math-pedagogy

[turbid zenith]

And I’ll probably put contradiction into the first course and make the argument it belongs there

[tall bolt]

Yeah for sure, I don’t doubt you’ll come up with something good, but I do just think it’s a bad idea generally, and that will make your life harder

[turbid zenith]

The nice thing about being such a tiny school is we can try stuff and course correct more easily than if we were a giant R1 or something

[tall bolt]

FWIW I did take an intro to proofs course in a sense, but we spent a week going over logic, quantifiers and sets then a week on proof methods, then the other 8 weeks were spent applying that to get a taste of a variety of different areas of maths (some combinatorics and set theory, very basic analysis and we teased the idea of groups)

I think this is the way to do a proofs course if you insist on having one

[turbid zenith]

My plan is actually to introduce it through combinatorial game theory

Do lots of different games for Mathematical Thinking, then a deep dive on one game for Proof Techniques

[tall bolt]

My general position is that a proofs methods course is a complete waste of time though, because the actual logic behind why the like 5 main methods of proof work is not hard, the hard thing is knowing how to use them and that’s subject specific

It’s just important to then spend time explaining how to reason with those proof methods in introductory analysis and algebra, because they both have their own unique ideas and bags of tricks

[tall bolt]

Ooh that sounds quite cool, that could be fun

[turbid zenith]

And draw connections to other disciplines when possible, but I figure CGT gives a purpose to the proofs rather than just doing set theory or number theory in isolation

[tall bolt]

The one exception to this being a more general intro to university maths type course which covers some logic and set theory and then just gets you to taste a bunch of different stuff, I don’t think this is a bad idea. Though it’s probably best to do mainly combinatorics and set theory and then maybe a week each on analysis and algebra

[tall bolt]

That does sound like an interesting approach, I would be happy to hear how it goes as you’re doing it

[cosmic ibex]

That sounds like it would make an ideal occasion to present (informally) a game semantics for logical connectives ("or" and "exists" means you get to choose, "and "and "forall" means your opponent gets to choose, "not" interchanges the roles; when you reach an atomic formula its truth value determines who wins).

[turbid zenith]

There are some Boolean logic games out there but I’m actually gonna use Clue (the card game version) to introduce logic 😛

(I know it’s not a combinatorial game but it’ll be a good intro)

[feral swan]

This sounds like a fun course to teach & learn.

[fiery quiver]

Darn tutoring is revealing a lot of gaps in my knowledge. I can work backwards from memory of the definition of the derivative to explain the secant line stuff but I cant start the explanation the other way. Ig this sorta thing comes with experience but man I feel bad screwing up/giving a poor explanation when students got imminent testing.

Anyway, is setting up a resources site for various courses a good idea? Im thinking of just linking some youtube channels and Paul's math notes up there

[fiery quiver]

Fr. Also from personal experience as someone whos at a uni with a mandatory proofs course: it somewhat helps to familiarize myself with proof language but I think it was missing motivation and fun. My uni moved over to an upgraded version of the old class with some extra combinatorics and graph theory in it which I think is a smart idea

I did do quite bad with our first direct proofs in that class but we were standards based, meaning that we had a weekly quiz that covered a standard. Each quiz has an opportunity to redo a standard we could not pass from the week before with a chance to redo any standard at the end of the semester

[rapid tusk]

in the same way that HS geometry as it’s taught in the US is fucking terrible as an “””””intro””””” to proofs specifically

the results are for the most part insultingly trivial

“wow I don’t need 5 lines to prove vertical angles”

and none of the actually interesting stuff gets covered

[fiery quiver]

Ima be honest I barely remember anything from my high school geometry class. It was over covid. I just remembered being quite bad at it but literally everything past it was smooth sailing, including trig.

[winged maple]

Thanks for this, by the way. I had a think about how I interact with students and have made the environment more informal and welcoming to questions. Hope that helps them at least a little bit.

[turbid zenith]

Current draft of the proofs courses!

[zinc dove]

I see proofs by contradiction at the end of 195

[turbid zenith]

Yeah I want them in there :V

Now that I’m planning it I think it’s good to see them earlier instead of waiting until 196, but it’ll also come up when we prove R is uncountable so it’s hit then too

[tawny slate]

can i take this course for free somewhere

[abstract grove]

Hackenbush, hell yeah!

looks like a nice plan

will you cover the Schroeder-Bernstein theorem somewhere around bijections/cardinality?

[turbid zenith]

Maybe?

I would have to remind myself what that is again and how to prove it

[tall bolt]

It’s one of those things that you don’t need that often but is very occasionally quite helpful (IIRC it’s also part of ordering cardinals so I guess it gets used implicitly a lot)

I don’t remember the proof but I’m pretty sure it’s some choice nonsense, possibly not ideal for an intro to proofs

[halcyon glade]

the proof is actually quite elegant, and the result gets used all the time without thinking about it

it doesn't require any axiom of choice

it's one of the proofs that you can just draw a picture for and it makes sense

[tall bolt]

Oh is there a choice free way to do it? I’m pretty sure I just saw it as a corollary of well ordering tbh

[vocal phoenix]

Surprisingly no, there's a very "caveman" argument (you basically construct a bijection out of the two injections)

[tight star]

yeah i think schroeder-bernstein is a big result in "set theory without choice"

it's used extensively in famous papers like "Division by three"

https://arxiv.org/abs/math/0605779

[vocal phoenix]

Reminds me of one of my favourite paper titles: https://arxiv.org/abs/2309.11634

[vocal phoenix]

https://en.wikipedia.org/wiki/Schröder–Bernstein_theorem#Proof Here's an overview of the proof that doesn't require choice

[tall bolt]

Oh that’s quite slick actually, much nicer than having to do anything with choice

[midnight scarab]

A less slick (more straightforward) but still efficient proof:

#math-pedagogy message

[worldly sigil]

The fixed point proof is really nice

[green valve]

I do pure math, and has not been involved in competition math since elementary school. I just accepted to tutor a kid for my country's olympiad. Am I cooked?

[abstract grove]

uh, you might want to look at some past tests and see what you’re in for

[green valve]

Yeah these are difficult questions. Luckily there are some solutions online.

But yeah how should i go through this should we just start answering the past papers?

[cosmic ibex]

That's the usual advice, yes. But use those past problems wisely -- olympiads tend to avoid "type problems" where one can just drill a solution procedures, so the greatest value of old problems and their solutions is to be examples of which tools one should be sure to have available for the competition in question.
For example, you might end up teaching some elementary number theory from scratch, if the past problem sets are full of occasions to use the Chinese remainder theorem, or the theory of linear recurrences.
Depending on which kind of school system your student has been through, they may have formed an expectation that mathematics consists of type problems, in which case your chief goal is to help them unlearn the bad habits that come with that. (And better now than when they hit university and run into think-for-yourself problems there).

(There's also #competition-math).

[tight star]

I have heard that students struggle with the concept of unfolding and using definitions - do people have experience with this? What strategies tend to resolve this issue?

[austere delta]

I have experience with students asking for help about some exercise claiming to be completely lost.

Then I just ask, well, what's the definition of X? A few times and they solve the problem.

I like to think that having this happen a few times functions as a learning experience, but that I can't speak to.

[tall bolt]

I always think back to my like early days in physics and the tool box method, where at the start of a problem you write down every piece of information you currently know in the corner of your page. If that was a projectile motion question you’re writing gravity, launch angle etc, but I think it works similarly well in maths.

Whenever I’ve been tutoring first year maths students I’ve told them like “If you’re reading let f be a smooth function then I like to think ok what’s the definition of smooth let me write that down because I’ll probably need it, and it’s easier to use when I have it right in front of me”

I think when you have all the pieces laid out in front of you like that it can be easy to spot where you can plug in part X that you know and spot part Y that you don’t know yet, and I’ve found that generally seems to work

Another good thing to do is genuinely just take introductory algebra, do much of the basics just come down to applying the definitions of the objects you’re working with and that just gives you loads of practice

[halcyon glade]

Yeah I think it's just practice and emphasizing it as a skill that they can practice and become proficient at

[cloud spear]

So what I realized is that many students have an intense emotional experience associated with mathematics. For many, just the sight of a concept that they believe themselves (key term here is believe) incapable of understanding sends them into a kind of psychological paralysis, in which they are incapable of recruiting prior knowledge and making associations between information that would otherwise be completely obvious. Essentially past experiences and the intense fear of failure send many students into a state of "fight or flight". What I found as a tutor is that giving them room to address the emotional state and co-regulating with them through the experience allows them the mental bandwidth to break through. It's always important to emphasize that the implicit narrative they've constructed around not being capable of understanding is completely untrue. Typically once they become aware of their emotions surrounding math and have a couple breakthroughs they develop a transformative confidence in their abilities. Unfortunately, many teachers (and parents) lack the emotional intelligence to recognize that negative reinforcement (scolding, criticism, etc.) are completely counterproductive to someone trying to gather an understanding of logical abstractions. Hopefully that makes sense.

[tight star]

Oh this is fascinating to me actually

[cloud spear]

We all have experienced this. The dreaded impostor syndrome. That emotional location where despite all our achievements, none of us is good enough. It's a fantasy constructed by the mind in order to manage some unexpressed psychological strain

[tight star]

I’ve been thinking about the extent to which students’ difficulties with math are psychological vs conceptual or computational

I know that for myself, a lot of the difficulty I had in learning higher math was more psychological

[cloud spear]

Same. Funnily enough I had a really funny moment with a student where I asked her to express her emotions when I:

1. Opened her textbook
   (PANIC!)
2. Closed her textbook
   (Calm...)
   We spent the next 30 minutes talking about her emotions and unpacking them. She was then able to solve the entire homework ~30 questions in about 13 minutes flat 🤣🤣🤣

[tight star]

That’s pretty incredible

[cloud spear]

I mean we all know about fight or flight and what it does to the brain but I feel like we never apply this knowledge in areas like teaching. Developing emotional intelligence is probably just as important as technical mastery when it comes to mathematical communication. That's just my two cents.

[atomic pulsar]

this is smth i think about all the time abt the ppl in my class

the ppl who dont perform well do so because theyve drilled it into themselves that they arent gifted or are stupid

i dont really have any scientific basis for this but based on my own experiences i feel like 95% of getting good at anything is mindset and mindset alone

you need a good mindset to dare to tackle hard problems and when things dont work out you need a good mindset to find out why things arent working out instead of just resigning yourself to self diagnosed stupidity

[spare void]

I love this so much!!! thank you for sharing.

on courses I've taken for overall test and singular item construction, math sometimes necessitates that word problems be very long and wordy just to ensure zero ambiguity. it can be a really simple problem actually but if it's sufficiently wordy students WILL shut down 9/10 times. so it is funny, a bit, but also really sad. so math pedagogy doesn't end at education, it extends to addressing the social and cultural perceptions around math as a school subject.

big fan of taking a step back and co-regulating the math anxiety with your tutees. the struggle now is, how can we replicate this feasibly for a teacher inside a 30 student classroom working under a paced syllabus and limited hours per week, right?

[spare void]

my preferred approach is a very uhhh butchered and babyfied version of socratic dialog

[tight star]

Socratic dialog?

[spare void]

yesss

I learned that from. law school shdkfjfjdn

instead of info dumping everything you ask a lot of structured questions to guide the student to the answer through their own critical thinking

questions that adhere to the principles of the socratic method are meant to be like reallyyyy open ended but also thought provoking

wah I'm bad at explaining this it's been a while since my last ethics and philosophy class

and then the discussion is just like a series of throwing questions back at the student and then waiting for them to find the answer on their own

um but for the purpose of math I use a lot of leading questions and yes/no right/wrong recall/state/define type questions

it takes a while to get it right though. because it can also come across as just you asking for the sake of asking instead of trying to help

which can make a student cry... that's bad

[spare void]

but I like this a lot even if it took me a while to get it right. it's already at evaluate on bloom's taxonomy! pretty hard to beat

[cloud spear]

I know exactly what you mean! I've only ever tutored math so I don't have the requisite classroom experience to speak to this.

I feel as if the emotional regulation aspect is lacking in our general approach to education. Our shared experience with it comes down to us all being "math people" 😅 and the fact that math deals more so in abstractions that any other subject.

Maybe there should be some dedicated classes on the emotional aspect of learning. Not for educators but for students.

Again, I'm not qualified to say either way. I'm really just speaking from my experience tutoring and navigating my own emotional struggles throughout grad school

The research on the connection between emotional states and learning ability is pretty well established, it's more a statement on the dysfunction of our education system that this goes completely unaddressed.

*Rant over

[turbid zenith]

Anyone here use PreTeXt?

[feral swan]

How do you actually talk to them?
I don't know how to get them down the tree. I just see they're on the tree.

[cloud spear]

I started off by asking her how she felt in the moment. I told noticed she seemed a bit exhausted and overwhelmed and we barely started.

I listened to her describe it on her terms

And explained to her that everybody else has felt the exact same way.

That even big brain nerds with PhDs feel like they're not good enough or they don't understand. #Imanimpostor

It's part of it.

So after we recontextualize the experience of doing the homework because we called it out

And she actually spoke her feelings about the situation in words

It became easier to manage the anxieties and she was able to progress (quickly)

But again that's just anecdotal.

