math-pedagogy

[boreal agate]

adding cross products isnt very intuitive

adding bivectors on the other hand has a clear geometric picture

(at least in 3d, where all bivectors are simple)

[turbid zenith]

So I'll return to my question about the TNB frame

[boreal agate]

the TNB frame does require metric information

and it uses it a lot

without fully learning exterior algebra you wouldnt benefit for this specific thing from it

[turbid zenith]

So what does that imply for multivariable calculus students?

Well so far I've grabbed two examples of things that are using cross products in the Stewart book

And so far we're 2 for 2 on exterior products not being beneficial

I want to see how students in a basic multivariable calculus course would USE bivectors. Not give (or follow) lofty abstract arguments about them, but put pencil to paper and answer a problem.

Picture related in spirit.

[tight star]

as someone who’s been trying for a while to see how cat theory can be used pedagogically, I’ve had to get very good at figuring out how to make it help with actual concrete computations

[turbid zenith]

(And the hidden panel!)

[tight star]

because the formalism is so abstract that trying to argue it’s useful means I need to figure out how to concretize it as best I can

luckily this has also helped me understand cat theory better

[tawny slate]

yeah im just going to point out that if i were a test dummy, all of this exterior product stuff just sounds like gobbledy gook to me

i have no clue what any of the discussed stuff means

[austere delta]

For how you can use it, instead of calculating with a big 3x3 determinant rule you can multiply things of the form
(adx + bdy + cdz)
with the rules dxdx = 0, dxdy = -dydx etc

Then end up with something you can take the length of in the normal way.

Not sure whether it's a lot more useful, I'd say it's probably easier to memorize, though may or may not take up more space on paper.

I'm not convinced any first time students are getting anything out of the determinant definition of cross product

[tawny slate]

i am currently very motivated now to stay in the realm of what i know with cross products than touch that stuff right now

[tight star]

i do genuinely think that 3b1b's explanation helps the cross product make sense

[tawny slate]

id like a simple example, side by side with a classical example, to highlight the superiority of one method over another

[tight star]

personally after that video i never had an issue with cross products

[tawny slate]

yeah idk where all the dislike for 3b1b is coming from all of a sudden

[turbid zenith]

Eh? Is there dislike for him recently?

[tawny slate]

like yes it cant supplant a textbook or an entire course but like even grant himself says dont do that, use all of the resources and practice

[tight star]

i think some people have valid concerns about learning just from videos as opposed to videos + exercises

[austere delta]

I'm not convinced there is any meaningful difference for simple examples

[tight star]

you have to remember it's edutainment, and optimised for that

[tawny slate]

sure but thats not like a reason to hate on 3b1b when he himself says in his videos not to do that iirc

[tight star]

nonetheless i do think there's some amount of elitism present in the way people reflexively dislike 3b1b

yeah

[turbid zenith]

I didn't realize there were people who reflexively dislike him wow

[tawny slate]

like who else has spent that much time to make such pretty animations on a topic that no one really bothered to animate and explain in that way before

[tight star]

#math-discussion message

[turbid zenith]

I've disagreed with him on a video or two but nothing like that

[turbid zenith]

Wow. That's asinine.

[tight star]

[turbid zenith]

Anyway yeah I'm also not convinced yet that using bivectors instead of cross products would help my students with computations

And that makes me sad because I actually really like bivectors conceptually

Like it always does seem that using cross products to do things when you actually want to use area seems roundabout

[austere delta]

Why is the end all be all to help with computation though?

[turbid zenith]

I didn't say it's the end all be all. But it's the first step.

[tight star]

i think the utility of cross products is that they're a way to represent an area with a vector

[tawny slate]

sudgylacmoe got me hooked in his expository video but i have since completely forgot the details and i havent had the time to watch the full courses

[astral sinew]

Personally I was teached multivariable calculus via differential forms (and this was a real analysis course) and not the usual approach.
Many students I know complained the way it was teached, especially those from a mathematical background, but those who were from physics and were sharing courses, the differential form approach was neat. I personally loved the differential form approach this professor provided.
I didn't do the exterior calculus explicitly, but I did use the wedge product and Poincare lemma for my computations, and treated divergence, curl and so on via differential forms only

[austere delta]

Wouldn't understanding something conceptually be a good goal?

[turbid zenith]

Of course it is. But you need to also be able to play with the thing to be able to understand it.

[tawny slate]

uh yeah but having a concrete example helps with that meta goal

[tight star]

for example, this allows you to add parallelograms in 3D!

[tawny slate]

understanding is the meta goal, concrete examples that help with that are the subgoal

[tight star]

you just add their corresponding cross product vectors

[austere delta]

Right, so you're not saying that it just isn't helping, you're saying it's hurting computation?

[tight star]

whereas i think a priori it is not at all obvious how one adds parallelograms in 3D

[turbid zenith]

So far it seems to be yeah

I'm thinking in terms of students learning to work with increasing levels of abstraction

[tight star]

this is another reason why cross products are helpful for torque, because it lets you add the effect of multiple torques on an object

which i think is not at all obvious in the area picture

[turbid zenith]

And trying to go too abstract too quickly is a great way to make people not understand the concept because you don't have something to hook into yet

[tight star]

so i think that's one place in which cross products win over bivectors - they allow you to make an addition operation on oriented areas, by just doing vector addition on the cross products

if that makes sense

[austere delta]

I'm not sure thinking about rotations as happening in planes as opposed to around an axis is necessarily more abstract, even though it allows more abstraction.

[turbid zenith]

That video was excellently done

And I really want to try to find a way to make at least some of it work for my courses

[tight star]

perhaps this kind of hybrid approach could be useful?

[turbid zenith]

When I say I'm not convinced, I'm not saying I can't be convinced

[tight star]

geometrically, you can certainly introduce the idea of an oriented area

but the usefulness of the cross product is the ability to represent an oriented area by an ordinary vector

[turbid zenith]

I think that's only one use of the cross product though

[austere delta]

What kind of computations do you need to be doing with them?

[turbid zenith]

Another is for normals

[tight star]

yep yep, and i think you can motivate this by understanding the way in which the cross product vector represents the oriented area

in particular that it's necessarily perpendicular to the oriented area

which means you can derive from that a method to produce orthogonal vectors "for free"

[turbid zenith]

The last time I taught through the calculus sequence, they mostly showed up in:

• TNB frame in Calculus III
• Surface integrals in Calculus IV
• Curl in Calculus IV

[tight star]

so in this way, you can get the geometric idea of bivectors as an oriented area, but you also get the computational advantages of the cross product

[turbid zenith]

Also I suppose in at least one of the formulas for curvature

[tight star]

in this way, the cross product is viewed as a computationally convenient way to represent an oriented area

[astral sinew]

Is it an argument on bivectors versus cross products or differential forms versus the classical approach?

[tight star]

some mixture of the two i think, though what i'm starting to lean towards is having both bivectors and cross products

[turbid zenith]

Oh, also I suppose in finding the equation of a plane containing two vectors, although again that's one of the places where passing through the cross product feels somehow roundabout

[tight star]

essentially, i think i agree that there are many instances where a concept arises naturally as a bivector, i.e. an oriented area, rather than a cross product

however, bivectors alone can be awkward to work with computationally

[turbid zenith]

Yeah that's pretty much where I'm at right now

[tight star]

so one could motivate the cross product as a way to represent oriented areas by vectors

in a way that makes computation more convenient, and also gets you other things for free (like normality)

[astral sinew]

I think both are important, even though the theory I got was purely differential forms, I remember we also did the R³ case and treated cross product as a particular example (mind it was on a real analysis II course, but it basically is equivalent to some Calculus III or something)

[tight star]

doing cross products without bivectors is like skipping the oriented areas part, and i agree this makes it confusing

but i think doing bivectors without cross products handicaps you computationally

[turbid zenith]

If I could make bivectors just as computationally convenient as vectors, I think that would help

[tight star]

but i think that's exactly what the cross product is for

[astral sinew]

For example, mixed product identity is kind of hard to remember without the cross product iirc

[tight star]

it's a computationally convenient representation of 3D bivectors

[boreal agate]

https://www.desmos.com/3d/xdjy7xdyzk

here

i made an example graph

[turbid zenith]

I guess probably the simplest example might indeed just be using three points in a plane and finding their equation, it doesn't get simpler than that

[boreal agate]

this shows the geometry behind bivector subtraction

[boreal agate]

who said people shouldnt learn about cross too?

[tight star]

uh

.

wasn't this literally you

[turbid zenith]

This guy I think

[boreal agate]

thats just because i am disgusted by it (and anything else gibbs is responsible for)

[tight star]

anyway i think category theory has made me understand bivectors and cross products even better :D

[boreal agate]

i do think that skipping over teaching the cross product is dumb

[tight star]

i'm gonna try using this going forward

[turbid zenith]

Let's say we have:

• P = (3,1,1)
• Q = (5,2,4)
• R = (8,5,3)
  And you want to find the equation of the plane that goes through all three.
  The way it's usually taught, you can easily define u = PQ, v = PR, get a normal n = PQ × PR, and then if you have n = <A,B,C> your equation for the plane is A(x-3) + B(y-1) + C(z-1) = 0.

I would be genuinely interested to see what this would look like with bivectors instead.

[tight star]

in particular it gives a really clean way to talk about curved bivectors

[boreal agate]

its very helpful for the simpler scenarios, and not teaching it would be a disgrace seeing as how much of the literature uses it

[tight star]

which is something that does actually pop up in physics all the time

specifically with current loops

[boreal agate]

wdym by that?

[tight star]

oriented areas whose shape is not a parallelogram

[boreal agate]

btw does this graph make bivector subtraction clearer?

[tight star]

in magnetism you get oriented areas whose shapes are little disks

these represent current loops

[boreal agate]

bivectors dont have to be parallelograms

[tight star]

nor sums of parallelograms

i mean a very general notion of "oriented area"

[turbid zenith]

So seriously, I really want to try to carry this forward and see if we can reverse-engineer a bivector method for this, and see what it would take to actually teach it.

[boreal agate]

funnily enough, they are described relatively well as "magntitude with a 2d orientation"

[tight star]

so how would you describe a disk-shaped bivector?

[turbid zenith]

So you've got PQ = <2, 1, 3> and PR = <5, 4, 2>, so the normal would be <-10, 11, 3>

I believe the Hodge star of that normal would be a bivector living inside the plane we want

[boreal agate]

this plane is (Q-P)^(R-P)^((x,y,z)-P)=0

same idea, but somewhat cleaner

no metric data needed here

[turbid zenith]

99% of my students have not heard the word "metric" used in this way

So talking about something being metric-free or whatever is not going to be particularly relevant to them

[boreal agate]

its still just a bivector, you can morph it around as much as you want as long as you dont change the area or orientation

[tight star]

so what is a "bivector"?

[turbid zenith]

Anyway so what would this look like?

