math-pedagogy

[turbid zenith]

But wait, how do you earn the fourth Star on Topics 1-4? Well, on the cumulative questions on the Final Exam, you can earn more Stars. Let's take Integration for example:

• If you have 3⭐, then you need 70% of the points on the Integration questions on the final to earn the 4th star.
• If you have 2⭐, then 70% = 3⭐, 80% = 4⭐.
• If you have 1⭐, then 70% = 2⭐, 80% = 3⭐, 90% = 4⭐.
• If you have 0⭐, then 70 = 1⭐, 80% = 2⭐, 90% = 3⭐, 100%(!) = 4⭐.
  The idea is that if you didn't get something the first time, but you study it and show you've learned it by the final, you get credit for it toward your final grade. If you try to push it off, you CAN just earn it all on the final, but you've gotta do really well to be able to do so, so you're still incentivized to try during the semester. For Topic 5 (Limits), since it's only tested on the Final Exam (as it's the last thing we do), you can only earn 3⭐ in it.

. . . okay, all of that went through. I have no idea what happened.

[feral swan]

I think there's a term for this, achievement based learning or things like this.
it sounds pretty interesting I've always wondered if it can work in practice.

[rapid tusk]

mastery based?

[midnight scarab]

Emoji limit maybe?

[turbid zenith]

That’s a thing?

[turbid zenith]

Yeah it’s been called Mastery Based Grading, or more recently Grading for Growth

[quasi maple]

Sorta? Emojis typically take up more than one character unit

[turbid zenith]

Hmm I see

[midnight scarab]

I think so, yes

[dapper flume]

My calc class in college identified 14 Core Learning Targets which were graded similarly. It was so effective that it inspired me to do standards-mastery grading at the middle school I work at. I'm a huge advocate of mastery grading.

[rigid coral]

Looking for feedback on this video. https://youtu.be/J6KNoGZJj98

I'm trying to explore various ways that math can be enjoyed by people in a more casual sense, and I am planning to make another one of these for calculus (with more explanation on what I'm doing and why), but this is just to see what you all think about the format.

[white fulcrum]

I like the idea of a speedrun, it'd be better if you could get someone to race against you

also I've gotten reduced hours at my tutoring job, and I suspect it's because business isn't doing well 🙁 . I hope I can keep doing it

[turbid zenith]

https://bill-j-shillito.github.io/calculus-scbi/online/sec-approximating-the-circle.html

Milestone reached: finished a draft of the introductory section!

Any feedback or questions are very much welcome

And suggestions for exercises relevant to the section are very much welcome

[zinc dove]

For how many years have you been teaching Calculus?

[turbid zenith]

Tutoring it since about 2010, teaching it in the classroom since 2013

I will say, I had considered having some trig-based exercises, but I'm waiting until Chapter 4 to define sin and cos in terms of the harmonic motion ODE

So I'm keeping this one to just stuff you can do with the Pythagorean Theorem alone

[tight star]

Putting myself in the mind of a student learning this for the first time, it’s not immediately obvious to me why infinitely many wedges would produce a rectangle? I feel like I could imagine having an infinitely wiggly top and bottom instead of the straight sides of a rectangle. Though I guess that may not change the area…

Another thing I might worry about is that if we use infinitely many pieces, they all have infinitely small width. So it feels to me like I could “compress” the rectangle like an accordion to change the apparent area?

In the area formula, I think it’s fine to use pi r^2 since students have probably met pi by that point, and might not immediately recognise 1/2 C r. When I first saw it I honestly thought C was a constant and got confused since area should be proportional to r^2

[turbid zenith]

...ope yeah I really should have instantiated C as the circumference 😛

Making the jump from "lots of sides looks a lot like a rectangle" to "infinitely many sides makes it exactly like a rectangle" is definitely hand-wavy, deliberately so

Since limits come in at the end of Part I to make those ideas more precise

[tight star]

Also somewhat related but have you come across this meme

[turbid zenith]

Of course 😛

[tight star]

What would your response to it be

[turbid zenith]

"Good question, we haven't developed the tools to deal with that yet"

[tight star]

Ok what about to a fellow educator

[turbid zenith]

Do you have an answer in mind yourself already?

But I guess to a fellow educator, it would be that the limiting processes don't commute

The perimeter of the limiting shape doesn't equal the limit of the perimeters

[tight star]

But you used inscribed polygons as one of the exercises right?

So when does it work and when does it not?

[turbid zenith]

What would you suggest as the best answer here?

[tight star]

I’m not sure since I don’t have as much experience

I have an answer but idk if it’s the best

In part that’s why I was curious about your answer

[turbid zenith]

I'm not sure mine is the best either

I'm not sure what the 110% rigorous real-analysis approved (TM) answer is

Probably uniform convergence or something like that idk

Most of my answer to that question has usually been of the form "if that were true then this absolute nonsense would also be true"

e.g. the diagonal of a unit square is 2 by the same argument

[zinc dove]

Are you currently using the structure of this book as a basis for you classes?

[turbid zenith]

About to in the fall, yes

I used something similar the last time I taught through the whole sequence

So instead of having to grab something from here and something from there, I'm putting it all in one place

[tight star]

Sure though this just makes me curious what exactly fails in that argument

[turbid zenith]

Thinking about it some more, one thing about the pi = 4 is it DOES show that the circumference of the circle is AT MOST 8

But the bounding regular polygon process (numerically at least) brings that upper bound down to 2π

And the inscribed regular polygons bring the lower bound up to 2π as well

But I don't have a good answer for "why doesn't the other way work" past what I've already said

[zinc dove]

Yes indeed. Thank you!

[rigid coral]

Yea that's what I was thinking too, something like an MIT integration bee but with much easier questions so it's more accessible. It's something I want to do in the future once I get more recognition and I can host these bigger events

I simply think there is too much of a notion that math is for "smart" people and not enjoyable for many

[turbid zenith]

This is cute! I like it!

The split lol

[rigid coral]

thanks!

[quasi maple]

"my new circle is jagged af"

[pure light]

Video on this and also a surface area calculation with similar issues https://www.youtube.com/watch?v=yAEveAH2KwI

at the end he explains that to converge to the interior size (area, volume, etc) you just need to converge to the shape, whereas for boundary size (arc length, surface area, etc) you also need the tangents to converge. Which is at least a reason although it was lacking in justification

[turbid zenith]

In what sense does the tangent need to converge?

As in how would we measure convergence in this case?

[tawny slate]

man i thought of an analogous scenario but i think it's more complicated not less, but i guess i'll share it in case it inspires any good ideas

[rough kettle]

yaaas share it i'm all ears

[tawny slate]

what's the probability that if you select a random real number between 0 and 1 uniformly that you get a rational number?

selecting a random real number is something you can do in a handwavy way, but if you wanted to be more rigorous, you can for instance first split the 0 to 1 line into 10 parts, and pick one at random (this is like selecting the first digit at random). then by repeating this process n times, you get random decimals with n digits. taking the limit as this approaches infinity gets you a real number

but at any point of this process, the probability of obtaining a rational number from this random process is always 1. taking n to infinity therefore doesn't help, the probability is still 1

unfortunately, not only does this contradict the conclusion that the probability must be 0, in order to understand the proper correct answer, students need to first understand that the cardinality of the continuum is larger than the cardinality of the countables, which i think is one of the main reasons this is more complicated not less

[stuck bramble]

I'm hoping it's okay to ask here, but what are people's thoughts on delaying learning calculus in favour of other topics such as linear algebra, abstract algebra, and some basic combinatorics?
For some background, I've found myself in a mentoring position where the student is interested in those subjects. They took calc 1 quite a while back, but nothing beyond that. Their ultimate goal is to get a computer science degree, but they can't pursue that for a few years. Just wanting some opinions before giving potentially harmful advice. My background is in number theory and algebraic geometry, so I'm a bit biased in my views.

[tawny slate]

my experience is also going to be quite biased, being mostly a freelance math tutor with no formal math degree, but i think this is fine

firstly, most people (at least in the US) lack even basic math competency, so i think any student that reaches this level is already doing quite well, relatively speaking

secondly, i think while calc does have a lot of very useful and interesting ideas, calc 1 and calc 2 are really where most of the "meat" is, and you can just pick what you need from them as you need. especially if you're doing computer science, i think the topics you mentioned are generally speaking higher priorities, assuming we don't know what they are going to be working on specifically

in calc, the big ideas i think that need to be understood are mostly: definitions like continuity and differentiable, the derivative/integral (and basic applications like polynomials, trig funcs, chain rule), Taylor series, and analysis like radius of convergence and such. the rest is not of critical importance I think

but given my particular bias, i do hope others chime in and offer their thoughts

[midnight scarab]

The construction obviously gives an upper bound, but it's not a priori obvious the limit/infimum coincides with the circle's perimeter
The reason beinf the same as the fact that if f_n -> f uniformly, then int f_n -> int f so you get the correct area, but int |f_n'| does not necessarily converge to int |f'| (here I'm imagining the simplified situation with just a horizontal zig-zag)
All of this can also be seen using epsilon-delta (and is very easy for the horizontal zig-zag: there are order 1/epsilon triangles of area ~ epsilon^2, but perimeter ~ epsilon)

I'll also add that these difficulties can be circumvented by also having an approximation from below

And verifying upper bound - lower bound -> 0

[tight star]

why does it give an upper bound?

[midnight scarab]

Well, ig it's not as obvious geometrically as for the area (just an inclusion)

[tight star]

i'm not even convinced you can prove it geometrically

[midnight scarab]

4 > pi 😛

But you're right, and that's not really important to my point

Which was that there is no actual reason to believe this quantity somehow controls the perimeter of the circle

[tight star]

then why do the inscribed polygons work?

[midnight scarab]

For areas, it does provide upper and lower bounds

And they converge to each other

[tight star]

yes i agree with the area argument

i'm thinking about perimeters

[midnight scarab]

First of all, you need a definition of curve length

Which will basically be sum lengths segments joining points on the curve, with steps going to zero

All based on intuition from

[tight star]

Interesting

First time I’ve seen this result

[midnight scarab]

And lemma 2 is also how you prove the perimeters of enclosed perimeters are smaller than those of broken segments approximating the circle, ans thus smaller than the circumference

[tight star]

Thanks for sharing!

Where’s the screenshot from?

[midnight scarab]

Kiselev (Givental's translation)

It was the official (middle school...) geometry textbook in the ussr in the 1920s-30s

[vestal tangle]

i think it makes a lot of sense

linear algebra seems like an easier thing to move into having just finished elementary (high school) algebra

and i think knowing linear algebra makes calculus, especially multivariable/vector calculus, easier

[turbid zenith]

I made some edits that I think will address some of your notes!
https://bill-j-shillito.github.io/calculus-scbi/online/sec-approximating-the-circle.html

This especially

[midnight scarab]

I find it funny that you're computing the circumference by approximation rather than the area, since the former is much harder to justify (see the discussion above)

It's also true that it likely doesn't matter to your students

Btw, dropping the edges (which protrude beyond the triangles) is actually pretty easy to justify: you can include each wede in a small square of side epsilon, so the error is order 1/epsilon × epsilon^2

On an unrelated note, I think geometry is a better intro to limits than decimals, because students seem much more ready to agree that the area or circumference have a definite numerical value

So like, the process VS limit value distinction is more manifest

So you avoid the 0.999... nightmare

And when you get to it, you can also use this geometric intuition to explain the distinction

Great exercises btw!

[turbid zenith]

Yeah, for the students' sake, the main point here isn't actually to justify the area formula OR the circumference formula — in neither case do I have enough for a rigorous proof

The point is to plant the seed of "curvy things look straight when you zoom in close enough"

And to get them thinking of infinitesimal pieces (though also to plant a little bit of doubt on whether that's valid, so that can be resolved later)

[midnight scarab]

I do think it'd be cool to revisit at the end (after proving the FTC). Maybe not in class but just for anyone curious

[turbid zenith]

Perhaps! Would be a cool way to come ... full circle

[halcyon glade]

I don't think it would make sense to do abstract algebra, but linear algebra first makes perfect sense and is probably more widely applicable in this day and age

basic combinatorics I think is already done as part of the algebra curriculum in US high schools? maybe this varies from school to school

I would personally prioritize both statistics and linear algebra over calculus but I understand some people want to do physics :<

[rapid tusk]

its not covered very well from my experience

anything actually substantial with binomial coefficients gets handwaved away by "just look at pascal's triangle bro"

permutations and combinations are just presented by rote, their formula is kinda just dumped there

as a footnote to binomial theorem etc

[tawny slate]

i took an accelerated route and combinatorics was barely touched
not sure if its changed much since, but i doubt it

we didnt even cover pascal's triangle

[white fulcrum]

There was a thm that was proven by showing that it's true if RH is flase and also if RH is true, so by LEM it's true. which one is it?