[tight star]

Is it accurate to say that, when limits are introduced to people in school, it’s presented primarily as a way to extend arithmetic (and general functions) to infinities and infinitesimals?

Given how much of limits at that stage seems to be about “manipulate the expression until you can plug in the limiting value”

In a sense it’s a very “algebraic” approach to limits

[cosmic ibex]

I doubt it's often deliberately motivated that way, but it does seem to be what many students take away from it.

[tight star]

yeah, that seems to be the case

[cosmic ibex]

I think it's an unintended consequence of how the concept has a fairly convoluted definition (certainly compared to anything the students will have seen yet), together with the fact that "manipulate until just can just plug in" is -- when it works! -- so much easier than going to the definition.

[tight star]

i wonder if there's any other way to present it so that students don't just take that away

[cosmic ibex]

Also, it seems to be somewhat tricky to find good accessible examples of a finite limit at a point, other than taking a nice elementary function and punching out a hole in it by introducing a 0/0 somewhere in the algebra.

[native iron]

I'm often disappointed (perhaps unduly so) that students don't understand that, when we write $\lim_{x\to 0} \frac{x}{x} = \lim_{x\to 0} 1$ we can do so because for all $x \neq 0$, $\frac{x}{x} = 1$. Instead, students seem to think something like "in limits, we can perform more algebraic manipulations that normal" or something like that.

[burnt vessel]

Luna Victa

[native iron]

I personally try to emphasize that limits concern the behavior of a function near but not at a point.

[tight star]

Yeah that’s exactly what I’m worried about

That students come away from limits thinking they’re just spicy algebra

[native iron]

However, I don't think I've found a way to actually explain that to students.

I also think this confuses students because they feel rightfully uneasy about those algebraic manipulations.

[tight star]

Mhm

[native iron]

"Can I actually do this manipulation" is, IMO, a good mathematical instinct (in moderation).

[tight star]

And the allowed manipulations for limits can feel… quite arbitrary?

[native iron]

Yeah.

[tight star]

What’s wrong with this explanation?

[native iron]

Uh, nothing I'm aware of?

[tight star]

Hm then why this

[native iron]

Oh. I think there's a gap between the concept I want to convey and my expository skills.

But to be honest what makes explanations "work" to non-mathematicians is pretty hazy to me. I don't feel like I've been able to figure out how to fully empathize with introductory calculus students. Not in the sense that I can't imagine being confused and frustrated, but I don't really have a sense of what they don't get.

I know that if I were in their shoes, epsilon-delta proofs would help, but that kind of rigor tends to scare most students.

[tight star]

It seems what would be ideal is some way to get across the epsilon-delta def without needing the full rigor

[wary rapids]

Drawings. Lots of them

Probably not a miracle answer obviously but it's what made it click for me and dozens of other people if i'm to trust the reception to 3blue1brown's videos in general

[halcyon glade]

I think one confusing thing is the use of the = symbol, when what is really meant is that something is getting arbitrarily close to a value, not that the function is ever exactly equal to that value. US students at the high school level are already notoriously not great at understanding what the = symbol means (a lot of people just view it as meaning "find the answer", etc.), so overloading it here feels like it just confuses things more. One could try writing everything in words first, and then introducing the lim and = notation as a symbolic convenience ("as x gets closer to 3, f(x) gets closer to 1", then writing "as x → 3, f(x) → 1" as shorthand, then finally introducing the "lim" notation).

I think this would also help with students getting confused about the concept of limits not existing

Just a guess though, I've never taught limits.

I don't think it's worth talking about the epsilon-delta definition except as a fun bonus to people that want more information (probably the easiest way to do it is via game semantics of quantifiers?)

[native iron]

Does this achieve anything that showing them examples where $lim_{x\to a} f(x) \neq f(a)$ doesn't?

I've also not taught limits, only tutored students who are learning them. However, next year, I will probably teach limits, and I'm not sure how I'm going to approach that. I think calculator games were reasonably helpful to me when I was learning, but they don't necessarily help with the concept we're talking about here.

[burnt vessel]

Luna Victa

[native iron]

I don't, for two reasons.

1. The audience for 3b1b is self-selecting.
2. I think some people are good at deluding themselves into thinking they "get" something without actually being able to use that knowledge to solve problems. I think 3b1b is rather bad in this regard. I like 3b1b; I think he does good things, but I don't think any of his videos are useful for actually teaching math other than maybe as an accessory to a course.

[wary rapids]

Absolutely true. I still believe pictures can provide good motivation and enable building intuition about what limits are and how to deal with them conceptually. Sure, getting good at conceptual and algebraic manipulation, in the sense of being able to output correct reasonings and answers, can only come with some time and struggle -- where the student truly engages with the material. But psychologically this can be well eased with the help of pretty pictures and some sort of narrative to put the imagination in the right context

Because showing students examples where something is true and others where it isn't may help them gather some bits and pieces of what's happening, but they may still feel like they don't have the whole picture inside their brains

[gloomy rampart]

Hello 👋

I have a nephew who is currently in third grade. I was asked to tutor him in math. I really love my nephew, but I have zero experience teaching kids. Besides that, there is another problem. Not only does he have much less capacity to focus compared to adults, as a child, but even among his peers he stands out as a troublemaker who can't stand still.

I'm reaching out because I want to help him succeed, but I'm honestly not sure where to start. Does anyone have experience tutoring young children, especially ones who struggle with focus and sitting still? I would be very interested in:

• Ways to make math engaging and hands-on for an active child, but at the same time
• Work on improving his ability to focus, his patience.
• Techniques to manage his energy and keep him on task without getting frustrated
• Absolute do not's when teaching a child
• How to earn his respect

I want to be patient and supportive, but I'm very worried that things are going to get south really quick because of my lack of experience. Any advice would be greatly appreciated.

Thank you!

[lethal hornet]

hmm. just a thought, you could try asking for some advice from a local mathnasium or kumon or something like that. they usually have good input on these situations.

[gloomy rampart]

Your local city, but no Mathnasium/Kumon branch listed here.
🙁

[elfin sleet]

Regarding the limits conversation just above - I recently tutored someone learning limits and at first we had all the issues that were mentioned but I eventually stumbled across a strategy that seemed really effective at fixing them, which was to introduce it in terms of a vague notion of continuity. What I did was draw (but not actually define, to avoid getting caught up in manipulations) a function that was continuous except at one point, and I said "this function is weird at this point, right? We would want the value to be [limit at the point]. In order to have a nice smooth well-behaved function, it should be [limit]. But instead this function goes and puts it way over at [value at the point] for no reason! Now, what exactly do we mean by 'it should be [limit]'?" At this point I got some input from her, in which she mentioned the function being close to the value, and I latched on to that and explained that that's what we mean by limit - the function gets really close to that value as we come from either side, so that's the value that the function "should" have. Then I went on to draw the continuous version of the function and point out that continuity means the limit equals the value, and then I showed functions without limits and explained that there's no limit because there's no value we could put for that point in order to make the function continuous

[tight star]

i actually have an article coming up which deals with trying to formalise this notion of "closeness"

[tall bolt]

I still think this to myself literally every time I see a limit, granted I don’t work with them often but I never remember what is legal and illegal under which sets of assumptions

[cosmic ibex]

= is its own can of worms, but I think it does connect here to some degree. I've sometimes seen mistakes that strongly suggested that people were mentally parsing $$\lim_{x\to 3}\frac{x^2-9}{x-3} = 6 \quad\text{as}\quad \lim_{x\to 3} \left[ \frac{x^2-9}{x-3} = 6\right]$$ and in retrospect I think such a misconception can also often explain why it's difficult for people to internalize limit rules correctly, even when it isn't immediately obvious that's what their problem is.

[burnt vessel]

Troposphere

[native iron]

I'm struggling to imagine what that could mean. Which I guess is how those students feel.

[turbid zenith]

I think in at least some cases, the issue is that the student is confusing the limit as a process versus the limit as an object. And a lot of educational materials don't help with that distinction.

So they think of the limit as "the process of getting closer and closer" rather than "the thing you're getting closer and closer TO"

Which is why you hear people saying "the limit approaches 6" instead of "the limit IS 6"

[rapid tusk]

precision of language is not covered nearly enough as a meta skill from what I’ve seen 😔

[native iron]

Students always say things wrong and I never know how to insist on precision without being pedantic.

[halcyon glade]

I think it's always a give and take, you can acknowledge what the student got right while also kindly suggesting feedback

[feral swan]

It's also the first university-level operation they are introduced to and is in some ways the most complicated machinery they have seen so far.
It's also the first time where you have a new concept which also introduces new manipulation rules so they need to both understand the concept and that you can black-box the concept and just manipulate it, only that some manipulations are not allowed but some aren't, with a logic that is distinct from what they saw so far.
Expecting total understanding at a first course is too much imo.

[turbid zenith]

This is why I teach limits at the end of Calculus I, after they're already familiar with derivatives and integrals.

[tight star]

Hm, how does that go?

[turbid zenith]

We start with intuitive infinitesimal definitions so we can solve real-world problems from the get go and students see calculus as useful

[tight star]

infinitesimal definitions?

[turbid zenith]

Then we bring in limits at the end to clean things up and talk about how mathematicians formalize intuitive concepts

Yeah. Derivative is tiny change in y over tiny change in x. That sort of thing.

[tight star]

hm, i think i've always felt uncomfortable with these kinds of definitions..

plus arithmetic with infinities and infinitesimals has lots of strange edge cases

[turbid zenith]

Something the students can viscerally see.

[turbid zenith]

The way I talk about them, I'm sort of thinking like a limit anyway without actually call it that — for example when I first introduce the derivative it's in the context of Kleiber's law (caloric needs of an animal), and we talk about sensitivity to changes. We have a change of 0.1, of 0.01, of 0.001, notice that the values are approaching something, and then think "what if we had an 'infinitely' small change?"

There are also moments where I'll call out "that thing we did looks kind of uncomforable even though it seems to work... we'll tidy that up later in the course"

[tall bolt]

Isn’t this the approach usually taken in the UK anyway? We learn derivatives and integrals before limits in highschool at least as far as I remember

I don’t honestly remember if I ever covered limits in highschool but we certainly did a pretty reasonable amount of calculus, most of what I’ve seen people call calc 2

[turbid zenith]

How did they "define" derivatives and integrals?

[tight star]

Yeah I guess I don’t know what an infinitely small change is supposed to be

Archimedean property and all

[tight star]

Hm I thought limits came before

[turbid zenith]

Most students in my Calculus I course have never heard of the Archimedean property.

[tight star]

I’m aware

[turbid zenith]

And an infinitely small change isn't hard to conceptualize — lots of students seem to be able to at least imagine it

[tight star]

How, exactly?

[turbid zenith]

Ask anyone who's insisted that 0.999... < 1

Sure it's not going to be """rigorous"""

[tight star]

In part I started thinking about this because of the difficulty people had with infinities and infinitesimals

[turbid zenith]

But it tends to be useful, and we can add an appropriate amount of rigor later

[tight star]

#calculus message

[tight star]

Yeah here’s another example

[turbid zenith]

I'm thinking next time I do it I want to also mention hyperreals at the end as another way people have figured out how to formalize what Leibniz was originally thinking of

[tight star]

I would feel confused simultaneously thinking “0.999… = 1” and also “infinitely small changes exist”

[turbid zenith]

Yeah, those can't both be true

My answer to whether 0.999... = 1 is "it depends on what number system you're working in, and what you mean by ..."

But it's reasonable enough to start off in a world where nonzero infinitesimals do exist, do some useful stuff, then zoom back out to the real numbers

[tight star]

I guess I’d have to take your word for that

I’ve always approached calc with finite quantities and not infinitesimals

[turbid zenith]

Part of this comes from the idea of "if your number system doesn't have what you want, invent something that does," just like we do with i

[tight star]

Right, but usually I think of that as another example where students get confused

[turbid zenith]

If you can have numbers less than 0, and numbers that can't be represented as fractions, and numbers whose square is negative, there's no reason you can't have numbers that are infinitely small but not zero

[tight star]

In terms of why we’re allowed to add a solution to something that didn’t have one before

[turbid zenith]

Well to be fair in the case of i, nobody ever tells the students how it's useful

It's done in the abstract with zero application

Not to mention zero geometric intuition either — I've met SO many students who learned complex numbers but never graphed them

And I show them the whole bit where multiplying by i rotates by 90º, and how that generates the powers of i, and they invariably go "oh my god, we just had to memorize that, that makes so much more sense"

Teaching complex numbers without the complex plane seems like a crime to me

[tight star]

Oh of course, but this to me seems like a point against taking a purely “algebraic” approach

[turbid zenith]

What do you mean?

[tight star]

In terms of teaching complex numbers purely algebraically not being a good idea

[tall bolt]

Possibly, I guess the Scottish and English systems are actually different so maybe you did, but from looking it up to refresh my memory it seems the only discussion of limits was for linear recurrences and that hardly counts because there’s just an explicit formula

Possibly there is more covered but I don’t remember it and it didn’t show up when I googled quickly

[turbid zenith]

Oh, yeah

Unfortunately that's precisely how it's taught in the US

[tall bolt]

That’s a very strange decision because so much of their power and clarity comes from the geometric interpretation

I probably would still introduce them algebraically though because that is literally what i is, but maybe that’s my algebraic bias showing lol

But yes I would certainly then show all the other, very often most useful, ways of looking at them

[turbid zenith]

It seems to be along the lines of:

"Well, this quadratic equation has no solution, but if we invent these weird numbers, it does!"
"Why would anyone want to invent those weird numbers?"
*crickets*

It also kinda seems to be part of a larger trend where people seem to hold the belief "pictures bad, symbols good." Like, pictures are for kindergarteners, to do real Mathematics with a capital M you need symbolic representations.