[boreal agate]

i was just saying that to express how no hodge dual is ever required here

[turbid zenith]

I guess if you do basis vectors x, y, z, the normal I'm working with is -10x + 11y + 3z, so the bivector would be -10yz + 11zx + 3xy

But I have a hard time seeing how you'd visualize that.

Although I guess I can kind of visualize it

[tight star]

ok so i think i've realised a way i would like to think about this

it's in terms of oriented areas $S$ and the corresponding vector areas $\vec S$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

and with is-does duality

[boreal agate]

its an algebraic way of expressing an oriented area (at least simple bivectors)
here an orientation is understood to be (after fixing an ambient orientation) a 2 dimensional subspace with either a conceptual +1 or -1 (this is judt to express that at this point exactly 2 orientations exist)

[turbid zenith]

I just said, I'm trying to do it based on how I understand it so far so I can figure out what it would take to start with bivectors instead.

[boreal agate]

another way of understanding specifically bivectors, is as the type of object rotations happen in

[tight star]

so how do you say it's disk-shaped?

[boreal agate]

this generalizes to non simple bivectors too

[boreal agate]

thats the neat part, you dont! the shape isnt encoded in the data of a bivector, and all shapes are considered the same so long as they have the same area and orientation

this is the same as how you can consider a magnetic field as a bunch of circular loops or a bunch of square loops

[tight star]

ok but i don't actually like this

i would rather be able to actually say what the shape is

[boreal agate]

it turns out that for a lot of things, the shape doesnt actually matter. in particular, if you have a differential bivector, you sure dont care what the shape is

[tight star]

i'm not entirely sure what you mean by "consider a magnetic field as a bunch of circular loops"

ok but i'm in a situation where i do care what the shape is

[boreal agate]

then a bivector is not the object you seek

an embedding is

[tight star]

ok i guess bivectors aren't for me

[boreal agate]

wdym?

[boreal agate]

eyup

to visualize this, start with an yz square (parallelogram)

flip the orientation and enlarge tenfold

[turbid zenith]

What I mean is I'm trying to reverse-engineer the bivector representation of the problem, so I can think, if I were to retool my Calculus III course to use bivectors as an object instead of cross products, how would I teach my students to use them to solve this problem?

[boreal agate]

now take a zx square and enlarge it 11 fold

then change the side lengths for one of them such that the area is preserved, and change it such that you end up with a common side for the zx and yz rectangle

then you add them just like vectors: put one at the end of the other one, consider the line segment you end up with, and draw the parallelogram with one vertex at 0 and one side this segment

[boreal agate]

:(

[tight star]

it's ok i figured out a replacement

[boreal agate]

lol

[turbid zenith]

Yeah so far they're not really working for me either

[tight star]

and it uses is-does duality :D

[boreal agate]

ill draw a picture

[turbid zenith]

I get the picture

[tight star]

(solid angles have you been following what i've been doing?)

[turbid zenith]

What I'm not seeing is how it will be useful pedagogically

[boreal agate]

ah

[turbid zenith]

Not really, I've been working in Desmos

[tight star]

would you like me to try and explain it?

[turbid zenith]

I've managed to get a decent picture of a cross product versus a blade

But I'm right now trying to construct the blade out of basis bivectors visually

[boreal agate]

it would mainly be useful as a tool for first and foremost intuition seeing as we are now literally describing the plane rather than some funny looking vector (and this works just as fine in 2d too), but the lasting effect would be a certain nudge in a the direction of a better geometrical understanding of linear algebra, vector calculus and potentially generalizations to higher dimensions where other tools dont lend themselves so easily anymore

[turbid zenith]

But so far you haven't really given me much replacement for how to use them in actual problems that my students would do

Your response has been "bivectors aren't useful here because there's a metric" or something

[tight star]

-# i think my pov has a chance for using stuff in actual problems...

[turbid zenith]

I'd like to hear it soon

[boreal agate]

i also have another philosophy, that choosing the right object to describe something is incredibly important to understanding what that object holds in terms of available information, which ultimately is critical to understanding it

[turbid zenith]

Right now I'm just figuring it out with this problem

[tight star]

okies but i do also need to go to sleep soon :P

,ti

[turbid zenith]

Feel free to PM it to me I guess

[burnt vessel]

The current time for pseudonium is 10:47 PM (BST) on Tue, 12/08/2025.

[boreal agate]

in 3d its not going to be anymore convenient for the most part

,ti

[burnt vessel]

You haven't set your timezone! Set it using the interactive timezone picker with ,ti --set.

[turbid zenith]

So far it seems actively less convenient

[boreal agate]

less convenient how?

[boreal agate]

this is the picture i wanted to draw

[turbid zenith]

As in it would take much more setup to be able to get my students to do it

Have you taught this btw?

[boreal agate]

yes

several times already

[turbid zenith]

In what setting?

[boreal agate]

however i do change my approach depending on who im teaching

[boreal agate]

informal

1 on 1, 1 on 3 at most

[turbid zenith]

So just like in one-on-one conversations like this?

Not in a classroom?

[boreal agate]

not a classroom no

[turbid zenith]

And with what kind of students in your case?

[boreal agate]

physics students and math students

maybe one chem guy

[turbid zenith]

With what existing background though?

[boreal agate]

all i know on a personal level

[turbid zenith]

Like, are we talking about people who have had linear algebra, multivariable, etc already?

[boreal agate]

varies quite a bit actually

[turbid zenith]

Because right now I'm not convinced this would help when I have a classroom of 16 students of various majors who have never seen this before

I'd think I'd like to hear your idea though @tight star

[tight star]

yayyyyyyyyy

ok so

for now, forget about cross products and bivectors

[boreal agate]

if the end result is to get all of the material out in time, without getting sidetracked at all, then no this wouldnt help

[tight star]

what we're going to focus on is the concept of an oriented area

[turbid zenith]

Okay then. Thank you.

[tight star]

the simplest examples of these would be planar oriented areas

[turbid zenith]

I'm looking for practical solutions.

[tight star]

this is just a planar shape along with a choice of a "positive" and "negative" side (equivalently, a choice of "outward-pointing normal")

[turbid zenith]

Okay this I can get behind so far, that's simple enough. That's what I'd want to kind of get at with bivectors anyway.

[boreal agate]

if the end goal is instilling a great intuition and deep understanding for the kind of concepts presented in linear algebra and multivariable calculus though, i believe this would help

[tight star]

given two vectors, you can define such a planar oriented area

but there are lots more you might want to consider!

maybe a disk-shaped one, or a triangular-shaped one, or a hexagonal-shaped one

[turbid zenith]

Intuition and understanding are the goal, but there's only so much time we have in the semester, and I have to reach as many students as possible.

And I'd rather 90% of students be at 70% understanding than 10% of student be at 95% understanding.

[tight star]

is the concept of a planar oriented area clear so far?

[turbid zenith]

Yes, I think it always was

[tight star]

yay :)

let's denote this by $S$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

the idea is to use a form of is-does duality

[boreal agate]

then it might not fit, yeah. personally i do think this should be taught, but leaving it to one course already short on time is definitely not the right move; imo students should be engaging (even just a little) with these concepts over multiple sememsters, through several courses

[turbid zenith]

And I get that you can represent that planar oriented area with a vector with the same magnitude

[tight star]

what the planar oriented area "is" is a little 2D shape in space with an orientation

[boreal agate]

bestagons!

[tight star]

hang on

[turbid zenith]

And vectors are easier to work with computationally

[tight star]

we're getting to that

but we need to go through something else first

ok?

[turbid zenith]

Sure.

[boreal agate]

using bivectors isnt harder computationally than using the cross, the computations themselves end up pretty much exactly the same

[tight star]

the idea is to then consider what a planar oriented area "does"

how does it interact with other things?

[turbid zenith]

I agree there. We split our multivariable course in two at my school — Calculus III is vector-valued functions, partial derivatives, and multiple integrals, and Calculus IV is coordinate transformations, line/surface integrals, and their fundamental theorems

[tight star]

in this case, we can use flux/surface integrals as a motivation!

namely, what a planar oriented area "does" is define an associated volume functional

[boreal agate]

youre about to reinvent differential forms i think

[tight star]

given a vector $\vec v$, we can consider the prism $S \times \vec v$, right?

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[turbid zenith]

Yes, this is sounding like differential forms, but yes

[tight star]

and, given such a prism, we can take its volume!

so, we have a map $\text{Oriented Areas} \times \text{Vectors} \to \mathbb{R}$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

given $(S, \vec v)$, we spit out a real number $\text{Vol}(S \times \vec v)$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

the (signed) volume of the prism S x v

so this gives us one candidate for saying what an oriented area "does" - it interacts with vectors to produce real numbers

make sense so far?

[turbid zenith]

Sure

[tight star]

we can also consider this as a map $\text{Oriented Areas} \to (\text{Vectors} \to \mathbb{R})$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

This sends an oriented area $S$ to the map $\text{Vol}(S \times (-))$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[turbid zenith]

But again it mostly makes sense from the point of view of someone who's already seen these things

[tight star]

hm, what parts of my explanation do you think could potentially be confusing?

[turbid zenith]

Well a "functional" is already something I didn't get until I was in grad school

Despite having been told its definition

[tight star]

ah sure, i think this map should hopefully be understandable though?

[turbid zenith]

And I guess I'm also still thinking about my super-basic problem

I'm not seeing how it would help there

[tight star]

like i don't think the concept of "volume of a prism" should be that difficult to grasp

[turbid zenith]

I might try this out

[tight star]

ok wait at least let me finish

[turbid zenith]

ok

[tight star]

patience, is all

[boreal agate]

the "idea" behind this btw is that v1^v2^...^vk is 0 iff {v1,...,vk} is linearly dependent

[tight star]

then, the key observation is that $\text{Vol}(S \times \vec v)$ is linear in $\vec v$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

and this means it is possible to represent it via taking a dot product with a fixed vector

i.e. there's a unique vector $\vec S$ such that $\text{Vol}(S \times \vec v) = \langle \vec S, \vec v \rangle$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

this is the area vector associated to S

using geometric properties of the dot product, you can then show that $\vec S$ is necessarily perpendicular to $S$ and has length equal to the area of $S$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[turbid zenith]

I think it makes sense

[tight star]

this is the sense in which $\vec S$ "represents" $S$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[turbid zenith]

Again I don't see how I would use this as a starting explanation 😛

At least not as written

[tight star]

well, i think the idea is that

[turbid zenith]

But I can see how I'd simplify it perhaps

[tight star]

oriented areas arise naturally

area vectors give you a way to represent these oriented areas via ordinary vectors

at least in 3D

so for your 3 points in space, a natural oriented area to consider is the triangle formed by them

[turbid zenith]

But I'm trying this, and thinking what, say, a Stewart-style textbook explanation would look like.