[pure light]

from Ireland and Rosen, A Classical Introduction to Modern Number Theory, 1990 (2nd ed.)

[white fulcrum]

I'm looking to give this as an example for my students, is there a way to make the result even remotely accessible?

[white fulcrum]

also the three results that say if GRH is false, where's the one that says if GRH is true?

[pure light]

at the top

[white fulcrum]

oh I see I missed that

[white fulcrum]

thanks! The next step: what's the name of the result, and is there any way to make it accessible?

even a little bit accessible is fine

[pure light]

the leadup to that if you find it helpful

[white fulcrum]

is the 163 related to ramanujan's number? exp(pi*sqrt(163))

also I guess I'd have to study this first before I could make it accessible

[late vortex]

Im tutoring some highschool students that graduate next year. I think whatever they are doing in class is not preparing them for the math in a stem degree. By extension, im fearing that this is the case for the subjects they are more passionate about and maybe want to pursue in university aswell.
Atleast for maths (in germany) i dont think its a controversial statement that the level has gone down (atleast in highschool, idk for university).
However this is not what i want to focus on (unless someone heavily objects to that statement).
For me its quite upsetting that someone can get a grade that makes them feel "everything is going well/amazing, no need to worry or work harder" and then they go to university and learn that they have to play catch up for a while (or in a worse case the amount of catching up to do feels/is too big and they get discouraged/fail and quit).
Again, feel free to correct me if you think im beeing too dramatic or that this is not the reality.
Anyway, are some of you feeling similarly? If yes are you openly talking about it with your students when youre more familiar with them? And if you talked to them about it, did they appreciate it?
Like i was very clueless in highschool about such things and I think I would have appreciated it if someone would have told me this in a nice way.
In my case the students are doing well in their other classes, so I feel like its not like i would cause extra stress to a person that is already having a hard time in highschool.
On the other hand I dont have that much experience with highschool students, so id be interested in what you think about it.

[warm gazelle]

Guys ,You all pedagogies which book would u guys like me to recommended or CHALLNAGE specially?

[midnight scarab]

I would've appreciated it, yes. Well personally I knew I was not satisfied, and I did take some steps to do more. Though present me thinks I should've been even more unsatisfied, and perhaps also done things somewhat differently, but hindsight is just hindsight

Anyway, I don't think there's anything wrong with showing them some of the things you think they should know but don't, and why they're important

Obviously, I don't know what personality your students have, and how'd they'd take it

[surreal lily]

I don’t have enough experience with high school students to answer your question. As someone who teaches algebra to college students, where I am (in North America) we definitely have the same problem. There’s been a lot of grade inflation in high school and it means that students come into college, expecting to be able to get good grades in math and get really frustrated when they discover that they have gaps. Some students who want to be engineers have to start well below calculus and even though we offer half semester courses for people in those situations, they’re still in a really tricky situation. Parts of the system don’t work very well and it’s frustrating for me as an instructor and I imagine it’s worse for students

[turbid zenith]

I think the grade inflation is a result of grades becoming the all-important metric for success on top of the fact that if you make a mistake it sticks with you forever

Teachers are afraid to give bad grades because it’ll destroy the student’s chances at whatever

And as a separate but related issue, if an institution is using grades to rank and sort students rather than reflect achievement it becomes a really screwed up arms race with perverse incentives everywhere

(In my opinion if mastery based grading could become more universal it would fix at least some of these problems, both at the high school and the college level)

[turbid zenith]

But you’re in a good spot as a tutor because you can call out their weaknesses and help them overcome them without it directly affecting their grade

[quasi musk]

I've also heard for Letters of Recommendation, that the institution will ask "Is this student a top 20%, 10% 5%, 2%, or 1%?"

And if you don't write top 1%, then your student doesn't get the position, so even letters of rec become arms races of "Best student; great thesis" etc.

[harsh zenith]

I have a question for US teachers specifically, because from what I hear and have seen, this is not really an issue abroad.

Why aren't proofs taught in US public schools?

[quasi musk]

They are taught in the US public schools, especially during the geometry curriculum

[harsh zenith]

I mean for stuff like calculus, and even past geometry it isn't really taught

Like Algebra II and Precalculus, there were barely any proofs

And we never had to write any

[harsh zenith]

Those were Honors classes too so it's not anything to do with skill

[harsh zenith]

And the proofs taught for geometry aren't formal or typical for what is done at a higher level

[quasi musk]

Ok, so the question isn't "Why aren't proofs taught?" The question is "Why are proofs only relegated to the geometry class". On this, there are a few reasons (1) It doesn't help students practically solve problems and (2) They're hard to teach

Whether or not you think these are good reasons, is a different matter

And even in the geometry class, they don't play a central role, they largely play a background/nuisance role for most students

[harsh zenith]

Yeah

Which in my opinion, takes away from a lot of the wonder and intuition of math

And this is mainly a US thing

In Romania for example, they start proofs in the 6th grade (according to my math teacher who is native to Romania)

[quasi musk]

Sure, and I think most mathematicians are inclined to agree. So if you want to convince math teachers otherwise you have to solve (1) How does this help students develop their problem solving skills and (2) Make them easier for teachers to teach to students

[harsh zenith]

Fair

[pure light]

the goal of math classes at the high school level is generally to prepare students to be engineers or scientists, not mathematicians, so from that perspective proofs are kind of a distraction from the main goal which is to give a problem-solving toolkit

[quasi musk]

For calculus, I can understand; the proofs are largely too difficult for most students to follow, especially the limit ones involving epsilon-delta. I do believe the proper way to do calculus is a computational approach first

Solid Angles has been a proponent of this, to which we largely agree (which is quite rare, I think we disagree a lot on how to run a class)

[harsh zenith]

I guess, but, at least for me, understanding proofs has made me not just a lot better at math, but also more able to internalize the abstract mathematical concepts that we cover

It's made me able to walk into a testing room and just finish the test within half the time we're given I'd argue

I can explain why Calculus things (for example) work after actually doing the math

[quasi musk]

Sure, but you must understand that for most students, this is certainly not the case. In fact, proofs mostly confuse students

[harsh zenith]

Why is that?

[quasi musk]

In the US, there seems to be a large set of detractors against memorization of basic facts. So, in Geometry, students can hardly state definitions of things, or Euclid's Axioms/Postulates

[harsh zenith]

Obviously you don't have to literally explain the concept within proofs only, I think that's excessive

[quasi musk]

How can you understand how to prove that the sum of triangles is 180 degrees, if you can't even label corresponding angles? Or recite that vertical angles are congruent? What is the 5th postulate again?

I've taught this to middle schoolers before, but it's a lot of work - and they were motivated students

[tardy ember]

i think in some sense this is in itself an argument for why teaching students proofs is hard

you don't get that understanding from learning proofs because it happens for free, you get it because it is pretty much required to prove anything nontrivial and so you will either find your own way to it or fail

[quasi musk]

I don't think the lack of teaching proofs is a problem in US middle school/high school system. I think people just get caught up on "Oh I struggled with proofs, if only I went to a math specialty school which got me started on this earlier"

[tardy ember]

and so it's hard because the end result is a completely different way of looking at maths, which you just have to somehow figure out

[quasi musk]

Yeah, believe me. I've tried to teach students how to think more precisely with the rules of math. It went well with maybe the top students in the class, the middling students were trying - but to no avail. The worst students were completely left behind by this approach

A lot of times, in Romania, they'll just keep filtering students by their ability. They'll take the creme de la creme, and teach only those high level math; then, they will claim their math education is far superior to the US, while ignoring those that aren't so talented

I have a lot of respect for the US system trying to get an education for everyone, even with its drawbacks

[harsh zenith]

I feel like this is less dependent on trying to teach proofs and more dependent on an earlier struggle, because in the US people still severely struggle with understanding math. I have seen people get left behind by the system, and it doesn't really function much different here. We teach the top students Calculus or Multivar (in my school), and the other students are left back in lower levels

Hell, I had to pay money to take an extra course to get up

I was initially supposed to be in Geometry freshman year, but then I took Geometry over the summer so I could skip it

And I still have people ask me for help on why something works

Obviously I don't give them a proof

But I feel like at worst proofs will just show what was already true, that people who are at lower levels of math knowledge struggle to understand

[quasi musk]

Huh my msg didn't go through just now

[harsh zenith]

Weird

Yeah I saw a notification

Just didn't show up

[quasi musk]

Oh I know now lol - I mentioned a banned general intelligence test, forgetting that it always leads to a terrible discussion lol

Basically, I think students in the US that are left behind mathematically, are largely not even trying. There are exceptions, of course, but if students care to try, then they will move up

We even let students into honors/AP classes, wtihout regard to their ability in the subject, more about their responsbility to get tasks done

Which isn't necessarily unreasonable, but you do end up with a lot of students in AP Calc that have no business being in Calculus

[harsh zenith]

Where?

In my school it's dependent on grade & teacher recommendation

[quasi musk]

Oh I'm american, so I say in the US. The grade & teacher recommendation is often a poor indicator of actual mathematical ability

[harsh zenith]

I should specify rhetorical where

[harsh zenith]

Fair enough

[quasi musk]

I understand and empathize with a frustration of the state of math education, which seems in perpetual decline. There are after school programs like Art of Problem solving that do a lot of good on this front

[harsh zenith]

I mean that technically applies to any subject

[harsh zenith]

Like I would argue my skill in writing essays is poor largely because I honestly just don't care

But that doesn't mean we stop teaching essays

In fact, essays are very unapplicable to most jobs, because a vast majority of writing ends up being emails and presentation writing

But we still teach the skill of writing an essay because it teaches reasoning skills

[quasi musk]

Many high schools are moving away from fiction/literature, to non-fiction and "jobs" programs; this happened during my senior year of HS english class.

[harsh zenith]

Ah

That seems bad, actually imo

[quasi musk]

So, it's not just a math thing. It's everywhere in the curriculum.

[harsh zenith]

Ok wait I'm gonna pause here because this discussion has ended up going into other subjects

Which are, in fact, not related to #math-pedagogy

[tawny slate]

this is another reason why i think discrete math and combi are really important to teach high school and prior, as opposed to or in place of calculus

i think people really underestimate the value of proofs in counting. we take it for granted, but consider how often a student confidently gives an incorrect answer in a combinatorics problem, and almost no student actually understands how to rigorously justify the answer to a counting problem

in addition, often when a geometry proof is wrong, it's usually because the student has what they personally believe to be a correct intuition but just can't find how to express it in proof form, or they have no intuition whatsoever and so the proof is difficult to relate to, so it makes it even harder for them to care about the proof. it's either an annoying obstacle they don't need, or it's a bunch of obtuse steps that can't comprehend anyways. by contrast, in combinatorics, students will often have some conception of an answer in their minds as an estimate, so when the proof is wrong, it usually actually leads to a wrong answer, which makes it more powerful in demonstrating the value of proofs

[rapid tusk]

geometry “proofs” as presented in a typical hs class are also usually on insultingly easy/“obvious” results

do we really need 5 lines to show vertical angles ?

when 3 of them are boilerplate nonsense just made up to increase busywork for high school freshmen?

[pine kayak]

I currently student teach 3 geometry classes and 2 intro to stats classes. I think we need more geometry proofs.

A lot of students can't visualize - they literally can't imagine a triangle and an altitude of that triangle or why you would ever want to. Proofs really help with that visualization. Also it is a justification/way to derive the formulas. If you want proofs to be more tangible, you can add values to the angles and lengths.

Also, a lot of my intro to stats students will get answers off by orders of magnitude and have no idea that the number of ways you can select 1st, 2nd, and 3rd from 196 different countries isn't 6... Combinatorics and permutations is really really hard for them and making it more rigorous with proofs will only benefit higher level students since the proofs seem so abstracted unless you're doing really small examples, which should be done regardless of level.

