math-pedagogy

[next relic]

I've been TA for ODE for two semesters and here are my two cents.

(1) I assumed that students are comfortable with quadratics but NOT with cubic/higher order poly, and that usually works for me. I spend half of the first tutorial going over integration techniques - one example on substitution, one example on IBP, one example with trig integrals, one example on partial fraction decomposition.

(2) Just throwing it out but would it be a good idea if you let them know what's gonna go on the quiz for the following week? Make it really explicit to students what prerequisite skills they need to have, because many times instructors just blow on without really checking if students are following.

[vagrant meadow]

i would only find out what's on the quiz a day before it. i eventually i started a discord server so i can give them as advanced notice as i could.

[next relic]

Ah I see, so you're in quite a pickle there. I guess you can anticipate the skills they'll need based on the content on the subsequent tutorial? Eg: giving them an example on solving simultaneous equations when dealing with local extrema in multivar differentiation?

[elder timber]

I'm thinking of making math YouTube videos, and weighing my options for software to do the visuals. I've been playing around with Beamer but there's a certain style I'm going for that I'm not sure Beamer can do efficiently. What I want is to write proofs where a line of text shows up on the slide, in large print, then shrinks down to make room for the next line of proof. That way, it's always clear which line I'm talking about at any moment, but also the previous lines are still visible.

Beamer can resize text but -- as far as I've seen -- only in the sense that you can set the size of a block of text. That means if I want text to be one size in one frame and the same text is a different size in another frame, you have to basically re-write the whole frame.

And then that causes editing nightmares as you try to write out your proofs, then look over them and change things, and so on. And the placement and sizing takes a ton of fidgeting, which sucks if you're trying to make this stuff kind of fast.

Anyway, anyone know of either a better way to do this than Beamer, or a way to make Beamer do the job?

[open sparrow]

im also doing it

and i think that u can just use like whiteboards

that is also pretty feasible

[turbid zenith]

I use PowerPoint.

It's served me well for many years and it's surprisingly good for math videos.

[elder timber]

After some digging, I'm thinking Quarto + Manim. Looks better and more readable than my handwriting, for sure. Fast to edit the simple stuff. Manim can be futzy, but if something's worth it, then you make a nice graphic every now and then.

[turbid zenith]

https://vxtwitter.com/solidangles/status/1733185220814278968?s=46&t=oOrJqXqWA6UuDrnj-ULIIA

Poll, would be interested in your schools’ policies

[long pelican]

good poll!

I've learned that the topic of grade distributions has a lot of competing philosophies

I get so much cognitive dissonance thinking about it

[elder timber]

Yeah, I think a class getting all A's is a good thing, as long as the standards of the class are high. The idea that you have a normal distribution of grades, or any specific distribution, just seems like a "weed out" strategy which I don't think is good.

[turbid zenith]

This also depends on what you think an A should mean

Or even what grades should mean, what their purpose should be

[elder timber]

True, I think it should mean "there is a standard body of knowledge and the student has strong proficiency in it". It should be a qualification to move on to further studies, and a certification of knowledge acquired.

[turbid zenith]

I’m still trying to get my grading system where I like it.

My Calculus III students REALLY are struggling with their final exam, but the way the system ended up working out many of the students had “safe” A’s going into it because of their work and revisions throughout the semester, and that feels like I need to make adjustments for the future.

Like you ask something a slightly different way and they’re lost. We did at least a few in class activities that involved using an elevation contour map, but as soon as the problem involved isotherms instead students were lost despite the calculation being exactly the same

[elder timber]

Yeah, I think lots of opportunities to bring a grade up is good, so long as the score correlates to learning. My main frustration is with students who do not exert effort and still want good grades. For them I would like to bring back corporal punishment.

[turbid zenith]

Yeah. They learned it in the moment, but it seems they can’t USE it in a situation.

And yes that’s the worst

[turbid zenith]

What do y’all do when students do poorly on an exam or a final?

My other section of Calc III just did their final and they had just as much trouble

[long pelican]

When this happens to me I think about whether the exam was too hard and the students still did a good job with problem solving

If yes, I adjust what score gets an A

If no, tough luck for them

[turbid zenith]

How do you judge if they did a good job with problem solving if they’re not getting it right?

[long pelican]

here's an example rubric I used for a problem on a final exam

But in addition to this rubric I carefully note if the student went a wrong route but showed a good understanding of the math involved in their solution path and whether there was a lack of misconceptions

[cosmic ibex]

I suppose you can to some extent judge from their responses to the things they did get right whether they did those by understanding or just by following steps.

[long pelican]

Similar rubric for the last problem

What Troposphere said helps too

[turbid zenith]

I see

Small detail but personally I'm not a big fan of the idea of the rubric being all about losing points

I know it's mathematically equivalent to start off with 0 points and then gain points for what you showed you did understand, instead of starting off with full points and watch them slip away

But something about the psychology of everything being about this ding here and that ding there gets to me

[long pelican]

Unfortunately gradescope has no option to just say "x points" which is what I would do on paper

[turbid zenith]

Ahhhh I see, so that's what that program is

I swear I've seen that before

[long pelican]

Gradescope is pretty nice

one thing is it does all the data and item analysis for you

[turbid zenith]

I mean ... I'd love to see understanding but I've got students who can't even follow the steps 😛

[long pelican]

Well, students who can't even follow the steps won't be expecting A's, right?

[turbid zenith]

Wellll you'd be surprised XD but that's a different issue

But I'm not focusing on the grade right now

[long pelican]

I want to understand the mindset of someone who doesn't understand math but thinks they deserve an A

[turbid zenith]

Usually it's because there are other classes where if you "put in the work" you get the A

[long pelican]

Ah-ha, maybe they have never felt what understanding math feels like, and you need that to understand that you don't understand math

[long pelican]

Yeah unfortunately...

[vagrant meadow]

it does

[vagrant meadow]

i agree in theory, but when i moved to negative grading, the scores went up.

it might have just been the structure of the quiz, though.

[long pelican]

You'd be talking about positive scoring

[vagrant meadow]

i've only just started using it this quarter, but there's point adjustments

unless i'm misunderstanding what you mean by "x points"

[long pelican]

Completely breaks statistics

A grader used it so that's why I know

I didn't bother to tell him not to

statistics are cool but not having them wasn't the end of the world

[lethal leaf]

IMO positive points > negative points

Encourages me to actually look and make sure they hit everything when grading

[winged urchin]

Well positives are greater than negatives

.>

[long pelican]

Very interesting thing in my experience: Positively graded questions with very detailed rubrics got a lot of "Why didn't I get more points" and "What did I need to write so that next time I can get those points" type questions (I hate answering the latter type of question, because it's as if every possible answer makes them less of a mathematical thinker), while using the simple "correct/almost/something relevant/something correct" for 2 semesters I got no such questions

The reason underlying that is the one thing that makes me lean away from positive grading

The underlying reason being: with positive points, you get points based on whether you write enough of the correct keywords or key phrases according to a predetermined rubric, while with "correct/almost/something relevant/something correct" it's clear that you get full credit if you completely solved the problem and partial credit based on presence of correct ideas and partial progress

[elder timber]

The Reddit math subreddit talked recently about the idea of getting rid of grades.

[turbid zenith]

This is why I also use holistic rubrics

[long pelican]

Nice

[turbid zenith]

It's reduced the point-grubbing just like you said

Though after the clusterfuck that was this last final exam, it's become clear that my students aren't retaining anything

So I need to very much modify how my grading system works

[long pelican]

Hmmm, I'd wager retention is a large unaddressed issue in math classes in general

mainly because we don't really have any solid idea what makes kids retain their math knowledge after final exams or even unit exams/midterms. Lots of people have their own theories of course

[turbid zenith]

In this case I think they had no motivation to do so

They learned something well enough to do it on the autograded homework and then flushed it from memory

So it was like they'd never done anything like it when it came to our tests

That final should have been straightforward, but I think the median was 55 / 120

I think I was saying earlier that a lot of students feel that if they "put in the work" they should get an A

So I need to be a LOT more explicit about what students need to do.

[long pelican]

The yikes-ness of median is tempered by the fact that I know you made your final questions un-scaffolded ;^)

[turbid zenith]

Well here's the thing

I offered "buyable" scaffolding

They could "buy a hint" in exchange for capping the max points for a question at 15/20

And even then they couldn't do it

[long pelican]

I see, interesting

[turbid zenith]

So ... yeah. Complete disaster.

[long pelican]

So I need to be a LOT more explicit about what students need to do.
What's that?

[turbid zenith]

What I mean is, I need to let them know that on tests they'll need to use the techniques and concepts from the homework

That they can't just flush cache

Yes that should be obvious but my grading system was working against that I think

[long pelican]

Ooh I see. So they genuinely thought homework was just busywork

[turbid zenith]

So I'll be tweaking it next semester.

[long pelican]

What in your grading system contributed to that?

[turbid zenith]

Yeah. And a bunch of stuff I'd set up to try to be helpful had inadvertently exacerbated the problem

Like how that homework had a "practice similar" button which would shuffle the constants and would let them see my own written solution

So ... instead of using it to understand, they used it to pattern-match and immediately forget it

[long pelican]

This reminds me of when question 1(a) on my complex variables final exam was meant to remind the students of a neat result about partial fractions they proved on the 7th homework assignment. After this conversation I'm now considering the possibility that some of the students who missed that problem had the same view of homework!

[turbid zenith]

Soooo not doing that again lol

[long pelican]

(They had to use that result to solve it efficiently, otherwise it was brutal complex number division calculation)

Did you code the "practice similar" stuff yourself?

[turbid zenith]

Yup. In Edfinity (WeBWorK)

[long pelican]

That is pretty nifty

well the result wasn't

[turbid zenith]

5 questions per assignment. 20 assignments total.

[turbid zenith]

@long pelican I've been thinking about some of the things we've been talking about, and I think I'm going to start giving tests that are just about solving problems and not aligning to a specific list of standards.

Hopefully separating that out will make it so I can be more holistic about it

And test them over whether they've learned to USE what they've learned.

[long pelican]

👍 I like that change

[turbid zenith]

What I'm struggling with is you know how you said students shouldn't just be learning "problem types?"

[long pelican]

Yep

[turbid zenith]

In my case the students didn't even learn the problem types :/

Like that's not the end all be all but it's at least something

I don't know if it would have been any better if I'd said "find the local max of this function" and nothing elsle

[long pelican]

I'm trying to think of what the crucial ingredients are to have students learn and retain techniques and ideas from solving homework problems

One thing that comes to mind is that you need to approach a problem like a puzzle, rather than like busywork, to have a chance at retaining the lesson of the problem

This correlates well with my own recollections of what I tend to remember and what I tend to forget

[turbid zenith]

Question is how to get students to see it like that 😛

If you're interested it's a puzzle. If you're not, it's busy work. If it's just a hurdle for you to get over to get to the rest of your major classes, it's busy work.

I have almost no math majors in my classes.

And I try to show how it's relevant to many different fields but at the end of the day it's just a box to tick.

[long pelican]

I believe non-math majors have just as much potential to be interested in puzzles as math majors

[turbid zenith]

Sure, and a handful were.

[long pelican]

One obstacle to approaching a problem like a puzzle (you are probably familiar with this in your own grad studies) is being confused about the contextual material

For example, a student that's scared of functions because they don't have a good mental model of what a function is, is not likely to treat a function problem as a puzzle

A modern math research problem sounds more like word salad than a problem until you are familiar with basically everything surrounding it!

(as an analogy)

[turbid zenith]

Mhm.

[long pelican]

Another obstacle is sort of orthogonal, which is a history of math being taught in non-puzzly ways for 12 years

[turbid zenith]

Part of it sort of comes from being used to symbol pushing

Students almost universally failed problem 4 on my final

Because there were no symbols to push around

[long pelican]

Hehe

[turbid zenith]

One student estimated the partial derivatives correctly and got it right.

Out of 30.

[long pelican]

did you analyze the wrong answers and get a feel for what happened?