Personally I blame Bourbaki.

[tight star]

What would your answer to this be?

[turbid zenith]

My go-to example is AC electricity. Multiplying by i repeatedly changes the current from positive to stopped to negative to stopped to positive again, and certain components of a circuit (capacitors and inductors) are out of phase with the current.

So I tell students, yeah you can't have "i apples" in your hand, but that's not what complex numbers are useful for. Complex numbers are useful for things that rotate and things that fluctuate (among other things).

I do want to workshop my explanation more to give an ELI5 of why capacitiors and inductors are out of phase though

[tight star]

Mhm, I see

I’ve had some success in the past motivating complex numbers through a geometric interpretation of real number addition and multiplication

With a categorical flavour

[turbid zenith]

How so?

[tight star]

#linear-algebra message

You’re using a form of is-does duality here, in fact very similar to the kind that gets used in the yoneda lemma

Back then I was prototyping this explanation, I’d probably phrase it a little differently now

[turbid zenith]

One of these days I’ll understand enough category theory to get this 😛

I do get the is/does thing but I have no idea how the yoneda lemma works

[tight star]

The yoneda lemma is essentially how is/does duality manifests within category theory

[turbid zenith]

Other than that the hiragana よ gets involved somehow XD

[tight star]

In this case the “does” of an object X corresponds to the function Y -> Hom(Y, X)

Or really, the Hom-functor Hom(-, X)

And yoneda, from a certain (and perfectly rigorous) perspective, asserts that X is isomorphic to Hom(-, X)

[turbid zenith]

Here’s a question

[tight star]

Quite literally, what X “is” is isomorphic to what X “does”

[turbid zenith]

What would you say to a student who asks, “why can’t you think of multiplying by -1 as a reflection instead of a 180 degree rotation”?

[tight star]

Oh, you absolutely can

I would make an appeal to the fact that it’s easier to implement a rotation of a string irl, then reflect it, since reflection involves passing the string through itself in some way

Another way to say this is that reflection is a “discrete” operation, while rotation is a “continuous” one

[turbid zenith]

Fair enough

[tight star]

You can smoothly rotate the number line by 180 degrees, but reflection is necessarily more “snappy”

[turbid zenith]

So it’s not immediately clear how to “halfway reflect” something

[tight star]

Yes, because of the discreteness

[turbid zenith]

That’s a good way to do it, I like it

[tight star]

[tight star]

Is/does duality isn’t a fundamentally categorical idea though, not even a fundamentally mathematical one

It’s often possible to classify perspectives on a thing as either “passive/intrinsic/internal/prescriptive”, or “active/extrinsic/external/descriptive”

[turbid zenith]

I wonder how infinitesimals might be conceptualized using is/does duality

[tight star]

Generally speaking, “does” in maths refers to some kind of function

If elements of a set A interact with elements of a set B to produce elements of a set C

[turbid zenith]

You could think of multiplication by an infinitesimal o (omicron) as shrinking down the whole real line so that it fits close to 0

[tight star]

So you have a map A x B -> C

Then you can view this as a map A -> (B -> C)

An element of A can be viewed as a function from B to C

This is essentially the “is to does” translation

[turbid zenith]

(I use o and ω as my go to name for infinitesimal and infinite quantities)

[tight star]

Then you can start asking questions like:

• Do different elements of A produce different functions?
• How can we recognise if a function B -> C came from an element of A?

[turbid zenith]

Makes sense

So I use that in my explanations all the time without thinking of it in those terms

I often say “if i is a number, what happens when we do numbery things to it?”

Although I guess this is more like IT doing the numbery things

[tight star]

mhm, i don't think it's a particularly esoteric concept

In category theory, the pairing we use is Hom : C^op x C -> Set

In linear algebra, you might have an inner product V x V -> R

though i have found it helpful to give a name to it

[turbid zenith]

By the way, just curious

Have you ever heard of * (star) in combinatorial game theory?

[tight star]

no, i haven't!

oh wait, hm

i did watch this video on hackenbush a while back

[turbid zenith]

It's one of my favorite examples of an infinitesimal!

Yeah it's mentioned in the Hackenbush video — the value of a single green branch

[tight star]

https://youtu.be/ZYj4NkeGPdM?si=9UYe3K8afGUv4I61

yeah

[turbid zenith]

Yup that's the one

[spare void]

nice to consider. limits-derivatives-integrals calculus is, in its broadest sense, the very first time most students will be seeing math be stretched outside of its well established territory in high school. where all processes and all numbers are well defined, and there are rules that will always be followed. until they aren't, and now you can divide by zero because a constant/0 = inf. among other examples...

I understand the compulsion for rigor for most people in this chat. but students simply won't have enough prerequisite knowledge to understand the necessity for rigorous definitions. when you explain that an integral is the sum of an infinite number of infinitesimally small subdivided areas under a curve, they'll be understanding the vocabularies infinite and infinitesimally small from the perspective of what those words mean in basic english. and that will be enough for the express purpose of introducing a new lesson. they can relearn the appropriate vocabularies of what infinite infinitesimally etc mean in the mathematical sense as they go further along in college, when they develop better content literacy.

[spare void]

"infinitely small change" is hardly passable language, if at all, in math. like how do you even begin to define small, how small should it be, infinitely as to what sense ... but, just intuitively, we see that it could work when students are trying to visualize the general behavior of limits

[turbid zenith]

That's something that can start off fuzzy and be made progressively more precise when students are ready for it.

You don't have to start with the most pedantically precise covering-all-pathological-cases definition on Day 1.

[spiral elbow]

I feel like arbitrarily is a better choice of word than infinitely in this case: the limit of f(x) when x -> a is determined by the behaviour of f(x) when x is arbitrarily close to a. Not a huge difference, but maybe it helps to avoid some misconceptions that can be hard to correct later on

But then again, I'm not sure how many students in the target audience are familiar with the word arbitrarily 🤔

[turbid zenith]

Yeah. Infinitely is a good bit more visualizable offhand.

And "arbitrarily" is one of those words that has a different English meaning than mathematical meaning

[boreal agate]

is it though?

if we look at the definition, "forall" is remarkably similar to "arbitrarily"

i think the main difference is when we say "arbitrarily small"

[cosmic ibex]

I'm not a native speaker, but my impression is that in non-mathematical English, "arbitrary" has some fairly negative connotations.

[turbid zenith]

Yes, this

"The professor just gave us grades arbitrarily"

[quasi maple]

Roughly, "with little if any consideration given"

"arbitrarily small" - "as small as we bother to make it"

[quasi maple]

And in this example, "The professor just gave us grades without caring to check whether they were reasonably assigned"

[elfin sleet]

Yeah, I think it just means "without any particular reason" which is similar to what it means in math. It's often used negatively but I think that's just because we don't usually like it when things are done without reason

[zinc dove]

Hello, do you know any server that has a "CS" pedagogy channel, similar to this channel? I looking for ways to make programming labs that can be solved by students, but not readily solvable by LLMs. I'm looking for papers and already found some that deal with cheating in CS1 courses (introduction to programming).

[rapid tusk]

see if the cs server linked in #old-network has anything?

[zinc dove]

No, it doesn't

[rapid tusk]

ah rip

[cloud zealot]

it's worth mentioning for the sake of a balanced view that the hyperreals are not complete, which is unintuitive in its own way, and properly defining and working with the hyperreals just pushes the abstraction buck of epsilon-delta to these new numbers so that students don't get too enamored with this idea when they do get to epsilon-delta proofs.

https://math.stackexchange.com/questions/2097694/why-is-non-standard-analysis-not-fully-or-at-all-integrated-in-our-current-schoo

https://math.stackexchange.com/questions/1609463/what-are-the-disadvantages-of-non-standard-analysis

of course, that doesn't mean i'm specifically discouraging their study

[dim depot]

I believe offering a standalone "Introduction to Proofs" course has these three advantages:

1. It provides systematic training in logic and proofs, preventing students from "groping in the dark" during analysis or algebra courses.
2. It helps students understand fundamental structures like propositional logic, quantifiers, proof by contradiction, and mathematical induction.
3. It allows instructors to thoroughly explain proof strategies (such as constructive proofs vs. counterexamples).
   Therefore, I believe such a course can effectively help students adapt to the transition "from computation to proofs."

[rapid tusk]

i think it's just a shitty bandaid

for proof writing not being well integrated into lower level courses

that's also a symptom of the death of honors versions of said lower level courses at many schools

[abstract grove]

my version of that class was called Introduction to Higher Mathematics and all math and physics majors were required to take it their first quarter

it went over various proof techniques and some of the major developments of modern math and open problems

it was very good

going from AP calculus to that course was a huge change and definitely helpful in my case

[tall bolt]

My uni had a similar course, called proofs and problem solving and it was an intro to set theory, propositional logic and proofs but we learned all of the proof techniques by doing very brief introductions to combinatorics, analysis and a little bit of group theory. I think this is a good way to do it if you’re going to do it

But I’ve always been of the position that it’s kinda a waste of time. You need to introduce people to set theory logic and the techniques, fully agreed, but the actual logic behind proof methods is incredibly simple and I don’t think that’s what anyone gets stuck with (like what’s the most complicated thing? Induction possibly, but that’s very straightforward, maybe proving the contrapositive but again, not that hard)

The actual difficult thing is knowing when and how to use the techniques, but that’s actually different from subject to subject. Analysis has its own set of ideas, as does topology and algebra, and you only learn those from doing those subjects

My position is just that introductory algebra and analysis should take more time to develop those skills, and remember that people are new. It’s been my experience that even though people took the intro to proofs courses they still struggled in those classes because it’s just more complex and requires subject specific knowledge, and there’s only one way to get that

[tight star]

One thing that students struggle with is the more technical/computational/formal aspects of proofs, I think

Conceptually a proof isn’t necessarily that difficult to grasp

But knowing how to actually write down a proof, and translate something from mental reasoning to symbols on a page, is hard

I can certainly remember times where a student could explain to me, in words, that a sum of even numbers is even

But then was unsure how to actually write that down as a formal argument

A good analogy is the difference between being able to describe an algorithm to solve a problem, and being able to actually program it correctly

(Indeed the curry-Howard correspondence makes this precise)

I could explain the idea behind mergesort, but I’d have to think quite carefully to implement it correctly as an actual program

<@&268886789983436800> spam

(they seem to be hitting multiple channels)

[tall bolt]

Oh yeah I fully agree with this, and reading first years proofs makes this incredibly clear, but I don’t think proofs classes solve this issue at all

I think the way around this is a lot of feedback on the proofs they write, teaching them how to properly read the proofs in their books (because this is a source of hundreds of theoretically well written proofs) and just getting them to practice and t a lot

Students definitely need help with writing proofs, that’s clear, but I think the solution is just better teaching, I don’t know how an intro to proofs course solves that problem

Like I think analysis is particularly bad for this, because how you solve a problem in analysis and how write a proof are essentially the complete opposite of each other and if you only read the book I think it’s hard for anyone to grasp how to actually do it themselves, it’ll feel like big N values just come out of nowhere

So this should really be emphasised and taught in lectures, having a lecturer just run through a proof and declare some bullshit bound works isn’t helpful for anyone. Students have access to books for the technical stuff, the lectures should really be focused on the actual skills required

[abstract grove]

I mean, if it has them attempt to write a bunch of proofs for the first time, that's pretty helpful for that

and by the first time I mean, since high school geometry

or 8th grade or whenever they took it

[spiral elbow]

I like having a discrete maths course function as an intro to proofs. Then you can spend the first couple of weeks going through the basics of proofs, then you can proceed by applying them to concrete stuff like recurrences, combinatorics, graphs, simple number theory, etc.

[tight star]

I wonder whether putting more of an emphasis on the syntactic nature of proofs could help at all

[tall bolt]

How do you mean?

[tight star]

Something like natural deduction

[cosmic ibex]

I feel that would actually be counterproductive in many cases.

It would encourage some students to think that all that matters is following the symbolic rules, and lead them to attempt to avoid having any intuition for why and how those rules work ...

Sure, there are other students who would be completely ready to deal with formal proof theory right out of high school; they should be free to seek it out themselves for the intellectual satisfaction of it. But I don't think it would help those who already struggle to follow along with prose proofs.

[tight star]

Hm, then what would you suggest?

[cosmic ibex]

I don't really see any alternative to engaging heavily with ordinary prose proofs.

[tight star]

Hm, I guess I was trying to see in what ways you could improve the existing approach to teaching proofs

[cosmic ibex]

Well, the most common approach seems to be "close your eyes and hope students figure it out themselves based on homework feedback".