[turbid zenith]

Do we really disagree that often? 😂 I’m agreeing with a huge chunk of what you’ve been saying! But I’m not keeping score lol

[turbid zenith]

THIS. Lockhart’s Lament gives a quite damning view of what is presented as “proof” in high school geometry

Here’s a little more background on why they became like they did

[turbid zenith]

I would also say that one issue that students have with proofs is they often don’t see an intellectual need for them.

[harsh zenith]

One could argue the same for essay writing, yet we still write essays consistently, because it does provide value. As much as writing essays is a pain in the butt it helps me understand the book better, even if out of force (I haven't seen any large-scale evidence to suggest that essay writing requirements are decreasing across schools outside of the SAT Essay no longer being mandatory)

[harsh zenith]

This I do agree with though

[turbid zenith]

I don’t disagree! But I think math class could do a better job providing that intellectual need rather than just saying “trust me, it’s good for you even if you hate it now”

[harsh zenith]

Oh no for sure

I think math teachers in general need to be a lot more clear on why certain things work and getting there on your own in general

And proof isn't necessarily the only way to do that

But I believe it's a way that shows the most mistakes, which, although it will make teaching harder, is inherently beneficial because encouraging mistakes helps to benefit the learning enviroment imo

[turbid zenith]

I really like Dan Meyer’s framework

Find the headache for which the thing you want to teach is the aspirin

[midnight scarab]

That sounds like making everyone sick in order to sell them your medicine

[harsh zenith]

Tbh a lot of what I'm saying is inspiration from 3Blue1Brown💀

I should specify inspiration because he doesn't exactly do specific proofs but he does a very similar form of explanation that builds the intuition

[turbid zenith]

This sounds like the least generous possible reading yes ;P

[quasi musk]

This is an approach that I employ a lot, present a natural question about this math thing, show how the reasoning with words or with symbols can lead to diverging answers, and then show how having a clearer definition/framework makes it go away

For instance, saying zero is "nothing, no value" doesn't tell us how to add, subtract, multiply, or divide

[quasi maple]

I mean idk, making sure kids understand that maths even has a damn SYNTAX is apparently already difficult

The incessant "viral maths problems" that all boil down to a fuxking solidus being poorly used really highlight how that's not doing so well

[halcyon glade]

Isn't that an example of the opposite, people applying the syntax they learned to resolve an ambiguity?

I feel like that one has more to do with misunderstanding math as an intelligence check instead of communication

[quasi maple]

No I mean that they're not learning correct syntax

I can only speak for the UK, but generally the reason BODMAS/BIDMAS (in the US, PEMDAS) even exists im school environments is because they're given as a shorthand often to primary school teachers, for whom formal maths education isn't a requirement, meaning they're not that likely to have seen any maths beyond GCSE

But what that means is that it's a tool that gets applied blindly

It's like leaning "whom" exists, and then not teaching where it goes, so some stuck-up inevitably tries to correct a sentence like, "The man who saw me yesterday wants to interview me for the job", despite the fact that "whom" is in fact ungrammatical here

In much the same way, some people will see " 6/2 (1+1) " and either apply BIDMAS blindly to get 6, or PEMDAS blindly to get 1 - instead of acknowledging that the expression itself is poorly written

[tawny slate]

genuine question, i still dont see how this qualitatively differs from what eric said

do you mean that even understanding that order of operations serves the end goal of clear communication, they still incorrectly apply it in a way that should make it clear?

[quasi maple]

Sorta - i.e. that they incorrectly learn that "because there is something like BODMAS, any mathematical expression has no ambiguity because you can always apply it"

[turbid zenith]

That’s the take I do see often online

Even by math teachers

Just shut up and follow the rules

[quasi maple]

It's akin to learning that because grammar exists, the sentence, "This rides horse he well." must have a correct interpretation (and it might do) but without acknowledging that the sentence itself is poorly worded

[turbid zenith]

Like what they’ll do is just replace implicit multiplication with an explicit symbol and then go left to right

[tawny slate]

"and it might do" lmfao

[quasi maple]

Of course, we use human/natural language FAR more often than mathematical language, to the extent that we can spot that this is ungrammatical (and that a better rewrite would be "He writes this horse well."

[turbid zenith]

Failing to acknowledge that the fact that the multiplication is implicit is what’s causing the issue

[quasi maple]

But that doesn't detract that the original statement isn't clear

[quasi maple]

It's also why I personally just teach that if you're violating an order, you need extra brackets

All that an "order of operations" does is tell you where it's safe not to add brackets

[turbid zenith]

Yeah for sure

[tawny slate]

i wonder if "linguistic cherries" are a thing because i love this kind of extra flavor

[turbid zenith]

All for the sake of good communication

[quasi maple]

So for instance, if I mean by 2 + 3x4 the calculation "add 2 to the product of 3 and 4", I don't need to add brackets

If however I meant "Multiply 4 with the sum of 2 and 3", I need brackets around the addition

[turbid zenith]

Personally I think those rules can be broken if you’re clear about it and set a convention

[quasi maple]

Generally yes, and it's the same approach I have with any language learning lol

But essentially, paramount is that you have to understand the rules first

[turbid zenith]

We do it all the time

[quasi maple]

cf. using trig without brackets

[turbid zenith]

Yup 🤣

I made a blog post about this years ago and called out the same issue

[quasi maple]

(I've seen someone in one of the help channels the other day erroneously enter into a calculator the equivalent of tan(-π/4 - 1))

-# You can probably guess what they MEANT to calculate, but that has to come with experience

[turbid zenith]

Like is sin 2*π equal to 0?

What if I write it as sin 2 * π?

[tawny slate]

so this video is relevant https://m.youtube.com/watch?v=DEc03_qsQho

it mentions all of the points here and also a section on dimensional analysis. i think that labeling each number with what it represents also helps provide context as to what the intended calculation should be, even if the syntax is not ideal

i dont know if educators have enough time and bandwidth to go over something like this but i think dimensional analysis really should be squeezed into the curriculum somewhere

[turbid zenith]

https://www.solidangl.es/post/when-does-nothing-mean-something

[quasi maple]

Me whenever explaining this sorta thing to students

[boreal agate]

that one young sheldon episode:

[quasi maple]

that "there is no zero" scene is something that still bugs me lol

[swift hatch]

I don't know which I hate more, that scene or the fact that I know what it is

[quasi maple]

It will have been pushed to YT shorts and Instareels and shit, dw

[swift hatch]

Ok but the missing context is that I have watched all of young sheldon with a friend

[zinc dove]

You have sin 2u in the formula and then you write sin 2x later, is that a typo?

[turbid zenith]

Oops looks like it

[quasi maple]

Incidentally, damn you've acc written a fair bit about this sorta thing before this post

[turbid zenith]

Yeah

Never got much of an audience so I haven’t written much in a while 😛

[astral zinc]

Does anyone know some motivation for Brzozowski’s automaton construction? I don’t think I’m doing a good job motivating it for the students. So far the best explanation I have is the recursive nature of $\alpha^{-1}(L)$. Taking one letter at a time until left with the “atomic” question “Is it true $\varepsilon \in \alpha^{-1}(L)$?”

[burnt vessel]

Ted

[halcyon glade]

You might ask in #theoretical-cs too

Is that the one about translating regexes to FSAs? It might be instructive to try a couple examples first to build some intuition and then introduce the general algorithm and apply it to the examples.

[rigid coral]

https://youtu.be/LW4MMBfRweg

I made this video for Pi Day so I would love some feedback on this!

[mossy gate]

is there a channel for using lean

[rancid wigeon]

try #computing-software?

[mossy gate]

ty!

[turbid zenith]

Happy Pi Day fellow educators! 😄

[quasi musk]

"Hi I haven't been to class for 3 months -- can I do the make up work now???"

• one day before the term ends

[turbid zenith]

Did this happen to you 😭

[quasi musk]

Many such emails in my inbox

[turbid zenith]

o o f

[quasi musk]

The thing is, I've had students say "I have X medical issue that is chronic; here is my doctor's note, I need XYZ support" and they did this like two weeks into the class

So my sympathy level at the end of the term is about zero

[turbid zenith]

Yup, I feel that.

It's amazing the Hail Marys that students will try to pull.

[quasi musk]

I have to finish most of my grading today, and my inbox is full of makeup worksheets

[vestal tangle]

surprised you still give out worksheets

with AI as powerful as it is

[tall bolt]

Many of the solutions I read are sufficiently bad that either

1. AI isn’t that powerful
2. Students aren’t using it
3. students really don’t know how to use it

So there’s still value in that. And even if most students are using AI to cheat through all their homework, the feedback is still valuable to those who aren’t cheating themselves and I don’t think they should be deprived of that

[quasi musk]

I'm a lowly TA, the worksheets are done in class, under my supervision

[turbid zenith]

What kinds of worksheets?

I'm curious what yours looks like, if you'd be up for sharing one

[quasi musk]

Unfortunately they're not mine to share, and are under copyright

[turbid zenith]

Ah. :/ Okay.

[turbid zenith]

Can you at least describe what they tend to look like? What kinds of activities or something? Is group work built in somehow?

[quasi musk]

One of the classes is just worksheets. The other is a support class for students that got a C-, C, or C+ in the first term

The support class has everyone go to the whiteboard and work in groups on the problems

[quasi musk]

So for the non-support class they're just standard problems

For the support class, they're problems that they do together at the board. These problems tend to be more creative, and asking them to build a mathematical function that describes the height of someone on a ferris wheel

[turbid zenith]

Ahh okay. So this is Precalc?

[quasi musk]

yep

I have stuff for real analysis that I personally made, which is shareable

[turbid zenith]

That I would love to see if you’re willing! 🙂

[zinc dove]

Hello, did you noticed a change when you changed the question from "DOES anyone have any questions?" to "WHAT questions do you have?"

[turbid zenith]

Not particularly. :/

Hence the things I pointed out right after

[little drum]

this look AI generated to anyone else?

[native iron]

My concern with AI use in things like this is that there's a well-documented phenomenon of AI encouraging or worsening delusions of grandeur in its users. While that's not necessarily what's happening in your case, you should be cautious if a chatbot is telling you that you're onto something but the humans around you are telling you that it doesn't make sense.

Have you talked to any mathematicians you know in real life (say, an advisor) about this? They might be able to help you better exposit what you're saying and to understand the limitations of your new ideas.

Also, I'm not sure this is germane to this channel. Is the idea that your system helps you create new pedagogical techniques? It might help if you explain more explicitly what this has to do with teaching math.

[midnight scarab]

Well, the difference is that Tao knows the math, he's not reliant on the IA

Also formalisation is an entire different problem from learning (or even coming up with proofs, but anyway)

But I agree this doesn't seem to have much to do with the topic of the channel

[little drum]

definitely

honestly as soon as someone says "exocortex production" I'm just zoned out lol

[wicked kelp]

is there any channel which talks about phd jouneys

[halcyon glade]

#advanced-lounge is prob suitable

[quasi musk]

All or nothing grading for parts of a pre-calc exam for freshman

Can't say I 100% agree with this grading decision, but it is nice to hold students accountable to being able to solve basic questions without partial credit

The median, of the 15 pts I graded, was 5/15

[astral zinc]

Its better than the median being 0 🤡

[turbid zenith]

What kinds of questions?

[quasi musk]

cos(x) = -1/2, what are the possible values for sin(x)

tan(x)

what is cos(x+pi)

[dim blade]

I think it's better this way

[quasi musk]

there were a couple limit questions where people would write down the wrong number

e.g. the answer would be like -2027/3, but a student might mess up and write 2026/3

Just by adding or subtracting a one, where normally I could give some credit

But this was no credit, even for rather simple arithmetic mistakes

[dim blade]

That's unfortunate

But I think it has some pedagogical value, not a lot though. I prefer to not punish arithmetics

[quasi musk]

I think it definitely has value. At what point should you be expected to solve a simple equation with no mistakes?

Can you get a C, or B, in a class, without solving a question completely correctly?

Is such a student actually ready to move onto calculus?

If not on the final, then when?

[dim blade]

Ye, I think arithmetic mistakes on simple equations should be rare once you’re past early algebra

A surprising amount of high school students have seemingly never bothered to learn the multiplication table, but will penalizing arithmetics in an exam setting help address the issue or will it just discourage?

[quasi musk]

I think passing on students that aren't ready is a bigger problem than discouraging students

[turbid zenith]

Depending what their mistakes are, they could be.

If you set up a trig identity correctly but did 2 + 3 = 6, then I would say yes, you understand the trig.

If your mistake was integral to the problem at hand, like if you said sin(a + b) = sin a + sin b, then no, you don't understand the trig.