[turbid zenith]

Lots and lots of just wild guessing and trying to cobble something together

The only coherent issue I saw was a surprising number of students just took the temperature difference between the two points (along a diagonal line) and divided it by both Δx and Δy

Instead of seeing "what happens if we change JUST x? What happens if we change JUST y?"

[long pelican]

Dang my first instinct was to estimate the distance from the 230 isotherm to 240 isotherm as being about sqrt(2) so you get about 10/sqrt(2) as the rate of change

Using distance between adjacent level curves to estimate directional derivatives is a multivariable calc thing I remember from TAing it a while back

You probably gave partial (or even full) credit to such answers

(if there were any...)

[turbid zenith]

Of course

I’ll still give credit for a reasonable answer

[long pelican]

Yep

[vagrant meadow]

it also sometimes encourages students to vomit keyword over their answers in a desperate attempt to hit the ones in the rubric. which only demonstrates that, though they didn't actually learn the concepts, they were at least listening enough to pick up the names of many of the concepts they were supposed to know.

[vagrant meadow]

the arguments i've seen for abolishing grades makes more sense for humanities classes. but i'm not convinced there shouldn't be some necessary demonstration of learning the necessary course content in math. especially for the lowest levels.

but now in grad school it seems that professors give anyone who actually tried a B+. and, honestly, it's been kind of nice to be able to just focus on doing my best on the homeworks and to understand the material without stressing about passing. still haven't finished my first quarter yet, so i'll probably have a proper opinion on it after i get my final grades.
still, do you have a link to that reddit discussion? i'd at least want to hear what they have to say.

[long pelican]

Exactly

[winged urchin]

Though in grad school the standards are far greater to be honest

Many students in my grad program considered B+ a 'failing' grade

A- was struggling, A is acceptable, and A+ is doing well aha

[elder timber]

I mostly agree, but I do worry about how grades cause a Goodhart's law effect. I wonder if perhaps some kind of pass/fail system wouldn't help, which is still a grade but maybe takes focus off of the exact grade point and maybe refocuses students on the content.

I also kind of think that we under-use interviews to determine proficiency. I can usually get a sense of how much someone understand by talking to them, better than how well they can cram formulas for a test.

https://www.reddit.com/r/math/comments/18e05ir/anyone_feel_that_grades_and_exams_sometimes_take/?utm_source=share&utm_medium=web2x&context=3

[vagrant meadow]

wouldn't pass/fail only just be an even worse Goodhart's law effect? i don't see how we escape it.
though, i can see that for many unmotivated student who only care about passing with the necessary grade, we're basically already there...

it's a very interesting point

[vagrant meadow]

i think more oral exams would be good though

[elder timber]

I can't promise that it wouldn't be, but my hope at least would be that once you've made it over the pass/fail hump, then you'll just focus on understanding.

I'd want data to be sure of anything though.

[vagrant meadow]

just make anything less than an A a fail /s

one of my primaries has an interesting grading scale that i actually kind of like

it makes me feel a lot less bad about giving lower quiz grades, because it makes failure more acceptable. and one learns far more from failure than success.

[elder timber]

I do remember talking to a literature professor about abolishing grades, and I expressed concern that people would claim proficiency that they didn't have. She got extremely pissed at me. It feels like for some on the liberal side, abolishing grades is becoming a bit of religion.

I just frankly don't know what to do about grades.

[vagrant meadow]

lol

[elder timber]

But I'm willing to experiment.

[vagrant meadow]

indeed. no way anyone would try to take advantage of such a system. even though people have been taking advantage of every system since the dawn of humanity.

i would be very interested in seeing experimental grading systems implemented.

[elder timber]

I am finding that everything on Windows ****ing ****s. This whole separate environment for programming, that practically has a wall between it and the rest of the OS, ruins everything. I need to switch to Linux.

[turbid zenith]

I've been experimenting with "ungrading" in my liberal arts math class

Still trying to decide whether I like it

[viral pike]

what's ungrading

[turbid zenith]

Instead of giving grades back on assignments, you give qualitative feedback

The only time there's a grade is at the end of the semester, when students have to create a portfolio and argue how well they've met criteria for a grade

[tacit adder]

We’re also trialling that next semester for my algorithms class, will be interesting to see how students feel about this

[vagrant meadow]

i've always felt uncomfortable with that system. because i feel like there's a motivation to just always say A+ and then the professor will just give the highest grade they think they actually deserve. because if you lowball due to humbleness, you might end up with a lower grade than you could have actually gotten.
how do you actually plan to deal with those situations where a student gives too high or too low a grade?

to play devil's advocate, though: i'm just not convinced conventional grading in mathematics is really flawed beyond it "feels" bad sometimes. having a qualitative measure and numeric score for how well you executed solving the problems is useful for telling you what you did and didn't know and what topics you need to study to improve.
perhaps we could incorporate some of aspects of systems like ungrading and separate the score one has in the class from the final grade they get from the class?

[lethal leaf]

in my physics class in HS we did something similar

HW wasn't graded, quizzes weren't graded, unit tests weren't graded

but we did them and got feedback

and the only graded things were some midterms and the final

which in theory would be fine since we would have seen the material in exam like settings multiple times (quizzes, unit tests)

but in reality what happened was that no one tried for the HWs, quizzes, and tests

and so every bombed the hell out of the midterms and you can't fail everyone so curve carried everyone to high grades anyways

I think it really really depends on the motivation of the students to learn vs get a good grade

[vagrant meadow]

yeah i've been very happy with this final grade is separate from class performance in grad school, but that's also because we're all here because we need to know this stuff and we know that. doing that in high school seems like asking for trouble.

[turbid zenith]

I do it by basing it on evidence.

The students have to show evidence of understanding when they submit their final portfolio.

And I use my professional judgment on whether that evidence does demonstrate the grade they claim. If it doesn't I contact them and we agree on something.

There's a decent bit of discussion of the ins and outs of ungrading and other schemes at this blog: https://gradingforgrowth.com/

[vagrant meadow]

very nice, okay that definitely addresses my concerns. that sounds like a very interesting system, especially for students in the liberal arts. doing this kind of thing feels like it's more in their wheel-house so that seems like a great system on paper. i'd love to know what your thoughts are after it's concluded.

[turbid zenith]

We'll see what the quality of the porfolios is like this semester, but I've had very strong classes this semester!

But the more I've thought about it, the more I feel like a quantitative "points" system is really just objectivity theatre.

[vagrant meadow]

have you gotten any feedback so far on their thoughts on it? like throughout the quarter i've had some feedback on my worksheets, and i imagine if i was in their shoes i'd be very enthusiastic that the math course i was forced to take isn't in the scary form i think a lot of people envision.

[turbid zenith]

"You got a 79.24% on your test, that's ⭐OBJECTIVE⭐"

[long pelican]

What I'm curious about is how well students who are not understanding know they're lost

[frail summit]

I'd say using what you learned is the best way to retain that knowledge. The brain gets rid of what is not using (or makes it unaccesible as I know from personal experience that learning again things I knew well in the past, but forgot due to no usage of it, is way much easier than learning something totally new)

[cosmic ibex]

Asking "which grade do you think you deserve" is going to be hard on students with impostor syndrome.

[turbid zenith]

In that course, a lot of students have reported that they feel like they can just concentrate on learning and not the grade

[turbid zenith]

That's why they're given particular criteria to make that decision.

And why I reserve the right to mark their grade as higher if they're rating themselves too harshly

[vagrant meadow]

that's how i feel about my grad courses, and i like it. very interesting.

[turbid zenith]

When they try to do something in class but aren't getting it, or when they turn something in and it gets marked incomplete with a whole bunch of feedback on what needs to be revised

[vagrant meadow]

so rather than grade a poor assignment with a low grade, you mark it "incomplete"? do they have to redo it?

or, at least, are they encouraged to do so by you i think i should say.

[turbid zenith]

Yes. If it's incomplete, you have to revise it. If you don't revise it, then you can't really say you've shown you understand it.

[vagrant meadow]

was this grading system chosen by you? and, at least right now, do you think you would choose it in future courses (just like your gut opinion)?

[turbid zenith]

So instead of trying to calibrate the proxy points so that the students who do understand the material get lots of points and the students who don't get few points (and then dealing with those miscalibrations that always come up because of a particularly hard question or a technicality you didn't think of), you just directly answer the question holistically, "did they demonstrate understanding or not"

Yeah, I decided to try it out for the first time last semester, and I've been doing it again this semester. It wasn't perfect but it definitely seems much better

But next semester I'm going to be trying out IBL problem sets and I haven't yet decided whether this system will play well with that

[vagrant meadow]

the more i think about it, the more appealing it becomes. because giving feedback on the parts where the student was wrong is happening anyway, you just no longer need to assign a numeric value to it anymore. i might imagine writing constructive feedback on a problem that would recieve a 0 in normal grading might be difficult, though.
my main concern would be a sea of students coming to me asking "is how i've been doing enough to pass the class/get an A?"

[elder timber]

Yeah, I think this philosophy of "You're going to succeed if you try and fail if you don't" works well with people who choose to be in the classroom. It probaby will always fail with compulsory high school classes.

[lethal leaf]

this was mildly compulsory

in that it was honors instead of normal

but yea I think everyone could see from a mile away it was gonna be a disaster

I remember walking into class the day after the exam with a picture of the dunning-kruger effect on the board 💀

[elder timber]

I have no idea how this could play out in practice, but I feel like somehow the incentives for doing work and focusing need to be different. I think we just see that for most kids "If you don't learn the material you get a bad grade and that probably means bad school and job prospects" just seems to distant and abstract. It also seems to have a lot to do with culture -- if the parents and community cares about academics then usually the student does too. But I meet a lot of students whose parents care about money, status, or other things, and get upset that their kid doesn't do better in math. I'd guess kids pick up pretty easily on what really matters though, by watching what their friends and family actually care about day-to-day, and it usually ain't math.

[digital coral]

I agree with you a lot that marks just aren't a good enough motivator. For some students, it will be enough. However I know a lot of students who will not start studying mathematics at all until their later years. Right now I have a student who I'm tutoring who has not passed a math exam for the last 3 years. She didn't care about mathematics and her parents have gone through countless tutors. It was only now that she's in year 11 - 12 that she's putting effort in, however this is out of necessity. Put simply, the "marks" motivator has motivated too late. Nowadays I relate everything back to business and economics (things she enjoys), which allows her to care about it more. But inherently we're doing 4-5 years of math content in 1-2 years and her teachers and mental health issues aren't making it easier. But it's out job as educators to be able to motivate, and this isn't as easy as relating to what the student enjoys every time.

[elder timber]

Yeah, I've worked with business students, compsci students, physicists -- some of them can "muscle through" their math course work, but if they resent the fact that they have to do math in order to do the thing they really care about, they never make good scores.

[vagrant meadow]

the vast majority of the students in my linear algebra TA section are CS or business majors. and just throwing out stuff like "this is actually super important in application if you're in computer science because of blah blah" seems to keep them somewhat tuned in. it seems my enthusiastic delivery helped too. even if i didn't take a lot of time to explain the connection, it seems just knowing "oh hey this is actually useful for something" seems to motivate them a bit. "search algorithms", "markov chains", "image processing/compression", etc. things that they may have heard and sound kinda out there

[fickle nebula]

Guys what do you think about rigorous math courses that focus on proofs are converted from exam-based to written project-based

I saw that a couple unis have this initiative, I was wondering if these writing intensive courses are viewed inferior to exam-based ones

Written projects include a portfolio of proofs, writing peer reviews for fake journals, etc. It also includes a small-weighted written final exam

[austere delta]

I don't think their viewed as inferior. The main argument I've seen against is that it requires more resources to grade.

[fickle nebula]

Have you heard of arguments regarding comparison of academic outcomes?

I was curious if they established that which one is better, or one is better under some specific circumstances

[austere delta]

It's my impression that spread out and more active grading is better for outcomes, but I don't know the research on the subject.