[tight star]

given this triangle, you can get the corresponding area vector

[turbid zenith]

So far, this makes sense conceptually

[tight star]

which is guaranteed to be perpendicular to the triangle

[turbid zenith]

Where I'm not sure how to proceed would be actually computing that relationship with w

[tight star]

so this is how to actually compute an area vector given an oriented area

[boreal agate]

wdym?

[turbid zenith]

As in, okay I've done that, now what

[tight star]

for parallelograms, you can use the determinant, since the volume functional can be described by a determinant

[boreal agate]

im still confused

[turbid zenith]

So the original problem was to find the plane between P(3,1,1), Q(5,2,4), R(8,5,3)

I want to end up with -10(x - 3) + 11(y - 11) + 3(z - 3) = 0

[tight star]

well it's $(\vec v - P) \cdot (PQ \times PR) = 0$, right?

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[turbid zenith]

But using wedge instead of dot and cross

[boreal agate]

right

[turbid zenith]

All I've been wanting to see this whole time is what that would look like for a student doing the problem

[tight star]

i guess in volume functional land this says $\text{Vol}(PQR, \vec v - P) = 0$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[boreal agate]

ohhhhh ok

[tight star]

here PQR is the triangle viewed as an oriented area

[turbid zenith]

I feel like all of what's been said would go SO far over the heads of living breathing students in my classroom

[tight star]

what part of what i've said would go over their heads?

talking about functionals?

[turbid zenith]

Yes. All of this is-does stuff.

It's great ... AFTER you understand it

[boreal agate]

i dont actually think i know, i think i may be too familiar with these ideas already to put myself in the shoes of a student unfamiliar with them

[tight star]

ok but i don't think is-does duality is that hard to explain

it's not even a fundamentally mathematical concept

[tardy ember]

write $u \wedge v$ as a sum of basis bivectors $a(\vec{x} \wedge \vec{y}) + b(\vec{y} \wedge \vec{z}) + c(\vec{z} \wedge \vec{x})$, and then think about what $(a(\vec{x} \wedge \vec{y}) + b(\vec{y} \wedge \vec{z}) + c(\vec{z} \wedge \vec{x})) \wedge (i\vec{x} + j\vec{y} + k\vec{z})$ is

[burnt vessel]

bee [it/its]

[tardy ember]

(if you can't tell, i am kind of running out of variable names here)

[turbid zenith]

I don't really want to get TOO far into educational theory

[tight star]

this is true of literally everything

[turbid zenith]

But, in a nutshell, I'm looking for stuff that would get my students to the Action stage in APOS theory. What you're describing is more Process or Object stage.

[tight star]

@.@

i don't know what your acronym means

[turbid zenith]

In terms of students dealing with abstraction.

[tardy ember]

most of the terms vanish because they contain two copies of one of the basis vectors ($\vec x$, $\vec y$ or $\vec z$), and so you're left with just $ak(\vec x \wedge \vec y \wedge \vec z) + bi(\vec y \wedge \vec z \wedge \vec x) + cj(\vec z \wedge \vec x \wedge \vec y)$

[tight star]

but i don't think that, if you explained it well, is-does duality would be unapproachable

[burnt vessel]

bee [it/its]

[tight star]

certainly not something that would go over everyone's heads

[turbid zenith]

Maybe at some point I should explain APOS theory to people in here in case they might find it useful 😛 I ended up using it for my PhD dissertation, it's a really useful framework for how you fit math into people's brains

[tardy ember]

you can rearrange these by antisymmetry so this whole thing is actually just $(ak+bi+cj)(\vec x \wedge \vec y \wedge \vec z)$

and we want that to be $0$, so $ak + bi + cj = 0$ and there's the solution

[burnt vessel]

bee [it/its]

bee [it/its]

[tardy ember]

if you fix the weird choices of letters i think this is an equivalent answer to what you get out of your existing method

[turbid zenith]

APOS = Action, Process, Object, Schema.
Four mental constructions students make. This is horribly simplified for a single Discord message, but what can you do.

Example: Let's say we're talking about students' understanding about averages.

• A student at the Action stage can take the average of a given set of numbers, say by following a set of steps they learned.
• When a student has done that multiple times and reflects on those computations, they can interiorize the Action and reach the Process stage, where they can mentally imagine taking the average without having to actually be given specific numbers. They can reverse the process, coordinate it with other processes, etc.
• When a student can think of the result of the averaging Process as its own mathematical entity that can be transformed, they've encapsulated the Process and reached the Object stage. So, such a student would be able to answer a question like "If you were to add 10 to all the numbers in a data set, what would happen to the average?"
• As a student constructs various Actions, Processes, and Objects, they arrange them in a mental framework called a Schema, where they basically tie different concepts together. So a student's Schema may relate averages to standard deviations, or the mean value theorem, etc.

This is a framework that's used a lot in math education research to describe how students learn math

[tight star]

Ok so if I understand you correctly, the "action" part could be done by doing a specific example of an oriented area to area vector translation

[turbid zenith]

Yes

[tight star]

i agree that it'd be good to lead with this

[turbid zenith]

But the thing with APOS stages (at least the first three) is that generally you can't skip them

[tight star]

yeah the way i explained it was not the same way i'd teach it to a student

[turbid zenith]

Expecting students to function at the Object stage when they haven't even gotten to the Action stage is a recipe for disaster and memorizing

[tight star]

more about the way i'd explain the reasoning to a fellow educator

for a student i'd certainly start with a concrete example of an oriented area

you can even do some fun stuff with Cavalieri's principle to calculate the area of the corresponding prism!

[turbid zenith]

I did see this though

And I think that's what I was looking for

[tight star]

which provides a good motivation for the area vector

[tight star]

the idea is to chop up the prism into a bunch of cross-sections, and then "slide" the cross-sections over so they form a non-slanted prism

this has been around for thousands of years actually, so there's some fun history you can go into there too

anyway doing this shows that to calculate $\text{Vol}(S \times \vec v)$, it's enough to project $\vec v$ onto the line perpendicular to $S$, and multiply the length of the projection by the area of $S$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

i.e. $\text{Vol}(S \times \vec v) = \text{Area}(S) \times \text{signed length}(\text{proj}(\vec v))$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[turbid zenith]

Okay I worked this out

And it is exactly what I was looking for

Thank you

[tight star]

if $\vec n$ is the the normal to $S$, then this is $\text{Area}(S) \times (\vec n \cdot \vec v)$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

which is $\vec S \cdot \vec v$ for $\vec S = \text{Area}(S) \vec n$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[boreal agate]

oh you wanted the computation?

if so, i entirely misunderstood you lol

[tight star]

so you could do your "action" stage for an oriented circle and do the cavalieri's principle story to get the area vector

[boreal agate]

sorry

[turbid zenith]

Yes. I literally just wanted, if a student was going to solve this problem using wedge products or whatever, how would they actually do it

[boreal agate]

ohhhh

[turbid zenith]

I don't know what I could have said to make that more clear

[boreal agate]

i thought you were asking how might a student figure it out from 0

which is why i thought i couldnt answer it, seeing as i feel very comfortable with these concepts already

[tight star]

i think you could've just said "what are the steps a student would follow to solve this for this particular plane"

a concete application of bivectors

[turbid zenith]

I thought literally giving an example problem with numbers would have made that clear

[tight star]

idk the APOS stuff seems like unnecessary jargon

[boreal agate]

not to me lol

its definitely my bad though

[turbid zenith]

That was in response to you asking why my students would have trouble

But it's okay, now I've got what I was looking for

And the stuff you said would probably be useful later in the course

[tight star]

sure but you could've just said "my students would need to see a concrete example first"

[tight star]

.

[turbid zenith]

I feel like I said that like 10 times

Sorry if that wasn't clear

[tight star]

i don't think using an acronym i had no idea of helped, to be fair

[turbid zenith]

I even said "to solve this problem"

But okay, we're talking past each other at this point

At this point I have a better idea of how I might teach it

[tight star]

i will also try the is-does stuff and see how that goes

i'm a little more optimistic than you about students' ability to grasp it

[turbid zenith]

...okay

[tight star]

?

[tardy ember]

tbf i understood what you were looking for pretty quickly and was just stuck on figuring out how to actually do the computation, in part because i hadn't quite figured out why the method using a tangent vector actually worked

[tight star]

I was more exploring my own method than trying to figure out bivectors

[tardy ember]

this was what made it click for me, combined with the half-formed ideas i already had

once you've figured out that you can test if a vector is on the plane defined by a bivector by just taking their wedge product, you can just write everything in a basis (which was the part i had already sort of figured out) and then compute and it works

[boreal agate]

i really think youd enjoy taking a full look at differential forms

[tight star]

Idk I’m suspicious of them

[boreal agate]

the is-does duality is very prominent in them

[boreal agate]

how so?

[tight star]

People glaze them too much

[boreal agate]

hey - if theres a donut, you gotta glaze it

[tight star]

Like I have been very successful doing tensor calculus without differential forms all my life

[boreal agate]

they are really just that clean

[tight star]

So I get suspicious when people overhype them as the One True Way to do diffgeo

[tight star]

See I don’t believe this

[boreal agate]

in coordinates though, right?

[tight star]

Yes

Though Einstein notation is kind of “coordinate-agnostic”

Since you’re not making reference to a specific coordinate system

[boreal agate]

not when it comes to derivatives though

[tight star]

What are you talking about

[boreal agate]

once you want to start taking derivatives and integrals of your tensors, shit hits the fan

[tight star]

Again, I’ve done that my whole life and managed just fine

[boreal agate]

(ok not integrals of your tensors thats horribly phrased)

[tight star]

Index notation is very convenient for this stuff

In fact I’ve often found the coordinate-free versions to be more confusing

E.g. for navier-stokes

[boreal agate]

what i mean to say is that taking partial derivatives of the components is a coordinate-dependent operation

and it is far too easy to do it using index notation

[tight star]

So is taking coordinates of vectors

I don’t see your point

[boreal agate]

no no, taking partial derivatives of components is really bad, when you add back the basis vectors, you get different results

thats why we have connections

[tight star]

We do this all the time in fluid mechanics

It’s only when you have a nontrivial metric that you have to worry about this

[boreal agate]

youre secretly using a connection, you just always use coordinates where that connection trivializes

[tight star]

That’s just dumb

[boreal agate]

or some other type of connection :)

[boreal agate]

nuh uh

:3

[tight star]

I’m not secretly using anything

I’m just doing tensor calculus

[boreal agate]

one of the nice things about differential forms is that they use no additional structure beyond the differentiable (usually smooth) structure of whatever manifold youre working in

[tight star]

I don’t care

[boreal agate]

tensor calculus does include this though

[tight star]

I’ve never had to worry about connections in fluid dynamics

Nor in QFT

[boreal agate]

your loss; personally differential forms are super enlightening to me, and considering how you enjoy "is-does" duality id think you would enjoy them too if you gave them a chance

[boreal agate]

wait what?