I would let the former student move on to calculus and most likely not the second.

[wicked kelp]

phd is like being a paid slave of someone

[quasi maple]

Wait till you hear what jobs are

[wicked kelp]

i would be a school teacher

[turbid zenith]

.......ewwww

What is honestly the point of putting derivatives like that on an exam

[tight star]

Why not

[turbid zenith]

To me that's like an unrealistic chess position that could never show up during an actual game

Yeah you could analyze it but what good is it

[plain pebble]

At some point you’re not testing a student’s ability as much as you’re testing their self-hatred

[turbid zenith]

This pretty much

You're not testing whether they understand derivatives, you're testing whether they can trudge through a minefield

[tight star]

Don’t people do those tho

[turbid zenith]

I guess sometimes? I'm thinking of that one study

[plain pebble]

Not really
Most chess positions you’ll teach are like
Things you could reasonably expect in a game

[tight star]

Check if you know how to keep a level head and find complicated derivatives

[turbid zenith]

Where they compared how chess masters did compared to novices

[vital tulip]

solid angles

do u need help with ur limits hoemwork?

[tight star]

The latter is also an important skill

[turbid zenith]

Unless "keeping a level head" is in the learning outcomes for the course, I think that's a bit much 😛

[tight star]

Though for an exam this may be a bit much to be fair

[vital tulip]

solid angles, do u need help with ur limits homework, also do u like chess

[pure light]

i don't think it's necessarily a bad thing to put one or two really nasty problems just for students to show they can apply the derivative algorithm recursively. i would definitely not include this many though, maybe a couple as extra credit problems

[turbid zenith]

Are you trolling 😛

[tight star]

No

[turbid zenith]

(I meant randomPhysicsKid)

[vital tulip]

me?

no. Im in middle school and have a passion for math

[tight star]

Yes that seems sensible

[vital tulip]

im doing multivariable calculus, ee, and some other things

[turbid zenith]

Here are the two exams these come from btw, from a Reddit post.

[vital tulip]

is this from a college

or highschool

this is really simple

[turbid zenith]

On a homework assignment I'd agree with you honestly

On a test, not so much

[vital tulip]

first of all, the derivatives u just need quotient rule, product rules and other intgration techniques. for the 1, its mainly just some factoring techniques and knowing lhospitals rule.

for number 4 its just a formula like f(x+h)-f(x)/h

its mostly just like AP calculus

[turbid zenith]

Ah yes, integration techniques for a test on derivatives, why didn't I think of that :V

[vital tulip]

I meant diffrentation techniques

my bad gang

[tight star]

Mfw fractional derivatives

[vital tulip]

also is this at a highschool or college

[tardy ember]

i mean i think there is something meaningful being tested here in terms of like, being able to differentiate anything, instead of only being able to manage simple cases and getting stuck when presented with something complicated

[vital tulip]

tbh ill do it righ now

[tardy ember]

...but also yeah these questions do seem like a bit much for an exam

[vital tulip]

but i need to finish soldering on this PCB im working on

[turbid zenith]

You should do #6 on Exam 2

[vital tulip]

Is that just MVT or am i dumb

[turbid zenith]

Well you said it's easy

[vital tulip]

i said the first one

also im in middle school

i dont get taught these things, i self study

#2 is mostly just related rates problems

[turbid zenith]

Yeah that's fair. And some nesting is fine yeah. But at some point these look written to deliberately be Hard Mode.

Like, less to test students' knowledge and more to separate the wheat from the chaff.

[vital tulip]

solid angles

what grade are you in

[turbid zenith]

I'm a math prof.

[vital tulip]

Dang

can you lpease teach me in multivariable calc

[turbid zenith]

That's what this channel is for.

Talking about pedagogy, that is.

[vital tulip]

also how are you taking a test when your a math prof

[turbid zenith]

I'm not taking a test. I saw this online and posted it in here.

[vital tulip]

oh ok

[turbid zenith]

To discuss with other educators. Because that's what this channel is for, math educators discussing how to teach math.

[vital tulip]

im in 8th grade

can u pls coach me in multivariable calc

[turbid zenith]

Not for asking for math help.

[vital tulip]

what is ur specialty

[turbid zenith]

You're welcome to ask questions in the help channels or like in #multivariable-calculus

[turbid zenith]

Combinatorial game theory!

[vital tulip]

oh wow

thats not a pretty common thing people are special in

can I call you to disscus about math

[turbid zenith]

I will decline, thank you 🙂

[vital tulip]

ok

btw, are you getting anoyed by me?

[turbid zenith]

All I will say to that is to please pay attention to what the different channels are supposed to be used for. 🙂

[vital tulip]

ok

[tight star]

Isn’t it just wrong what

[turbid zenith]

Oh hey, here's something from the prof who made these. I at least see what he's doing, even if it's not at all what I would do.

[turbid zenith]

Yeah it's just not a true statement

[tight star]

What was the intended statement

[turbid zenith]

https://www.youtube.com/watch?v=dUc49Jw5KQ8
This video spends most of the time going over it

Apparently it would be correct if you also had g(a) = g(b)

Or something I dunno

[vital tulip]

CALCULUS IS EZ
\

oh is that where you got the 2 test from

[turbid zenith]

Yup.

[vital tulip]

are you good at number theory?

@vital tulip

[vital tulip]

are you working on the calc test?

[spiral elbow]

One such question might be fine, but having four questions of needlessly complicated derivatives on an exam is bordering on sadism IMO. And having it as homework is just disrespectful of students time. It's like insisting they do RREF on a 7x7 matrix by hand, it's just mindless computation, and they could use that time for actually productive learning

[vital tulip]

im rightnow doing the test

im going to send a picture once im done

[lost mauve]

@vital tulip the purpose of this space is to allow educators to discuss pedagogy with each other, it is not for you to solicit tutoring/math help. As such you have had send message perms removed in the advanced channels to allow this space to function as intended. If you have questions about MVC feel free to ask them in #multivariable-calculus, the people there tend to be pretty good

[trim wing]

tbh I'm surprised nobody went to the dept chair to complain about that exam. That would likely have happened at any university I've taught at, and it would have been appropriate for students to complain about questions that absurd.

Too many people teaching math courses forget that the purpose is to help students learn, not to stunt on them

I also flat out do not believe the claim that students did well in that class. Anyone who's taught calc will know that 95% of students would do miserably at such questions.... unless there was a huge curve, but if you have to put a huge curve on an exam or on final grades, that means 100% of the time that you fucked it up

[rapid tusk]

one time someone showed me a practice calc1 exam from the classes at my uni and like fully 25% of the test was just differentiating horrible nasty tedious functions

like my first reaction upon seeing it was an audible “wtf”

[halcyon glade]

I see some value because some people can do one-step stuff but break at the sight of anything more than like two steps and say it's impossible

I probably wouldn't put this many of them though...

[feral swan]

e) is pretty nice

[midnight scarab]

~~Ask them to compute ~~

[tight star]

Taylor series time

[midnight scarab]

Yeah, the story is that you need to go to quite a high order if you want to do it bruteforce, but actually the answer is just one, as can be seen geometrically (kinda...)

This teaser is due to Arnold (V. I.)

[tight star]

How geometrically

[midnight scarab]

https://mathoverflow.net/questions/20696/a-question-regarding-a-claim-of-v-i-arnold

[copper root]

cute one. Nice exercise for me who am just learning calculus/analysis

[turbid zenith]

Solving stuff like this“quickly” is overrated 😛 certainly not the goal of a general calculus class

[midnight scarab]

I obviously agree, that was just a joke

[turbid zenith]

Yeah I know, I’m saying it to agree with you and disagree with whoever Arnold was 😛

Since I’m pretty sure there are still places who really do believe that

I do think there’s still a place for problems like that though! Just haven’t quite decided what that is, precisely

[midnight scarab]

I mean, Arnold wasn't advocating this for a general calculus class. Not to mention he's always extra provocative.
But really the point he's trying to make (imo) is just that sometimes geometric intution and perspective can save a lot of effort

(Slightly triggered by "whoever Arnold was" /j)

Also the context was this:

He did put it in his "trivium" though

https://www.physics.montana.edu/avorontsov/teaching/problemoftheweek/documents/Arnold-Trivium-1991.pdf

[plain pebble]

If you’re saying a few minutes I’d expect most mathematicians could spam l’hopital or series expansion enough to solve it

[midnight scarab]

Well, you do have to expand each term to x^7 😂

[slim path]

Could anyone here informed about American math pedagogy tell me why books like Dolciani's Modern Introductory Analysis sort of never became the norm? Unless that got sorta absorbed into the whole AP Calc system but that doesn't appear to be the case since it lacks emphasis on proofs.

Dolciani's book and to a lesser extent Olmsted's Prelude to Calculus and Linear Algebra seem like fantastic high level Precalculus + Intro Calculus text with a focus on logic and proofs as opposed to the vanilla books that we tend to have today that focus too much on computations.

So what went wrong here. Unless I'm missing something and am terribly off base.

[turbid zenith]

Probably at least partially because proofs at the high school level and scale are harder to grade, harder to test, and harder to teach.

But also I wouldn't advocate for Dolciani being the standard anyway.

Starting with logic and abstract axioms may be mathematically "rigorous", but that doesn't mean it's pedagogically effective for the majority of students.

[slim path]

Hmmm. I'm considering whether it would be applicable to use such material within the Indian system.

First couple of years in secondary school here typically introduces students to polynomial factorisation, elementary trigonometry with heights and distances, basic synthetic euclidean geometry and the co-ordinate plane.

The next two years is a hodge podge of Precalculus and Calculus. It does start with sets and relations to quite some degree, upto equivalence classes and introductory mathematical logic but is paid very less attention to. Then it quickly moves onto more advanced trigonometry and algebra and analytical geometry. There's also a little combinatorics and probability there. And finally Calculus in one variable.

I feel like within this curriculum a book like that for reference is nice. That said, pedagogically even I'm not sure. It's only my second time teaching a hs class.

I would ideally like to strike a balance between calculations and proofs but lean on the latter more. The norm here is the other way around. In fact attention to proofs is practically non-existent. The point for me is not to be able to write rigorous proofs (barring some cases) but rather have good reasoning while working on mathematical problems. Writing a proof, even an intuitive one at this level comes a long way towards that imo.

[turbid zenith]

I'll say, if you want the students to learn about sets and logic and such, you need to think of WHY they should know it.

And as for proofs ... proofs should still be part of the mathematics curriculum, but again, you need to be explicit about WHY the proof is needed

If students believe something after a few examples, they're not going to see an intellectual need for proof

If you show them something genuinely surprising, they're going to want to prove it (maybe)

Or if you manage to sow some doubt

[rapid tusk]

that i think is the main issue with hs geometry "proofs" lmao

those "results" are usually not very interesting

and more often than not just substitution bash

now smth like inscribed angle theorem is potentially an actually surprising result

[turbid zenith]

Giving my students extra credit if they use \varepsilon instead of \epsilon on their LaTeX assignment

[cloud zealot]

how much? an epsilon's worth?

[slim path]

Aye, that's a given.

[midnight scarab]

\let\epsilon\varepsilon

[quiet vortex]

I am a programming instructor based in Russia. Recently, I decided to create a summer bridge course for high school graduates — a transitional math course designed to ease their way into the first year of university. I wanted to share my plan with you and, if you have a moment, ask for your thoughts.
The Course Plan

1. Logic — especially quantifiers.
   Many first-year students stumble when they encounter the formal definition of a limit, and I believe the root cause is that they have never properly worked with quantifiers. I want to build comfort with ∀ and ∃, negation of quantified statements, and perhaps go as far as first-order logic.

2. Types and sets.
   I want to introduce the ideas of sets and types: that a function has a domain and a codomain, that expressions can be typed, and that when we build complex expressions from simpler ones, we must make sure the types "match up." This connects naturally to their programming experience.

3. Circles and arrows.
   Domains and codomains can be drawn as circles, functions as arrows. Arrows can be composed to form new arrows. Every circle has a special loop — an arrow from itself to itself — such that composing with it changes nothing. And yes... this is secretly category theory. But I wouldn't tell the students that right away! Only at the end would I introduce inverse arrows and isomorphisms, and reveal the name.
   My Concerns

Is this plan realistic? Will it actually help, or will I waste my students' time and money — or worse, confuse them before university even begins?