[feral swan]

As a student I slightly dislike this idea as it will make me think but on the whole it sounds an amazing training

[turbid zenith]

Yeah XD I tell students that ungrading can sometimes be more difficult

Because you can't get by with pointsmaxxing, you actually have to understand

[feral swan]

Grading is stressful but it's easy and I'm part of the student crowd that is good at this system

But in effect

I spend a lot of time revising little points of material and checking boxes of materials so I can answer every question

And the "reaching the 100" grind is exhausting. Getting from 90 to 95 to 100 is an exponential useless grind but I have nothing else to do that will improve my status in the system itself.

[winged urchin]

This is reliant on useful comments from grading I suppose though

I was helping a student in a proof course this fall and the comments on assignments are abysmal. Just question marks, very short remarks, or no comments

[feral swan]

I mean, woe is me, yeah, I know I'm the lucky one here, and In effect this makes me learn by myself little by little. But it just feels like feeding my effort into a soulless machine

[turbid zenith]

Anyone up for looking at a game-theory-related IBL activity I'm working on and giving feedback? It's pretty self-contained

[feral swan]

If you'd like a student's review I can look at it by the weekend

[vagrant meadow]

had an idea; would like to hear thoughts on it.
basically, the determinant was properly discovered by Cramer in terms of solving systems of equations (through cramers rule). what if we defined determinants through cramers rule? like we can show the solution of Ax=b will be xi=Di/D. can we just define det(A)=D?

what do y'all think of how that might do in a linear algebra class? it would at least be more concrete than "here's cofactor expansion it has all these weird properties and I can't really explain why this definition is what it is very well"

and I guess maybe we define the det to be zero if A has linearly dependent columns to make the definition fit in general? idk ignoring the smaller details I'm just curious if any of y'all think this might be a path worth looking into

[austere delta]

I feel like this just pushes the problem onto why Cramer's rule works, which might be even harder to motivate then the determinant.

I think a good definition of the determinant could be to motivate it as how the transformation changes the volume of the unit cube.

Then you could define it on elementary matrices, define it to be multiplicative and show that's well defined. I think that motivates the definition much better.
This also better reflects how you actually calculate the determinant, through row reduction and not cofactor expansion.

[cloud zealot]

You may be interested in texts which defer treatment of the determinant. See Meckes or Hefferon.

These books are also designed as first courses.

[quasi tartan]

if a=b, then ab=b^2. True or False. I am really struggling with this.

[cosmic ibex]

Unless you're trying to teach that fact to someone, this is the wrong channel.
I'd suggest #proofs-and-logic, or possibly #prealg-and-algebra.

[vagrant meadow]

oh interesting. so basically like

1. define it as volume.
2. determine the values for elementary matrices
3. it should be multiplicative just intuitively (the scale factor is multiplicative in nature) so we can lump that into the definition
4. since any invertible matrix is a product of elementary matrices blah blah now we just need to show the result is the same for any given elementary representation
   that's very interesting. and then you're saying to show that this leads to cofactor expansion, then?

[austere delta]

Nah, I'm saying forget cofactor expansion. That's not good for anything. Like you could have a little technical lemma that says it's equivalent to cofactor expansion, but that's all it should be: a technical footnote.

[vagrant meadow]

oh so like literally row reduction only? hmmmmm

that's a lil against my current worldview, but imma think on it. because it's actually not that ridiculous. I already avoid determinants at all costs or use calculators to do it for me. and I never really do cofactor expansion anyway... that's pretty radical

there are cases where it's useful, though. like when there are a lot of zeros. but that's arguably niche enough to be footnote worthy...
anyway I'll keep thinking about it. thanks a lot @austere delta (:

[austere delta]

If there really is a lot of zeros, then just row/column reducing to triangular form should be pretty quick

Imo cofactor expansion can in some cases give simpler rigourus proofs for things, but at the expense of deleting any intuition.

[cosmic ibex]

How do you feel about Leibniz expansion?

[austere delta]

I've never had it come up for me, but I think I'd place it on an equal footing as a technical description of the determinant.

[vagrant meadow]

the way I do determinants when I have to is I cross out and bring out the entry from a row/col with only one nonzero entry. it makes evaluation very compact and quick when possible

ex

[vagrant meadow]

this is the main issue with it imo. it's the most straightforward computational algorithm, but it's completely unintuitive.

[cosmic ibex]

(My immediate favorite would be to_define_ the determinant as linear in each row, alternating in rows, and det(I)=1, then use Leibniz expansion to show that such a function exists, and row reduction to show it is unique. The Leibniz expansion also shows it is linear and alternating in columns, and row reduction + elementary matrices shows it multiplicative).

[turbid zenith]

What’s wrong with cofactor expansion?

That’s what I learned and it made sense

[cosmic ibex]

It works, but it's not particularly intuitive -- the n×n determinant arises as the result of defining 1×1, 2×2, 3×3, ... determinants recursively each in terms of smaller ones, so the definition doesn't paint any clear picture in my mind of what actually happens to the entries in an n×n matrix.

[turbid zenith]

I suppose it depends on what class it’s meant for

I didn’t need to know the exact inner workings of the formula to be able to use it and understand its geometric meaning.

That got added in later when it made more sense to do so and I had a good feel for it

[vagrant meadow]

currently reading Hefferon's determinant chapter and it's very interesting. thank you for suggesting it.

[vagrant meadow]

maybe i'm just dumb but i cannot at all see the geometric meaning in the cofactor expansion definition. the fact that it describes signed volume still boggles my mind years after i first learned about determinants

but the context i'm looking at is for a linear algebra + intro ODEs course. in DE at least, determinants are primarily used to determine linear independence like a nontrivial kernel detector.

[turbid zenith]

I didn’t see the geometric meaning in the cofactor expansion definition but I understood the geometric meaning of the determinant in general

And I’m very much against the idea that you need to “rigorously” define something in its full formality and prove all its properties before you’re allowed to play around with it

[vagrant meadow]

well so that's what i'm trying to avoid. the idea of

here are two seemingly completely different things: volume and cofactor expansion. they are both the determinant and entirely equivalent. don't ask me to prove it.

[turbid zenith]

I don’t think you need the proof at the very beginning.

[vagrant meadow]

of course not. but i don't like "just trust me these are the same" with no good intuitive explanation given ever

[turbid zenith]

https://math.stackexchange.com/questions/590164/geometric-interpretation-of-the-cofactor-expansion-theorem

This has a few looks at the geometry behind it though.

[vagrant meadow]

sure. but i've never seen a linear algebra prof ever actually talk about these in a class. not to mention that these explanations are either

1. only for 3d space or
2. use geometric algebra which would be absurd to bring up to intro linear algebra students
   i'm not trying to be intentionally dense or infuriatingly contrarian. my point is just that we have about three or four definitions for the determinant that seem completely separate and different but (seemingly) "happen" to be the same. and direct proofs for their equivalence are lengthy and not particularly elementary.
   some level of "just trust me" seems to be necessary, but i'd like to minimize it as much as possible.

[turbid zenith]

I don’t see a problem with showing in detail how it works for 3D space and then saying that higher dimensions work by analogy.

Some people will call it “hand waving” but I believe in judicious hand waving if it means students “grok” it and can fill in the details later.

“3D volume = 1D height * 2D base, so it would make sense to say that nD volume = 1D height * (n-1)D base” is good enough for me.

At least because it points by analogy to something that students do understand more intuitively.

But I’m also pretty sure I’m in the minority so take what I say with a grain of salt :V

[vagrant meadow]

no i think that's a very good point. and in practice i think that's the way to go

[turbid zenith]

I keep coming back to saying this, but kindergarteners don’t need to know Peano axioms to learn to count, and elementary school students don’t need to know ring theory to know how add and multiply. I see this as analogous.

But of course it doesn’t mean that you can’t allude to what’s going on at a developmentally appropriate level, and by developmentally appropriate I mean less in terms of brain development and more in terms of that nebulous thing called mathematical maturity

[vagrant meadow]

in most cases i agree, but the determinant feels like a special case imo. there seems to be no truly elementary way to "discover" the determinant in its full generality. a lot of students ask "why is this the definition and what does it mean" about the determinant, and i've never heard a very good short/intuitive answer. i think comparing this directly to counting, adding, and multiplying is definitely appropriate in your point, but i feel like the level of complexity is much different.

[turbid zenith]

Looking over the last (very long) answer on that page I linked, I could actually see this being a really interesting IBL activity

Like creating a set of leading questions that has the students see how it works in the 3D case at least

[vagrant meadow]

i think that would be really cool!

[turbid zenith]

It would probably also lend itself well to a video that I do not have the 3D animation capabilities to make :V

But for the activity what I’d probably do (and maybe I’ll do this if I actually get to teach LA sometime) is once the students have done the 3D part, the next part would say “okay how could we extend this idea to 4D? What would be the same, what difficulties might we run into?” Something like that. But just to give the idea that we’re reducing a larger problem to a smaller problem to justify the recursive nature.

And doing it on a particularly well-chosen example, I think, doesn’t detract from it working in general.

[cosmic ibex]

The only way I think cofactor expansion really makes sense to me is something like: We know the determinant is linear in columns, so if we pick a random column, it has to be a certain linear combination of the entries in that column, with the coefficients somehow depending on the entries in the other columns. Those coefficients we then choose to call "cofactors". If I further know Leibniz expansion, it is then easy to see that the coefficients happen to be ± determinants of particular (n-1)×(n-1) matrices with column and row deleted.

[cloud zealot]

It is true that students, especially young children, should not start from absolute first principles. However, there should be more of an effort to engineer university-level mathematics into something developmentally appropriate for K-12 students that still gives mathematical rigor and formalism, especially unambiguous definitions, an essential role. I first heard about the math-pedagogy-as-mathematical-engineering idea from Hung-Hsi Wu. Here is a quote from this talk he gave that summarizes this idea:

Regarding the nature of mathematics education, Bass (2005) made a similar suggestion that it should be considered a branch of applied mathematics. What I would like to emphasize is the aspect of engineering that customizes scientific principles to the needs of humanity in contrast with the scientific-application aspect of applied mathematics. Thus, when H. Hertz demonstrated the possibility of broadcasting and receiving electromagnetic waves, he made a breakthrough in science by making a scientific application of Maxwell’s theory. But when G. Marconi makes use of Hertz’s discovery to create a radio, Marconi was making a fundamental contribution in electrical engineering, because he had taken the extra step of harnessing an abstract phenomenon to fill a human need. In this sense what separates mathematics education as mathematical engineering from mathematics education as applied mathematics is the crucial step of customizing the mathematics, rather than simply applying it in a straightforward manner to the specific needs of the classroom.

For example, one of Hung-Hsi Wu's favorite examples to illustrate the math-pedagogy-as-mathematical-engineering idea is fractions. Yes, we should not teach fractions as equivalence classes of pairs of integers. However, we can define as a primitive notion what 1/n, n some integer, means. There is no need to appeal to vague analogies to pies. Also, equivalent fractions are less obvious with discrete items. Half of a pie is visually the same as two-fourths of a pie, but is one green Jolly Rancher out of two the same as two green Jolly Ranchers out of four? And how does one do arithmetic with these analogies? A complaint raised by Wu is that textbooks gesture towards what a fraction is, but shortly after plop seemingly arcane rules for adding and multiplying fractions onto students. Analogies might help illustrate what a fraction is supposed to represent, but it's not very good at illustrating fraction arithmetic. Further details may be found in Understanding Numbers in Elementary School Mathematics by Hung-Hsi Wu. He also has a website.

[wise onyx]

Oh. Actually you can turn this into a complete proof.

[cosmic ibex]

Yes.

[wise onyx]

Well, provided you know determinant is linear in columns and rows

[cosmic ibex]

I'm championing that as a definition. :-)

I think it should also be possible to use the core of that argument to derive the cofactor expansion directly from linearity (and alternatingness), but I'm not entirely sure of the details.

[long pelican]

The linear algebra instructor I stole notes from uses the universal property as a unique multilinear map satisfying certain properties. I used the exterior algebra (top exterior product, specifically)

[long pelican]

I remember reading about math-education-as-math-engineering, and it's good to be reminded that this idea came from Wu as well!