[tight star]

I suppose I’ll keep experimenting to see if there’s a better way

[spiral elbow]

I learned a lot from playing around with Lean before uni actually, I wonder if that could be helpful for others. Lean makes it easy to see what your goal is, and what hypotheses you have to work with, then it becomes almost like a game trying to get from the hypotheses to the goal. I guess this would be somewhat of a syntactic approach, but I don't think that's necessarily bad. I think a lot of students struggle with knowing what you can and cannot do in a proof, because the rules are never clearly stated

[tight star]

Yeah I had a similar experience, which is exactly why I was considering going more syntactic

[spiral elbow]

I don't think teachers should shy away from making things formal for fear of making things difficult for students - a lot of times, things are difficult precisely because it's not formal, instead things are vague and handwaved. Atleast this has been my experience

[native iron]

I wonder if Lean could be especially helpful for people that feel they're good at programming and bad at math.

[halcyon glade]

I feel like what students would benefit from most is involved feedback on their proof attempts, but unfortunately, there's often not much capacity for that since it's a lot of work

[spiral elbow]

I guess Lean could help in that respect, since you get immediate feedback on whether your proof is correct or not. Unfortunately, if your proof is not correct, you don't get feedback on how to correct it

[quasi musk]

Lmao

[tight star]

Curry-Howard correspondence and all

[halcyon glade]

In the end that won't teach them how to write proofs in natural language though, that's partly understanding logic and partly developing specialized communication skills

I wouldn't mind if a course leaned more on those soft skills of understanding how mathematicians communicate

[quasi musk]

I do think there is an over emphasis on rigor in undergraduate mathematics, to the detriment of students learning math

[tight star]

An attitude I take towards this is that coming up with the “natural language” version of a proof is intrinsically hard

[quasi musk]

It takes too much time to get to the frontier in math when you're constantly building everything from the ground up

[tight star]

But the process of converting natural language to a symbolic proof is far more mechanical

[halcyon glade]

Yeah I agree Pseudo

[tight star]

I think it might actually be worth emphasising this part, since it’s “unnecessarily hard” for a lot of students

That way they can devote more brainpower to the parts which are intrinsically hard

[tardy ember]

the problem i've seen sometimes with more syntactic approaches is the opposite of the "not formal enough" issue - where someone can see intuitively that something should be true, and they're completely right, but they just haven't been supplied with a rule that says that thing, and so it's difficult-to-impossible to prove that that holds within the system for no good reason

[halcyon glade]

There's certain things about the way you actually write a proof, like how to structure it to be readable, emphasizing the main point, what details to omit, etc. that can be really difficult even if you have the logic in your head

[tight star]

Hm, do you have an example?

[tight star]

Exactly, but I think these kinds of “structural” parts of proof-writing might also be easier to teach

[quasi musk]

I think using the archimedean principle to show that for any $\varepsilon > 0$ you can find a n large enough so that $1/n < \varepsilon$

[burnt vessel]

MoonBears-C-

[tight star]

As others have pointed out, the natural language part of proofs varies quite a lot between different math fields

[quasi musk]

I think a lot of students just intuitively think $1/n \to 0$

[burnt vessel]

MoonBears-C-

[native iron]

A student I was working with had this exact problem.

[quasi musk]

Which they're right, but of course they don't know really how to work with

[native iron]

He just couldn't get to the point of flipping $nx > y$.

[burnt vessel]

Luna Victa

[quasi musk]

I was working with a student that claimed $$\bigcap_{n=1}^{\infty} (-1/n, 1/n) = {0}$$

[burnt vessel]

MoonBears-C-

[quasi musk]

I was trying to get them to prove this carefully, and they just said it's obvious

[native iron]

People starting in real analysis want to call everything infinite a "limit" and apply intuitive ideas from calculus to them.

[quasi musk]

So it's definitely true that emphasizing rigor does teach us something, but I do have to wonder how our colleagues can teach applied math/physics, and they seem to do just fine

[tight star]

I think this is to do with the standard of truth in different fields

[native iron]

Can you expand on what you mean by this?

[tight star]

In physics the standard of truth is “agrees with experiment”

It doesn’t matter how logically airtight your argument is - if it doesn’t match experimental data, it’s bunk

Rigor can sometimes be helpful in physics, but it’s never the end goal

Modelling reality is

[halcyon glade]

Reminds me of a physics professor I had where any time someone would ask a question about where he got a formula, he would be like "from friends in the math department" and then move on

[tight star]

Yeah I’ve had professors who’ve said similar things, at every level

[quasi musk]

I find my applied mathematics course that's not focused on developing everything from the axioms carefully, but rather just trying to solve problems & model them, very refreshing now

[tight star]

I’m biased because I’m a physicist, but I can certainly sympathise

There’s a joke about how “in mathematics you have to prove yourself to other people, in physics you have to prove yourself to God”

[quasi musk]

Granted it's a 3rd year graduate class, assuming we've taken PDEs, Real Analysis, etc. but it's just been very eye opening that my skills in real analysis are actually useful at these modelling stuff

I do agree that first & second year grad courses (at those pitiful publics that have to teach in the first two years) should have a huge emphasis on rigor

And it's hard to prepare students from undergrad going into graduate school for that if you don't

[tight star]

Tao has written quite a bit on this I believe

Something about “post-rigorous math”

At a certain level, even if there are technical errors in a math paper, it’s much more likely to be “morally” correct

Than technical errors in a proof given by an undergrad

[quasi musk]

I know there used to be an advanced calculus class before taking real analysis

[halcyon glade]

I think there's definitely value in learning to work within a deductive system

[tight star]

It’s probably the part of proofs that are most novel to students, I believe

The natural language formulation of proofs has analogs in other areas, just as the concept of a logical argument

And everyone has to deal with those to some degree

But maths is kind of unique in the way it writes down/formalises its arguments

[halcyon glade]

I think people have difficulty conceptualizing the fact that they have to think about that while doing math, since so much of their previous education is rote calculations, at least in the US

[tight star]

Mhm mhm

Partly I’m wondering whether you could lean into that to make the transition to proof-based math easier

It could certainly backfire, but perhaps if students were first introduced to the “calculational” parts of proof-writing, they could gradually see how their calculational skills can now be used to make logical arguments

[tawny slate]

weird, i interpret it the opposite way. only humans care about whether the results are pragmatic and observable. only god would care about whatever abstract concepts exist in the ether of formalized logic

also im pretty sure physicists are less religious than mathematicians

[tardy ember]

i think this #discrete-math message is the case i was thinking of

[tight star]

Yeah I would phrase this in terms of intro and elim rules

Good way to start talking about universal properties too

[tired belfry]

Gang I impressed a girl with math
What is happening

[halcyon glade]

this isn't the right place to talk about this, please go to a discussion channel

[halcyon glade]

That seems like it's the fault of the professor for not actually teaching what to do tbh

[boreal agate]

idk lean, but from my limited experience with coq id argue it will probably not help that much considering how slowwwwwww proofs become
everytime i have a situation where i need dependent pattern matching but i cant because of equality i want to punch the screen

[rancid hull]

i was actually hooked into math by agda before anything else, at a time where i did feel good about programming and bad about math }:). previously i couldn't wrap my head around proof, and it really helped to have an interactive system whose rules i understood.

[boreal agate]

i suppose proof checkers can be useful as motivation for students more comfortable with programming

but i think they dont make math "easier" in any case

just more approachable to certain people

[tawdry venture]

chat im writing a unit test for my grade 10s
im trying to decide whether part c of this question is best valued at 2 or 3 marks

i can come up with a 3-mark scheme like:

Calculates distance from R to _their_ 1(a), or square thereof  1
Compares against _their_ r^2 or r from 1(b) as appropriate     1
Correct conclusion -- dependent on prev marks                  1

but i think part c is not difficult enough to justify being a 3-marker

and if it's relevant (most likely is) im prepping these kids for A-levels math papers

and i want to keep my mark assignments roughly in line w/ those

[austere delta]

Have they seen a lot of examples of determining if points are outside or inside a circle before, or do they sort of have to realize that it's inside when the distance is less than the radius themselves?

[tawdry venture]

we've covered circles and gone over why the equation of a circle in the xy plane is what it is, but the realization to compare the distance to the radius will have to come from them.

[austere delta]

Then I think 3 points is fair

At least in the sense that it should be worth more than b

Not sure b is really worth more than a though. The fact that you need to do a to do b already gets you that extra point

[midnight scarab]

I assume it's 1 mark for finding r and 1 for knowing the eq of a circle

[tawdry venture]

yeah p much that

[quasi maple]

I'd make this 2-marker, personally

From what I've seen in mark schemes, for AL maths you don't get ECFs for the Accuracy marks, only for the Method marks

I'm not a test-maker, though; so I'd consider looking through mark schemes (specifically with the A/B/M mark breakdowns)

[tawdry venture]

ECFs?

[tardy ember]

i think it's "error carried forwards"?

[quasi maple]

They're more common in the other science A-Levels, and more generous, but from my experience less so in Maths

(I know I use them too but yeah abbreviations in mark schemes are awful)

(like, I still forget what AWRT - "answer which rounds to" - means, because I only see those in stats papers)

[tawdry venture]

im planning to make a question on function transformations and asking the students to apply a sequence of 3 of those (translation, reflection, stretch) and im thinking of awarding 2 marks per step for a total mark of 6 for the question

is this too much

[tall bolt]

I would probably say so, but it’s going to depend on the overall paper really

You could possibly do like 1 mark per and a bonus mark for getting them all correct to make it a 4 marker, but again I think it depends on the specific question and the larger test in context

[turbid zenith]

I've started using them myself, but I give students a guide to them

[astral agate]

Does anyone know how to make geometric figures such that they show their sizes

Like in textbooks, when they show you a square and under it you see its size measure

The only thing I’ve figured out is to go to geogebra and make screen shots

[austere delta]

I mean you can make them in Tikz, or just in some external drawing program.

[rapid tusk]

i use asy

[gentle talon]

I'm reading about cognitive psychology and thinking back to how I learned math in both high school and college makes me realize that following textbooks like Stewart Calculus forces a lot of massed practice. Have any of the teachers or professors here found a way to deal with this?

Coming up with more of an interleaved approach with different lessons and even chapters. Instead of assigning 1-whatever odd for homework, pick a subset and then have students go back to some of those problems at a later date? So it could look like 1.1-1.3, 1-9, then 1.1-1.3 10-20, 1.4-1.6 1-9. What do you think? I'm also thinking what if the whole chapter was presented before homework and then you start assigning problems from the whole chapter so that the students start to develop the discrimination skill for different problem solving methods.

[gentle talon]

Stewart was just an example, really any textbook has this problem.

[rapid tusk]

some lower level books will have "mixed review" exercises in each section to try to keep older material fresh

(i dont know how well it actually works in practice)

[pure light]

well if you have more frequent shorter homework assignments then stufents have to revolve their schedule around your homework schedule which they might not like. with longer more infrequent homework assignments many will do it all in one go last-minute but they could choose to do it a little at a time if they wanted to

also if you wait an entire chapter before starting homework (if it's a book with long chapters like stewart) then you risk people forgetting the start of the chapter by the time they start doing exercises on it

it could be a decent idea to start introducing some questions from previous sections each assignment to keep students thinking about previous topics, although ofc that leaves less room for the current topic

[gentle talon]

this could be turned into a low stakes quiz at the beginning of every class. Forgetting is not that big of a problem, research has shown that the more effort required to remember the better.

This is the book I'm currently reading about the topic: https://www.makeitstick.com/about-make-it-stick

[pure light]

sure, as long as you don't mind losing class time to it

[cloud zealot]

https://www.rxddit.com/r/math/comments/1o38thg/are_there_any_books_or_resources_that_collect/

[fickle quiver]

I'm curious if there is any research regarding when would be a reasonable time to begin various math topics with a child. I really see no reason a fifth grader who's competent at standard arithmetic should not begin algebra, and with that starting point, calculus would not be unreasonable by grade 8. Also curious about the more fine-grain topics appropriate for kindergarten onwards.

Sorry if that's quite vague.

[rapid tusk]

heavily depends on the child

i could probably have done calculus in 8th

[vapid trench]

I love this book!!!! As a potential future educator it gave me a lot of insight, and it also has helped me frame my own learning in a new way as I continue my math education.

[fathom cobalt]

does math pedagogy stand for teaching math or for math with children?=

[tall bolt]

(See ⁠get-advanced-access to use this channel) For teachers, tutors, TAs, and professors to discuss math teaching techniques. This is not a channel to ask for math help.

[fathom cobalt]

okay i get it so it is more like the first one but still covers a bit of both right?

[spiral elbow]

Pedagogy means the study of teaching, it doesn't matter if it's for children or adults. But interestingly, pedagogue comes from latin and means teacher of children, so you're not that far off

[cosmic ibex]

(Greek not Latin, though).

[spiral elbow]

-# same thing

[fathom cobalt]

its probably due to a lack of translationskill on my end, sorry

[vocal phoenix]

It's really not

[native iron]

Pedagogue originated in Greek, but the Romans used the word too, I believe.

[native bay]

General question: Those who have taught math at community colleges, what’s that like in terms of classes, student quality, etc

Also is it generally acceptable to teach math at a CC with a MS in Statistics but BS in Math?

