difficult to verify

[turbid zenith]

Perhaps it is difficult to do perfectly

But my gut is that if the exam is well proctored then the risk of getting caught outweighs the potential reward

[zinc dove]

What about the HP calc?

[valid snow]

back in high school our math teacher would let us use geogebra on our phone as long as it was in exam mode, it's pretty hard to falsify that one

not sure if it also applies to desmos or to laptop calculators in general

but it surprisingly worked out until we had to get our own graphing calculators and such

unrelated; im trying to tutor a couple of high school "kids" (lol), specifically about limits/derivatives and such. I've done so temporarily for free now to a couple of friends who don't do terribly in math but have been lagging behind on the last year's curriculum but it seems that no matter what I do there is something about limits specifically thats hard to convey to them, more specifically about the definition of a limit using epsilon/delta

I'm starting to think that maybe this is too out of reach for me? I'm barely like 1/2 years older than most of these people but this was something I was really good at in Calculus and when I've done tutoring to much younger kids about more "simple" concepts this didn't seem to be the case

I was trying to maybe look around for some more intuitive and general approaches to teaching this concept but I think the fundamental issue is I'm maybe not even supposed to be teaching this

I think a factor that may make the definition more confusing is the fact that you have a lot (4) cases to deal with and you're not always going to know which one to apply

this also applies to the application of limits themselves I feel like, because iirc this is also kind of the first time in the average Italian math curriculum where you're presented with "something" where you need to match patternS and test what works and what doesn't, as opposed to literally everything else before which is mostly a matter of applying a pattern, singular

it's also hard to do guided exercises for limits without just spilling everything when the person you're tutoring is completely unaware of where to go

maybeee I'm over complicating this a bit but high school calculus is kind of the thing I excel at so not being able to teach it removes a big part of my "toolkit" or whatever

[midnight scarab]

Which four cases? Like limit at infinity vs at a finite point?

[valid snow]

yeah

finite limit at a finite point
finite limit at infinity
infinite limit at a finite point
infinite limit at infinity

[midnight scarab]

Yeah I've been unironically thinking that maybe it's easier to talk about neighbourhoods

(Or rather, epsilon-neighbourhoods, at least for a start)

This way you have just one definition, which uses just words

And then you just have to distinguish nbhd of a point vs of pm infty