[turbid zenith]

Why do I recognize that name

Is he the guy who talked about "school math"?

Ahh, yup, looks like he is

[turbid zenith]

I’ve been thinking about this, and I think I’d be interested to see what commonly used textbooks actually do today when introducing fractions.

I can’t speak for textbooks at that level, but the teachers I’ve seen who talk about how they teach fractions (or the professors who teach teachers about how to teach fractions) certainly don’t “plop arcane rules” with no meaning.

As for Jolly Ranchers, I would argue that trying to do arithmetic with certain analogies would be awkward, but that doesn’t keep them from being useful in their own contexts.

[long pelican]

I think the plopping of arcane rules is never what a teacher aspires to do but rather what they have to do when nothing else works and standardized tests are coming up

[turbid zenith]

Oof this is FELT

[cosmic ibex]

Right. I don't know the facts on the ground either, but the anti-pie rhetorics sure sounds like trying to hide a strawman.

[long pelican]

You do have to ask why does nothing else, including presumably the teacher's first choice of lesson style with amazing analogies, work?

[turbid zenith]

Yeah. I don’t think the pies are vague, even if they’re not always the representation that works best in every situation

[turbid zenith]

Sure, but I doubt “it’ll all be clearer if we make them points on the number line and lay out a precise definition” is what’ll make it click with those students

I think more often it’s because of much broader issues

[long pelican]

Oh I wasn't thinking of Wu or his work when asking that

[turbid zenith]

Ahh okay

Yeah it is definitely a question worth asking

[long pelican]

I like to phrase it as a disconnect between what the teacher thinks will reach the students and what actually does or doesn't reach the students

Teachers are generally very skilled at working with kids (I think) so why is there this disconnect

[turbid zenith]

Some teachers at least, and in some subjects

It’s unfortunately not universal and I do think that bad teachers can do real lasting damage

[long pelican]

The biggest share goes to unknowingly bad teachers, rather than intentionally bad teachers

[turbid zenith]

And I do think there are a number of teachers who plop down arcane rules, unfortunately

[turbid zenith]

This

[turbid zenith]

In at least some cases because they don’t know another way

[cosmic ibex]

The elephant in the room is the hypothesis that the teacher simply don't know any better than arcane rules themself because their education failed ...

[turbid zenith]

I’ve seen this happen personally.

I used to tutor a student who was in a program to become an elementary school teacher, and they REALLY struggled with anything non-rote with math

If you gave them 3x+1=10 they could do the steps and get the answer, but anything off script would get a deer in headlights look

[cosmic ibex]

It depends a bit on whether middle-school teacher duty assignments even take into account subject-matter knowledge, or adminstrators just think "well, they're functioning adults so they all know the relevant facts, they just need to have pedagogy knowledge".

[turbid zenith]

So with fractions the textbook had all sorts of things about benchmarks and ways of comparing fractions that didn’t rely on an algorithm, and it was all I could do to try to make anything stick … they just viewed it as more to memorize

Things like being able to tell why 5/8 is bigger than 9/20

They couldn’t notice that one was a little more than 1/2 and the other was a little less

[long pelican]

Deep subject knowledge should be part of learning on the job, but somehow professional development is all about exciting new ways to dress up rote problems, new grading schemes, flipped classroom, and such

That seems like a community / culture problem now

[turbid zenith]

I mean this is useful but it shouldn’t be everything

But at least in my state at our annual teacher conference we had both.

Content and pedagogy development, that is

[long pelican]

Another thing is it might be that many teachers haven't seen many great math problems, by which I mean problems that are good puzzles in and of themselves

Like maybe they know rote ones in the textbook, projects with real life stuff, and word problems

But math isn't fun without feeling puzzled and solving it

[turbid zenith]

What would you give as examples of the kinds of great math problems you mean?

[long pelican]

well we can start with the 5/8 and 9/20 example actually

like say the algorithm of cross-multiplying is illegal for now

5/8 and 9/20 can be compared by comparing to 1/2

but if you change it to 5/8 and 11/20

that's no longer available

what logical explanation can you find now?

This would be targeted at a 4th or 5th grade level

I was looking at old AMC problems and problem 2 from 2009 is this. The key is not to use the problem verbatim but think of something based on it. Here, I noticed that you can probably ask students to observe that if you continue this process you'll get fractions with Fibonacci numbers. Explaining that would be a good problem

[turbid zenith]

Oh yeah, that’s more of the sort of stuff that was in the textbook my student was learning from!

[long pelican]

(second one probably higher grade level)

[turbid zenith]

They could not wrap their head around it

[long pelican]

Oh sounds like your textbook is a lot better than the ones they had in school!

[turbid zenith]

And this was an undergraduate :/

[turbid zenith]

It’s the sort of thing I see in most math textbooks for elementary school teachers

[long pelican]

That's different from the math textbooks used in schools, right?

[turbid zenith]

Yes. I don’t really know what actual elementary school textbooks look like these days

So all I can speak to is the way they’re trying to prepare teachers

[long pelican]

Seems like it's currently very hard for the prospective teachers to shift their mindset in how they think about math even with these textbooks and good teacher teachers

(in case of misunderstanding, your anecdote just now is an example of what I mean)

[turbid zenith]

Yes i figured

It’s evidence of a widespread issue

[long pelican]

mm-hmm

[cosmic ibex]

So is your student in the anecdote going to pass (even if with a so-so grade) and then actually become a math teacher without ever seeing the light?

[turbid zenith]

They did pass.

This was about 10+ years ago

[cosmic ibex]

Figures.

[turbid zenith]

I believe they're a kindergarten teacher now.

[cosmic ibex]

At least fractions are not expected there.

[turbid zenith]

Whee planning for the spring :V

Teaching vector calculus (what we call Calculus IV) this spring and it's a much smaller class than I've had because most students only have to take up through Calculus III ... so maybe I can use this to figure out how to have more success with non-rote problems

(And then figure out how to do so for the next calculus cohort when I have all levels of students again)

[elder timber]

I'm trying to think of an example integral $\int f(x) \ dx$ which is hard-ish to integrate using standard rules, but easy-ish to integrate if we can express $f(x)$ as a MacLaurin series and then exchange the integral and sum. Anyone know an example off the top of their heads?

[burnt vessel]

addem

[turbid zenith]

How about $x^2e^x$?

[burnt vessel]

DM Ashura

[turbid zenith]

Like you could do integration by parts twice, or you can do a Maclaurin series and just use the power rule twice

[elder timber]

I like it! Really maybe even $x^{1000}$.

[burnt vessel]

addem

[turbid zenith]

Unless you’re saying you want a closed form at the end

[turbid zenith]

As in $x^{1000}e^x$?

[elder timber]

Yeah, it's especially compelling if it has a nice clsoed form at the end. But I was considering $e^{-x^2}$ which is even worse kinda.

[burnt vessel]

DM Ashura

addem

[turbid zenith]

I think something like $e^{-x^2}$ is a great example because it shows that Maclaurin series give you a way to handle functions that wouldn’t even be nicely integrable otherwise

[burnt vessel]

DM Ashura

[elder timber]

Yeah, I would think you just distribute into the summation for $e^x$ and then use the power rule for each term indexed by $n$.

[burnt vessel]

addem

[elder timber]

I'm probably going to shotgun a bunch of examples, so I'll probably just include all ideas.

[turbid zenith]

Good idea!

Meanwhile I’m trying to think of what would count as non-routine problems for vector calculus

Since those already get so involved anyways

[elder timber]

Any particular topic within the subject? Any particular effect you're trying to have on the audience?

[turbid zenith]

Just starting to plan in general, but I guess we could say curved coordinate systems (polar, cylindrical, spherical)

Since that’s where I’m going to start

I just know I want to give students opportunities to both get used to the mechanics and also have interesting problems to solve … and that the difference between the B students and the A students should be their ability to “take it further” and use their knowledge in new situations. That makes sense to me but I’m not really sure how to pull it off.

[elder timber]

Hm, nothing comes to mind. I do think it's a subject that is best explained by routines at this point, although that might just be my teaching style. One thought is to relate everything to the origins of calculus which is the modeling of celestial motion.

For instance, I try to explain calculus by holding an object in front of me and releasing it into free fall, and saying "Figuring out what just happened is basically why calculus was invented."

[turbid zenith]

Mhm. We’ve talked a LOT about applications in Calc I-III

Though for vector calculus the applications get a good bit narrower in field

[elder timber]

Ah, maybe I'm not clear on what is in vector calculus -- I tend to think of it as basically the same thing as Calc III.

[turbid zenith]

Calc III at our school is vector-valued functions (think space curves), partial derivatives, and multiple integrals. Calc IV is coordinate transforms, line integrals, and surface integrals.

So the applications are pretty much physics and engineering now

perhaps computer graphics if you want to think about light intensity etc

[elder timber]

Yeah, nothing very interesting springs to my mind for those subjects.

I do view them as the tools we use in daily life, so it's kind of like teaching the times-tables. An advanced college version of it, but a little similar in flavor.

I taught someone contour integrals last semester and she benefited a lot from basically getting a kind of "algorithm" of parameterizing curves, because no resource out there really just lays out what we realistically do when paramterizing curves. If it's a line segment, do like this. If it's an ellipse, do it like that. If it's a function curve, so on.

She was smart and interested in learning, she just needed a pretty straight-forward and assertive "here's what this is" kind of lesson. So I don't think there's anything wrong with giving clear, routine exercises for that kind of stuff.

[tawny slate]

I'm looking to compile any applications of "advanced math" into everyday life, if anyone has anything

Here are some examples of what I'm looking for:

• I need my back scratched at a specific location, so I define a coordinate system with my wife to target a location with arbitrary complexity without needing to measure or count anything (practical use) (I have actually done this with my wife and system works beautifully)
• using Boolean algebra and logic to distill complex conversations about politics, philosophy, morals, into its syllogisms and axioms (abstract use)

It can be something that pervades your way of thinking in an abstract way, like in the second example, or a direct use, like the first example

[quiet jackal]

something tells me you might have more luck asking on https://www.reddit.com/r/math/

[turbid zenith]

I will start declaring and using variables when I am trying to explain something and using too many pronoun referents 😂

Maybe that counts?

[pastel sundial]

when grading, is it better to be harsh but uniform, or generous but less uniform

like, when giving partial credit, there are number of "edge cases" that don't fit neatly into my rubric. In these cases, it feels like the student is demonstrating more understanding than would be reflected in their score if I just went with the rubric. So I want to award extra partial credit, but I also don't feel like I am capable of doing so in a way that's fair across all students in that situation.

Like if a student "deserves" n points by harsh standards, or m points by more generous standards, it feels like my choices are to either give them n points or \approx m points. And I'm not sure what is better.

One way to put it I suppose is that it's fairly easy to sort students into the categories "entirely/almost entirely incorrect", "entirely/almost entirely correct", and "partially correct", but subdivisions withitn "partially correct" are much harder

I don't want to have the lingering worry that how much partial credit I'm awarding depends on my mood at the time. But I also am a bit bothered by giving the same score to a student who did almost everything right but made a basic algebra error in a crucial step, and a student who failed at step 2

usually my rubric does distinguish those students, but sometimes it feels like I'm giving the same score to students who demostrated very different levels of understanding

[lethal leaf]

how advanced is advanced?

lots of cool algebra in the theory of error correcting codes

and some cool probability/stats as well

nvm I read the example you gave and my example may be too complicated lol

I guess dumb it down to polynomials and sweep the finite field stuff under the rug

[tawny slate]

Yeah I wanted like

Examples that the regular everyday person might think about or use

So maybe something that for instance has to do with cooking or driving or something

[lethal leaf]

driving, shortest path algorithms or something

cooking, scaling recipies (home vs restaurant) idk

[tawny slate]

Scaling recipes uses fairly basic math, simple ratios and proportions, so it's not a strong example, not what I'm looking for

[lethal leaf]

I know in busisness it's common to keep a spare set of funds for invoices that just never get paid

that's an expected value thing if you sweep away enough details

idk how advanced that is

[tawny slate]

Driving and shortest path stuff could be relevant except gps services like google maps and Waze exists now

So it's not like people would ever actually use the concept

[lethal leaf]

yea

[tawny slate]

I want something of actual tangible use for the layperson

Business example same thing, most people don't do accounting so its also not great

A better example of expected value would be, for instance, knowing that gambling is always a losing situation, and that's why you should never gamble

And if you ever win, stop immediately, because any bets following that can only go downwards, not up

Knowing whether or not gambling is a good or bad idea is something that literally everyone could use

[turbid zenith]

This is why I’m a fan of grading more holistically and answering the question, “how well do they understand what I want them to learn?”