For context: the Russian math curriculum is, at least on paper, quite demanding. In the final years of school, students study trigonometry and introductory calculus — derivatives, integrals, and so on. (Is that comparable to what American high schoolers encounter? I'm curious.)

Should I include calculus in this course? I must confess: I personally find the standard epsilon-delta approach rather off-putting. All those limits, all those ε's and δ's...

However, I have recently discovered nonstandard analysis, and I find it genuinely exciting! The idea that actual infinitesimals and infinitely large numbers can be made rigorous — that analysis can be "turned into algebra" — is deeply appealing to me.

Here is my thought: what if the first lessons introduced calculus concepts through the nonstandard approach — using hyperreals and infinitesimals — and the formal epsilon-delta definitions were brought in only later, once students have already built intuition? Do you think this could work, or might it do more harm than good?

[cloud zealot]

well, infinitesimals might be easier to work with (at least for proving certain theorems in calculus), but the trade-off is that you have something that's ontologically more mysterious. for example, the hyperreals do not form a metric space (and hence can't be cauchy-complete). epsilons and deltas might be harder to work with, but they intuitively capture notions of closeness. also, there's something to be said about appealing to a new kind of number system to justify something that seems like it could be justified purely by appealing to the reals

https://people.math.wisc.edu/~hkeisler/

keisler, a logician, did write a calculus book using infinitesimals if you want that as a model

[midnight scarab]

Mixing non-standard and standard sounds like a recipe for confusion. And naturally once they get to university they'll need standard

[cloud zealot]

https://mathoverflow.net/questions/470599/does-every-series-of-hyperreal-numbers-converge-to-some-hyperreal-number

[midnight scarab]

Anyway, I personally think epsilon-delta makes a lot of sense, and is the natural formalisation of what you'd do physically

[cloud zealot]

here's a weird fact about the hyperreals

[quasi musk]

More harm than good. I think you should create specific goals for what you want them to learn, and what kinds of problems you want to solve

[midnight scarab]

And epsilon-delta teaches students about quantifiers, how to "finitely encode infinity", ideas like the fact that it's a forall epsilon > 0, but really it's just a statement about small epsilon etc.

[quiet vortex]

Am I right in thinking that non-standard analysis is a bad idea? Maybe we should first try teaching non-standard analysis, then show them the Weierstrass function, and then say, "Well, that's where mathematics broke down, and here's how Cauchy and Weierstrass ~~completely ruined it ~~fixed it."

[compact isle]

Making nonstandard analysis rigorous requires a lot more effort than epsilon delta

and epsilon delta type arguments are the bread and butter of all analysis, it's a key skill to develop

I also don't see the benefit in teaching students about "circles and arrows" to hide category theory, really I don't think you should teach students category theory in an intro class at all

Same with type theory

[turbid zenith]

You don’t need to start with the formality of ANY of these

You can get students thinking about types without “teaching type theory”

And yes epsilons and deltas are very important in real analysis

…so teach them in real analysis

[warm valley]

I agree with saying "this function has this type" in an informal way, aligning with programming and to prevent a lot of mistakes surrounding not knowing what objects are. In fact, I think you can do this in quite a short time, especially if the students have programming experience

[turbid zenith]

Yes. Just saying things like “the floor function takes in a real number and outputs an integer” or “the dot product takes in two vectors and outputs a scalar”

[warm valley]

Notwithstanding nonstandard analysis discussion (I agree with the chorus), I think the biggest problem with what you suggest is that I don't see anywhere things the students already know and might find fun to learn more about (apologies if you have planned for this but not mentioned it). Learning about sets and types and quantifiers is very dry, and often in my experience students have trouble with it until I put it in a setting (e.g., elementary number theory) where it is both useful and can help them solve fun problems

[quiet vortex]

Look, it seems to me that type theory and category theory are very, very large and abstract areas of mathematics. But I'm not going to cover them in their entirety!
I'm going to cover the very beginning. Up to isomorphisms, at most.
It's the same with types. I'm not going to touch lambda calculus at all. I'll simply explain that expressions can be parsed into a tree, and nodes have their own types. And that when constructing more complex trees from simpler ones, these types must match.

The same goes for functions! They have a domain and a range. And functions can be typed!

This will allow students to parse complex expressions and avoid confusion when composing them.

[quiet vortex]

Hmm... Can you offer me some simple theory so I can study all this using it as an example? Something really simple, not too stressful.

[turbid zenith]

Are you convinced this will help your students understand it?

Do you know how comfortable they are with abstract mathematics as opposed to computations and applications?

[compact isle]

This is fine the problem is that this is just not how math is done at the university level unless the curriculum is different where you are

nobody really uses type theoretic language in pure math so I feel as if it would be confusing

maybe if people are going into theoretical computer science I guess

[compact isle]

(well, not nobody obviously but it's not used in the standard undergraduate curriculum)

[turbid zenith]

Personally I think a lot of the intuition I’d want them to gain by thinking about types might be better taught through thinking about dimensional analysis

Like getting students to think about these questions:

1. What is 2 feet + 3 feet?
2. What is 2 feet + 3 inches?
3. What is 2 feet + 3 grams?
4. What is 2 feet x 3 feet?

[quiet vortex]

@warm valley Hello! Can you give me example of theory for learning logic and so so?

[midnight scarab]

What about 2 feet × 3 grams (joke but also not)

[cosmic ibex]

If we're speaking about feet, then I suppose grams might also be a unit of force, and then that makes a perfectly cromulent torque. :-)

[midnight scarab]

To think I tried so hard to not complain about the stupid units...

[midnight scarab]

Have you heard of... the pound-foot 😂

Not to be confused with the foot-pound

[cosmic ibex]

What's that in euro-meters?

[tawny slate]

im still trying to figure out a useful way to interpret square dollars

[turbid zenith]

Oh speaking of units

How mad at me would people be if I used $[x]$ to represent the units of $x$ and $[![x]!]$ to represent the dimensions of $x$

[burnt vessel]

Solid Angles

[turbid zenith]

😛

e.g. for some length $x$ you might have $[x]=$ meters and $[![x]!]=\mathsf{L}$

[burnt vessel]

Solid Angles

[midnight scarab]

Hm, yeah I've always found it annoying that square brackets are used for both

I've also seen {x} for the value of x, in some units

So like x = {x} [x], of [x] is the unit of x

[turbid zenith]

I haven't decided yet whether I even want to include dimensions as well as units, the units are usually what matters for what I'm teaching

Like I kinda hate that radians are dimensionless but they're really useful as a unit

[midnight scarab]

Also I'm just realising that notation like [x] for units of x is kind of suggesting that units are a well-defined function of the physical quantity, which is obviously wrong (unless you choose a system of standard units)

[turbid zenith]

Well for me it's mostly pointing out things like $$\qty[\dv{y}{x}]=\frac{[y]}{[x]}$$

[burnt vessel]

Solid Angles

[midnight scarab]

Yeah, radians should be dimensionful (though e^it etc would be a bit annoying), and stuff like moles shouldn't

[turbid zenith]

And how units can show you why the naïve guess for the product rule doesn't work

[turbid zenith]

I don't actually know how to interpret the dimensions of a function like that

I know that sines are ratios so they probably shouldn't have dimensions

The example I used to give in my Chain Rule lesson was if h = 10 + 50 sin(θ) where θ = 0.6t or something, you have [t] = seconds, [θ] = radians, [h] = feet

So you have feet per radian times radians per second = feet per second

[midnight scarab]

Yeah, it's more that it's secretly exp(2 i pi angle/full rotation)

[plain valve]

Why should radians have dimensions

Aren't they by definition a ratio of lengths

[midnight scarab]

I mean, angles should. The question is whether a radian is an angle or a ratio of angles (or lengths)

[turbid zenith]

The confusion over radians having no dimensions is why torque doesn't get measured in Joules

[midnight scarab]

Is that really the reason? I'd say it's more just semantics

Just like you have different units that just mean s^-1

So even if you did have some quantity with units of Joules, you wouldn't use them unless it was somehow an energy

[turbid zenith]

I would rather measure torque as Joules per radian!

[midnight scarab]

Fair enough: torque times angular displacement = energy (work)

[turbid zenith]

See it makes sense

[midnight scarab]

(How badly am I outing myself as a physicist?)

[midnight scarab]

Yeah, I agree

[turbid zenith]

With as big of a proponent as I am of treating dy and dx as infinitesimals you’d think I’m also a physicist

[midnight scarab]

Sacrilegious

To me dy's are just delta y's with implicit little o's

[turbid zenith]

I mean under the hood I think the interpretation that seems to most make sense is coordinates along the tangent line or plane at a point

But I’m not sure if there’s a way to reconcile that with the dx in the integral

[midnight scarab]

Sounds like Arnold's exposition in his ODE book

[turbid zenith]

But I morally think of them as infinitesimals until I have to do otherwise ;P

[cosmic ibex]

As a practical matter, if we're using dimensional analysis to catch unintentional errors, it can be useful to consider "angle measure" to be a dimension, even if your computations at the end of the day are going to be in radians. You do need a way to intentionally pass between "angle" and "dimensionless", of course.

[midnight scarab]

What you're saying is essentially just treating dx and dy as differential forms. And you can integrate those

[tawny slate]

you dont even need to get that esoteric, just notice that 2 radians + 14 is kinda weird and arbitrarily decide radian is its own thing

as people have mentioned here already basically

that's the nice thing about dimensional analysis, you can pick and choose whatever dimensions you want, any way you want to define it

I can even give dimensional treatment to combinatorial objects if i wanted to

[midnight scarab]

In fact, I always make my integration variable dimensionful. Very helpful

[turbid zenith]

How does the tangent space definition connect to differential forms in integrands?

Wanting to know this comes from a series of articles I read where mathematicians were trying to reconcile what the dx means in derivatives vs integrals and eventually the consensus was essentially “eh you just gotta vibe with whatever you need it to mean and that’s okay”

[midnight scarab]

In essence, define dx(v) and dy(v) to be the x and y components of a vector v. Then for any curve y = f(x), if v is tangent to the curve at some point x, then dy(v)/dx(v) = f'(x). So you've realised the derivative as the ratio of two finite numbers: the components of any tangent vector.
Then define a differential form as a omega = dx + b dy (with a and b some functions on the plane). This is thus a function of a point of the plane and a vector (or better: of a vector applied to a point). Then the integral of the form along some curve Gamma, admitting some parametrisation gamma(t), is defined as the int omega(gamma'(t)) dt, where gamma'(t) defines a vector (1, dgamma/dt) at the point gamma(t)

By change of variables in ordinary integrals, this integral does not depend on the parametrisation

(You can also define the integral of a diff form directly via Riemann sums)

[halcyon glade]

Nahh radians are a ratio

[halcyon glade]

As long as you implicitly convert between tangent vectors and covectors it's okay

You can think of dx as a little vector and then integrals dx as doing a dot product with dx

[midnight scarab]

Yesn't

radians are a unit of angle, such that the numerical value of the angle in radians is a ratio

But anyway, this is just vocabulary quibbling

[midnight scarab]

This ^ is the main point

[pure light]

keeping radians consistent is useful when you're working with angle measures only but you run into confusion if you try to keep radians intact when multiplying by dimensional quantities

[turbid zenith]

Okay this is really cool :3

I'll tell you the other thing I've been wanting to figure out, because I DO like to end my Calc IV class with a tiny peek into differential forms and exterior calculus, basically juuuuust enough to be able to state the Generalized Stokes' Theorem in its succinct glory

I've wondered for the longest time if there's a way to reconcile being able to swap dx and dy in a double integral without changing the sign, whereas if you swap them for a 2-form you do get a sign change

I sorta have a heuristic way of thinking about it right now where if you're integrating dx dy over the rectangle [0,a]x[0,b] , if you move from (0,0) to (a,0) and then to (a,b), therefore respecting the order of which integrals you're doing first, you're traversing it counterclockwise; if you move from (0,0) to (0,b) and then to (a,b) you're moving counterclockwise and that cancels out the sign change you'd get from switching dx /\ dy to dy /\ dx

But of course that's entirely handwavy nonsense unless there really is a way to make that precise

(Because I can anticipate a student saying "wait but why didn't we ever have to switch the signs in Calc III for double integrals?")