I’m getting mixed signals, some want Math/Adjacent Fields but others say Physics is acceptable and other CCs say just MS math/applied math/mathed

[tight star]

A kind of separate question, but what are people’s preferred ways to teach the concept of a quotient (say specifically a quotient set). From what I’ve heard, students can get confused if there’s not some canonical choice of representatives, or the way in which equivalence classes overlap

[halcyon glade]

Wdym equivalence classes overlap

I haven't taught anyone about a quotient in a while but I usually think about it as a way to ignore details you don't care about or zoom out on a set, like maybe you have a set {apple, pear, celery, spinach} and you quotient it out to {{apple, pear}, {celery, spinach}} because all you really care about is whether you're talking about a fruit or a vegetable

Z/nZ is probably the classic mathematical example that I would start with after that

[cosmic ibex]

The blessing and curse of Z/nZ is that there's a natural choice of canonical representatives of the residue classes.
Blessing because it's easier to think concretely about that way.
Curse for the same reason because it permits students to cling to thinking about representatives instead of entire classes.

[halcyon glade]

Yeah I think then it's just a matter of having multiple examples to think about

[tawdry venture]

do yall think it is a good idea to suggest my students play around with function graph transformations on Desmos, on their own time, to get a better feel for how these things behave

[lethal hornet]

i’m curious as to why you think it might be a bad idea

[rapid tusk]

if done with sufficient structure it could be cool

[tawdry venture]

cutting into students' free time even by implication

[abstract grove]

what if you suggested that they would get higher grades if they spent more time studying?

like

that doesn’t seem like a concern you should harbor as their teacher

[tall bolt]

Very good idea, it’s just a useful study tool that’ll help them visualise functions and understand how they behave under scaling and translation

I don’t see how it’s any different to suggesting they should be studying outside of class or doing their homework etc, you’re just recommending a study technique/tool

[tight star]

I have lots of fond memories playing around with desmos

Students hardly get to experiment in math, so I think desmos is a great idea

[tawny slate]

especially in an age where students are very likely to reach for chatgpt almost on reflex, and search engines are becoming worse by the flood of AI generated content, i find it supremely important that students know where to reach for core tools like desmos and wolframalpha

in combination, they are the greatest (afaik) free calculators in existence, accessible with only internet/smartphone

they wont remember to use them if it is merely mentioned, but once they play with it and get familiar with it it will embed into their memory and i think will dramatically and drastically alter their course of math education for the better

i wont go into technical caveats here because im assuming we are all competent educators so i dont need to state the obvious

[long pelican]

The free time point is the most interesting one here

My math interest and skill seems to require that I thought about math in a lot of my free time. This is true of many others here no doubt

Question: do you think it’s possible to cultivate true math interest and skill if you suppose the student in question never thinks about math outside math class or homework time?

(I am not proposing forcing people to think about math in their free time. In the worst case scenario we would have to give up trying to get the student to be interested in math or to truly understand the concepts and big picture of the subjects at hand)

[long pelican]

Reworded: is thinking about math for 45 minutes a day on average enough to get deep understanding of, say, high school math concepts? Separately, is it possible to develop an interest in math with only 45 minutes a day being spent thinking about math? Assume best case scenario for what goes on in the 45 minutes

[boreal agate]

yup

also desmos is peak

and i absolutely recommend for students to play around with it

[tall bolt]

I can’t say I ever did in highschool, I always liked maths but I didn’t really know anything about what maths “really was” or develop a real interest in it until I got to uni (I started uni studying physics)

[elfin sleet]

yeah same

[austere delta]

I think if you cultivate true interest, then that would lead to the student thinking about math outside of class.

I certainly thought a lot about math outside of class, but I don't think it's a prerequisite for learning the math curriculum well. It is if course necessary for going beyond the curriculum, but I don't think that's an expectation you need to have of the students.

[abstract grove]

It might be possible for a very small number of students who are, for lack of a better term, naturally gifted; they can just show up, pay attention, and get everything they hear, and do all the homework without problems. Very few kids are like that

And to "cultivate true interest" while also catering to the background abilities of each student, and adhering to a strict curriculum and, most likely, a strongly encouraged set of conventions in the classroom, is an almost impossibly difficult task

[tepid smelt]

I think general interest plays a major role in a students ability to do anything. It's much easier for a student to do math and focus when they have an interest in what they are learning. Some kids no matter what you do will not have that interest and learning will be harder for them.

I think there are strategies with how you present material to allow students to have success but general interest I have found is an intrinsic quality that is hard to give students.

[tight star]

<@&268886789983436800> spam

oh nice

[zinc dove]

Hello, which topics should a discrete math course for CS cover? I see that the banner for the #discrete-math channel has some topics, but I would like some opinions on which other topics to include.

[mossy tendon]

p and q truth tables, binary numbers, logical operators, sets/unions/venn diagrams, Θ(n) notation, psuedocode, contrapositive and similar words are what i remember from when i took discrete math.
The stuff in the first video or 2 of this playlist would've been nice to know while learning discrete math but it is kind of off the topic of math, but it is applied discrete math and helps give context to why we were being taught what we were being taught. https://www.youtube.com/watch?v=HjneAhCy2N4&list=PL9vTTBa7QaQOoMfpP3ztvgyQkPWDPfJez

[gentle girder]

I’ve prepared a video to make the famous Brouwer Fixed Point Theorem accessible to students. I hope you enjoy the work. Have a great day, everyone!
https://www.youtube.com/watch?v=HLxhV6wNTas

[pure light]

nice paper on making vector spaces with operations that aren't just "do addition/multiplication on R a bunch of times", for the sake of introducing vector spaces in LA:
https://doi.org/10.2307/2687687
follow up which extends the technique to algebras:
https://doi.org/10.1080/0020739790100203

[cosmic ibex]

Unfortunately paywalled. I wonder what the people who set these price points are even thinking ...

[austere delta]

Presumably, if they set the individual prices high that works as a negotiation tactic for the package deal they do with universities where the actual money is made

[turbid zenith]

Here’s the piece that’s doing most of the work

[abstract grove]

that’s pretty cool

I’d rather have students just prove that as an exercise than concoct some wacky examples with it, but I could see it being beneficial

maybe do a couple wacky examples leading up to that general result

[tight star]

Yeah I think this would be better

[errant iris]

https://www.reddit.com/r/math/comments/1obkocy/the_failure_of_mathematics_pedagogy/

[tight star]

I agree with a lot of the observations in this post

One thing I have observed a lot more in maths is that a failure of pedagogy is usually attributed as a skill issue on the student’s part

Rather than something the teacher could improve

It really contributes to the sink or swim attitude, as well as the obsession with reinventing everything yourself

If students don’t understand what you’re saying, it’s their fault for not being smart enough, or not figuring out the material on their own anyway

Suffering builds character, so it’s better actually if the student struggles for hours without being able to ask for help

Every subject has teachers like this, but ime it’s a lot more prevalent in math

[tall bolt]

The point they make in the post about a lecturer just reading out the book and not actually pointing out how maths is done I fully agree with and it’s part of why I think intro to proofs courses are a waste of time and why people tend to struggle so much with analysis. That’s absolutely bad lecturing.

But I do also think that being stuck and struggling is unavoidable and not a bad thing. I’m not sure I have your same experience of this being put as some failing of the student or that they cannot ask for help.

I had a lecturer who always loved to say “stuck is the natural state of the mathematician”, and I think that’s pretty much right assuming you’re actually pushing yourself and learning. So in a sense you have to become comfortable with that and you have to learn how to navigate out of it, and I think this is one of the most valuable things university teaches you.

There’s been a fair bit of discussion in research about this idea that the struggles students will face in the transition to university is actually just unavoidable, and it forms a sort of rebirth of them and their understanding of maths, and it’s important that rather than trying to avoid this (as was the goal for a long time in research, and proved to not be particularly possible or successful) you kinda just have to embrace it, but support students through it

If you don’t support them you get situations like in Germany where the drop out rate in year one is like more than half (granted this is also because of the free education and such over there, but it’s certainly a factor) but all of that is to say, I don’t think struggling is bad, and I don’t think the goal of teaching should be to remove the struggle from the students. It is however important that students know it’s ok to struggle, know how to help themselves through the struggle, and have access to resources for more support after that

[tight star]

I think the martial arts analogy they give in the post is apt

Some struggle is good, that doesn’t mean all struggle or all types of struggle is

Students can struggle because of an intrinsically hard topic, or because of a bad explanation

[vocal phoenix]

There are a number of issues here, one of them is that university education definitely expect the student to do much more work on their own, and in shorter time; in Polish high school students spend literally 37 class hours on quadratic functions; and then the typical first-semester calculus course is 60 class hours (30h lectures, 30h interactive problem-solving classes) and covers basic general notions related to function; then sequences and their limits; then function limits and continuity; then derivatives; then indefinite and definite integration.

[tight star]

Yes, though I’ve found that the rate at which you can learn does accelerate over time

Mathematical maturity plays a part in this I believe

[tall bolt]

Of course, and I certainly agree, I’ve had plenty of bad lecturers who just read from the book and gave no help as to how one actually does this stuff. I always mention analysis with this because I think it’s particularly true there since the actual work done and the proofs presented actually happen in the opposite order

[vocal phoenix]

Yeah, that's very bad practice, when I do proofs in lectures I focus heavily on why certain things are done the way they are.

[tight star]

To an extent I think this stems from the standard of truth in math

Physics is empirical, so other physicists have to actually test your theory to decide if it’s true

This produces a kind of natural motivation to give better explanations

But in maths something is true if it’s proven

And as long as at least one person understands the proof, your lack of ability to understand it is a skill issue on your part

[tall bolt]

I don’t think that’s the case, because the standard of truth in maths is the same, it should be sufficiently rigorous to convince your peers that it’s true

I think it’s just a lazy and antiquated lecturing style from the times before it could be assumed that everyone had the textbook or the grand sum of human knowledge in their pocket

[tight star]

Hm I don’t think maths is empirical in the way physics is

[tall bolt]

There’s a lot of discussion on what a proof actually is in maths and how this understanding differs between mathematicians and undergrads etc

[turbid zenith]

Yeah this is usually a good way to do it

Lead up to the abstraction rather than jump right into it

[tall bolt]

In a sense, sure? But UGs in physics also aren’t doing all of the experiments to test the theories they’re being taught so I’m not sure this matters much

[turbid zenith]

Also ooh I'm gonna need to read this article

[tight star]

Well they do labs

[turbid zenith]

Because it matches my experience too thus far :V

[tall bolt]

all

[tight star]

And my argument is more that this produces a natural tendency for physicists to value exposition

Moreso than mathematicians

You can see the prevalence of pop physics vs pop maths as an instance of this

[tall bolt]

I think yeah the fact that all of the maths you’re doing should be carrying some physical interpretation does require more exposition, sure

[tight star]

Physics is more than just maths?

[tall bolt]

?

[tight star]

Physics is not a subset of math

[tall bolt]

Please reread my message

[tight star]

I have, and I don’t understand what you mean

My argument is that physics being empirical requires more exposition

[tall bolt]

When you do maths in physics it carries physical meaning, hence you have to give some sort of explanation of how this maths corresponds to the physics phenomena that have been observed

[tight star]

I don’t mean explaining the maths in physics

I mean explaining physics generally

[tardy ember]

...thinking about it i think physics might be, like, inherently easier to write good exposition for...?

[tight star]

Idk this just sounds like cope to me

[tardy ember]

like, if you're talking about physics then you're talking about the universe and the thing you just naturally end up doing is at least somewhat tying whatever you're saying to actual physical phenomena that the reader is already familiar with

[tall bolt]

At an introductory level, sure maybe, but I don’t think this is true as the physics progresses

[tardy ember]

ok maybe this was misphrased

i think it's harder to write bad exposition for, unintentionally

[tall bolt]

Like you should have intuition for Newtonian mechanics but I’m not sure that holds true for quantum or whatever

[tight star]

And at least, I think it’s not enough to account for the expository gap

[tardy ember]

if you just don't care at all, then you get better results in physics than you do in maths

[tight star]

I don’t mean this as a moral judgement or anything

[tall bolt]

Is there one? There does exist plenty of good maths exposition and plenty of bad physics exposition

[tight star]

It’s just a comparison of my experiences with physics pedagogy vs maths pedagogy, as someone who’s had a lot of both

[tall bolt]

I feel like this is just becoming maths bad physics good

[tight star]

Yes but the degrees to which these exist are less

[tall bolt]

As have I, I studied physics for 2 years at uni, I’ve had pretty similar experiences with both

[tight star]

I don’t think physics is the field that has worried about pedagogy problems

We’ve got our own problems

[tardy ember]

because maths is purely abstract so it's very easy and still entirely rigorous to throw concepts around without any connection at all to anything comprehensible

[tall bolt]

It is certainly true that physics has studied pedagogy for longer than mathematicians have, we did discuss that in my education course

[tight star]

I just think you’ve gotta recognise the problem before you can start solving it

[tall bolt]

In fact the entire department of maths education at my UG was spawned in light of the successes of the physics education department

[tight star]

3b1b has talked about this for CS

How CS pedagogy seems to not have nearly as many issues as math pedagogy

[tall bolt]

This is true, but pedagogical problems have definitely been identified in maths, it’s an absolutely massive field and there’s a lot happening in the area

[tight star]

Sure, I think the next step is to take them seriously

[tall bolt]

I also think they are

[tight star]

Culturally, I think physics places a lot more importance on pedagogy than math

Perhaps my standard for “taken seriously” is just different

[tall bolt]

I don’t even really know what we’re discussing anymore this more just seems like physics good maths bad

[tight star]

This isn’t a value judgement on the fields

I’m just pointing out differences in their culture

Physics has tons of problems with crackpots for example

[vocal phoenix]

I don't know much about physics, and also I don't really know much about how math and the teaching thereof is approached in general; but I will admit that at my university there isn't much focus/discussion of how we teach students.