[cosmic ibex]

I mean, there are many other forms of entertainment where you fully expect to leave the venue with less money in your pocket than you arrived with, and they don't seem to be regarded with the same amount of scorn.

[long pelican]

Here’s one I thought of. If headphones sound wrong out of the box you can test which frequencies are affected by playing pure tones. The very idea of doing that uses the concept of Fourier analysis

Also the same principle behind hearing tests, and how hearing aids work for people with hearing loss which typically is not equal in all frequencies

Central limit theorem has applications to how to think about statistics and randomness and even multiplayer games

[long pelican]

I have never thought about it that way before!

Well in a way I have. I've proposed before that gamblers are paying money to experience a primal form of excitement and anticipation, but to them that is not what is going on

[long pelican]

This never happens to me and I grade basically on a scale of "Nothing relevant (and usually full of nonsense)" and "Almost there"

[tawny slate]

Ok sure, I guess I just mean that this helps people have an accurate understanding of reality in order to properly set healthy expectations

I know plenty of people who go into gambling expecting positive expected value because they believe they are inherently lucky people or happen to be lucky at that moment, and that's dangerous imo

[tawny slate]

This one I agree is good, but I'm looking to compile some specific examples, or if they are possibly abstract like this one, a more detailed explanation on how it affects how we consume media like games

I'm trying to think of a game in which my understanding of the central limit theorem actually affects how I play, and I struggling to immediately find something directly explainable

[long pelican]

Speedruns with rng elements, you can estimate if it's worth trying to beat the record 😛

[long pelican]

there's also that whole Dream debacle

[tawny slate]

Speed runs would be borderline what I want

Its not something everyone does, but it is something that a layman might do

The dream debacle is also borderline I think, because it's more abstractly imo about how to evaluate the reasonableness of others' actions, evaluating the truth

The dream example I guess a bit niche but I guess it works

[long pelican]

If you really wanted where this century's advanced math figures into things everyone does, you're going to be waiting thousands of years

or hundreds if you're more optimistic about the future trajectory of humanity

[tawny slate]

I patently disagree, which is why I am gathering responses

I know that I subconsciously do a lot of this stuff myself, but I sometimes take it for granted because I already know it

[long pelican]

Ok so do you consider using a computer which is built on advanced math, while not activly thinking about it, an example of such a thing for computer science?

[tawny slate]

Knowing the math pervades your subconscious

[long pelican]

there's probably a misalignment of definitions in this argument

[tawny slate]

I do not consider that one an example of what I am looking for

The motivation for this is this:

Imagine a student asking a teacher: "why should I be learning this? Why is this useful or helpful to me?"

[long pelican]

Oh I think I got where the misunderstanding is

My claim was for what people already think about

But you are looking for possible new ways to think about current things

new to the layman

[tawny slate]

Doesn't matter if it's old or new, as long as it demonstrates the practical value of learning math, through concrete or abstract examples

Everyone understands that basic counting and arithmetic is important

But why should someone care about group theory?

Why would laymen bother to learn calculus?

I'm trying to find reasons for why even laymen will regularly use and apply these ideas on a daily basis

For instance, I often think about calculus while driving in order to maximize my fuel efficiency

[long pelican]

Why will laymen use group theory today? False premise. But there can be good answers for why laymen of the future will use group theory

that's kind of why we learn group theory and other advanced math. To empower us with tools to build things no one has ever thought of before

[tawny slate]

Again, I reject your premise because I know there have been plenty of cases where I have visualized group theory to understand something mundane

I simply cannot remember the specific example

[long pelican]

Ok there's a communication error

You asked why laymen will blahblah

Whereas what you meant is why a hypothetical layman can blahblah

rather than actual laymen

[tawny slate]

I want to provide examples to formulate a convincing argument for why everyone can and will find value in these math topics, yes

Assuming they understand it and know where to use it

[long pelican]

Ah that last assumption is why I was confused when you said laymen

[tawny slate]

I mean laymen not as people who don't understand math but laymen as common everyday applications

[tawny slate]

Again, this is the motivating question

[long pelican]

I will say that what you are looking for is kind of reductive

ok I can't really explain the reductiveness right now but I have another line of thought

Imagine you are in algebraic geometry class and asked me "Why am I learning algebraic geometry? Why is this useful or helpful to me?"

And say I personally am studying algebraic geometry not because of real life applications so I don't really have any in mind

How should I answer?

[tawny slate]

Just that

Not everyone is going to have the same feelings or motivations for math, but from my own experiences, I have found all kinds of advanced math to be meaningful in everyday usage, it's just that I haven't logged them so I have since forgotten many of them

I want to make a convincing argument that all kinds of math are useful for just about everyone, not just mathematicians

The convincing argument is my concern, not anyone else's

I'm simply compiling examples, which almost anyone could provide

[long pelican]

ok so you want us to take for granted that this person you are presenting the argument to will certainly be most convinced by the presence of examples of where math is used in everyday life

I can try to do that...

for now

[tawny slate]

The argument is my responsibility, I just want examples from other people if they have it

[long pelican]

So the ones I already gave didn't quite work, meaning I misunderstood the requirements a little bit

What are they again?

ok so if I understand correctly, you want a situation that everybody encounters and in which an layman educated in advanced math will be more empowered than the layman who is not so educated

emphasis on a situation that everybody encounters

My problem is that the set of "situations that everybody encounters" is not too large

We listen to music, we have fights with friends, we have fights with family, we have celebrations with family, we go shopping, we eat, we do work (but nothing can be said about what work),...

we drive (most of us)

we walk, we go traveling sometimes, we sleep

we use phones, we use computers, we go to school, we shower, we open doors, we spend money, we save money

That is one reason why I made my claim earlier that you said was false

Regardless, there is your example with driving and calculus
My example with being annoyed at headphones and diagnosing them using pure tones covers the listening to music part

the set of things that all laymen do now is constrained by the layman's knowledge

In the future when laymen have more knowledge there will be more advanced things in the set of things laymen do

[long pelican]

In the "using computers" bit there are a lot of things that an educated layman can do for anything they might be annoyed at with the computer. Probably some examples there

[tawny slate]

I don't need perfect examples

I just need examples

I appreciate anything anyone comes up with

I was merely explaining the motivation to help clarify what I'm looking for

[quiet jackal]

seems like a communication breakdown. maybe you should try using boolean algebra to distill this conversation into its syllogisms and axioms so that we can all be on the same page.

[long pelican]

This but more serious (aka just the first sentence)

[sharp beacon]

Seems clear to me

Ways that someone other than me can use math to improve my life (e.g. by writing algorithms for my computer to use) don't count

Entertainment or training to enter a field that uses lots of math won't apply to many people

Math is probably good at training logical reasoning abilities in general which is something everyone can use but not necessarily a "practical application"

[cosmic ibex]

For much of high school math, though, the honest answer to "why are we learning this" is "because you'll need it to be able to learn whatever math will be actually useful to you if you enter a technical or otherwise number-heavy field".

[turbid zenith]

The way I see it, there are four reasons to learn things in high school math:

1. Because it has a direct application
2. Because it explains, connects, and extends previous material
3. Because it lays groundwork for future material
4. Because it's just plain interesting

(#4 doesn't come up often but every once in a while it does!)

Also woohoo, just finished (drafts of) two of my syllabi for next semester!

[lethal leaf]

#5 it builds your analytic thinking and logical reasoning

[turbid zenith]

Truth! But that's true for all the things

[lethal leaf]

Yea but always good to state that as a reason (especially since 2 and 3 most people don't care about)

"I didn't like math before so why do I care that this builds off of it"
"I don't like math now so why do I care that future math builds off of this"

[cosmic ibex]

Those who demand a reason at all won't care about "building analytic thinking" either.

[lethal leaf]

yea fair

[vagrant meadow]

sorry but i feel this is relevant 😆

somehow i've managed to escape teaching the lower levels of math but it's these kind of questions i'm most scared of. i much prefer higher levels and more motivated students

[vocal phoenix]

same

I could never teach children

at least at uni when a student actively refuses to learn I can just fail them and move on

[quasi musk]

I used to have this mindset before I actually taught children. Your impact on their experience in 5th grade is way more impactful than teaching them in College. There are a lot of cons to teaching kids, but there's a lot of pros as well

I'd say don't knock it until you try it

Now, I don't want to be a full-time k-12 teacher in a public or private school setting, but it's an eye-opening experience

[fallen needle]

idk if its the same for tutoring, but after i showed them some applications for what i taught them, or just cool stuff that could be done with it, they seemed a lot more open to learning. actually, i had one kid take up competition math after learning algebra cuz he found it so interesting.

[vocal phoenix]

Yeah, I'm not entirely serious, but on the whole I don't get along well with kids so it's just as well

[inland pier]

I know this is kind of a late reply but one time my dad was building a set of stairs to his porch and asked for some help calculating length of wood based on angle of elevation or something, don't remember exactly, but he texted me later after he built it and it came out perfectly

Just some simple trig but it helped him a lot

The layperson's average math fluency is such that even simple stuff can seem like wizardry, and therein lies part of the issue, there are plenty of people who have learned math concepts in school etc, but there's no guarantee they have the ability to recognize when they can be used "in the wild"

[tawny slate]

I totally agree, thanks for the reply

Examples of straightforward applications of relatively simple math I think are pretty easy to find and construct even

These I have a lot of, I'm trying to find stuff that is a little bit more niche

Using more advanced, abstract ideas

But nonetheless I appreciate the comment, always good to have explicit reminders so that you don't lose sight of what's important

[wide ice]

I don't mind low or high level, but I get irritated a bit when I fail to explain something after 15th try

[inland pier]

I guess it depends how abstract/niche you're looking for. But I also think by restricting yourself to niche, you're naturally going to find less stuff that can be used everyday. Like, just per definition of what "niche" means

[tawny slate]

For instance maybe using group theory in cooking

Or using knot theory to figure out how to arrange the wires of your PC setup

Basically I want all kinds of examples, but the more abstract and advanced the math, the better

The more commonplace/mundane the application, the better

But I'll take any examples of any kind

Anything may spark some kind of idea

[turbid zenith]

I feel this.

Some topics are just really hard to teach though. You try to come up with the perfect explanation or activity that’ll make it click and then it still doesn’t, and you’re like “what am I not saying or doing here?”

Recent example: sampling distributions.

There’s something about moving from taking the mean of a particular sample to considering all possible sample means and thinking about THEIR mean and standard deviation that seems to lose students

[long pelican]

Everyone's got their own blocking misconceptions and you can only do your best break down the most common ones you can identify

[vocal phoenix]

Yeah, I've been teaching for 15 years and I still get caught off guard by some novel ways people misunderstand stuff

I don't think you can ever be prepared for everything

[turbid zenith]

Which do y'all prefer: thinking of the gradient as a row vector and writing the directional derivative as ∇f u, or thinking of it as a column vector and writing it as ∇f · u = ∇f^T u?

My instinct is to do the latter, but one of my friends has talked about how "the gradient is REALLY a covector so it should be a row vector"

And it would be kind of nice to have the gradient as a row vector because then the Jacobian of a multivariable function is just the gradient rather than its transpose.

[frigid crypt]

ive always interpreted the directional derivative as a dot product and not as a matrix multiplication. but aside from that i like thinking of it as a lineair combination of the standard basisvectors

[cosmic ibex]

Making a gradient a row vector makes most sense theoretically, but I'm afraid it wreaks havoc with the idea of nabla as a "vector of differential operators" that's inherent in the conventional notation for curl and divergence...