[quasi musk]

The transposition map $$(x,y) \mapsto (y,x)$$ is orientation reversing. So if you compute the Jacobian, you get a minus sign in the Change of Variables. So the two negatives make a positive

[burnt vessel]

MoonBears-C-

[quasi musk]

When teaching that Calc IV Class do you normally cover Jacobians? I'd imagine so, since it's just how change of variables

[turbid zenith]

Yeah, that's precisely when we do Jacobians

[quasi musk]

It makes for a precise, but not very satisfying answer

[turbid zenith]

Well, in the change of variables formula for integrals, they usually take the absolute value of the determinant

I've always wondered whether that was really necessary :V

[quasi musk]

Oh interesting

I haven't taught multivariable calc in a while, so I'm not so up on it right now

[turbid zenith]

Lemme find what Stewart says

So when it says "Jacobian" it means the determinant:
$$\pdv{(x,y)}{(u,v)}=\begin{vmatrix}\pdv{x}{u} & \pdv{x}{v}\[2 ex] \pdv{y}{u} & \pdv{y}{v} \end{vmatrix}=\pdv{x}{u}\pdv{y}{v}-\pdv{x}{v}\pdv{y}{u}\text.$$
And then it mentions to notice that $\Delta A=\abs{\pdv{(x,y)}{(u,v)}}\Delta u\Delta v\text.$
Finally for the integral it gives
$$\iint_R f(x,y)\dd{A}=\iint_S f(x(u,v),y(u,v))\abs{\pdv{(x,y)}{(u,v)}}\dd{u}\dd{v}\text.$$

[burnt vessel]

Solid Angles

[turbid zenith]

So yeah, it's taking the absolute value of the Jacobian. Aaaaactually, come to think of it, now that you mention the sign change, I wonder if that's what they're doing to not have to deal with that sign change and they're just not mentioning it.

So basically you're saying that when we switch the order of integration but don't change the sign, it's because "under the hood" the Jacobian is canceling out the negative from the anticommutative wedge product?

[halcyon glade]

This is two different types of integration. In the first kind, dx dy is a measure and you're integrating a scalar whereas in the second kind, dx dy is an oriented surface, and you're integrating a vector field

It's just that both types of integration reduce to the same thing in 1D

[turbid zenith]

There's no way to go between them though?

Like to say that one can be translated into the other?

[halcyon glade]

You compute surface integrals by turning them into area integrals so that's the link

Like int F dot dS = int (F dot n) dA

where dS is your oriented surface element and n is the normal vector that points in the same direction

If you swap the variables in your dS, then the n flips direction, and the dot product changes sign, but your dA is unaffected and can be written as either dx dy or dy dx as desired

That's how I think of it

The reason you need the absolute value when you do change of variables is that when you do an area integral, you only care about the size of the area, not the orientation

Secretly you do this in 1D too but it's obscured by our notation for 1D being different from our notation in higher dimensions. You can call attention to it explicitly like this by writing the integral over a domain like we do in multivariable calculus rather than with a start bound and end bound though: compare
\int_0^1 -x dx = \int_0^{-1} u (-1) du
vs
\int_[0,1] -x dx = \int_[-1,0] u |-1| du
where x = -u.

For the latter notation, you need the absolute value for it to be correct whereas in the former notation, the sign change is implicit in that \int_0^{-1} is the same as - \int_{-1}^0

@turbid zenith IDK if this fills in the gaps you were wondering about or if I'm just rambling?

Terry Tao has a nice article on higher dimensional integration that maybe you've seen https://www.math.ucla.edu/~tao/preprints/forms.pdf

[turbid zenith]

I think it makes sense

Any way I can tie these together is good, basically 😛

So thank you!

[halcyon glade]

A kinda fun consequence is that you can do one type of integral on an unorientable surface like a Möbius loop but not the other

"What's the area of the Möbius loop?" is entirely okay but "What's the flux across the Möbius loop?" makes no sense (flux in what direction at each point?)

[cosmic ibex]

Hmm, perhaps we can invent some kind of projective flux for that?

[halcyon glade]

As in like sticking an absolute value in the integral to make it make sense? Haha

[cosmic ibex]

Hmm, no that won't really work; "projective" erases magnitudes as well as signs.

[halcyon glade]

Maybe this would make a fun demonstration for students to illustrate the difference between area integrals and flux integrals? IDK

[midnight scarab]

$\diff x \wedge \diff y$

[burnt vessel]

afqt

[midnight scarab]

(More seriously, https://en.wikipedia.org/wiki/Density_on_a_manifold)

[midnight scarab]

Oops, I meant $|\diff x \wedge \diff y|$

[burnt vessel]

afqt

[quasi musk]

Yeah, this is the way I think of it

Of course there's the interpretation given above, with dx dy, and dy dx integrals be with respect to different measures; especially when doing surface integrals, however, I believe this misses the easier point at hand

[turbid zenith]

Yeah, the students would likely not have learned measures by this point

It is interesting to think of the xy plane as an oriented surface because it does have a canonical “up” direction

[halcyon glade]

I don't say measure to mean the mathematical sense (though it is), but you can tap into their intuitions about calculating areas (areas can't be negative!).

[turbid zenith]

Wait… wouldn’t swapping the variables change the variables throughout the integral?

Like I’m talking about integrating, say, e^x^2, where one direction isn’t tractable and the other is

Is there a change in variables you’re technically doing when you decide to integrate it dy dx instead of dx dy?

[turbid zenith]

This is the kind of problem I'm talking about ... I don't see anything that suggests a change of variables and a Jacobian

[midnight scarab]

Indeed, this isn't a change of variables but Fubini

Note that the integral with dx wedge dy is fundamentally the 2d integral over D, not one of the iterated ones

[turbid zenith]

I think I see the issue

The dx dy and dy dx are next to each other but they’re not actually interacting with each other in any way

They only do so when you explain what dA is

So Fubini is essentially telling you that the integral with dx /\ dy can be “resolved” into iterated integrals either way

[turbid zenith]

But also I found this

If this is too far from pedagogy let me know 😛 but for me, I’m using this to get a feel for what I could put in my textbook to explain the connection as simply as possible

Source: https://pi.math.cornell.edu/~sjamaar/manifolds/manifold.pdf p 64

[turbid zenith]

This seems more promising (https://math.jhu.edu/~brown/courses/s19/Lectures/Lecture24.pdf). I’m gonna try to parse it.

[midnight scarab]

Hm, I think it's somewhat misleading, in that it's mixing up different things

1. Yes, dx wedge dy = - dy wedge dx can also be viewed as a change of variables x' = y, y' = x, in which case to preserve the integral you need to apply the change of variables formula, except if that's the logic, which is what your second source seems to say, then really that'd give you int_R f(x,y) dx wedge dy = int_R' f(y',x') dy' wedge dx' = int_R' f(y',x') Jac(T) dx' wedge dy'

With R' the image of R under T

And I don't think that's what you want

[turbid zenith]

I think I'm going to just have to work through this myself to see what's going on

[midnight scarab]

But the main point I want to make is that I don't think that's the right pov

Really, the fundamental thing is integration wrt a measure

[turbid zenith]

Because yeah when I think of a change of variables my first thought is, isn't that going to turn it into f(y,x)?

[midnight scarab]

So an unsigned thing

For which you have Fubini, and also a theorem about change of variables with abs val Jac (which has nothing to do with Fubini, at least at first glance)

With Fubini saying that a 2d integral over the product measure is equal to the iterated integral either way

And then integrating forms is a layer on top of integrating wrt a measure

[turbid zenith]

I see

You have to admit it's annoying that it looks like there should be something going on here 😛

Like I feel like it's such a natural question to ask ... "why switch sign now but not switch sign then?"

[midnight scarab]

Namely, you take a form of degree that of your manifold, and you turn it into an integral with respect to a measure by choosing a "canonical" nowhere-vanishing top degree form (the volume form)

[turbid zenith]

And it feels like the answer should be something elegant like "because there's another thing that's actually compensating for the sign switch" instead of "those are two completely different unrelated things that just happen to look related"

[midnight scarab]

In particular, picking an orientation

[midnight scarab]

Switch signs where exactly? (I get your question in essence, but I just want a concrete thing to be able to give a concrete answer)

[turbid zenith]

Why switch sign of dx /\ dy to - dy /\ dx when doing forms but not switch sign of dx dy to - dy dx when doing Fubini

[midnight scarab]

But what I'm saying here is that to integrate a form on R^2, you first pick what you call a volume form. This is your dx wedge dy. You postulate that integral f dx wedge dy is _by definition integral f dmu (where mu is your surface measure)

And then if you have any other 2-form, it can be written as a function times the volume form dx wedge dy

And this is why integral f dy wedge dx = integral (-f) dx wedge dy, which is by definition integral (-f) dmu

[midnight scarab]

Because dx dy in a measure is not anti-symmetric

[turbid zenith]

"Well you're not using forms" "Okay but I've heard it said that all integration 'really is' integrating differential forms"

[midnight scarab]

Fake news

[turbid zenith]

XD

[midnight scarab]

Fundamentally, an integral of a form needs to be reduced to an integral wrt a measure

And a measure if, fundamentally, a thing that doesn't care about orientation: measures are defined on subsets not oriented subsets

the measure of a rectangle is the products of the sides

[turbid zenith]

So in that sense, even in Calculus I, every Riemann integral can be thought of as an integral of a form that reduces to an integral over a measure?

[midnight scarab]

You only start caring about orientation when you get to forms, because the orientation is what tells you how to assign a value to an integral of a form (by reducing to an integral wrt a measure)

[midnight scarab]

Yep

[turbid zenith]

Like good ol' $\int_0^1 x^2\dd{x}$?

[burnt vessel]

Solid Angles

[midnight scarab]

I mean, good ol integral can also just be thought of as an integral wrt a measure

There's 2 povs

[turbid zenith]

But are those alternate pov's just as good as each other, or does one lead to another?

[midnight scarab]

1. this is the integral of x^2 over [0,1] (an unoriented(!) subset of R) wrt the Lebesgue measure

In this pov, under y = -x, you're now integrating over the image of [0,1] which is [-1,0]

And the measure transforms with the abs val of the Jacobian

In other words, you never get any minus signs

Because there's no discussion of orientation

[turbid zenith]

I wouldn't be expecting minus signs here anyway

Since I'm only talking about minus signs that explicitly come from interchanging variables

[midnight scarab]

But in pov 2), that of differential forms, you have two minus signs cancelling each other out

[turbid zenith]

Right now I'm zeroing on this

[midnight scarab]

Because Jac = -1, and your integral is now on the oriented segement from 0 to -1

[turbid zenith]

Because this to me sounds like "it starts off as an integral of a form over an oriented interval, and it TURNS INTO an Lebesgue integral with respect to a measure over an unoriented interval"

Not "you can think of it either way" but "A turns into B"

[midnight scarab]

Right, so in the 2nd pov, to assign a value to the integral, what you're doing is picking a "volume form" dx, and declaring that this is what should be converted to the Lebesgue measure

[midnight scarab]

You can either think of B directly, or think of A, but A needs to be turned into B

The reason why A is convenient is because it's algebraically nicer

[halcyon glade]

^

[turbid zenith]

Would anyone be up for giving feedback or suggestions on the exercises in one of my early textbook sections?

[rapid tusk]

id be down sometime in the next few days

(probably cant today bc im making last minute preparations for a math contest im hosting tmrw LMAO)

[turbid zenith]

No problem! Anything helps.

https://bill-j-shillito.github.io/calculus-scbi/online/sec-the-derivative.html

At this point no derivative rules are introduced yet — the focus is mostly on building numerical and graphical intuition, and interpreting the meaning of the derivative in real world situations.

Once I hit the next section there will be no shortage of exercises I can come up with ;P

[midnight scarab]

I think they're great! Some comments:
For 4b, it's maybe a bit confusing whether the "real-world significance" must say something biological, or just be "it's the time when the concentration stops increasing and begins decreasing" or something along those lines
For 6c, I like that the wording is more open-ended, which lets the students (if motivated) to think about how good the approximation is

I do think that it might be a good idea to emphasise the notion of linear approximation more in the main text

Which works very well on the quadratic: you have a remainder, you can see it's small, can discuss how small h must be for h^2 to be negligible (and compared to what)

I would put a lot of emphasis on this, which is very important both in applications and when generalising to multivar

I'd also put this before the sentence about infinitesimals, and introduce infinitesimals as a mental simplification for the idea of linear approximation

[tight star]

As I’ve gone further in mathematics I’ve increasingly appreciated the linear approximation point of view

[turbid zenith]

I was planning to do that in Chapter 2, but maybe it’s a good idea to bring it in earlier, so it’s immediately useful?