[tight star]

The extent to which you’re glazing them has turned me off them even more

[boreal agate]

but theres a connection in yang mills though?

[tight star]

How much QFT and fluid dynamics have you actually done

[boreal agate]

again, thats just your loss. i tend to glaze anything i particularly enjoy, i do the same with cat theory (probably more)

[tight star]

Maybe you should try to work on that

[boreal agate]

fluid dynamics none, but i do know what the navier stokes equations are and i looked around at some fluid dynamics on wikipedia some time ago, so i recognize what youre currently talking about

[tight star]

sigh

[boreal agate]

i have learned a bit of QFT, but i really need to get my shit in order. i wanted to get through a textbook this summer but im not sure if that will end up happening. TQFT on the other hand...

[tight star]

I’m gonna keep doing tensor calculus the same way I always have

I don’t care if mathematicians view me as lesser or stupid for it

[boreal agate]

why? is it hurting anyone? why shouldnt i be obsessive about the topics that excite me?

[tight star]

Or if they get upset when I take partial derivatives

I could not care less about that

[boreal agate]

never said you shouldnt

[boreal agate]

they do not

[autumn tangle]

True quantified boolean formula? 😜

[boreal agate]

where is the last t

[autumn tangle]

shhhh

[tight star]

maybe consider the effect your words have on people

[autumn tangle]

it works out trust me

[tardy ember]

yeah that would be TQBF, not TQFT

True Quantified Formula Thing

[autumn tangle]

exactly!

So I was right 😛

[boreal agate]

pov: youre reading a pdf and you got to the point where the proof should be:

[autumn tangle]

Oh this one's even better because the statement is actually of decent length

[boreal agate]

im sorry if thats the way my words came across, i didnt mean so. i just wanted to "show where the connection is hiding" in the tensor calculus youre used to. that doesnt make anything you wrong, it just aims to show another perspective

[tight star]

I’m just too used to being looked down upon by diffgeo glazers for working in coordinates like a dirty physicist

[boreal agate]

coordinates are very helpful at times

and its very reasonable to work with coordinates

i just dont like to lmaoooo

[twin lichen]

This is not a good place for this convo

[boreal agate]

yeah this swerved off topic

[turbid zenith]

Pretty much yeah, and the algebra of working with the wedge is pretty easy to learn, and might even be a bit easier to teach. I’m still not 100% yet on deciding what it would be like to teach with this but I’m closer.

Though I guess cross product works the same way come to think of it if you split it into components and do the distributive property but everybody switches to the determinant method because it’s quickest I guess?

Though apparently not everybody learns matrices in high school anymore

So a number of my students hadn’t seen expansion by minors, and so they were like “why do you need the minus sign”

How do you even answer that question for someone who’s never done matrices? I have yet to come up with a standalone answer that I’m actually satisfied with. I think of it in terms of cyclic ordering of variables but that seems to beg the question.

[boreal agate]

im actually a supporter of defining the determinant in terms of the exterior algebra

you can easily derive all formulas from there

and the geometrical interpretation is instantly apparent

(for reference, said definition is:
part 1: outermorphisms
given a linear map f:V->W
its is possible to extend this map uniquely to a linear g:E(V)->E(W) where E(V) denotes the exterior algebra of V, such that g(A^B)=g(A)^g(B). because the outermorphism g extends f, i will also denote it by f.
part 2:
let V be an n-dim space.
then det(T) is the unique number such that T(w)=det(T)*w for any n blade w

existence is promised due to the space of n forms being 1d

[turbid zenith]

So, this works.

It's not as fast as finding the cross product using the determinant, but maybe the wedge product could be used to develop that, I dunno.

It's very consistent, but that's a lot of places to go wrong.

Also if you did the same thing with the cross product defined in terms of x^, y^, and z^, you'd have almost all the same steps.

And yes I forgot a 3 in my transcription at one point.

So I wonder if there's a way to clean things up?

[boreal agate]

yup, the computations line up pretty much exactly

thats because ultimately the cross product in 3d is defined as the hodge dusl of the wedge of a pair of vectors :)

[boreal agate]

well you did write the computation out very verbosely

its possible to reduce the size of it (hence making it seem cleaner) by taking shortcuts, like never writing x^x in the first place, and reflecting y^x immediately to -x^y

another shortcut that can be used is recognizing that in 3d, the single coefficent of u^v^w is the determinant of the matrix with columns u,v and w

other than taking shortcuts however, i dont think it gets much cleaner

[turbid zenith]

Hmm I see

How about when doing a surface integral? The factor of $\Vert\mathbf{r}'_u\times\mathbf{r}'_v\Vert$?

[burnt vessel]

Solid Angles

[turbid zenith]

Clearly that's conceptually a bivector

But for a student actually doing an integral with it, it feels like it would be quicker to just use the cross product "determinant"

[boreal agate]

it isnt actually! in many casez including this one, the cross product is directly replaceable by the wedge, so youd just have |ru^rv| in this case too. th eky thing to notice is that the norm removes orientation, so it cannot return a bivector in the usual settings.

computationally, we dont gain anything in 3d from the wedge

ultimately |ru^rv|=|ru×rv|

[turbid zenith]

So instead of $\iint_S f(x,y,z),\mathrm dS=\iint_D f(\mathbf r(u,v))\Vert \mathbf r'_u\times\mathbf r'_v\Vert,\mathrm dA$, it would just be $\iint_S f(x,y,z),\mathrm dS=\iint_D f(\mathbf r(u,v))\Vert \mathbf r'_u\wedge\mathbf r'_v\Vert,\mathrm dA$?

[burnt vessel]

Solid Angles

[boreal agate]

however what the wedge doesnt improve in terms of computation, it improves in terms of clarity. |ru^rv| works in 2d and nd in general for surface integrals, no modification needed, because it directly works with the concept of an area element rather than resorting to a vector

[turbid zenith]

And they way they'd compute that last bit would probably just be the same?

[boreal agate]

exactly!

[turbid zenith]

Sure, I mean like I said I like the idea of a wedge as an oriented area

That makes lots of conceptual sense

So at least in this case the computation isn't worse, and the concept is more clear

[boreal agate]

yup

[turbid zenith]

I'm working through all my use cases to see what it would take to actually convert to using wedges instead of cross products, rather than just using wedges as an extra enrichment.

[boreal agate]

also in terms of concepts that get explained far better, the idea of a pseudovector gets removed completely!

[turbid zenith]

To be fair, "pseudovector" isn't mentioned in most calculus texts

That seems to be a physics thing

[boreal agate]

thats reasonable

[turbid zenith]

So, surface area elements seem to be pretty much the same, and basic plane stuff isn't TOO bad though there are probably shortcuts

The last big thing is probably curl

[boreal agate]

it is something you need to take into account when changing a basis though

[turbid zenith]

Which has always annoyed me

[boreal agate]

it has annoyed me too lol

[turbid zenith]

Because strictly there's no 2D curl, even though it's easy to see what it should be

[boreal agate]

and it does indeed get replaced by wedging with nabla to get a bivector (and if we already have a bivector, wedging with nabla gives us its "divergence"!)

[turbid zenith]

It seems like the curl could just be defined as $\nabla\wedge\mathbf F$

[burnt vessel]

Solid Angles

[boreal agate]

and it is indeed!

[turbid zenith]

And the divergence is still $\nabla\cdot\mathbf F$.

[burnt vessel]

Solid Angles

[boreal agate]

for a vector field F, yes

[turbid zenith]

But those should work in both 2D and 3D.

[boreal agate]

they do!

and 4d

and 1d

[turbid zenith]

Well, vector fields are where we tend to use divergence and curl so yeah 😛

At least for anything I teach.

[boreal agate]

mhm

here the curl does get a nicer geometric interpretation too imo

we can notice that nabla=
"1/V*(dydz,dzdx,dxdy)"

hmm wait im not saying what i want to be saying

curl is freaky

yeah its more nuanced than that

i might expand later

[turbid zenith]

Thiw would be what I want to make sure I can express.

That's supposed to be in 2D, so strictly curl F is supposed to be just the z component of the curl when brought into 3D. That never sat right with me.

So when I taught Calc IV, I told my students that N'_x - M'_y essentially was the "2D curl".

Because then the Kelvin-Stokes Theorem and Divergence Theorem are just instant generalizations to 3D.

So it seems that in this case, using the wedge to define curl really would make that work?

[tight star]

You do sometimes want the oriented area element dS and not just its magnitude

[boreal agate]

yup, although you do need to end up with a scalar to integrate

[halcyon glade]

I think it's the same calculations, no? You'll notice calculating u wedge v the way you did is the same as calculating the cross product via expansion by minors

So I think it's just a question of what you're used to, IMO the bivector approach is more conceptually clear, but I can see why people teach the cross product so that they don't need to talk about any additional topics

Ah I see necro already responded, my bad

[tight star]

i guess i just fundamentally don't understand what makes bivectors conceptually clearer

[halcyon glade]

Yeah I would say the commonality is that in both 2D and 3D, you're taking the exterior derivative of a 1-form, and it's only really proper to talk about curl when you're in 3D. I think you already know this, but the issue is in the correspondence between a 2-form and a vector. In 3D, you can do some conversions (in more detail, the Hodge star and musical isomorphism, but terminology isn't important here) to turn a 2-form to a vector field, but in 2D, those same conversions give you a scalar field. So it's more direct in my view to just think about the 2-form. Admittedly this is a decent amount of overhead just to make things conceptually more tidy though (but you could replace the conceptual overhead of teaching the cross product with this?).

[halcyon glade]

If you're only ever working in 3D, it doesn't really matter because every bivector uniquely corresponds to a vector (although there's some complication here because you really have to think of it as a pseudovector or else you'll transform it incorrectly), but if for example you want to work in 4D (as is the case for relativity), this perspective completely breaks down.

[tight star]

Ok so suppose I just want to work in 3D

Why bivecrors

[halcyon glade]

There's a unique correspondence so you can either think in terms of pseudovectors or bivectors, there won't be a difference for calculations. Although there is the benefit that you never have to explain why pseudovectors are different from vectors (you can just manipulate the bivectors directly).

[tight star]

.

[halcyon glade]

...? I'm trying to answer your question

Also though some students might only ever work in 3D but some might work in other dimensions later on, and the bivector approach works equally for both students, whereas a student who learned vector calculus via cross products and wants to learn something further not in 3D will have to relearn the subject anyways once they take more classes, so there's something to be said for that too.