[turbid zenith]

Oof, yeah. And I am teaching curl f and div f as ∇ × f and ∇ × f respectively.

[marsh compass]

Huh, why do you think so?

[cosmic ibex]

In order for ∇·f (with a dot) to make sense as a divergence, it seems that ∇ needs to be thought of as a column vector of partial-derivative operators. How would you then explain that ∇f (without a dot) makes a row vector?

[marsh compass]

I would explain that ∇f is a column vector.

People confuse (total) derivatives and gradients a lot...

[cosmic ibex]

Hm, then I don't see what you're disagreeing with.

[marsh compass]

The (total) derivative is a covector, but the gradient is a vector (its Riesz representant).

[quiet jackal]

vectors are conventionally column vectors

in undergrad math courses, and everywhere else

[marsh compass]

Yes, vectors are conventionally represented by column matrices.

[quiet jackal]

the pairing (u,v) = u^T v

is what you are using

to represent df as a “vector”

so the thing representing df should be a row

nvm

hmm

[long pelican]

Does multivariable calculus also traditionally conflate 1-forms and 2-forms?

(in R^3)

[cosmic ibex]

That's basically what the cross product does, isn't it?

[long pelican]

Yeah, it is

[cosmic ibex]

Except we can also say R³ with cross product is the Lie algebra so(3), and then it's suddenly legit to say the inputs are of the same kind as the output. (But I don't think one does that in multivariable calculus).

[long pelican]

That's my first time hearing that perspective. Tell me more!

[cosmic ibex]

I don't really know much Lie group theory, just that this isomorphism exists.

And it's somewhat intriguing that it appears to provide an excuse for mixing up vectors and pseudovectors in R^3. But my actual knowledge stops at "hmm, that's intriguing". 😆

[pastel badger]

Sorry if this is the wrong channel but has anybody here ever worked as a PASS/Supplemental Instruction leader before? I’m starting in that role spring semester and was wondering if anyone has advice on how to do well.

[marsh compass]

Multivariable calculus silently uses Hodge duals

[long pelican]

Yeah the 1-forms and 2-forms conflation is hodge dual, which makes sense to me. The so(3) reinterpretation also makes sense on the surface (because there’s a natural bilinear form on any Lie algebra + the Lie bracket) but I still don’t understand it on a deeper level, for example generalizations to other Lie algebras

[wispy slate]

hye

[quasi musk]

https://math.stackexchange.com/questions/54355/the-gradient-as-a-row-versus-column-vector

TL;DR I don't think it matters. If you feel a row vector is more rigorous then you can go with that

[marsh compass]

What are you referring to in that MSE post?

The distinction is important.

[cosmic ibex]

What I see (both here and in the MSE post) is several people insisting that the word gradient and the notation ∇f should be reserved for the column vector form, but I don't understand why you think that form is important enough that it should get first rights to the name and notation. Is there any need to have it at all, other than "so we can write all our vectors vertically"?

[quiet jackal]

what do you mean by “form” here

as in 1-form

or just the english meaning

[cosmic ibex]

No, just the ordinary-English "shape". Sorry for the ambiguity.

[quiet jackal]

isnt a consistent convention enough justification?

[cosmic ibex]

What would become inconsistent if we only ever talked about the row-vector form shape for the derivative?

[quiet jackal]

the point is that df and \nabla f are not the same object

[cosmic ibex]

I'm trying to understand why you think that is an important distinction to have.

[quiet jackal]

in calc 3 its not

but

in life i think it is

[cosmic ibex]

I don't know that "calc 3" refers to.

[quiet jackal]

wherever students first meet gradients

[austere delta]

I guess talking about the length of the gradient, or the vector field it defines is more convenient if nabla f is a column vector

[tight star]

I mean df doesn’t depend on coord choice

Whereas nabla f does

[cosmic ibex]

Why?

[austere delta]

Then you can always think of the associated operator as
nabla f dot

[quiet jackal]

no it doesnt

[tight star]

I mean it is a different column vector in different coords

[quiet jackal]

it is defined through the pairing

[tight star]

Ofc it is related by the jacobian but

[austere delta]

So that nabla f lives in the same space as the domain of f

[quiet jackal]

it is coordinate free given a riemannian metric

[quiet jackal]

but it doesnt

[marsh compass]

How you call and denote them doesn't really matter but there needs to be a distinction between the derivative as a linear transformation and the gradient as an element of the original vector space.

[tight star]

Ok but you were talking about calc 3

[quiet jackal]

df is also coordinate-dependent in calc3 lol

[tight star]

Wait it is

?

[austere delta]

It's not

[tight star]

For me i just remember df = nabla f . dx

As a way to define gradient in any coord system

[cosmic ibex]

Isn't that just "such that we can write all our vectors vertically".

[marsh compass]

No

Also, in curved spaces, the gradient is not just the transpose

[cosmic ibex]

What I'm asking is, why is it important to have a concept of "the <whatever you want to call it> as an element of the original vector space"?

[austere delta]

Not really, it's to make a distinction between vectors and linear functionals

[cosmic ibex]

Huh? How does it help you to make that distinction if you instist on writing the derivative as a column instead of as a row? That seems to be conflating the linear functional with a vector rather than make any distinction.

[marsh compass]

For example, if you want to perform first-order optimization, you start with an initial guess and move in the direction of steepest descent (which corresponds to the negative of the gradient). Moving in the direction of a linear functional (different object than elements of the original vector space) does not make direct sense.

[austere delta]

Because one depends on a choice of basis (or inner product) while the other doesn't

[cosmic ibex]

But the "direction of deepest descent" depends on a choice of coordinate system!

[austere delta]

Exactly

[cosmic ibex]

So why insist on teaching the one that depeds on a choice of basis, instead of the one that doesn't?

[austere delta]

I don't insist on teaching just one, I'm saying it's useful to have both and to distinguish them

[cosmic ibex]

But you don't seem to be revealing why you think it is useful to have both.

Rather than just the row form which is nicely behaved and doesn't depend on a choice of basis.

[austere delta]

To talk about the direction of steepest descent for example

[tight star]

It’s always better to have more perspectives?

[cosmic ibex]

But the direction of steepest descent depends on a choice of basis!

[tight star]

And?

So do vector components

[cosmic ibex]

Teaching a concept that hides that dependence seems to do a disservice to the intuition.

[tight star]

Idk I find it intuitive l

[austere delta]

Yes, exactly. What are you not getting?

[tight star]

It’s an arrow in space pointing in the direction of steepest descent

[austere delta]

I don't feel that that hides it

Would [df]^T be any better?

[cosmic ibex]

What I'm not getting is why you feel it is important to keep teaching the concept that depends on a choice of basis as the natural generalization of derivative from the R->R case to R^n->R case.
It would make so much sense to teach that the natural generalization of a derivative of an R^n->R function produces a row vector.
No matter which words we use for it.

[cosmic ibex]

The "direction of steepest descent" does not make any senses until you choose a coordinate system.

[tight star]

Sure it does

You draw a graph

You have a ball

It rolls

[marsh compass]

(Btw I don't think that the gradient depends on a choice of basis, its coordinate representation does, sure, but other than that the gradient only depends on the choice of Riemannian metric / inner product)

[tight star]

Direction of steepest descent!

[cosmic ibex]

Your ball will roll to different places if you drew your graph with a different coordinate system.

[tight star]

Ummmm

That doesn’t sound physical

The graph is like

Height

And when you fix a coordinate system you get a function

But the ball rolling doesn’t care about that

It just rolls

[cosmic ibex]

Where it rolls depends on the amount you have scaled and mixed your coordinate axes before you realize the function as a graph.

[tight star]

I feel like you’re conflating the graph with the formula for the function

[austere delta]

Probably teaching the derivative is something I think you should do. But it's not like concepts that depend on an inner product are unnatural and should be avoided at all cost.

Whether you start with the gradient, then introduce the derivative or the other way around, I think should depend on your end goal.

[tight star]

The graph is the graph

It just vibes

And then you can describe it with a coordinate system

By choosing axes for space

[cosmic ibex]

I don't think they should be "avoided at all cost", but I don't see what's good about teaching them in a way that doesn't show that they depend on the inner product.

[tight star]

Hmm well

I am a physicist

I like the steepest descent way

[marsh compass]

Yeah, there's nothing good about teaching them that way imo

[austere delta]

I'm not advocating for teaching them in such a way.

[tight star]

Idk I understood it

[cosmic ibex]

Whether you start with the gradient, then introduce the derivative or the other way around, I think should depend on your end goal.
It feels manifestly wrong to start by introducing a concept that depends on an inner product, and then further modify it by the inner product to get a row vector. In this case two wrongs do make a right, but it would be easier on the students to teach the right way without the detour through an inner-product-dependent concept.

[marsh compass]

It's easy to understand first (if you introduce the gradient as a column vector with the partial derivatives of the function) but then it gets confusing if you try to extend it to curved spaces.

[tight star]

Yeah but that’s like

Laterrrr

[quiet jackal]

preaching to the choir

[marsh compass]

I do advocate for teaching the total derivative before the gradient

[quiet jackal]

unfortunately the choir is often constrained by stupid curriculum decisions

that they didnt make

[tight star]

Mmm i guess mathematicians do prefer abstract stuff first

[quiet jackal]

so then the question is how to make the most of it

[austere delta]

I don't see why introducing the gradient as something geometric that depends on the inner product first, before abstractly defining coordinate free derivatives, is a bad thing

[tight star]

Mhm!

[quiet jackal]

i mean

[austere delta]

Especially if you mostly intend to work with a specific inner product anyway

[quiet jackal]

i learned it that way and was quite confused a couple years later when trying to learn calculus properly

[cosmic ibex]

It's a bad thing because it leaves the students unaware that the coordinate-free derivatives is a simpler concept than they one they're told the coordinate-free derivatives are defined through.

[quiet jackal]

the problem is courses being designed to simultaneously satisfy the needs of engineers and math students

[austere delta]

Yeah, balancing between math and engineering can be a concern. For a pure math class, it would probably be appropriate to teach the derivative first. But I think the other way is fine too. I think it depends what kind of things you want to do, and how far between you teach the concepts.

[cosmic ibex]

Suppose I have a 2-dimensional real vector space V with points O, A, B, and consider the affine function f: V->R given by f(O)=2, f(A)=1, f(B)=1.
If I graph f according to a coordinate system for V where A has coordinates (2,0) and B has coordinates (2,1), then a ball released on the graph at the point representing O will roll to A.
But if instead I graph f in a coordinate system where A has coordinates (2,0) and B has coordinates (1,1), then the ball will roll from O towards B instead.

[tight star]

Mhm

If you treat the function as foundational

Then sure

I treat the graph as foundational

[cosmic ibex]

In a practical optimization problem there generally won't be any "natural" coordinate system to choose for the goal function. Imagine an industrial problem where the inputs include parameters like "length of the discombobulator arm" and "rotational speed of the main shaft". If we consider that in units of meters and radians/second, the direction of steepest descent goes towards one part of the search space; but if calculate in inches and revolutions/minute then we get a different direction of steepest descent.

[tight star]

Mmm there’s still an abstract cost function though

Our choice of units then gives us a formula for it

[cosmic ibex]

It's relevant for the engineer who needs a solution to the problem to know that the performance of her steepest-descent solver can depend critically on linear coordinate transformations for the parameter space. This fact will be obscured if she was taught how the solver works by considering the column-vector derivative as a primitive concept.

[tight star]

Oh hmm sure

But this feels like a while away

And also a principle of

Choose gud coordinates

[long pelican]

Interesting that for a simple function like x^2+y^2 the direction of steepest descent only points toward the origin in the standard coordinate system

actually I need to work that out before claiming this

ok I can see it without working it out: if x is scaled but y isn't, level curves are ellipses, and the normals clearly don't all point toward the origin

[tight star]

Mhm!

[cosmic ibex]

Well, you can rotate the coordinates without ill effects.