Since Chapter 2 is when I bring in the second derivative to be able to talk about how good an approximation it is

[midnight scarab]

Yeah, I mean your exercises already use the idea of linear approximation

[turbid zenith]

Yeah, kinda implicitly

So maybe I can just be more explicit about it

[midnight scarab]

Yeah, which is why I was saying that the quadratic example is nice, because you can already do this

[turbid zenith]

Especially since that really interprets the “rates”

[midnight scarab]

Yep yep

[turbid zenith]

So I think I can do that yeah

[midnight scarab]

As a physicist, I think the idea of "approximately uniform motion" is a very good introduction to the idea

[turbid zenith]

If that’s the case, I can always revisit linear approximation later and make 2.4 more explicitly focused on quantifying error

Both with differentials themselves and with the fact that they don’t capture the curvature

[midnight scarab]

Btw, a great (open access) textbook is https://dn790003.ca.archive.org/0/items/HigherMathForBeginners/Higher math for Beginners_text.pdf

[turbid zenith]

How would you explain to students why we’d need a linear approximation when a calculator nowadays can do sqrt(24.999) to high precision?

[midnight scarab]

It has a nice intro to derivatives etc, and (even more importantly) it has plenty of physics illustrations

[midnight scarab]

Conceptually, that's how you define instantaneous velocity and other rates

It's also nice for mental computations, like 31^2 is roughly just 900 + 60

[tight star]

You do have to be careful exactly what you mean by the second derivative especially in multivariable calc

Derivative is equivalent to best linear approximation

However, second derivative is not equivalent to best quadratic approximation, but instead to “derivative of derivative”

[midnight scarab]

Well, it's also true that the quadratic part is given by the Hessian

(There's the slight caveat that existence of a quadratic approximation does not imply 2nd order differentiability, but nevertheless when the 2nd derivative exists, the relation holds)

[turbid zenith]

Yeah I’m trying to be careful of that

[deft ferry]

MATH PEDAGOGY IS VERY GOOD 👍👍👍

I think that math pedagogy is very useful

It is utilized across many fields with purpose

Math pedagogy assists in many fields

Many fields has math pedagogy as their foundation

I did not get my active role.

[spiral elbow]

Don't post in advanced channels just to get active

[deft ferry]

but it was a legitimate question

with a side intention

[spiral elbow]

What's the question?

[deft ferry]

oh I thought this was #groups-rings-fields

[white fulcrum]

I had to explain to one of my students today that shadows are on the ground

it was a classic pythag problem with a flagpole and a shadow, but he thought the shadow was the hypot

[deft ferry]

@white fulcrum have you ever entered any Scrabble competitions?

I loved it when I was in high school but eventually I started to get worse at it because of how much stronger the players were in college and just generally not playing much anymore

[cosmic ibex]

Now consider the explanation of a lunar eclipse as "the moon moves through the earth's shadow", in which case "shadow" means the entire region of space that can't see the sun ...

[halcyon glade]

That is kind of intuitive if you think of the shadow as the light path blocked by the flagpole tip

[halcyon glade]

<@&268886789983436800> spam

[midnight scarab]

Tbf in that case the shadow would be the entire triangle

It's true that shadow can mean both the volume and its section by the ground

[lyric fractal]

As someone told you, don't try to spam advanced channels trying to get active, do this again and you lose access to these channels.

[deft ferry]

I don't wanna hear it the third time but thanks moderator

[flint oriole]

hi

[white fulcrum]

no I just play against the computer

[deft ferry]

that's cool

[warm valley]

apologies for the late response. Here are some examples I've seen:

If you go far enough that you hit Turing machines, you can prove Godel's incompleteness theorem without a ton of trouble, which is fun (my undergrad did this in first year for us).
Hoare triples for program verification.
Using a theorem prover (I recommend Rocq) to get students to prove basic properties of, e.g., the natural numbers or set theory, so that it's gamified for them.

There's also room aplenty for doing basic parts of this in intro number theory and combinatorics (using induction to prove the fundamental theorem of arithmetic, exploring quantifiers by asking students if certain divisibility questions are true, like divisibility of linear combinations over the integers, explaining bijections on finite sets via combinatorial proofs) that students likely have seen some parts before in grade school, or are easier to motivate

My professors did all of these in my first year as an undergrad, and they worked very well. I've also TAed for such classes, and these kind of examples really help contextualize the abstraction that first year math students are otherwise chucked into

Not to claim all of this is particularly novel, I'd expect most classes of this form (in North America, commonly called discrete math) to do some of these

[white fulcrum]

I need false-but-convincing math proofs like the pi=4 one for april fool's day

I'll also take some paradoxes

[jaunty yew]

On a similar vibe, staircase paradox.

[dawn anvil]

hey guys I have a student that would like some proffesionals to look at his reaserch paper I have given him my feedback

if you are free please text him in @raw onyx

[marsh compass]

All horses are the same color

[hot warren]

I'm getting really frustrated. This isn't quite math but I teach High school physics.

Kids definitely get the topic.

And they ace their tests just fine. No cheating. I definitely know that.

But.

We have a common exam at the end of their final year and I just did a practice test, and they did terrible.

Now these questions do have typos since they are being tested by students for scrutiny for use on future tests.

But like some people did awful.

And idk why. Every test I do it's all been question from that common exam (Previous years).

Assignments and quizzes too.

And it's fine then.

Like.

I think for review, I am spending far too much time on theory and not enough time on actual practice.

We'll see.

[subtle swift]

hmm i'd recommend asking them what specifically was the issue with it

especially your high achieving students

it's easy many times to understand what part was the issue, like was it applying theories to actual problems, the wording, timing, calculations?

if theyve been doing well outside of it and did bad from this one, then they likely have a shared problem, which should be not too bad to fix with targetted practice on the subject

[honest walrus]

oh finally a break form the pre uni

[turbid zenith]

What do you mean?

[elder timber]

Logic should be taught so much earlier in math education. It's more useful for math, and broadly for life, than any other math subject.

[turbid zenith]

I'm not opposed in principle but you'd have to be careful how you do it

[elder timber]

Sure, I'm talking symbolic logic, discrete math, basic proofs. Not like... Godel's theorem. Stuff where examples are pretty real world, natural language, introducing symbols.

[copper root]

I find Gödels theorems relatively easy to understand (not their necessarily their proofs, I say) and consequently a potential way to catch interest.

[marsh compass]

I also think that learning the basics of mathematical logic (what propositions, propositional formulas, logical connectives, implications, equivalences, truth tables, quantifiers, etc. are and how to relate them to the real world) should be much earlier in math education. But I don't think that you should put the emphasis on symbolic logic or the sole use of symbols and focus more on the correct use of natural language in this context.

[copper root]

good consideration

[halcyon glade]

I don't... they're difficult to even state correctly without a good amount of background

[copper root]

hmmmmmmm

[elder timber]

Yeah, the technical version "There is no complete finitely axiomatizable first-order theory of arithmetic." is tough to say to a high schooler. I've told several high schoolers a fun version.

"No matter how you write a computer program, if it tries to list all the truths of arithmetic, there will always be at least one truth of arithmetic, that it systematically misses."

[cosmic ibex]

Or something it lists will actually be false.

But even in that formulation you're hiding rather a lot of technical complications in "truths of arithmetic", since we're talking specifically about truths that can be phrased as first-order formulas -- so to fully appreciate the result one needs to know what that means.

Say, a listener who thinks "truths of arithmetic" just means things of the same "the value of this arithmetic expression is such-and-such number" will get a wrong impression.

[elder timber]

Presumably you mean "we can't prove that it won't eventually list something false" right? Because I don't think it's guaranteed to list something false.

[elder timber]

Yes, I mean, every part of it is hiding a lot of complexity. Like "complete" and "first order" and "theory".

[cosmic ibex]

I'm saying the appropriate statement of the theorem in this interpretation is that every program that prints arithmetic formulas will either eventually print some falsehood or forever miss some truth.

[elder timber]

Ah, as in a program that deliberately lists non-truths as well as truths. Yes, fair enough.

[cosmic ibex]

Yeah, except it doesn't need to be deliberate -- the theorem in its modern formulations doesn't care about intent. (It proves just as well there there's no program that, even though it seems to make no discernible sense, accidentally happens to list all the truths and only truths).

[elder timber]

I figured you were going to nit-pick "deliberate". Yes, clearly deliberation isn't a part of the formal theory.

[cosmic ibex]

Just pointing out some subtleties I find interesting. We might perhaps substitute "provably" for "deliberate" to get some more technical claims. If you're entertaining students with facts about which programs can or cannot exist, they may know something like there is a program that loops forever but cannot be proved to loop forever. So it may be extra striking for them to be told that there isn't even a program that enumerates exactly the arithmetic truths but cannot be proved to do so.

[halcyon glade]

There's a nice connection between the halting problem and incompleteness too (as well as the undecidability of truth)

[elder timber]

I like the book by Boolos et al which shows all this stuff as the many faces of uncountable sets.

[tawny slate]

actually, on this topic, since im not an expert in this area, but at least acutely aware of the technicalities behind godel's incompleteness theorems, question:

what are some examples where godel's incompleteness theorems seem to be demonstrably false but really it is because of subtleties of the formal theorem statement, in various ways to show the nuances of the technicalities?

for instance, are there infinitely axiomiable examples (yeah true arithmetic)? what about second order arithmetic? etc

i would ask this in foundations but i want the focus to be pedagogical in nature here

[halcyon glade]

I feel like one of the most enlightening questions to ask a student learning logic is why the completeness and incompleteness theorems don't just contradict each other lol

Interestingly the completeness theorem fails in second-order logic https://math.stackexchange.com/questions/1446924/how-does-gödel-completeness-fail-in-second-order-logic#1446940

[elder timber]

Yeah, if you permit the axiom system to be non-computable (for instance, second-order) then you no longer get the incompleteness result. So the incompleteness result, insofar as it is a sad result, is sad only because of the desire for everything to be computable in finite time.

(One trivial version of this is to let your set of axioms be the set of truths of arithmetic.)

[astral zinc]

they should’ve picked a different name for one of the types of completeness

it’s very confusing if you don’t know the definitions

imagine seeing Gödel’s completeness theorem, Gödel’s first incompleteness theorem, Gödel’s second incompleteness theorem with no context 🤣

[valid whale]

Hi! I have a communication question. I want to invite friends and eventually students to play with math and to learn through exploration, but when I try to explain a math concept I frequently forget the context that I needed to learn that concept, and don't accurately estimate the context that osmeone else will need to learn a concept. This frustrates my friends and leaves them feeling exhausted and like they don't want to learn the concept. Do y'all have any resources for building up the broad strokes for complicated topics and then filling in the picture for people?

It also happens when I try to explain anything though. When I try to explain a concept that I'm struggling with or a problem that I'm struggling with I often don't know how much context someone will need to help me, and then they get exhausted or frustrated

[halcyon glade]

Maybe it'd be best to give an example. What's a concept you've tried to explain lately, and how would you explain it?

[valid whale]

A good example is that recently I've been learning about evolutionary population games, a type of game that is played constantly and continuously and allows players to reassess their strategies when a timer goes off. The game is defined like this: We have a bunch of populations, and each population is assumed to have an uncountable number of participants (so it's a continuum) as well as some mass. Each population has a bunch of actions it can take, and we define the social state of that population to be a vector containing the mass of each strategy where the components of this vector have to sum up to the mass of the population. We then define a payoff vector to be the vector containing the associated payoffs of each strategy. If we think about all the ways that the social state can move in space while still being a valid social state, then we reach a Nash Equilibrium (a place where nobody can change their strategy to get a better payoff) if and only if our payoff vector is at a right or obtuse angle with all of the ways that our social state vector could move

[halcyon glade]

Is that how you explain it to general people or to other mathematics students

Btw

[valid whale]

Mostly to other CS students

people who I think know what a vector is

[halcyon glade]

Ah okay, one thing I would suggest is that before you jump into defining everything, start with some concrete scenario or visual, and prompt them so that they can basically come up with the idea themselves, people get more invested if they feel like they're discovering it alongside the field

For evolutionary game theory the motivations are fairly concrete

And then try to omit as many implementation details as you can, and try to stay away from jargon

[valid whale]

What do you count as Jargon here?

[halcyon glade]

Like social state, payoff vector, Nash equilibrium, etc.