People just do their own things (mostly based on how they were taught as students) and no one cares so long as the student evaluations aren't too bad

[tight star]

I don’t think it’s unreasonable to think that different fields might have different cultures

This is true even within maths

[tight star]

And based on, for example, the prevalence of “pop physics” and physics outreach compared to math outreach, I don’t think this is an unreasonable observation

[vocal phoenix]

My gut feeling is also that exposition of physics has the advantage of it being more directly relatable to real life, so intuitions/illustrations would typically be easier to convey.

[turbid zenith]

Physics also has the added benefit of being ostensibly about the "real world"

[tight star]

Oh definitely

[turbid zenith]

It's a lot harder to see that for mathematics even though many of us in here know it's there

[vocal phoenix]

I would really struggle to imagine a pop-mathematical video about Baire's theorem

[tight star]

But I’ve seen enough of math culture where being a bit elitist about your own understanding is the norm, or even encouraged

[turbid zenith]

What do you mean, being elitist about your own understanding?

[tight star]

That it’s a point of pride for the things you study to be hard to explain to other people

[vocal phoenix]

I would say that's hugely variable, I've definitely met people with this attitude but I also know many who lament that their research is so difficult to convey to a layperson

[tight star]

Oh yeah I’ve definitely met those types as well

I just have not seen an equivalent in physics

[vocal phoenix]

One of my formative experiences as a student was asking a professor if his research has any real-life application, and him responding with an air of pride "Of course not"

I resolved to never become like that.

[tight star]

As fiction demonstrates, having real life application is not the same as being hard to explain

But it does seem that in maths, the two get correlated to an unhelpful degree

Where pride in not having real-life application (which I think is fine) becomes pride in being esoteric

[vocal phoenix]

I'm not sure I'd say that, fiction relies on being grounded in concepts people understand, and building something fantastic on top of those, and relating to those.

Fiction that's incomprehensible on purpose is notably less popular.

[tight star]

Sure but I don’t imagine people watching star wars do it for the real-life application of lightsaber fights

(Some do for sure, if they’re LARPing)

[vocal phoenix]

No, but they come into the lightsaber fights with preexisting concepts of knightly duels, and swashbuckling swordfights, so they know how to relate to a lightsaber fight.

[turbid zenith]

I won't deny that this exists in some parts of math culture, sure

It might even depend on the subfield

[tight star]

Do they lol

[tall bolt]

I don’t think it’s at all unique to maths research though

It’s certainly horrible but it certainly seems to be prevalent in all of STEM

[tight star]

I’m pretty sure I knew about lightsabers before knights

It was a while ago so I don’t remember

[tardy ember]

i think they come into the lightsaber fights with preexisting concepts of, like, people, and 3d space, and conflict, and violence

[vocal phoenix]

I mean, yes? Ultimately it's two people whacking at each other with pointy/sharpened objects.

It's a fundamental aspect of the human history and sometimes present

[tight star]

Yes but I think it’s more prevalent in math

This is part of what I mean about being able to recognise the problem

[tall bolt]

Perhaps not as much in the “softer” sciences but very much so in maths physics engineering CS

[turbid zenith]

Oh yeah agreed as well

At least it's believable, I don't know from experience

[tall bolt]

I just don’t know that I agree though, I think this is is you projecting your apparently poor experiences in maths. Without an actual study we can’t say concretely about anything but anecdote to anecdote, I’ve found about the same prevalence of this across maths physics engineering and CS

[turbid zenith]

But at this point, man there are a lot of anecdotes

[tall bolt]

Oh for sure it exists, I’ve met these people across all the fields and they suck

[turbid zenith]

Not enough to say it's more a problem in math than other fields

But enough to say that it's a significant problem in math

[tight star]

Every field has problems, and I think this is one that maths has more of

It’s not about “physics good maths bad”

It’s because I like maths that I want math pedagogy to be better

[turbid zenith]

Going back to the article itself, I agree with most of what it's saying

[vocal phoenix]

Yeah, same

[tall bolt]

I also haven’t denied that, and my initial points were really about the places that maths pedagogy fails in my experience but we seemed to have gotten quite far from that

[turbid zenith]

I think we're seeing more and more examples of professors breaking out of that model, thankfully

We're starting to see more emphasis on active learning

Not as universally as I'd like, personally

[tight star]

There’s an element of personal bias to be sure, but I also take cues from e.g. what 3b1b has said regarding this

I think being unwilling to admit that math has more of a problem in pedagogy is part of the problem

[turbid zenith]

(Gotta love how in my PhD program, my math education classes were all about how ineffective it is to just stand at the board and lecture ... take a wild guess what most of my mathematics classes were like)

[tall bolt]

I wrote my final essay for my education course on the effectiveness of examples as a pedagogical tool in the transition to tertiary maths, so I spent a lot of time looking at the shitty stuff a lot of professors do and how there’s such a breakdown of communication between mathematicians and new students

[tall bolt]

Yeah we joked about this quite a lot in my education class lol we tried to have as much in class discussion as possible but a certain amount of it does unfortunately just have to be someone talking

[turbid zenith]

Well yeah it can't be ALL active learning

Swinging the pendulum too far in the other direction is a problem of its own

[tall bolt]

Yeah my topology lecturer went in this direction, he’d start every lecture with “so do we have any questions?” And we’d just talk through stuff if people had questions, and this was generally helpful

If not, he’d just kinda rant for an hour, which was entertaining but maybe not helpful

Granted that was a bit of a special case, his higher category theory lectures were moderately more organised (he wasn’t technically supposed to be teaching topology, long story) but still it was his general approach

[turbid zenith]

The kind of model that I'm a fan of (and is what I try to do) is this:

1. Have students do some pre-work before class, either watching a video or reading a bit, with a few "check your understanding" questions
2. Start of class, see what questions people had on that pre-work
3. Introduce new if necessary stuff
4. Have students work through carefully sequenced problems in groups where they "notice and wonder", going around and giving them nudges if they get stuck
5. Discuss as a class, clarify things, formalize at the end

[tall bolt]

This is by far my preference as well

I pretty much don’t attend lectures unless they go vaguely like this, lectures which just read out that days section of the notes are entirely unproductive in my mind

[tight star]

Seems like a good idea

[turbid zenith]

Yeah, I had an abstract algebra class once where most classes were just the professor doing Definition-Theorem-Proof the whole time

And like maybe occasionally throwing a question out to the class for what the next step should be in a proof

[tight star]

(These were all my algebra classes)

[turbid zenith]

Then the exams were the sorts of things where you needed just the right flash of insight to know how to proceed and if you didn't get it you were screwed

So class time could have been better spent on how to build some of that intutition. The proofs we were doing were all in Dummit & Foote.

This one made me angry at the time

Because I remember thinking "How tf would I have thought to randomly look at the ideal (a²)?"

[tight star]

Yeah wait where does a^2 comes from

[turbid zenith]

You just have to know to try it

[tight star]

And this is why I suck at algebra

[turbid zenith]

These are the kinds of situations where if you have time, you can try different things! Try this, nope that didn't work. Try that, nope that didn't work either. Then suddenly oh try this, aha that worked and I've cracked it!

But on a 1-hour exam where you have to figure out which 3 of 5 questions you can scrounge the most points from?

[tall bolt]

This is why I’m just generally anti exams

[turbid zenith]

Same (although I've had to start giving them again because of AI...)

[tall bolt]

This is also like every problem I’ve done in exams, they’re pretty much always entirely unseen and I imagine that question would’ve been worth 8 marks rather than 20

[tall bolt]

Yeah… idk how that’s going to pan out in the long term

[tight star]

I wouldn’t say I’m anti exams

[turbid zenith]

Timed exams generally aren't the best medium for brand-new out-of-the-box thinking

[tight star]

Yeah

[abstract grove]

I remember when studying for a group theory final, and looking at past finals and quals and stuff, thinking the number of different tricks and ideas is just endless. Even for problems that only take a few lines to prove

[turbid zenith]

The only reason I passed that class? http://yutsumura.com

[abstract grove]

and there’s no way to get good at it by just comprehending the lectures

[turbid zenith]

Looking at all the problems and their tricks ahead of time

I'll also point out that the homeworks were mostly follow-your-nose problems

Manipulating definitions rather than those "flash of insight" problems

So it felt very unfair.

[zinc dove]

Thanks!

[quasi musk]

My experience in my PhD program has been largely the same. Professors lecture at you, and you're supposed to figure it out. However, my classmates and I regularly meet up to discuss, do active problem solving sessions, and go to office hours for guidance. I think for graduate students there's much more onus on the students to form their own little study groups in the office, and work through problems

I think most undergrads don't have the experience or maturity to run this very well

Math also has discussion sections at most universities, with ample resources for office hours. As far as blaming students for not learning, I think it's natural to think about the parameters we can control. When I'm a student in a class, I can't think about "Man I wish my professor would just teach it this way". It's largely my job to adapt to the class, rather than the class to me. Now, as educators we're starting to shift our thoughts to "Am I doing this in an optimal way for student learning

The more I've tried active learning in discussion or lectures, the more I see so many students that can't even do basic problems in class, and require time to process. Many times, I see students just sitting there unsure of what to do. The pre-lecture work is supposed to help push this along, but it just ends up being homework by another name. I'm not sure all this devotion to active learning is really achieving differences for student outcomes

Right now, my primaries give students worksheets with problems due at the end of the day. Students come, and I encourage them to work together on a specific problem for about 5-7 minutes. Then I go over how to setup the equations to solve, and rinse/repeat until it's done

[tall bolt]

Yeah I’m not sure that asking questions in lectures is the way to go, I’m personally not the quickest on my feet and usually need a minute to sit and chew a problem over.

What I do think is useful though is to assume students have at the very least read the notes about whatever you’re about to teach, and then spend the lecture either going over pain points they ask about or you expect, perhaps a tricky section of a proof, or just giving a more high level intuition of what’s happening. These are always the lectures I get something from, and as a result usually the only lectures I actually bother to attend

I also think if students come with a problem they’re stuck on and you live solve and give your thoughts as you go can be very helpful, but typically this is best if the lecturer has had some notice and actually thought about it so it’s not like 30 minutes of struggling through a proof

[turbid zenith]

Alright so coming back to this now

[turbid zenith]

I've found this same thing to be true, but I'm not quite ready to give up on active learning yet. My hunch is that one of the big problems with a lot of active learning implementations, including my own, is that there's not enough explicit instruction and practice to balance it out

Like you can't solve rich calculus problems if you can't compute basic derivatives

[quasi musk]

That's exactly what my issue is. The only balance that I have found is to tell students to directly work together for a few minutes, then I say how to solve the problem

Then move onto the next

[compact isle]

idk I feel like this is a bad example because to me looking at (a^2) is just really natural. If you want to show its a field, you have to show every nonzero element a is invertible. You have information about ideals, so its natural to consider an ideal related to the element a. The most obvious choice of course is (a), but there's no obvious way to use the fact that (a) is prime. Whereas (a^2) is also an ideal related to the element a and has a clear way to use the given property

I do agree in general that problems of this kind are dangerous on exams

but I also think you have to ask some questions that are going to push the intuitive knowledge and problem solving of students, rather than just give problems that are basically unwrapping definitions

the one take I have about math lectures is that going through every detail of a proof in a lecture is imo kinda a waste of everyone's time. I think that lectures should go over examples, big picture, and proof sketches (and then perhaps there's time for other kinds of learning as you've been discussing) and the details can be left to writing

When I'm in lectures I try to only write down the key ideas in proofs and fill in the gaps later on my own and I find that to be quite helpful in learning

[turbid zenith]

See, you've built that intuition.

To somebody learning it for the first time, it would really have helped to get some idea of why that's ✨NATURAL✨ and ✨OBVIOUS✨

There was very little done in the way of building that intuition in class nor on homework — the homework in particular was just unwrapping definitions

[turbid zenith]

So this kind of thinking was something we never really learned, and so the only people who could do well were those who could figure it out on their own

[cosmic ibex]

the homework in particular was just unwrapping definitions
Yeah, that's probably a major part of the problem then.

But there's also a weird gap in the offered proof:

By assumption, (a²) is prime.
But all that was assumed is that every proper ideal is prime, and it hasn't been argued that (a²) is proper -- in fact, since the ring turns out to be a field and a is nonzero, it definitely won't be a proper ideal and so by definition can't be prime.
So there should have been an remark in the proof that says that because of this assumption, every ideal satisfies the "interesting" part of the definition of prime ideal, namely xy in I => x in I or y in I.

(Or alternatively, "assume that a is a nonzero nonunit" and go for a contradiction).

[turbid zenith]

Question I'm thinking of for tonight:

Why bother teaching Calculus I students about the normal line?

Like, the derivative is the slope of the tangent line, and the tangent line can be used to approximate a function, do Newton's method, and so on. But what's the normal line good for?

[jaunty yew]

shortest distance between curves?