[tardy ember]

i mean i think the fact that the cross product only works in 3D is an element of what makes it conceptually weird even if you do only need to work in 3D

[halcyon glade]

Perhaps another benefit is that to visualize a bivector, you don't need to go through the intermediate mental step of dualizing a vector into a plane, but that one I'm less convinced about

[tardy ember]

like the fact that the cross product has this odd definition directly on the basis vectors that isn't even properly coordinate-invariant because it's orientation-dependent and so you have to remember that some things are "pseudovectors" but you still pretend that they're vectors

the cross product is the composition of the wedge product that produces a bivector and has none of this pathological behaviour, and a fairly arbitrary bijection between 3D bivectors and 3D vectors that's responsible for these oddities

[halcyon glade]

Yeah I think the pseudovector part is the weirdest quirk, you don't need to discuss any such thing with bivectors

[boreal agate]

i wouldnt say arbitrary, but definitely unnecessary

[tight star]

i'm not sure i quite buy this

[boreal agate]

ultimately you can get by quite well in 1d 2d and 3d without using bivectors when what you care about is a situation where a metric is present (even present in the sense that it is defined by a coordinate chart) due to how low dimensional the spaces are, they are quite tame and you can relatively easily separate anything youd do with the exterior algebra into cases according to degree, and reduce to scalars, vectors and covectors. there is nothing wrong with this, and as a matter of fact our ability to do this highlights important geometric properties of these low dimensional spaces, which can often be used to solve difficult problems or to prove powerful theorems specifically for these low dimensional spaces. also, these "coincidences" that happen in low dimensions arent "non geometrical" or necessarily conceptually unclear, as they often highlight exactly those geometrical properties that hold in these spaces and produce this important and detailed geometrical intuition for these properties specifically, that other approaches might not be able to replicate as easily.

using the exterior algebra really shines when you either want something more general than 1d 2d and 3d space with a metric, or you want to specifically focus on tge kind of things you can say without referring to additional structure such as a metric or a chosen basis

which might not be your goal, and thats fine

ultimately i find that using the exterior algebra helps me understand where certain properties are present and what defines them, and this is especially true after one reformulates traditional approaches such as the cross product through the exterior algebra, as this (objectively) produces a more complete understanding of what the operations are doing and how each structure of our space is used

and to me it feels more natural due to how everything becomes "straightforward" - you no longer use special isomorphisms and and low dimensional coincidences to describe these objects (pseudovectors, pseudoscalars), you just directly describe them using an algebraic structure that exactly encodes their natural properties and nothing more, and as such it also generalizes easier to more complicated cases

[tardy ember]

it's the same sort of thing as how choice-free constructions are often meaningfully nicer than constructions that use choice heavily, even if you only care about results with choice, because they're usually rather explicit and that's useful

working in arbitrary dimensions, even if it's not a property you care about intrinsically (because you only plan to work in 3D), is (at least often) indicative of a certain kind of underlying simplicity that makes the concept easier to work with

[turbid zenith]

Okay just waking up now… for me the reason I was thinking of using bivectors seemed to be because lots of stuff using the cross product seems to really want to be talking about area or oriented area, and filtering through the normal always seems like an intermediate step, especially in 2D when we have to go up a dimension and then get rid of it.

See this picture of either Green’s or Kelvin-Stokes’ Theorem.

It feels to me like telling somebody it’s not your unbirthday, instead of just saying it’s your birthday.

Weird analogy but it’s the first one I thought of. 😛

[warm valley]

If you find it helpful, here are some old notes a prof at u of a used to teach basic differential forms alongside calc 3/4, with examples of the kind it seems like you're looking for (looking again, not as many direct computations as I thought unfortunately but still some)
https://sites.ualberta.ca/~vbouchar/MATH315/section_hodge.html

I've used these to supplement teaching students differential forms (albeit with prior multivariable and vector calculus exposure)

[boreal agate]

cries in LEM

[tawny slate]

i just want to say that this is an awful attitude to have, especially in a pedagogy channel

you can glaze whatever you want, but the purpose of this channel is to figure out how to best explain it so others understand

[turbid zenith]

I’m not sure that’s really what they’re saying

I don’t think it’s not caring at all

But I do agree more attention should be paid here to whether this will fit better in the heads of novices

[boreal agate]

i do care, my tendency to glaze a concept and my ability to explain it are independent

[tawny slate]

retracted part of my statement

[boreal agate]

i do not wish to shove concepts down the throats of novices

[turbid zenith]

Fitting concepts into the heads of novices is literally my job

[boreal agate]

i do think including differential forms in the curriculum would be great, seeing as actually doing computations with differential forms requires nearly no theory and is quite easy to catch on to

(and differential forms are awesome and relevant)

[turbid zenith]

Last time I taught I did differential forms on the very last day to tie things together across everything we’d don in all four classes

[warm valley]

I agree, though I am somewhat of the opinion that there's something lacking in terms of materials to do it easily for undergrads. I think there's a reason that most people's first exposures to forms are infamously obtuse

[turbid zenith]

But I’m not sure doing them from the very start would be effective in terms of abstraction

[boreal agate]

thats a rough treatment, but it makes sense when time is the barrier

[turbid zenith]

Too abstract too quickly is a bad idea, and it’s a mistake I’ve made before

[boreal agate]

how so?

[boreal agate]

not the very start, 2 semesters in at least

[warm valley]

I think the full force of exterior algebra linear algebra is too much, but pulling formulas out of nowhere (namely, how they relate to the determinant is the hardest part IME) isn't great either

[turbid zenith]

I mean even at the beginning of vector calculus, like when I’m first doing divergence and curl

[warm valley]

I think Bouchard's notes above are really good though. Something to keep in mind though, when doing differential geometry, is that I still regularly use the cross product (on top of forms formalism, for what it's worth I think we should teach the Hodge star way faster in differential geometry)

[boreal agate]

how the determinant is related to the wedge product is something is see as relatively clear actually

[turbid zenith]

I can explain why those capture the physical essence of what we want to study without using differential forms

[boreal agate]

as long as you keep the oriented volume picture in your head, it ends up being pretty immediate

[turbid zenith]

This is useful to know

[warm valley]

I think I agree geometrically, I really mean proving it

[turbid zenith]

I have seen those notes, they first helped me make sense of differential forms

[boreal agate]

as far as im concerned, a valid way of doing determinants is starting from the exterior algebra

its definitely the quickest algebraic way of reaching the formula for the determinant

and it is quite intuitive, barring the mindless algebra

[warm valley]

I agree (and it certainly is valid), I think the hard part is relating it to the usual computational monstrosity students see in first year, without going somewhere into the full force of the exterior algebra (especially if you want to do it "right" and treat it as a quotient)

[turbid zenith]

I will point out that when I first learned the determinant there was NO mention of area or volume scale factors, and I think that’s terrible

Most precalculus textbooks say it’s “a number associated with a matrix”

[boreal agate]

i dont think so

[warm valley]

At least in my experience, I despise how the determinant was treated at my university's first year course, the students I've taught really struggled with this part

[boreal agate]

its a very clear process imo, and it seemed clear when i first learned it too

[warm valley]

like it was geometrically convincing, but when I gave the standard arguments people got extremely lost

[boreal agate]

yeah giving geometric "proofs" for the formulas is pretty difficult

[turbid zenith]

I would love if the determinant of a matrix was defined as the oriented area or volume of the blade defined by its columns

[boreal agate]

although i do know of a beautiful geometric proof of the laplace expansion

[warm valley]

like, to all the students I've taught, their definition of the determinant is exclusively cofactor expansion

[boreal agate]

ah but this is actually not quite it!

turns out there are 2 types of a "determinant"

one is the determinant of a linear transformation and the other is a volume form

[warm valley]

Like I guess I'm saying that I think this is worthwhile and I need to see where I can improve, but that I think you're really overestimating the linear algebra fluency at the level I'd like to aim for

[boreal agate]

thats completely fair, i mainly have in mind students post linear algebra 101

[turbid zenith]

Sure but if you think of the columns of a transformation matrix as where the basis vectors go it’s essentially the same thing

[warm valley]

The cross product has a similar "pulled out of nowhere" feel to it to me, but because it's been seen since high school physics, it's not a big deal to them

[turbid zenith]

At least to my understanding

[boreal agate]

kind of, except one is fully a volume while the other is a ratio of volumes!

personally this distinction evaded me for a long time

and it is important imo

[turbid zenith]

I’m not sure I see the usefulness in such a distinction especially for a beginning student

Both can kind of be taught at once

Though for students who haven’t had linear algebra, the volume one is more accessible

And a number of students in my class hadn’t had linear algebra yet

[boreal agate]

for a complete begginer it probably isnt too useful, but for an intermediate student this distinction really clears up how we get a determinant even in the absence of a meaningful notion of volume

personally when i saw det[g] in a bunch of expressions in general relativity i was freaked out, cuz g is NOT a map from some space to itself

[turbid zenith]

Then let that distinction come up when it first makes sense. I can see linear algebra being a good time for it.

[boreal agate]

i 100% agree

[turbid zenith]

Start with basic and then generalize when the context calls for it.

[boreal agate]

but i also think linear algebra should be taught earlier too

[turbid zenith]

At my school it’s a second year course but we have different students going in different orders because of various scheduling stuff

So we don’t have it as a prerequisite for vector calculus

[boreal agate]

i see

[turbid zenith]

Lots of “it would be so nice if students learned A before B!” gets stopped short by the reality of living breathing students 😛

The only prerequisite we have is multivariable.

[boreal agate]

learning multivariable before linear algebra is insane to me

[turbid zenith]

I learned linear algebra before multivariable and I did NOT understand why

For reference, at Georgia Tech (my undergrad institution), at the time you had:

• Calculus I: All of differential calculus and integral calculus
• Calculus II: L'Hôpital's Rule, series, and a whole semester of linear algebra
• Calculus III: Multivariable and vector calculus

So I finished Calculus II knowing how to find the eigenvalues of a matrix

And not having the slightest idea what the eigenvalues of a matrix are

[boreal agate]

lol

yeah thats a general issue with math pedagogy

its too easy to miss the motivation

what i find is that a lot of the time a historical exposition works really well to combat this issue

"what did the people who invented this lunacy want to achieve?"

[lyric ember]

quick question

would it be fine to gloss over defining some terms when it would be obvious what you meant?

[turbid zenith]

I think that highly depends on the situation 😛

[lyric ember]

for context, I'm writing notes on pointset topology and I just defined metric topology. Later in the notes, I wanted to clarify that not all topologies come from a metric

and I wanted to use the terms "topology generated by a metric"

[boreal agate]

and what do you wish to gloss over?