[tight star]

Mhm that’s one thing you will learn

About choosing gud coordinates

[marsh compass]

You would have to use a different inner product if you want to preserve the Euclidean structure under coordinate transformations.
Using the standard inner product in the transformed setting means that you work in a different Euclidean space entirely.

The gradient is invariant under coordinate transformations but its definition depends on a choice of inner product / Riemannian metric.

[astral zinc]

Hey guys!
I teach formal languages.
I have about 4 hours to solve some problems with the pumping lemma for context-free languages (which was proven in lectures), but before that I would like to give the students some intuition about the lemma.
But the best I can come up with is that big enough words will have repetition in their derivation.
I am wondering if there is another way to more simply explain the result.
Or maybe there is a good example grammar.

[lethal leaf]

Have they seen pumping lemma for regular languages?

[astral zinc]

yes

I did more examples using nerode classes though

[lethal leaf]

But then do they not have intuition from that?

Or are you looking for something new

Because the statement for CFGs vs regular languages is very similar iirc

[astral zinc]

they are similar but I feel like people get even more confused when you add more variables into the formula

they usually see that its a long formula and don't try to understand it

[lethal leaf]

I mean in regular languages the intuition is quite clear

[astral zinc]

I want something less scary 😄

[lethal leaf]

computing a string is a path on a graph, the graph is finite, so for long enough strings there must be a cycle, you can loop ("pump") that cycle

I guess for CFGs, the relevant automata has a stack which then you lose the cycle analogy

[cosmic ibex]

I would say the intuition is that if you have a parse tree and the generated string is long enough, then somewhere in that parse tree there must be a nonterminal that is its own ancestor in the tree. Pumping corresponds to cutting out the part of the parse tree between those two instances of the nonterminal, and stacking a number of copies on it instead.

So when using the lemma, we're looking for a string where any way to imaigine doing that will take us out of the language.

At least, for me, when I need to remember how all of the many substrings in the lemma should fit together, it helps to imagine some "generic" parse tree above the full string.

[astral zinc]

I guess I'll just tell them to at the very least remember something like this:

[cosmic ibex]

Together with this.

[elder timber]

I've designed a course in measure theory, where everything is motivated by the desire to interchange the limit and integral. So the whole reason for the Lebesgue integral is to try to prove the validity of the interchange under simple conditions, which then motivates the creation of the Lebesgue measure.

And basically all of that is motivated by the desire to find the coefficients of Fourier series.

Which is motivated I guess by whatever physics or information theory applications you might like.

Anyway, if you were going to try to similarly motivate Lp spaces, is there some big canonical result that motivates the theory?

Completeness is a big result, but how could one explain the desire for this particular completeness?

Are there particular Lp functions which we really want to be able to approximate for some reason?

[teal iris]

The question should be more whether there's motivation for considering the L^p norm, right?

Once you have a normed space, it's natural to want to take the completion to fill in “holes” (e.g. you might be trying to find a solution to something and you find a Cauchy sequence whose limit would work, but the limit might live in the completion rather than the original space).

(I don't know why general L^p norms are useful, so can't help with that.)

[vocal phoenix]

https://mathoverflow.net/questions/28147/why-do-we-care-about-lp-spaces-besides-p-1-p-2-and-p-infty

There's some discussion here

[elder timber]

Certainly if there's a motivation for the L^p norm then that would work. But thinking "once you have a normed space..." makes it sound like we got the space first. Which might be true, I dunno, but I don't necessarily assume it.

After some research, the best answer I've come up with is: some people discovered $\ell_2$ as a result of investigating quadratic forms, which themselves came from Fourier coefficients, which themselves came from (I think) some kind of heat problems.

[burnt vessel]

addem

[elder timber]

Then later, looking at differential equations which came from models of string vibration, Hilbert found that solutions were guaranteed to exist if we assumed that certain functions were "in $L^2$" (although he wouldn't have said it that way at the time). So I think that's the historical origin of the interest in these spaces, and from there people probably started noticing the similarities in the two spaces, and so on.

[burnt vessel]

addem

[wide ice]

Not necessarily the motivation, but an application that comes to my mind is Method of Moments

And for that one, the general L^p theory helps

[vocal phoenix]

The motivation for L^p is much stronger for p = 1, 2 or infinity 😄

I confess I don't know much about the usefulness of the other cases

In finite dimension L infinity is in a sense the limit of Lp, so maybe there's some approximative/interpolative motivation

[midnight scarab]

The idea of Fourier series was for the wave eq and due to Bernoulli. But yes Fourier himself was working on the heat equation. And then Dirichlet realised the relation to the theory of group reps after seeing Fourier's work

I wanted to say that L^p boundedness of a family of functions for p>1 implies uniform integrability, but I think most examples I've seen (from martingale theory) used specifically L2

[elder timber]

Yep, that's helpful! I suspected this was true and plan to look more into exactly how it's helpful to frame that in L^p space rather than Riemann integrals. Probably, at some point, it's useful to interchange a limit and integral -- it's just been too long since I looked at the method of moments, so I just need to brush up.

[elder timber]

Yeah, same here. I've seen a few people talk about the p-Laplacian operator and other niche PDE stuff, which is cool. But honestly I feel like the generalization from 2 to p to infinity is so immediate that it doesn't take too much motivation. Maybe that's just my personal taste though, I dunno.

[elder timber]

Interesting about the group rep thing, I wasn't aware of the connection -- I never went super deep into group theory. But I always love when things which initially seemed unrelated become related.

[midnight scarab]

Yeah the e^ikx are eigenfunction of the translation operator and its "generator" d/dx

It ties directly to quantum mechanics and the momentum operator

[quiet jackal]

Lp can be useful for more general contexts where you dont have a nice martingale structure

the most basic example that comes to mind is the kolmogorov continuity theorem

one can use it to prove existence of e.g. continuous versions of gaussian processes by first establishing an a priori estimate for high moments of a gaussian

there is also the burkholder-davis-gundy inequality

generally the L2 theory may not allow to take full advantage of high moments

[midnight scarab]

[wintry lichen]

#dynamical-systems

[vagrant meadow]

i'm sure deleted user will be very happy to know their question from 2019 was answered in 2024

[vocal phoenix]

I mean, some questions have waited for an answer for much longer than that 😛

[vagrant meadow]

better late than never! I just think it's really funny to answer something from so long ago, and from a deleted user who won't see it no less

[tawny slate]

So as stated from the previous channel #groups-rings-fields I'm trying to understand the motivations of the definitions of various abstractions of addition/multiplication

What is the common thread in all of the abstractions of these operations named "addition" or "multiplication", if there is one

I had 3 in mind, which are not necessarily distinct foundationally, but different enough that I think it's good to mention pedagogically

[brisk tundra]

I think you are really just looking at two notions: formal addition as codified as associative, commutative binary operations and things that derives itself from it. Your example of ordinal arithmetic can be understood as "set addition", where the sets have ordinal structure and then extending the added sets with ordinal structure

[tawny slate]

• mult is repeated add
• abstract properties like distribution/commutation
• linear transforms (like with complex numbers)

[brisk tundra]

Addition definitely isn't linear

0+a generally isn't 0

Nvm you meant multiplication ig

[wary ether]

Multiplication is very rarely repeated addition.

At least in rings.

I disagree with that part of the list.

If you look only at Z, then distributivity implies that multiplication is repeated addition — for that particular case, I remind you. So really you are just talking about distributivity again when you say that.

[tight star]

It’s an interesting property of Set that multiplication is repeated addition

But yeah I think matrix multiplication is called that cause it distributes over matrix addition

It also happens to be noncommutative which is probably why multiplication got adopted for other noncommutative operations

[tawny slate]

You make a good point about distribution, but what about ordinal addition and multiplication?

Also this is ultimately supposed to be a pedagogical list

So I think it's fine to keep the definition?

[tight star]

How do you decide when trying to make some specific piece of pedagogical content is worthwhile?

Like at one end of the scale, you have literal plagiarism where you just copy an article or reupload a video, which is just unethical and bad.

And at the other end, you have some completely original idea with a completely fresh take on a topic. This is almost certainly worthwhile pursuing but is also very rare to come across.

So what’s a good cutoff point? Like, it takes time and effort to produce teaching material, so it’d be good to know when you should expend these.

[quiet jackal]

whats wrong with copying teaching materials

like obviously you should properly cite things but that takes a couple minutes at most

[tight star]

Mhm hence why I specifically said plagiarism

[quiet jackal]

but like

the time difference is negligible

[tight star]

I guess then it’s like

How much is it worth putting something up if it’s already out there?

Is the kind of Q i wanted to have answered

[quiet jackal]

i mean if you make the material

then uploading it to your website or youtube is again a trivial amount of time

i thought you were asking about whether to recreate existing materials

[tight star]

Making the material is nontrivial effort

So the idea is what things to focus on

[tight star]

And is it worth investing time in things like this

[tawny slate]

I would say firstly, what's your goal?

Because if my goal was to reach the most students and impact the most students in the most efficient manner, I would be doing things very differently

I would probably be splitting my videos into shorts and putting them on tiktok

I would definitely cover other topics than what I'm currently doing

But because my goal isn't that, my goal is simply to take the intuition and understanding that I have, that I think is either unique or not found anywhere else, and record it somewhere for everyone else to see

I want to contribute any and all ideas that I think are unique to me, that allow people a portal into how my mind works

Simply because I don't have time to rehash what has already been taught countless times before, and I think this is the best and most efficient application of my abilities

This is clearly not going to be as mainstream as khan academy or 3blue1brown, but I'm okay with that

Secondly, you can ask what your content will add functionally speaking

Once you know what your goals are, evaluate how you're doing something differently

Are you taking an existing idea and just making it a video/interactive?

Are you explaining it in a simpler or different way?

It will be easier to decide if it's worth it if you compare it against your goals

Sometimes, you need more information, so it helps to just try and actually teach some students and see how well it performs

Do it and test it

Testing does more than just help you evaluate if it is worth it, it can also help you find weaknesses and mistakes to make you and your content stronger for next time or when you revise it

So I guess third would be sometimes you just take the plunge and find out later

Experience can help turn something less worth it into something more worth it

[noble hare]

I have to present this problem and the solution to a group of ~10-16 y/os tomorrow. I'm not entirely satisfied with the written answer and don't want to confuse the students. I particularly dont want to introduce or reinforce a misconception like "1^1/2 = \pm 1" (this is somewhat addressed in the solution) and I also worry the "represent a set of possible values" could be misleading/confusing. Any comments on what I could/shouldn't say would be appreciated

[quiet jackal]

this seems atrociously written

as (-1)^2 is not multivalued

maybe it makes sense if you read the equations from right to left

like emphasize that the issue comes from choosing the wrong value

but idk

[vagrant meadow]

I think a=1/2 and b=2 not the other way around

[vagrant meadow]

I think the main issue is that x^2 is not injective. idk if I'd say that x^1/2 is multivalued. the set of all preimages under x^2 (what we call square roots) is a set with multiple distinct elements, but x^1/2 (which we also call the square root function) is a function defined from nonnegative real numbers to themselves.

[midnight scarab]

This seems like just a stupid way of saying (-1)^2 = 1 yet sqrt(1)=1 and not -1

[noble hare]

I think the point they really mean to make is that x^(m/n) is only equal to (x^1/n)^m when n is odd (or something similar with more nuance)

they just seem to be trying to explain something to that effect in the worst way possible

[vagrant meadow]

I guess. but I think the problem is more generally that even if g is defined as an inverse function of f (where f is not injective) it may not be the case that g(f(x))=x
in this case, ((-1)^2)^1/2=1, not -1

[vagrant meadow]

this is probably more concrete, but the explanation hides the issues with injectivity under the hood

[noble hare]

i agree, issue is these kids will have no idea about injectivity and im not sure its something id want to bring up in whats meant to be a couple minute intro example

some of them will have possibly never seen a function really (UK)

[vagrant meadow]

you can show a graph of x^2 and show that two different points get you to 1

horizontal line test and all that

[noble hare]

yeah that could be a good

[vagrant meadow]

obviously don't use the word injective, I'm just using that to talk about the problem itself and leaving it up to you to translate it into layman speak

[noble hare]

yes yes

thankyou for your insight

the second problem is the "infinite chocolate trick" but presented as a series of 5 pictures , i will certianly be showing them the video instead

[vagrant meadow]

god yeah definitely

[quiet jackal]

yeah but then the issue is trivial: we cant make sense of sqrt(-1) a priori

so it is still pedagogically useless

imo this is not worth explaining unless the audience knows some precalculus

namely exp(ix) = cosx + i sinx

so then you can explain that one cannot define a^b due to periodicity

[noble hare]

yeah i think im happy to sort of black box it as "this power rule doesnt always work because taking powers isnt always a reversible operation"

[near oriole]

For the bare "why", I feel the injectivity explanation is the simplest, and I see no reason you need to involve complex numbers.