[turbid zenith]

So far any resource I find about finding the normal line is first about the tangent line and then it's like "yeah we can calculate the normal line too!" without any sense of why anyone would care about it

[native iron]

I never learned about the normal line in calculus.

[turbid zenith]

Stewart at least vaguely mentions optics but that's it

[rapid tusk]

can ask similarly about osculating and normal planes in calc3

stewart just mentions them without proper motivation

[turbid zenith]

At least with Calc 3 there's the Frenet-Serret frame

I can see the utility in turning that into a coordinate system

For Calc 1, I don't see the point

[turbid zenith]

I keep searching and find nothing XD

[jaunty yew]

^ shortest distance occurs when the curves are parallel, so the segment between them that gives the shortest distance is perpendicular to both curves (or along the normal line) @turbid zenith

Does this work?

[turbid zenith]

I guess I'd need to see it as an example 😛 because yeah things are closest when they're perpendicular but why does the normal line in particular end up useful here

[jaunty yew]

the easiest example I can think of (though this might be too trivial tbh) is the shortest distance between a straight line and a curve

the slope of the normal line directly follows from the straight line

so you can pinpoint where along the curve the distance is minimised relatively easily

[turbid zenith]

Okay that's not TOO bad

So I guess say you have the curve y = x^2 and the line y = x - 4

You could find the normal line to y = x^2 at the point (a, a^2) and figure out what value of a makes it have a slope of -1

That's simple enough that somebody could latch onto it

[jaunty yew]

I could've sworn I had a nice test question using the normal line but I can't dig it up lol

[turbid zenith]

I appreciate it anyway

I just like having a reason students should care about what I’m teaching ya know?

[long pelican]

do you not have control over the curriculum?

[turbid zenith]

Oh I absolutely do

So I haven’t even bothered with the normal line the past few times I’ve taught Calculus I

[long pelican]

Hmm what makes you consider doing it this time

[turbid zenith]

The fact that I’m also working on my own textbook

[long pelican]

Whoa that's a big project

[turbid zenith]

And so I’m trying to figure out if certain topics may be worth including

If they could be better motivated with a better narrative

[turbid zenith]

Oh yeah XD

[long pelican]

For what it's worth, I can't think of a good narrative for normal line besides optics (which I'm taking on faith...)

[turbid zenith]

Yeah and for optics you can just reflect against the tangent line instead

That’s how I think of it

[long pelican]

If you ever need a normal line, take the tangent line and find the perpendicular to the tangent line through the point in question... it's just 1 extra step

[quasi musk]

You can use the normal line to calculate distance between a point and a curve (I think)

It generalizes the "What is the distance between a point and a line"

And they see that again in Calc 3 when doing Distance between a point and a plane

As far as doing the normal line as a core topic, I'd say it's optional, it's a way of making a problem harder instead of asking for a tangent line, you get the normal line this time!

Gotta separate the A from the A+ student somehow

[jaunty yew]

@turbid zenith Ok it's not exactly a nice question but maybe the property in 4 and its relationship to normal lines might be interesting to you?

[cosmic ibex]

The question is perhaps not whether normal lines are ever good for anything (because of course sometimes they are), but whether they ought to be taught as a freestanding task in their own right.
In particular posing exercises of the "find an equation for the normal line of such-and-such curve" without any motivating context sounds like it could mislead procedure-focused students into spending way too much effort on the concept.
Rather than "here's a new concept you should learn", I think at most it should be presented as "here's something you hopefully already know from coordinate geometry that you can do, and it can sometimes be useful to do it to the tangent you find by differentiating".

[cosmic ibex]

What comes most directly to mind is that we imagine that some object (a vehicle!) moves along the curve and stays oriented along the curve, then the tangent and normal line together make a useful moving coordinate system for points on the vehicle.

Also, if you take normal lines at two points separated by an infinitesimal distance, then their intersection shows the center of the osculating circle.

But these are perhaps not very good examples for intro calc. They're both most useful in practice for parametric curves rather than graphs of functions, and the second one has a conceptually challenging nested limit.

[serene hull]

Hello, I am looking for advice for a friend. He is a programmer about to finish an intensive course on C, and also does Python. He knows basically NO math because he missed a big portion of highschool, but he does have the skill to learn it (He finished a sort of Associate-like programing degree with basically no help). He wants to learn math almost from scratch, but I have no clue where he should start. The most advanced math he knows basically is first degree equations and not much more.

Even though he does programming, he wants to learn math in a general sense too. What are some resources he could use? Because even though on youtube there is basically infinite resources, there is so much content to cover its hard not to loose track.

[austere delta]

Khan Academy is good resource I guess.

[turbid zenith]

Oh now THAT is interesting

[turbid zenith]

Yeah I think this is what I was getting at with my issue

because of course sometimes they are
I don't think I'd seen a way in which the normal line itself (and not just the concept of being perpendicular) was truly useful until you said the thing about the osculating circle

But again as a result I think I'm concluding it's not very useful for intro calc

[halcyon glade]

I didn't learn about normal stuff until multivariable calculus. But I expect the same sort of reasoning holds. If you're visualizing a level set (like x^2 + y^2 = 1), then the normal line represents the direction of fastest increase/decrease of that quantity. I don't see any reason to introduce that in Calculus I though. (In that case, the normal line is just the diameter line of the circle, which is nice and all.)

[cosmic ibex]

Hmm, it's interesting in this context that for a level set, the normal is what you can calculate directly (as the gradient, hoping it is nonzero) and you need to turn that by 90° to get the tangent.

(No wait, I'm silly -- that's just an artifact of mixing formalisms. If you have a curve given by an equation, differentiating that directly gives you what you need to write an equation for the tangent line. It the curve is given parametrically, differentiating it directly gives you what you need to write a parametric expression for the tangent).

[halcyon glade]

It makes sense because a level set is codimension 1 rather than dimension 1

Yeah whereas parameterizing is dimension 1

[turbid zenith]

So I found something about normals that I wonder might have been one reason it used to be included

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

This is a part of math history that's always been kinda mysterious to me ... we used to have all these things like involute, evolute, pedal curves, etc. People used to talk about cissoids, tractrices, and so on.

What happened? Why were these things relevant?

[tawny slate]

kimberling centers kind of mood

[turbid zenith]

Hmm?

[tawny slate]

https://faculty.evansville.edu/ck6/encyclopedia/ETC.html

there is an absolutely monstrously huge database of a bunch of obscure things about triangles

far more than we will ever need

[turbid zenith]

Oh shit I've seen those

Somehow the curves thing seems a bit less excessive than this

[rapid tusk]

so i somehow stumbled into the final stage interview for a math education company and im shitting myself bc from what i've read online this interview includes a mock lesson

that my teaching-experience-less ass will have to give

those of yall who have experience: what would you say is the biggest thing to look out for ?

[compact isle]

There are a lot but I'd say rule one is to speak clearly and not too quickly (especially if you are nervous in an interview)

[turbid zenith]

Going to agree here. It’s so easy to rush. Just calm down and slow down.

[halcyon glade]

Also remember to be engaging, make eye contact, etc., teaching is a multiple-person interaction

[turbid zenith]

Does anyone have any insight on my question about curves btw?

How all these named curves fit into the historical narrative of mathematics?

(Almost makes me wonder if a math history channel would be of any use.)

The next time I teach implicit differentiation or parametric curves, I want to include some of these as examples but I want to also show how they fit into the overall story.

[native iron]

If it's in person and you're teaching with a board, have a sketch on paper of what you want the board to look like. You don't have to reference it during the lesson but you should have an idea in your head.

Board work is important because students often write down most of what the instructor writes but none of what the instructor says.

When you're writing, you should periodically turn away from the board and explain it while facing the audience. The less time facing the board the better, but I'm pretty bad at that aspect, so I don't have the best advice.

[tall bolt]

Perhaps someone in diffgeo may know?

[cosmic ibex]

It's a good question. I tried looking around in my usual sources, but didn't find anything definite.
The general vibe I get, though, is that most of them enter the stage in the 1600s, in works by the same people who were working towards calculus at the same time. It seems that with the emerging idea of solving problems by breaking them into infinitesimal pieces, it was suddenly possible to say something interesting about curves other than the lines, circles, and conic sections they had been limited to since antiquity.
My guess would be something like the named curves are the examples that were analyzed so early that it was not yet clear what the organizing principles for the whole field would be, so the natural thing to name was the curve itself. Later on, when the consensus became that differential equations and functions were interesting things to study in general, those were the ones that received names when speaking about new work, rather than the curves they describe.

[turbid zenith]

That's a pretty compelling explanation.

[turbid zenith]

Update: I got some older books on analytic geometry, like from the 1950’s and I see these things getting talked about a lot more there

[cursive plover]

(disclaimer: I am not actually a teacher but I’m not sure where on here I would get better responses)

[cosmic ibex]

Hmm, then what is your relation to the question? Who would run a contest with such weird rules, and what's your interest in encouraging good-faith participation in the weird contest?

[cursive plover]

i was hesitant to post here for this exact reason

but if it makes you feel better i would run a contest like this after i get my degree

[cursive plover]

what’s your interest in encouraging good-faith participation in the … contest
more fun

[cursive plover]

anyways I think I have a solution to both problems
instead of points accumulating over the course of the semester I could award the prize to whatever bench got the most points in the least time

[cursive plover]

btw if you think the game is weird I think you’re a boring pedagogist! whatever level it is you may teach!

[zealous kettle]

If you only award a point to the single bench who answered the fastest, then the scoring is susceptible to the spoiler effect, and you'll have at most 3 benches seriously competing by the end of the year. I think to amend this, what you need is to track every bench's time to answer the question, and each bench's end-of-year score is the total cumulative time to answer all of the questions.

However, I think the idea is flawed because high school students are not gonna be motivated by a prize like this.

As a math teacher of any level before precalc, you have to understand: Your students are forced against their will to be in your math class plus ~6 other classes. If you get students who have already determined they're math nerds, then those kids will seriously try to answer the question. The rest will decide "It's not worthwhile to spend my energy to try to earn this prize, I'd rather read my book or scroll social media or brush up on the material for my next class."

That is unless completion of these end-of-class puzzles directly benefits their grade.

[rugged needle]

my stats teacher did this thing where he would spin a wheel at the start of class on some days and whoever the wheel landed on had to go up to the front of the class and answer a question

that worked pretty well

and then that person could phone a friend or get the class to vote on the correct answer

it's effective because instead of wagering points or some abstract nonsense, the teacher is exploiting social anxiety (but in a good way)

he did reward correct answers with poptarts

[turbid zenith]

So I'm thinking about the calculus textbook I'm working on in relation to AP Calculus.

The AP curriculum shoves polar coordinates, parametric equations, and vector functions into AP Calculus BC, and as a result lots of the standard texts dedicate time to those shortly after covering integral calculus. And then as a result of that, lots of integral calculus courses ("Calculus II" in the US) also put those topics in because, well, that's what order the textbook (e.g. Stewart Ch. 10) goes.

I'm playing around with the order of my own book, and I'm trying to build an overall "narrative" for calculus. It seems to me that parametric/vector stuff fits better in multivariable calculus (Calculus III), and that polar coordinates fit better when you're talking about coordinate changes with Jacobians (which my school does in our half-semester Calculus IV because we needed to split our multivariable course up to not be at a breakneck pace).

But the issue is, damn, College Board and the heavy-hitter textbooks sure have built a lot of momentum.

What do y'all think? I don't want to cave to that momentum, but is that ultimately setting up for failure?

[regal portal]

Are you intending to make a book that is split into semesters like calc 1, 2, 3, and 4?

[halcyon glade]

I agree that it is more logical for them to go in multivariable calculus

My guess is that it's taught in this order to make physics classes easier to follow while taking calculus concurrently

[turbid zenith]

It’s gonna be organized by how my school does those classes but certainly people could go in another order

The more important thing is the overarching narrative

[regal portal]

I think you should prioritize the narrative since unis or schools can decide the final order themselves -- so if someone wants to self study, they wont have to waste time being introduced to parametric equations twice (like it is (sort of) done in stewart calculus). im pretty sure the motivation of putting polar coordinates in calc 2 is as eric says, general physics 1 doesnt have a mvc prereq but polar coordinates help a lot so it is often introduced in calc 2

[turbid zenith]

There never seems to be a good way to make the Calc sequence work perfectly for the physics people tbh… I’ve tried

We have students taking E&M concurrently with Calc III

So they’re using div/grad/curl before learning partial derivatives 🙃

[tawny slate]

i think it might just be a reality we have to face, that people use things they dont ever really understand, like computers

[quasi musk]

I think a lot of this might be mitigated by AP Pre-Calculus

Some of the topics mentioned are in AP Pre-Calc

[quasi musk]

My physics prof "taught us" cross product in Physics, made zero sense how to take a cross product

[tight star]

How do you understand the cross product now?

[quasi musk]

Well I eventually took calc 3. There's the geometric sense of cross product, a vector normal to two given vectors (and the plane they span). There's the mechanical sense of cross product, by doing a determinant or some other calculation. There's the generalization with wedge product, although that's awkward

There's the physical sense of cross product, given by the right-hand rule

[quasi musk]

I think it's a great idea! But if you're not regularly teaching AP Calc or intro calc then it can be difficult to polish something to that level. I had a student in calc 3 yesterday that said words like "unit vector" and "normal" didn't really make much sense after AP Calc BC

[halcyon glade]

Wait, why is the wedge product awkward?