[lyric ember]

"topology generate by a metric"

[boreal agate]

ah

[lyric ember]

I think it's obvious since I already defined what a "metric topology" was

[boreal agate]

if you defined both metric topology and the general notion of topology this is reasonable

[vocal phoenix]

Funnily enough some languages do have double negatives which don't cancel out to a positive (although it's not a double negation or LEM thing, it's just that a sentence in the negative will have multiple "negatory" words); for example "nobody was there" in Polish would be "nikogo tam nie było", which would literally translate to "nobody wasn't there"

[boreal agate]

yeah i know this phenomenon lol

[lyric ember]

a similar thing happens in french

"the best thing I've ever seen" would be "la meilleur chose que j'ai jamais vu" which would literally be "the best thing I've never seen"

(side note, I've seen even french people confused about why it is that way LMFAO)

[quasi maple]

Technically, "jamais" should mean "ever"

As in, je n'ai jamais fait une telle chose" - I not have ever done a such thing" = "I have never done such a thing"

An actual example of "negative with no apparent negative meaning" is when you use "avant que" (before...)

e.g. "avant qu'il ne soit trop tard" - "before that it not be too late" = "before it's too late"

[quasi maple]

The reason for people reanalysing "jamais" to mean "never" comes from dropping the "ne" (cf. with pas and rien)

Je ne lui ai jamais dit la vérité entière ("I never told her the whole truth") thus gets rendered in speech for instance as J'lui ai jamais dit la vérité entière

[lyric ember]

I disagree.

take for example "une chose jamais réalisée"

or the expression "jamais de la vie!" or just "jamais!"

[quasi maple]

Those are consequences of leaving ne behind

[swift hatch]

doesn't mean it doesn't mean never

[lyric ember]

well, let me ask, does "rien" mean something in that case?

[quasi maple]

"une chose (qui n'a) jamais (été) réalisée

[quasi maple]

Not anymore, because of how often it got used in the negative

[swift hatch]

why not réalisée as an adjective

[quasi maple]

"rien" in old French means a "thing"

[lyric ember]

well, that's old French, in modern french it's a different story

[quasi maple]

Well, it is, it's a past participle

[quasi maple]

But this is exactly the point I'm making

These aren't two different languages; one evolved from the other

"jamais" still has some uses otherwise seen as archaic, which is why "que j'ai jamais vu" sounds weird

[swift hatch]

not the same thing though, because you can't just throw in words and claim they're being dropped

[quasi maple]

"rien" is one step further, where it NEVER means "something/anything" anymore, it got paired up in negative constructions so frequently

[lyric ember]

I mean, exactly, it evolved from it, therefore it's different

you can't expect things that worked in old french to still work in modern french

but ig I get your point

[swift hatch]

for what it's worth wiktionary just lists both 'never' and 'ever' as meanings for jamais

[quasi maple]

But that's... what I'm talking about

[quasi maple]

Number one I did say "technically ... should mean"

Not that it exclusively means

[swift hatch]

I would not interpret 'should mean' as non-exclusive

[quasi maple]

That'd be "must mean"

[swift hatch]

'should [also] mean'

I interpret 'should mean' as synonymous with 'ought to mean' but not 'must mean'

and all three implying exclusivity

anyway #math-pedagogy

[turbid zenith]

"PROCEDURE" IS NOT A DIRTY WORD

... thank you, I needed to get that out

We now return to your regularly scheduled math chat

[zinc dove]

What 😂

[turbid zenith]

. . . I've just gotten kinda sick of the idea that procedures and algorithms are the devil

And students should use ✨STRATEGIES✨ instead

[quasi musk]

Step 1, strategize the quadratic formula

step 2, prove the quadratic formula

step 3, solve

[rapid tusk]

strategize me some problem solving methods and then conceptualize apply blah blah blah

[tawny slate]

procedures and algorithms are very limited as there are uncomputable problems, thats why you must use creative strategies

[crystal vigil]

arguably there are countably many problems

[cosmic ibex]

I think the idea is that teaching only procedures misses much of the math that needs to be taught.
It doesn't mean that procedures shouldn't be taught at all.

One can and should do both.

[turbid zenith]

I agree. But I see things like this:

https://x.com/pwharris/status/1953924423834014012?s=61&t=xQsvdabii3pPOKmYR937jQ

Pam Harris (@pwharris)

Is it truly efficient to add things like 998 + 267? 4996 + 3529? 4.95 + 2.67?
Algorithms are terrible teaching tools because they can trap students into using less sophisticated reasoning than the problems are intended to develop.

#MathIsFigureOutAble #MTBoS #ITeachMath #MathEd

X · Aug 8, 2025, 9:00 PM

At some point the whole "here's an addition problem with one of the numbers nudged just below a really nice number with lots of zeros" gets tired

[vocal phoenix]

One of my most enduring pedagogical memories is when (in a calculus class in first year) the student got to the point where they had to solve the equation x^2 = 16, and proceeded to rewrite it as x^2 - 16 = 0, calculate the discriminant and so on.

But I do agree that discarding "algorithms", "procedures" and general rote exercises altogether is not the right approach.

[turbid zenith]

I've seen it recommended to write it as x^2 - 16 = 0 and factor

To keep from losing solutions

[tawny slate]

i know we are going a bit off the topic of the algorithms thing, but for that particular problem i also emphasize that you can just square root both sides as long as you are careful not to forget the plus/minus, but it is riskier

i say that the sanity check you should aggressively apply is the fundamental theorem of algebra: degree 2? 2 solutions. you dont have 2 solutions? check the hell out of your answer

[turbid zenith]

Good point

[quasi maple]

I tend to do this if I'm also the one teaching what a square-root is

Because that way I can at least ensure students don't just go "root 4? Okay that's just plus-or-minus 2"

[turbid zenith]

At least you can try to ensure it

Some students will latch onto it anyway 😛

[tawny slate]

i dont take any chances, i teach what a square root is on the spot

[quasi maple]

I can, but I'm just a tutor, so I get few contact hours compared to school teachers

[tawny slate]

but i feel like its only a small 5 min segue so i think its worth it

[quasi maple]

So I'd do this first if they made plus-minus mistakes like that

[turbid zenith]

It’s kind of amazing how quickly students latch onto anti-rules

[quasi maple]

Like, if they're fine on square roots and I can see that in their work, I don't bother

[turbid zenith]

Which is what I might start calling these, in the spirit of programming antipatterns

[tawny slate]

yeah, but the fortunate thing is early on, they usually just latch onto them without fully realizing why, they dont have a full foundation and just go by intuition

so if you can give a framing in which the correct explanation makes more intuitive sense, its easy to fix their thinking patterns

in this sense, i like "pretty" things more than rigorous things

theyre much more useful pedagogically

which seems kinda obvious when you think about it but w/e

[quasi maple]

I shitpost even during my tutoring

Peak memorisation tool

[honestly tho depending on what it is I'm teaching, it's profoundly effective ]

[quasi musk]

I don't think of this as a one way or the other

I like the function way of thinking, inverting; but the algebraic way is very good to get them to practice factoring

[midnight scarab]

I have 0 exp teaching at this level, but I imagine it's important to show that one can approach things from different perspectives

Both in math and in life, I suppose

[tawny slate]

well if we're talking about teaching what a square root is specifically, yeah I would obviously use an inverting squaring way of motivating it, but the key thing to emphasize and get into their heads is the convention that square roots are only ever positive

you want to students to understand that sqrt(4) is never -2, in most conventions

the way i do that is really simple, i simply ask them to compute sqrt(4)+sqrt(1)

and show that there are actually 4 possible different distinct values if you allow negatives, which makes it really unhelpful

[glossy tundra]

well also i wouldnt wanna box them in into thinking they can NEVER be negative because in some exotic number systems they are and you could just define your beginning semantics such that sqrt(z)<0 what i would try to do is get them to prove the theorem over the real numbers with some axioms and lemmas you give them. not like a full formal proof but a justification so they can really get more of a sense for actually doing math than just inhaling facts

[tawny slate]

that's the point of emphasizing it is a convention

as conventions are made up

[polar basalt]

What’s your go to method for getting the attention of the class?

Also do y’all prefer chalk or whiteboards?

[dapper flume]

Whiteboards have less dust and are a bit more visible from afar ☺️

Getting attention depends a lot on factors like age-group and class size. In general though, the rule of thumb is to practice a consistent routine right from day 1 where a certain cue (or small set of cues) that you use are signals for attention. Some people like using certain phrases ("one two three, eyes on me," or "raise your hand if you can hear my voice.") Some like a friendly audio cue (ringing a bell or clapping in a call-and-response rhythm). Some prefer a strictly visual cue (hands on heads, finger on nose, moving to a certain location in the room).

Whatever you choose doesn't really matter, so choose something natural and respectful. But it has to be a well-communicated and well-practiced routine to work consistently.

Also, it helps to make your class interesting. People pay attention to interesting things

[astral agate]

Hello there I need some advice

I feel demoralized because of feeling that I won’t be able to teach properly my student. I have a speech problem and I find it really hard to explain things in general too. I was feeling very enthusiastic about teaching this kid, but I felt like such a failure after realizing that he didn’t understand me very well on the first class we had. I’m afraid I won’t be able to teach him a single thing properly because of this.

Does anyone knows a way to improve this?

Also, how much does a kid need to be shown multiplication tables individually? Since he’s familiarized with multiplication I was thinking it would be more to productive to practice some simple exercises.

But I don’t know how much someone he’s age would benefit from learning that way. At least, as a math major student, I find it to be very productive to get to practice as soon as possible.

[dapper flume]

Your path to improvement is dependent most significantly on the data your assessment strategies. How did you determine that this teaching session did not go well? Was there an objective not reached? Can you identify what the child did and did not understand?

[austere delta]

Where I'm from
clap--clap-clapclapclap always gets the respond clap-clap followed by attention.

Drilled into people in elementary school.

Prefer chalkboards, whiteboard are too unpredictable.

[boreal agate]

????
wdym whiteboards are unpredictable????

[austere delta]

Pens run out with no warning.

And sometimes they're harder to clean than other times

If you walk over to a chalk board, you immediately see if there is chalk, sponge and squeece there.

And you can see how long it will last

Pens also fade slowly giving worse visibility over time, so it's a decision when to switch to another one.

So yeah, white boards are very unreliable

[boreal agate]

idk, except for the cleaning part i dont see these as issues

[austere delta]

You've never had white board a pen run out?

My experience with white boards is there always being a pile of half way useless pens you have to search through. Then when you find a good one it starts failing on you mid way through

[tight star]

Yeah I much prefer chalkboards for this

To be fair with chalkboards you do get a pile of tiny chalk pieces you have to search through

But it is easier to tell when you actually have chalk

(Also more satisfying imo)

[austere delta]

I mean you don't have to search through them. I can see if a piece of chalk is short or not

[boreal agate]

pens produce less dust, and they last a reasonable amount of time. if one runs out, just get more. they dont run out suddenly either as you claim, its very noticeable when a pen is in the last quarter of its life

[boreal agate]

sure, but there is usually another pen somewhere

[austere delta]

Maybe you just have a more organized custodian service at your institution

[boreal agate]

thats true to some extent, although its kind of fun to steal pens from adjacent classrooms :3

[austere delta]

Are you using dry sponges?

I haven't really had any problem with dust from chalk

[boreal agate]

we dont have any chalkboards (maybe some in the older halls but i dont enter those). i am mainly speaking from how i perceive chalk

maybe its not that much dust

i wouldnt know

[austere delta]

Like if you use dry sponges I would understand.