(-1)^2 = 1 = (1)^2.
But 1^0.5 evaluates to a single number (you can talk about how x^0.5 is a function on the non-negatives, if that's appropriate). So it can't be both -1 and 1.

Agree this is not particularly enlightening for a broader "why" but

The broader why wants to address why that identity even should hold in the first place... it's obvious via an inductive proof for natural a, b. Still alright for integers; negative means dividing. But then rationals, what was said above about the root functions

[tawny slate]

I approach this totally differently

I would generally use fewer words and more pictures

something like this helps highlight the idea that two different numbers square to the same result

or this

ask the students how you would reverse such an operation

they would likely say that you want just use both, so you don't leave out a solution

at that point I would give examples of why that's actually less useful

show them that an expression with n square roots in it has a maximum of 2^n possible different values (no need to be rigorous here, just show the 2 and 3 cases and they get the idea)

[tight star]

Ooh this looks like monads

Sort of, at least

[tawny slate]

and now they realize:

• why it's important to only select one value
• why it's not reversible
  and now they are oriented properly to think about the problem

the students now know what they should be looking for

so at this point, demonstrating that the error occurs here

is clear, because this is where we are trying to reverse something that cannot be reversed properly

and now you can abstract to the notion that x^(ab) is not the same as (x^a)^b

explain how it relates to this

I find that, in my own experience, separating each step clearly with very explicit examples and logic helps the students break it down and digest it

@noble hare

[plain valve]

this looks good

[near oriole]

Id be worried about it causing more confusion with regards to that error. Arguably, you have the same situation for equations having multiple solutions, but that's supposed to be "ok".

At a fundamental level one definition is not more "right" than the other --- it's about whether we want a multifunction or function. One is more convenient than the other in hs math.

=
I would prescribe the notion the square root function spits out one value (the non-negative root) without saying why.
That's how it was introduced to me at least.

=
I'm of the mindset that at an early stage, it's usually better not to bring up things having multiple possible definitions. This reduces confusion between them, and there is only one canonically "right" definition that needs to stick. Later on, they're free to find out that's not the case.

[tawny slate]

to be fair, at this point, I will have introduced to students the idea that when solving an equation, there isn't one correct value, but one correct set.

that students are looking for all values that can make the equation true

that may help

another thing that might help is to explain that it doesn't matter which value you're specifying, you can use existing language to do so

for instance, in the sqrt9 + sqrt4 example, suppose you wanted the sqrt9 to be negative but the sqrt4 to be positive

easy, just do -sqrt9 + sqrt4

all you need is a minus sign in front

I think my preference as an educator is i would rather explain it this way the first time

not explaining this correctly from the get-go can lead to some confusion down the road, when I won't be there to help clarify things

it also feels like a math lies-to-children if i don't take responsibility in clarifying that

there are some things I think you can and should skip at this level, for instance, using real analysis to explain what a real number is, but when it comes to notation, evaluation, vocabulary, some of the most fundamental tools components of math and communication, i think it should be explained as early as possible

I have personally not yet come across students that become more confused after my explanation than less, but I understand this is anecdotal

[noble hare]

thanks btw @tawny slate ! was after I had ran the session but i appreciate the input

[late schooner]

does anyone know of a desmos equivalent for plotting complex loci? e.g illustrating arg(z-i) = pi/4 as a ray

[elder timber]

I would expect that the best you're going to get for this, is a very manual approach using either matplotlib or shader programming. I doubt you're going to find anything that does this "out of the box".

[tepid smelt]

Recently, I've been inspired by Polya's 'Mathematical Discovery' and 'How to Solve It.' These books have been instrumental in shaping my understanding of teaching heuristics in mathematics. I'm eager to expand my knowledge in math pedagogy. Do you have any book or resource recommendations on this topic? I'm particularly interested in insights from fellow educators.

[late schooner]

yep, this is what i resorted to

[turbid zenith]

So I'm planning a lesson for my abstract algebra class about functions and symmetries and how they can form groups. I'd like to use the word "actions" to describe both of these in one fell swoop, with the idea being that you can compose two actions to get another action, whether that's composing two functions or composing two symmetries. Later on in the semester, I'll be introducing group actions, and I THINK that should be a seamless transition, but I want to make sure I don't inadvertently confuse the students. Any advice on what I should be careful of?

[minor turtle]

has anyone ever TA'd for calc II before? What was the hardest/easiest part of teaching it for you? and do you have any advice? I really want to do well this semester teaching

[vocal phoenix]

I'm not sure what exactly comes under the heading of calc 2 in your case (my experience is in Poland, so the range might be different), but I remember students struggling the most with:

1. determining the limits of integration in iterated integrals (and especially changing the order of integration)
2. the concept of series convergence (frequently confusing the limit of individual terms with the limit of partial sums)

@minor turtle ping in lieu of reply because I always forget to use the "reply" feature

[lethal leaf]

I think the use of the word "action" will just go over their head without introducing new axioms

At least if my prof did this, that's what I would think

You'd just be using the word action as another word for composition

And when you actually introduce group actions I won't recall the previous use of the word action

[long pelican]

Action as another word for group element, rather?

[turbid zenith]

I ended up just calling them all “functions” tbh.

I’m just emphasizing that I want them to think of functions as “doing” something

[faint cradle]

"Composing two symmetries" doesn't immediately make sense to me. Does it mean composing two symmetry-preserving transformations? If you have a group of functions G and a group of symmetry-preserving transformations H, you can consider their group action on some appropriate set S. It's slightly unclear to me what the "two actions" are under discussion as the unifying concept is a group action on a set S, where it's not so important whether the group G or the group H is considered.

[uneven juniper]

3x9

[cloud zealot]

Is there any literature comparing the pedagogical efficacy of a rings-first approach to a groups-first approach in an undergraduate abstract algebra course?

[bronze stirrup]

how do i get better at judging what hints are necessary to help others

i feel like sometimes when i help, it sometimes feels like i give the answer outright and feel kinda like cheating

[noble hare]

Try socratic questioning. Figure out what it really is you want to tell them, and try and figure out how you can get them to say that thing themselves through a series of exploratory questions

[bronze stirrup]

[noble hare]

Also if youre talking about helping on this server, maybe take some time to watch some other people help and see what they do. I'd say on average youre better off looking at a green but ofc there are like 200 of us so we vary

[bronze stirrup]

nah, it's for helping people ik

[wispy slate]

This is great advice, and what we are taught as math teachers. Asking them questions about their reasoning and how they understand the problem can uncover potential misunderstandings that you can then work to fix.

[winged urchin]

I do find it funny how in many tutoring sessions I am the one asking them more questions than the other way around ahah

But there are some students who are... stubborn to make attempts or otherwise are just looking for answers

As a last resort if you feel they aren't (or you can't get them to) follow a path of discovery then you can 'solve' a similar but different problem

Avoids the problem of just directly giving them the answer although of course it just enforces a sort of pattern matching mentality, which is not ideal

One thing you can do too actually, is try to find the simplest point where they can understand

If an equation has fractions and the fractions are seemingly tripping them up then perhaps going down a level and asking them more basic questions of fractions and see where they can do it and work from there

Sometimes this takes awhile though and they can feel like you are wasting time or just trying to get more money out of them if they are paying for your services

[sharp hawk]

https://youtu.be/oyCprtvkTQI

[long pelican]

Right thing to criticize, but unfortunately incorrect actual criticism 😔

I can say a bit more if you want but otherwise I'll let everyone else say their thoughts

[austere delta]

If only this was blasting some lil'Wayne in the background and explaining to me how RuneScape works this would take me way back

[sharp hawk]

Please go ahead 🙂

[long pelican]

It's a gimmicky problem, students quickly learn it as "just another" problem type to learn/remember, but the math is correct

Such a problem is good the first few times it's used but when it becomes standard you can't use it it reliably measure understanding anymore -- too many false positives, due to the fact that most students study standard problem types

[sharp hawk]

The first derivative is also for the left neighborhood of 0.
So no need to calculate the left hand limit. (Because it was already calculated by the first derivative)

Same for the right hand. It's already calculated by the 2nd derivative (the derivative definition is that limit; no need to calculate it again)

[long pelican]

ok I didn't expect that you wanted me to expand on why the math side of the criticism is invalid, I thought you wanted me to expand on why I think it's a bad problem haha

You can ask #calculus for the definition of the following words:

• function
• first derivative
• second derivative
• left hand limit
• right hand limit

#advanced-lounge might be willing to explain it to you as well

[sharp hawk]

Well, you said the math is correct.

[long pelican]

Yes, the mathematical aspect of your criticism is incorrect

For example, functions are not formulas

[sharp hawk]

In this case you can point out the mistake in these claims

Claims:

So no need to calculate the left hand limit. (Because it was already calculated by the first derivative)

Same for the right hand. It's already calculated by the 2nd derivative (the derivative definition is that limit; no need to calculate it again)```

[long pelican]

give a screenshot of the problem too

[sharp hawk]

Ok

https://media.discordapp.net/attachments/1070487342059634708/1203289887445491773/image.png?ex=65d08e09&is=65be1909&hm=8d9da4f5a09f5d159370c01f09ffd49ef70a07c2a20cdcca5460dbb87f7dadce&

[long pelican]

haha good I asked, now it clarifies you didn't mean f'' with "second derivative"

[sharp hawk]

Yes, of course. The problem was not about second order derivatives 🙂

[long pelican]

ok this specific screenshot's criticism is valid

[sharp hawk]

Ok

[long pelican]

this is not

[sharp hawk]

The other part is a bit subtler

I can explain

For example 0 should not be considered a number. It is a convenient symbol, with great utility, but it behaves differently than a number does.
That's the reason why division by 0 does not work: because it is not a number. It is not the same kind of mathematical object as a number.

And indeed that is one of the reasons many authors don't consider 0 a 'natural number'

0 is a different kind of mathematical object than a number is, so it behaves differently than a number.

[long pelican]

0 was indeed debated hotly in the 15th century (I might have gotten the century wrong)

[sharp hawk]

Here we are debating it 🙂

[long pelican]

I think the fact we accept 0 as a number and why we do is pretty rock solid

I don't really want to debate 0 right now

I don't think anyone here is interested in debating 0, honestly

[sharp hawk]

Are you sure ? That we accept 0 as a number? Think carefully.

For this claim to hold we'd need a precise definition for the concept of number, and then show that indeed 0 also satisfies the definition of number.

It's easy.

[tardy ember]

alright here's a definition then

[sharp hawk]

We certainly do say words such as 'number', and sentences such as '0 is a number', but this is just linguistically.
What I am saying, is that mathematically we are not working with 0 as a number.

[tardy ember]

a "natural number" is an ordinal that is an element of all limit ordinals

[sharp hawk]

What does 'all limit ordinals' mean ?

[tardy ember]

the sum of two natural numbers is the unique natural number equinumerous with their disjoint union, and the product of two natural numbers is the unique natural number equinumerous with their cartesian product

[tardy ember]

any ordinal that is not the ordinal 0 or the successor of any ordinal

an ordinal is a set that is transitive and which is totally ordered by containment

the ordinal 0 is the set with no elements

the successor of the ordinal a is the set that contains any set that is either equal to a or an element of a