[rain hawk]

I have now taught the seventh calc student in a row who didn't understand the FToC and I think I've diagnosed the issue. Students are basically taught that integration is backwards differentiation, do a ton of drills on that, and then when they're told this is a theorem they're confused because it feels like a definition

[tight star]

Yesssssss this is precisely my concern

To be clear, which part of the fundamental theorem of calculus are you trying to teach?

[tight star]

Also how else do you think it could be done

[tall bolt]

I’m not sure if it’s a perfect solution, but my analysis lecturer basically just made a point to say, “I know you’ve all thought of this as just backwards differentiation, but a priori there’s no reason this should be the case, it’s just a very very lucky coincidence and that’s what this theorem tells us”

[tight star]

I don’t quite understand..

[tall bolt]

Essentially just emphasise that at no point did anyone define it to be such, one’s about sums of rectangles and the other is just a limit. The fact that they work out to be inverses is practically a miracle, and that miracle is the FToC

[tight star]

Hm so you go with the Riemann sum form of integrals then

Also which part of FTC do you mean

1 or 2

[tall bolt]

The first part

This is essentially what tells you differentiation and integration are “inverses”

The issue I think just comes from doing a lot of calculus very mechanically before learning the rigour, so it’s easy to forget the difference between what you know and what you’ve proven

[tight star]

Hm I suspected as much

I think the second part might be more enlightening

[tall bolt]

How so

[tight star]

You can show the telescoping sum trick

[tall bolt]

I’m not sure I know what you’re getting at

[tight star]

So, the reason why FTC part 2 is true is because of telescoping sums

This is also essentially the reason why generalised Stokes’ theorem is true

In a sense, telescoping sums are the main trick we have to evaluate unknown sums

[turbid zenith]

Will point out there is no consensus on which is “Part 1” and which is “Part 2”

[tight star]

Ah I see

The way I think of them in my head is

“Every continuous function has an antiderivative”

And “You can use an antiderivative to evaluate an integral”

[tight star]

I think this part is more helpful as a theoretical tool

[tight star]

And this part is more helpful as a practical tool

In that - for computations, knowing that every continuous function has an antiderivative isn’t that helpful, but knowing you can use an antiderivative (which you can find algebraically, sometimes) to evaluate an integral is very helpful

In discrete calculus, you can build analogs of the fundamental theorems of ordinary calculus

But one of these holds essentially by definition, whereas the other one requires an actual proof - the one involving telescoping sums

[turbid zenith]

I’ve heard them called the “Antiderivative Construction Theorem” and the “Integral Evaluation Theorem”

I forget where

But I like that

[tight star]

In multivariate calculus, this generalises to poincare’s lemma

[tight star]

And this generalises to stokes’ theorem

[tight star]

Yeah that’s precisely what I’m going for

[turbid zenith]

Way more descriptive names!

[tight star]

And I feel like, for calculational math, the integral evaluation theorem is a lot more relevant?

[turbid zenith]

Certainly yes

And calculus students learn a whole lot more generalizations of it

[tight star]

Yeah, like with the divergence theorem, Green’s theorem, ordinary stokes’ theorem

[turbid zenith]

Gradient theorem, Green’s theorem, Divergence theorem, Kelvin-Stokes theorem

[tight star]

This is part of why I think “morally” you shouldn’t prove the integral evaluation theorem using the antiderivative construction theorem

[turbid zenith]

I don’t actually know Poincaré’s lemma

[tight star]

It’s a statement about differential forms

It’s essentially an “antiderivative construction theorem” for closed differential forms

[turbid zenith]

Interesting

[tight star]

In 1D, it specialises to FTC

[turbid zenith]

The Wikipedia page actually has a good explanation of why they’re really two different theorems even if it seems you can prove one from the other

[tight star]

The technique of “telescoping”, which allows you to relate boundary behaviour to interior behaviour, has lots of applications in calculus

[turbid zenith]

There are extra conditions that proving one from the other assumes, which aren’t really necessary

[tight star]

Mhm

Finding sums of “basic” sequences, like arithmetic or geometric progressions, can also be cast in terms of telescoping sums

It’s part of why I think it might actually be quite helpful to learn some discrete calculus before continuous calculus

[turbid zenith]

That would be pretty cool

[tight star]

It’s a lot more concrete, no worries about infinitesimals

[turbid zenith]

Though where would you fit that into the curriculum?

[tight star]

Whenever series are usually covered, I think

In many ways, discrete calculus helps you become an expert at summation

[turbid zenith]

I had a couple of semesters where I taught sequences and series in high school precalculus, and we actually hinted at discrete calculus

Though I stopped a bit short of teaching the discrete derivative and integral

[tight star]

There are cool ways to link it to combinatorial things like binomial coefficients/pascal’s triangle

It also helps explain why the trick of taking repeated differences works

To figure out what polynomial corresponds to a sequence

[turbid zenith]

We did just enough to show that degree goes down when you take a difference and up when you take a sum

[tight star]

It’s packaged in the Gregory-Newton interpolation formula, which is a discrete analog of maclaurin series

[turbid zenith]

Ooooo

[tight star]

The idea is to express polynomials in terms of binomial coefficients, because they behave very very nicely when taking differences and sums

And there are visual combinatorial proofs for those facts, which is cool

This lets you take differences and sums of any polynomial really easily

[turbid zenith]

Makes you wonder what a redesigned precalculus course might contain

[tight star]

I remember as a student wondering how you’d derive identities for the sums of fourth, fifth and higher powers

[tight star]

I think you’d be interested in David Bressoud’s work

[turbid zenith]

I love his stuff

[tight star]

He thinks a lot about alternative ways to teach calculus

[turbid zenith]

I’ve read his real analysis book and his calculus reordered book

[tight star]

Oh nice nice

[turbid zenith]

I’m not going so far as to put integrals first in my own curriculum

But we agree on putting limits after derivatives and integrals

[tight star]

It feels a little weird but I can understand that

[turbid zenith]

Embrace infinitesimals first, then use limits to clean up the “but wait” objections afterward

[tight star]

I’m still mixed on infinitesimals

They feel pretty unphysical and their arithmetic is weird

[native iron]

I might feel better about them if I understood them better, but the use of dx in 3b1b's videos has always driven me crazy.

[tight star]

It’s weird because he explicitly doesn’t use infinitesimals, he always uses finite changes

There are ways to do infinitesimals with ultra filters

[boreal agate]

"what? but theyre so simple!
just use differential forms over a smooth manifold..."

[tight star]

Differential forms already require some notion of tangent vector right

[tawny slate]

https://www.youtube.com/watch?v=rQzu5JUjaG0

video reference, published not too long ago, a fairly accessible introduction i think

[turbid zenith]

I feel like “imagine an infinitely small number” isn’t that different from “imagine a number less than zero” or “imagine a number whose square is -1”

[tawny slate]

the latter two are not problems because many people can conceive of how they will behave under common operations, they can visualize ways to handle them

the first one is just really difficult to handle if you're not careful, which is the biggest problem

"if c is infinitely small, whats 1/c? can i do 1/c + 4?"

[tall bolt]

<@&268886789983436800>

[halcyon glade]

<tangent> there's also a funny way to do it where all functions are smooth and infinitesimals are numbers that square to zero, but you have to reject the law of excluded middle for it to work

[halcyon glade]

if you wanted to, you could probably characterize them completely formally/algebraically </tangent>

[tawny slate]

is this more cursed foundations

[abstract grove]

I forget what source I read this from recently, but some author said something like "we essentially replaced the foggy notion of infinitesimal with the direction that that infinitesimal extends toward"

I thought that was pretty good

it may have been Spivak

Comprehensive Intro to Diff Geo vol. 1

[tight star]

Yeah I like this

[halcyon glade]

https://pi.math.cornell.edu/~oconnor/sia.pdf

[native iron]

My impression was that it only made any sense if the values became infinitesimals when you passed to the limit but it's been a while so I could be wrong.

[compact isle]

there are no infinitesimals in limits

working with just really small numbers is more or less fine so long as you remember what is going to zero and how fast its going to zero

I think its a pretty big misconception in calculus that infinitesimals are involved at all (of course they can be made rigorous, but in the standard approach nothing is an infinitesimal)

ultimately it comes from peoples misunderstandings about limits which are quite hard to quell

[quasi musk]

How do you define it so that sophomores/freshman can understand

It's not a good way to do it for them

[halcyon glade]

Maybe this is naive, but what's wrong with "when you wedge two vectors, you get an oriented area"

It feels geometrically intuitive that you can get a parallelogram out of two vectors

And then the basic properties like linearity and v ^ v = 0 follow from the geometric picture

[quasi musk]

I suppose you can teach it that way, but even the standard Calc 3 material they have a hard time digesting as it is

If you had good students, or maybe an honors class, you could do it

Or even a second pass at multivariable in undergrad

[turbid zenith]

Infinitely large and yes 😛

I saw another calculus book recently that actually uses infinitesimals throughout in its definitions which are NSA-based, I need to remember what it was

Aha, here it is https://www.amazon.com/Calculus-Set-Free-Infinitesimals-Rescue/dp/0192895605

He uses ω for a sort of "primitive" infinitesimal and Ω = 1/ω

Which I personally find kinda confusing, I'd rather have ω be infinite because I'm used to Cantor and Conway, but the idea makes sense

So yeah, if c is infinitesimal, I can place 1/c + 4 pretty easily: "zoom out" one level so you can see 1/c, then "zoom in" one level on the neighborhood of 1/c, then take 4 steps to the right. It's just like when you "zoom in/out" in elementary school when you're learning about decimals.

[tawny slate]

yeah I know you can do that, but students might find this unsatisfying

it's like more confusing and arbitrary and even less relating to real world applications, and it's all simply to create a framework just to get calculus to work

i personally love this stuff, but it's math for the sake of math for most people i think

[tight star]

What I always found weird is that oriented areas don’t have an obvious addition rule to me

[tight star]

<@&268886789983436800>

[quasi maple]

Subtracting 4 from 7 yields 3, though

[turbid zenith]

I guess I disagree 😛

I don't think it's nearly as difficult as it's being made out to be

[tawny slate]

all relative I guess, there is a part of me that is excited to teach it and thinks it would be easy to teach, and then there's another part of me that despairs at the level of the general public

[tight star]

i'm pretty anti-infinitesimal myself

[turbid zenith]

*cries in combinatorial game theory*

[tight star]

More accurately, I’m against them for the purposes of calculus pedagogy

[turbid zenith]

I know 😛 I just couldn't resist

And yeah, to each their own

[turbid zenith]

Oh wow I missed this XD

[turbid zenith]

Actually I’m thinking now… how confusing/arbitrary would you say imaginary or complex numbers are in comparison? Or what about relation to real-world applications?

[boreal agate]

imo complex numbers are way further removed from intuition than infinitesimals

while infinitesimals are by no means easy to grasp, complex numbers never fail to be absolute brainfucks, mainly because C->C functions require 4d to faithfully visualize (yes this applies to domain colorings as well - color is a dimension)

infinitesimals are relatively easy to "imagine" once you have the concept down with a few examples

i still dont get the geometric picture of holomorphic functions

also using complex numbers both as rotations and the things we wish to rotate can be confusing

its a very unnatural structure to someone that doesnt alrwady know about its properties

[tight star]

hm i disagree quite a bit

complex numbers feel pretty natural to me

[boreal agate]

was this always the case?

[tight star]

when i first got introduced to them no, but after seeing the geometric interpretation they've been pretty clear

on the other hand i still fail to understand infinitesimals

[boreal agate]

considering how much they are used within physics, it would make sense for you to have developed an intuition

[tight star]

you could say the same for infinitesimals

[boreal agate]

i suppose i could

[tight star]

though the difference is that thinking about actual infinitesimals is very rarely helpful in my experience

i work with "very very small" things, but always finite

[spiral elbow]

I don't know much about infinitesimals and the hyperreal numbers, but it seems to me that the hyperreals are much more than just "add a number that is smaller than every positive real number". Like, there are infinitely many infinitesimals, and if you want closure under division, you need to add infinitely many infinite hyperreals. How do you think about arithmetic on infinitely large numbers? There's something called hyperintegers, so you can have infinitely large integers; do they have anything in common with finite integers? And the construction of the hyperreals is pretty complicated, while in comparison the construction of the complex numbers is pretty straight forward: you can just define a multiplication on R^2, and show that this contains R and a square root of -1

[turbid zenith]

Well, with complex numbers, you don't just "add a number that squares to -1 either" — you want closure under your operations and again you see where that leads you

[spiral elbow]

It seems like if you introduce calculus using infinitesimals, you're shoving a lot under the rug, and I think it's a bit disingenuous to give the impression to students that you can just add a number with certain properties without making sure it behaves the way you expect

[turbid zenith]

Not to mention, the construction of complex numbers involves quotienting a polynomial ring by a maximal ideal

[spiral elbow]

exactly, the way I've been introduced to complex numbers have never been "just add a number that squares to -1" though, in my complex analysis course we just defined a multiplication on R^2 and proved properties about it

[turbid zenith]

You're shoving a lot under the rug without them too. That's why real analysis is a whole course.