But wet sponge + squeegee cleans the board so much better anyway. You just need access to water

[tawny slate]

i used whiteboards for several years and was extremely annoyed by the pens "running out"

once we opened a new box of markers and none of them worked at all

turns out the reason for this is because someone kept storing the markers standing up, which causes them to dry up

[swift hatch]

the correct answer is the boards in the big oxford mathematics institute lecture halls

[tawny slate]

on top of that, when markers don't quite work, in order to squeeze that extra bit of juice out of them, they make a squeaky sound

and oh my god that squeaky sound is my bane, i would rather use my own blood than have to hear that sound

[queen tartan]

The solution to this is to carry around an insane amount of whiteboard markers

[quasi maple]

Oh they're still shit believe me

They're not boards; they're just stretched laminated sheets of plastic

[gray smelt]

chalk feels better to get on hands than marker. cuz it's just calcium carbonate

[lethal leaf]

Yea when I've had to teach on whiteboards I brought my own markers

https://a.co/d/6gqOfNS

These are fantastic

[spark lantern]

Has anyone tested how good is Ai at validating undergrad math proofs?

[rapid tusk]

really bad.

!nogpt

[coral copper]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[spark lantern]

Why?

[tawny slate]

because the way LLMs work is that they are using statistics to predict the character or word that should come next, they are not actually reasoning or observing or computing anything in the content of their outputs

they were trained in a way that optimized to sound human rather than to be able to produce correct/accurate information (this is a very loose comment with a thousand fine print bullets but im not going to cover these here)

they have been very unreliable across the board since they were a thing, and while improvements and advancements have been made, it doesn't change the underlying fact that they are still horrendously bad at maintaining context, computation, and using correct reasoning or applying valid theorems

[spark lantern]

I know that but then how did they get a fold at the imo?

Gold

[tawny slate]

they didn't

im saying its a misleading statement/lie

[tawny slate]

https://mathstodon.xyz/@tao/114881419368778558

terence tao's comments on the claim

and again, the problem with LLMs is that they cannot formally verify their proofs. if they get the right answer, you need to have a human manually check

so either way you cant use them to check anything, because you still have to check it, on top of it only being able to solve a very small select range of problems

#math-discussion message youve got like 4 people all telling you the same thing

[spark lantern]

I asked a phd from Stanford they seemed to think for undergrad it should be fine, he was impressed by some of them. But i dont really follow about what you mean by megaprompting? I read Tao's post he only said there is no proof they didn't cheat not that they cheated

[tawny slate]

i think that phd is wrong, as an educator i find using it to validate proofs is both problematic and leads to bad habits and stems from a poor understanding of what LLMs are

the point isnt that they "cheated", the point is that its misleading about what the capabilities and use cases of the LLM are

if you really needed to verify proofs, either learn to read proofs yourself (which should be the point of studying math?) or learn something like coq or lean and formalize it

[turbid zenith]

AI can occasionally be good at generating plausible but wrong proofs for students to analyze

[spark lantern]

But then how did they likely do so well on the imo?

https://epoch.ai/blog/grok-4-math#grok-4-is-near-the-frontier-of-solving-proof-based-problems-but-much-headroom-remains

My experience is it generally seems to think my proofs are ok but keeps saying i need to make it more rigorous

[tawny slate]

you keep saying how how how and we are telling you it is not necessarily relevant nor true

read

this company has a track record of lying through its teeth about anything and everything and we know this is not how LLMs have ever worked

and all you ever respond with is "but how" ad infinitum

it doesn't, if you know how LLMs work, it can't

i dont have time to go over just how deceitful tech companies are when it comes to this kind of stuff

they stand to gain billions for tricking the public into thinking AI does more than it actually does, in particular LLMs, at the cost of public safety and integrity

i dont know how many times it needs to be repeated that LLMs cannot be used to verify proofs, period

they are a natural language model, not a specialized proof verifier

[quasi maple]

A lot of people in the help channels don't seem to get that either

[spiral elbow]

Also note that validating a proof is quite different from coming up with a proof. For the IMO I think they used a best-of-n strategy, where they generated a bunch of proofs for the same problem, and picked the best one. How do you use the same strategy for validation?

[quasi maple]

"bro you can't just use ChatGPT to check this for you"
"why not, is smort"
"because it's not a fokken calculator nor a mathematician, it's a mimicker of written language at best ffs"

[tawny slate]

this is also why tech companies want you to think its better than it actually is, its not only easy to fool you with it, its easy to lie about its performance, and people want to dump truly massive amounts of money into it

[spark lantern]

Wait but the link i posted to are phd mathematicians with benchmarks where they validate how well they do. You are then saying these mathematicians are crazy or what?

[tawny slate]

idk where you're getting phd mathematicians from, but phd mathematician does not mean they are experts in computation, machine learning, sociology, or educational pedagogy. i know phd mathematicians who are trump supporters who literally believe in miracles as divine interventions performed by the christian god. you have an entire community of math people and people in the specifically pedagogy channel who all tell you the same thing: that relying on LLMs as proof-checkers is ill-advised, to say the least

additionally, if you doubt the already fat section i pulled out that explicitly warned you about the representation of the results, here's another article written by the same person who says these results are not remarkable: https://epoch.ai/gradient-updates/we-didnt-learn-much-from-the-imo

[tawny slate]

additionally, none of these refute the core issue, which someone beautifully and succinctly wrote out here, which kills the notion that LLMs are a proof-checker. did you understand what was written here?

i know that you really really want this to be a convenient tool for whatever reason you may have, whether it is a sense of pride of our technological achievements or you just want comfort in knowing that there's an easy way to confirm your work or assist your learning, but unfortuntately it is not that, and there are existing actual tools that do the thing you want

[halcyon glade]

I'm also not sure what this has to do with #math-pedagogy

I mean you're not using the AI to grade for you, are you?

[spark lantern]

I read that article before, I don't believe it says anywhere there that llm's can't prove theorems just that there wasn't a way to detect any improvement from their 2024 imo to 2025 imo performance. So just because it didn't improve how does that say it can't do math with solving 4/6 imo problems? I haven't seen it questioning the truth of the claims that the problems were actually solved by Ai. Theres only 1 reason I am interested in this. I am a beginner who's 39 years old and first fell in love with math when I was in 7th grade doing some math competition. I regrettably never majored in math and right now having a 2 year old, a wife with health issues etc its unrealistic for me to go and major in math so I wanted to try to slowly learn real proof based math but need someone to check my proofs so I can fix my gaps

[halcyon glade]

there's lots of knowledgeable people in this server willing to check over people's proofs and give advice if you post in the right channels

[spark lantern]

Even if I do like full problem sets on a semi regular basis they can check all my proofs?

[tawny slate]

chances are getting feedback on a couple select proofs will help you figure out how to work on the others

you probably won't need every single proof checked, but you're always welcome to just ask as many questions as you like

it's a pretty big server

[spark lantern]

How do I know if the people answering know what they ate talking about if I dont see their background?

Are many math phds?

[tawny slate]

i mean, it depends on like what you're studying, but help here is generally pretty reliable? right now there's too little information to go off of, i think you should just start doing the actual studying and then come here as needed

either way, using LLMs here poses a huge risk, moreso than if you come here. without a doubt the responses here, while certainly not perfect, are going to be of higher quality and more reliable

there's also #study-discussion if you want to find people who will study with you

[turbid zenith]

We're about to have an "Academic Affairs Retreat" tomorrow before the semester starts, and we've been asked to read articles / watch a TED video about AI in education

So ... it's a big topic of talk at my uni

[autumn tangle]

LLM's are pretty good at spotting errors in undergraduate level proofs and I think they can help there.

Of course, it should go without saying that there is always a (rather high) chance that their output is complete BS, so be ready to verify their claims if it's important info.

[vagrant meadow]

i started tutoring again after a long time (i stopped in late 2023 when i went to grad school) so like it's been a huge adjustment in terms of how many students now just regularly have AI open in like another browser window. particularly for basic lower div stuff, it is pretty reliable. i did spend like a full minute trying to figure out what was wrong with someone's answer today only to realize that they had written (oo,-5/4] instead of (-oo,-5/4]. i expect i need to get back into the swing of doing these lower div problems, but i've found especially with like online homework systems where very tiny mistakes of inputting the answer can mark the whole thing wrong, the stuff i trained myself to look out for doing upper div/grad math coursework is often different from the more minor syntax issues most grad students don't struggle with (and even the classes i TA'd were usually not like early lower div so they were less prone to these issues too).

tbh, the student would have gotten a faster answer just taking a picture with their phone and asking chatgpt. i think AI is probably better at the minute syntax stuff now (which wasn't the case a few years ago, from my memory). like when i go to help a student and they have a huge page of scratch work, it's often hard to scan to find that tiny error, which may have only occurred right at the end when they were writing down their final answer. but, at least for me, i need to check the logic and process from the beginning. AI is fast and surprisingly reliable now for 99% of the content i'm going to be tutoring.

[vagrant meadow]

tbh i would probably trust it to point me in the right direction for finding the error in an undergrad proof (i would expect that often it gets the precise reasoning wrong, but sometimes it can point me to look in the right place and that's good enough). writing a proof from scratch is a different story. but generally, if it's something common enough to have a few posts on math stack exchange, it's probably more reliable than not (you just always have to take its output with a grain of salt).

but like i've found that explaining my thought process in full usually gives it enough info to point out where i'm wrong (haven't been using it for math much lately, but i've been relying on it for a data science-esque project and to learn more computer science, so a lot of the methods it's suggesting are completely new to me. so how i try to apply them can often be wrong).

i'm also someone that best learns by trying to articulate my thought process out fully and completely. and tbh AI is probably one of the best tools for that. most tutors/professors don't have the time for me to explain things in my own words and really unpack the issues or where I'm right/wrong, and it's often a bit of an imposition to post a wall of text in mathcord. using AI as a sort of kinda dumb sounding board has really helped me learn, at least. it's really not like it was a few years ago, where it was like a really overconfident but also very dumb person who feels like they need an answer to everything even if it's wrong.

[turbid zenith]

That I think is the big danger. If you know what you're talking about you can spot BS, but if you don't you can't.

So students who haven't developed their BS detector will fall for it because it sounds confident.