Every new word introduced takes mental processing and while they're trying to think through the concept, they're probably not listening to your explanation

I think the high level explanation you started with for the first sentence is good though, I like that you boiled the field down to the most important parts

[valid whale]

so maybe start by saying something like "Let's say we have a lot of different kinds of animals trying to get food, how would you model how their behavior will evolve over time?"

something like that?

[halcyon glade]

yeah exactly

[valid whale]

(or if I wanted to lead people gently into my research, a bunch of people sharing water)

[halcyon glade]

and then start introducing the core tenets, like each type of animal is individually trying to maximize the food they get, but they can adjust their strategies over time as they figure out what works

[valid whale]

and like some animals are taller or faster than other animals, so those animals will have different strategies available to them than other types of animals

How deep would you go with a general non-technical audience? I know I probably wouldn't get to the part where I give a definition for the nash equilibrium in detail

I don't necessarily like the way I've heard my professor say it "If the arrow of our rewards is pointing in a direction that we can go, then we will want to go in that direction"

(this in Convex Optimization when he was explaining that if we reach an optimum along a border, then the negative gradient must be pointing at a right or obtuse angle from the feasible set, and if we reach an optimum in the feasible set the gradient will be 0)

Maybe I'd say something like imagine you are a drop of water and you are trying to find the lowest point in a hilly area while not exiting a glass container, you'd want to go the direction that points you downhill the fastest, but if you hit the glass, and the arrow wasn't pointing directly outside of the glass, you'd want to go the direction that you can that goes downhill the fastest. The arrow still helps you find that direction, and when the arrow is pointing straight out of your glass enclosure then you know you've reached the lowest point

Or better yet, you're a drop of water in a hilly area trying to find the lowest point inside of your enclosure fastest, and you have an arrow that points in the direction you'd have to go to go uphill the fastest. How would you go about finding the lowest point? What if you ran into the wall? Where would you go then?

[halcyon glade]

I'd say it always depends on their time and their level of interest so you'd gauge it as you go, a good rule is probably to never define words in passing though

So like for example the Nash equilibrium is a really big and important concept

Take some time to motivate what the question is, why this is a natural definition, and then after you define it pause and then explain what it lets you do

I thought your explanation was overall very good though fwiw

[valid whale]

Thank you! I've explained it a few times at this point, and practice does make perfect

maybe that's also the key, I need to do a few dry runs and then have a guinea pig person to explain the thing to before I take it and try to explain it for real to someone else

[halcyon glade]

Yes I think practice is really critical

Especially if it's your research area then you get really sucked into it and take things for granted lol

A lot of schools have 5-minute thesis competitions or similar, where you're challenged to explain your research to a general audience really fast, maybe see if your uni has anything like that? There also tend to be people whose job it is to help with scientific communication working at the uni libraries

In my experience it takes a while to get a knack for when to create suspense, when to keep a rhythm, when to go into a story, etc.

[valid whale]

That's the big thing I want to explain, too. I don't want my research and my field to be some sort of unobtainable thing for people, ideally I want people interested in it as well so that sometime in the future it gets a little further along than where I get it. I want to start building something that will have a lot of small pushes rather than something that gets started and dies off

[halcyon glade]

It's also good to challenge yourself to various constraints: how would you explain it in one sentence? one paragraph? one page?

I think definitely with evolutionary game theory it's close enough to the surface that there are a lot of possibilities for how to explain your research

[Disclaimer: communication is always deeply personal and these are just my opinions]

[valid whale]

That's been the struggle with finding a research topic, I've wanted to find something suitably deep so that I can get very mathy, but suitably close to the surface so that it has an obtainable entry level

like EGT is a topic that anyone, even non-technical people can get a basic understanding of, but that someone can get a little bit lost in

[halcyon glade]

yea I agree

[valid whale]

Thank you for your help!

[halcyon glade]

You're welcome!

[quaint basalt]

am helping to be a TA for one of the courses, in the recent midterms, there was a question as follows. Let $f:(0,3)\to \mathbb{R}$ be defined by $f(x)=x^2-3x-1$ and $H(x)=\int^x_0 f(t) ,dt$.

[burnt vessel]

Former Rank 7 LLORT AJNIN

[quaint basalt]

the marking out of 2 was as follows:

1. H'(x) = |f(x)| (0 marks)
2. H'(x)= |x^2-3x-1| (1 marks)
3. H'(x)= expand the piecewise definition based on the roots. (2 marks)

would it be fair for me to give feedback to the prof? In my opinion, i thought all 3 should be equally correct

[halcyon glade]

yes

[quaint basalt]

how should i phrase it, i don't want to come across as rude

oops i realised in my orginal qeustion it should be $H(x)=\int_0^x |f(t)| ,dt$ *

[burnt vessel]

Former Rank 7 LLORT AJNIN

[halcyon glade]

Dear Prof X,

I hope you're having a good day. I am a little confused by the rubric for question X of the exam. I would have marked answers such as |f(x)| or |x^2-3x-1| correct since they demonstrate correct usage of the FTC, but on the rubric, they're awarded no credit or half credit. Could you explain the reasoning behind this? Thank you in advance for your help.

Best,

something like this

[boreal agate]

Prof X???? from X-men?????

[turbid zenith]

He studies algebraic telepathy

[quaint basalt]

thanks for this!

[turbid zenith]

What were they supposed to do with the question? In fact what was the question?

[midnight scarab]

I can understand the point of 3 vs 1/2 if you know the FTC as a theorem about integrals of continuous functions, so you want to actually bring yourself to that setting. 1 being 0 marks makes however no sense to me, especially if you award a mark for 2

[south ice]

Don't troll here

[turbid zenith]

So this question threw off a lot of my students on their most recent homework

[tight star]

oh how come

[turbid zenith]

So far for (c), pretty much everybody started with $\dfrac{y+1}{y-2}=x$ and solved for $y$, which they already did in (a), rather than starting with the expression $y=\dfrac{2x+1}{x-1}$ they got in (a) and plugging it in to show that it simplifies down to $x$

[burnt vessel]

Solid Angles

[turbid zenith]

I'm trying to show them the difference between doing your "scratch work" of figuring out the y and writing the proof showing the y works

[tight star]

hm

what's your intended solution

[turbid zenith]

For (c)?

[tight star]

for the Q in general

[turbid zenith]

So part (a) would be just solving and getting $y=\dfrac{2x+1}{x-1}$

[burnt vessel]

Solid Angles

[turbid zenith]

Part (b): if x = 1, then x - 1 = 0, so y is undefined

Part (c): Let $y=\dfrac{2x+1}{x-1}$. Then $$\dfrac{y+1}{y-2}=\dfrac{\frac{2x+1}{x-1}+1}{\frac{2x+1}{x-1}-2}\text.$$
Multiplying both numerator and denominator by $(x-1)$, we get $$\dfrac{2x+1+1(x-1)}{2x+1-2(x-1)}=\dfrac{3x}{3}=x\text.\quad\qed$$

[burnt vessel]

Solid Angles

[tight star]

hm..

it feels like there's some wasted effort happening in part c

[turbid zenith]

If they're trying to prove that there exists a y that makes the equation true, but then they manipulate the equation to get the y, they're assuming the thing they're trying to prove unless they explicitly use some if-and-only-if statements

We're trying to prove if y = foo then x = bar. If you start by assuming x = bar and prove y = foo, you've proven the converse.

[tight star]

yeah i think iff is the way to go here though

[turbid zenith]

I mean you could, if you're careful, yeah

If they did that I'd accept it

[dim blade]

I don't think it's a good practice for beginners though

[tight star]

why not?

[turbid zenith]

And it's not great preparation for, say, epsilon proofs in real analysis

Where you do some scratch work to find the right delta, but then when you write the proof, you start with "let δ = min(ε/2, 1)" or whatever

[dim blade]

Because it's easy to misuse iff when the step isn't actually reversible

[tight star]

oh i always hate this style of proof tho

[turbid zenith]

Well yeah, me too 😛

But I don't know any other way to do epsilon proofs 🤷

Every real analysis book I've ever seen does it that way

The best ones explicitly show the scratch work first, but then are clear where the scratch work ends and the proof begins

[midnight scarab]

I mean, in principle you just say "take delta > 0 to be fixed later", and then you collect all the requirements you encounter over the course of the proof, and at the end you show they can be satisfied by a suitable choice of delta (that doesn't depend on things it's not supposed to)

[turbid zenith]

I feel like I'd need to see what that looks like in practice

For a sample proof

[tight star]

i figured out a nice way to do it

at least when i was writing it in undergrad

i do:

\begin{itemize}
\item Let $\epsilon > 0$
\item Take an arbtirary $\epsilon' > 0$
\item (Rest of proof involving finding a $\delta$)
\item Finally, let $\epsilon'$ = (appropriate function of $\epsilon$)
\end{itemize}

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

i can go through an example if you'd like

[turbid zenith]

Yes please!

[tight star]

ok let's do "sum of limits is limit of sum"

suppose $\lim_{x \to a} f(x) = L$ and $\lim_{x \to a} g(x) = K$. show that $\lim_{x \to a} [f(x) + g(x)] = L + K$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

Let $\epsilon > 0$, take an arbitrary $\epsilon' > 0$. then there exist $\delta_1, \delta_2 > 0$ such that $0 < |x - a| < \delta_1 \implies |f(x) - L| < \epsilon'$, and $0 < |x - a| < \delta_2 \implies |g(x) - K| < \epsilon'$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[midnight scarab]

(Just to be clear, here I'm assuming sufficient maturity to know that if you've proved smthg < 3 epsilon (instead of epsilon) for all epsilon, then you've proven what you wanted. Otherwise yeah you'd need an epsilon')

[cosmic ibex]

How about (1) let delta>0 be arbitrary; (2) work out which epsilon' it works for; (3) invert the relation, making underestimates when convenient
(Oh, I see that was what Afqt spoke about).

[tight star]

take $\delta = \min(\delta_1, \delta_2)$. then $0 < |x - a| < \delta \implies |f(x) +g(x) - L - K| \leq |f(x) - L| + |g(x) - K| < 2 \epsilon'$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

finally let $\epsilon' = \frac 12 \epsilon$ and you're good

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

this way i never have to worry about constant factors

[quaint basalt]

checked the solution manual and got this, this is definitely not correct right? we do have the points inbetween at x=1 and x=2 too

[midnight scarab]

Yeah I never had to worry about constant factors because everyone understood they don't matter

[midnight scarab]

Well, what's the statement of the FTC as taught in the course?

[midnight scarab]

Yep, so the assumption is that f is continuous on the (closed) interval

Ah wait actually who cares, f is

Yeah idk what the prof's on about

[quaint basalt]

im guessing you were worried that the abs(f) was not cont?

yes i thought about that too but it should be okay

[quaint basalt]

thanks! was worried i was hallucinating

[midnight scarab]

Yeah I mean I was just thinking that's the only case where you'd need to split - if the integrand was only piece-wise continuous

Obviously abs(f) is continuous, it's the differentiability that's only piece-wise but that's not relevant here since you're differentiating the integral and not integrating the derivative

Anyway

[turbid zenith]

Hmmm, interesting... so what you're saying is you start with a $\varepsilon'$ (because I refuse to use $\epsilon$) and manipulate it. And then once you've managed to get it almost there, you modify your $\varepsilon'$ to get the $\varepsilon$ you need.

[burnt vessel]

Solid Angles

[tight star]

yep

[turbid zenith]

I get it, but I think a lot of students would have trouble making both the beginning and end meet at the same time like that

It works but it feels very ad hoc

LIke you're solving and proving at the same time

[midnight scarab]

I mean, yes! That's kinda just how it is

Though really for the constants there's really no reason to obsess over them

I really think one should ingrain into students the fact that "forall epsilon > 0 bla bla with < epsilon" is equivalent to the same thing but with " < c epsilon" with some c > 0

[turbid zenith]

Perhaps

Somehow it still doesn't feel quite as clear to me

[midnight scarab]

Imo it's a fundamental part of the intuition that the quantifier def is really just capturing something about limiting behaviour

There are two facts:

1. You can rescale epsilon
2. You can replace "forall epsilon > 0" by "forall 0 < epsilon < M" for any M > 0 of choice

[turbid zenith]

Okay that five letter joke is annoying, but I suppose I can wait to ask my question

Apr. fool pranks are okay except when they keep you from actually utilizing the server for its intended purpose