[tardy ember]

i think in some sense LLMs are just, getting better and better at slipping BS past increasingly knowledgeable potential readers

which in simple cases does mean that their output has to be reasonable, because there just isn't enough going on that you can put in a sufficiently subtle error

but it also makes it more of a problem, in the sense that more and more people can no longer tell that it's producing nonsense

[turbid zenith]

To be fair it's not always complete BS

I'm continually impressed by what it can do

People be like "oh it's a stochastic parrot, it's just putting words together based on probability" ... okay sometimes I wish my students would put words together with that kind of probability XD

[tawny slate]

there is also the danger that over-reliance on it will rot your brain

im less concerned with the student letting a couple of mistakes through in their exercises and more concerned with their mental wellbeing and habits

in this sense it's not actually about LLMs being accurate or not

at least when you use a tool like wolframalpha you have to actually think about what input youre giving it

[long pelican]

AI is useful to students at all levels
There's two main required steps to every problem

• Search for the correct direction/answer/solution/proof
• Verify that it has no errors

We can AI exclusively to help with the search aspect. Just don't forget that it does not help you with the verification aspect. For example, if the problem is to calculate the integral of e^-x^2 from -infinity to infinity, AI can help you see the polar idea, as a substitute for you spending 10 hours trying 100 ideas or as a substitute for you googling

Granted, some nontrivial search time does help a lot with mathematical development

as you learn something from each failed idea

[lethal hornet]

are students spending so much time looking up definitions and examples that they have to use AI? i just don't remember this being an issue when i was doing highschool level math

especially considering that there are so many resources online that aren't AI for this type of stuff

[plain valve]

I think AI just lets you use this sort of thing (even) more lazily

[halcyon glade]

It'd be interesting to see if in the future we might have AI personal tutors that would guess at when to withhold information, when to give hints, when to offer support or ask the student to think more, etc.

I imagine the real benefit that students see with using AI is the immediacy of getting a response customized to their specific situation

Which you can't get with static videos and resources

[swift hatch]

Is this not the subject of a cgpgrey video from ages ago (as cgpgrey videos tend to be)?

like "digitial euclid" or whatever

[tall bolt]

This is the only way I’ve ever used AI, I was very much against it for a long time, and for a lot of environmental and IP reasons I still am, but I’ve gotten back to learning German now and honestly being able to type sentences and verify my grammar is pretty invaluable

Like yes I have a book of grammar rules but it can be confusing, and it can be hard to tell exactly how some words work in context, but this is thankfully the thing that LLMs are fantastic at and I do find it to be a genuinely valuable pedagogical tool

[tawny slate]

the issue is you have to first teach digital literacy, responsible ways of using LLMs first

do students these days even know how to do web searches and determine source quality? I have found this has not been the case, they willingly believe a lot of what they see online without thinking twice

[lethal hornet]

yea, i remember learning this in school

[tawny slate]

if they dont even know how to use a web search or a library, do you trust them navigating LLMs?

[tawny slate]

i learned it in school too, but look how our generation turned out

[lethal hornet]

man

[heady dragon]

"as a substitute for you spending 10 hours trying 100 ideas" ie.... doing math?

[long pelican]

I covered that

[tall bolt]

Yeah I agree with this, I think the important context here is that I kinda know what Im doing and im being honest about my learning. I think my point is mostly just that I agree with eric in that the best place AI could go, is in some sort of tutoring direction, because its main power is the imidiacy of responce, and the ways in which it can tailor it to your specific issue

[long pelican]

I have a weird explanation for this, and it's not generational: This comes from a mindset where information is not easy for them to find on the internet, because of any of

• typing is hard
• using computers is hard
• English is not their first language
• using the internet is hard
• lack of experience searching
• they use computers for this kind of thing rarely.
  In contrast, at least for me, where using a computer is second nature to me, I can access 10000 different viewpoints on a topic easily so it's very obvious you should not take anything you read at face value

As a corollary, the most effective teacher for this isn't education or someone telling you "don't accept everything you read" but actually seeing 10000 different viewpoints on the topic

crucially, you have to see them yourself, they cannot be given to you by 1 person

[lethal hornet]

so by using AI, you can now limit yourself to having the information given to you from one source/point of view...

[tawny slate]

sure, in the ideal case, but that's assuming those 10000 viewpoints are "valid"
are you gonna teach students the viewpoints of conspiracy theories and psuedoscience? just in case they have a point? pitch evolution against creationism? pitch anti-racism vs racism?

[long pelican]

are you gonna teach students the viewpoints of conspiracy theories and psuedoscience

crucially, you have to see them yourself, they cannot be given to you by 1 person

[tawny slate]

sorry, my point is that students need to know, just as importantly if not more importantly, what not to consume vs what to consume

[long pelican]

AI isn't a person though, but it's a tossup whether it'll be a reliable way to see different viewpoints or whether it'll be just as biased as a person

[tawny slate]

this was different in the age where information wasn't so readily available, but in an ecosystem where it is now reversed, that AI slop and misinformation is more common than real information, they need to be more critical than less

[lethal hornet]

i never said AI was a person. but either way, like, why leave it to chance?

[long pelican]

What's wrong with consuming a conspiracy theory? Depends on the mindset you approach it with. Whether you read it because you want to find something to believe, or whether you read it because you want to see a viewpoint

[tawny slate]

conspiracy theories have led demonstrably to my life becoming more miserable due to forces outside my control, let's not go there

[lethal hornet]

what is wrong with consuming a conspiracy theory? have you seen the current political state in the US?

[long pelican]

You both are skipping a lot of steps

[tawny slate]

tbh that's just how this kind of conversation will go because it's complicated, i don't think either side here is intentionally wanting to be reductive

but clarification wouldn't hurt

[long pelican]

For example, someone who's curious about eastern traditions might read something on Buddhism while not being at risk of converting to Buddhism

Same goes for conspiracy theories

[tawny slate]

right, that I get, but i wouldn't call that "consuming a conspiracy theory", maybe this is just semantics, but sure I follow you

[long pelican]

I guess that you see "consuming a conspiracy theory" being reading it and then immediately believing it

That's sort of a caricature of how humans believe conspiracy theories, what actually causes the belief is not never single event like this

Most commonly, it's socially induced

In individual lone wolf cases, it's months of going down a rabbit hole + confirmation bias + being bad at finding diverse viewpoints

[tawny slate]

i completely agree but that's also not what I was thinking either

anyways that's a semantic point

[long pelican]

Ya, going back several messages, you're trying to say that being smart about what to consume vs. what not to consume is important

I'm disagreeing with that by saying, what's wrong with consuming a conspiracy theory, as long as you're in the right mindset of not taking a source at face value, just learning information about what people believe

In a very general sense, more data from an appropriately sampled distribution is not bad, filtering the data is what has issues

[tawny slate]

this is where im going to disagree, i think people well versed in that sort of information landscape take for granted just how much there is to actually learn in terms of information and digital media literacy

[long pelican]

Yay we reached a point of fundamental disagreement

[tawny slate]

there are plenty of people who have this exact same mindset and they end up consuming massive amounts of social media that is "curated" to them by big tech companies

the problem is structurally, we live in a society that is dominated by technology and the internet now, and so much of what we see on the internet is not in fact a balanced sampling of what we see in person, but a biased, and sometimes intentionally so, distribution that corporations and other big players want us to see

and then these people end up being exactly the kinds of conspiracy theorists and spreaders of misinformation i mentioned before

[long pelican]

Do you think this can be mitigated by not using 1 platform?

[tawny slate]

no, i think this is mitigated by improving digital media literacy

there is no substitute as far as I am aware

because imagine a teenager spending hours a day on instagram and reddit

that's two platforms, but his information quality probably hasn't improved much

[long pelican]

That's 2 platforms !

It's better than just one of those alone though

How about more than 2 platforms

[tawny slate]

ok and 1.0000001 is larger than 1

[long pelican]

10 platforms?

[tawny slate]

it's not about the number of platforms

it doesn't matter how many platforms you use if you still can't distinguish between quality sources and low quality sources

[long pelican]

Well for a more politically relevant example

one always says CNN + Fox News > just CNN or just Fox News

[tawny slate]

understanding how media outlets work + understanding sourcing is >> (any combination of any news outlets)

[long pelican]

How effective is trying to teach this understanding with no corresponding experience?

[tawny slate]

what do you mean "no corresponding experience"

[long pelican]

Like I suppose you think there should be a "digital media literacy" class where we teach principles of digital media literacy

For students who do not have much experience viewing anything other than Facebook reels I'm afraid of the possibility that this class will be very abstract to them, they will learn what is the correct answer on exams, they will learn what to do in hypothetical situations, but they leave the class still being mainly interested in Facebook reels

[tawny slate]

yeah, that's part of the reason why i think the school system is failing us

[lethal hornet]

abstracting the digital media platform is too abstract?

[tawny slate]

we definitely need to adapt our courses and curriculum and even our teaching policies and techniques to a modern age but these things move too slowly and society moves too fast

im arguably an "industry expert" on tech and im even finding it difficult to keep up

[long pelican]

Come again?

[tawny slate]

even the very big math channels on youtube, in the process of describing how AI and LLMs work, go through the "beautiful" math without talking about the sociological impacts of LLMs, which is not saying they are endorsing AI use, but they certainly aren't telling people to slow down

[long pelican]

Besides education (which I'm doubting is the silver bullet), I want to think about how to instill digital media literacy in everyone. To do this, it helps to understand how we ourselves have achieved it

[tawny slate]

i achieved digital media literacy by studying digital media literacy. I'm not saying education is a silver bullet here (since when has it been a silver bullet for anything), but it is, again, as far as i know, the most effective and most straightforward approach

[long pelican]

I reiterate that I believe this is important (in how we have achieved it)

You actually studied digital media literacy? As in took a class or what

[tawny slate]

yeah, i actually agree with the bullet points as you've listed them, though I don't necessarily have direct evidence, it does sound right though, I just disagree that people can just "feel out ideas and take them at face value" their entire lives

i did take some kind of computer or english class in school where a small segment of the course briefly talked about this thing, but it wasn't extensive, it wasn't a full class, and it certainly was not adequate enough to prepare me for a more modern information landscape

that i had to learn myself, by actually reading about and understanding digital media more, but additionally these ideas were shaped by a lot of other information from other fields, which were very difficult to find myself had it not been by communicating with other experts in other areas

for example, neuroscience informs a lot of why this is so important and the risks of getting it wrong

[long pelican]

It could just be number of hours put in. I easily have orders of magnitude on everyone else I know in real life in terms of computer usage and internet usage. Similar to how someone who chops wood in the forest every day passively figures out the best routes to the forest and the best trees to cut

[tawny slate]

my mom is on the internet a lot now, and she now believes a lot of internet misinfo, so I once again disagree for the same reasons

[long pelican]

how much we talking here

Also very important: your mom is Chinese I believe? English not her first language?

[tawny slate]

yes

[long pelican]

She probably browses Chinese websites like huaren.us?

[tawny slate]

idk what she browses but it's mostly all chinese yes

[long pelican]

This is not a lot of diversity in viewpoints (maybe I'm wrong here and Chinese in America are very diverse)

[tawny slate]

i only brought up family to make a point, id rather not go into details here

[long pelican]

It's funny because my mom is mostly the same

I still easily have orders of magnitude in terms of internet experience, though

[tawny slate]

yes but more doesn't mean it will eventually fix itself, is my point, and not everyone has the privilege to approach it this way, which is also a big part of my point