math-pedagogy

[ornate otter]

Ah, that might be true.

I feel like lots of people think of infinitesimals when doing analysis. Especially physicists like doing stuff with them that's forbidden but works

Like separating df and dx for rationalizing derivation or Integration rules

But it should probably be taught this way secondarily if at all

@zenith slate thanks for the feedback, but I feel you could be a little nicer sometimes

@long pelican @twin shell thanks for your thoughts, too

[zenith slate]

Fair enough, my bad

[long pelican]

One thought that just occurred to me while writing the above is that a good middle approach is to axiomatize the properties of limits that are used in practice (e.g.

1. Limit of a constant sequence is the constant
2. Least upper bound property
3. A list of functions that are "declared" continuous, meaning limit of f(x_n) = f(limit x_n) for all (x_n) in the domain)
   And the only thing that you leave unproved is "there is a unique and consistent notion of limit satisfying these axioms"

The upsides are as follows: students do not have to grapple with epsilon delta, but they still get treated to a view of math that is logical and axiomatic, rather than rule-based, procedural, and arbitrary

[twin shell]

i don't know anything about physics but when i think about integrals i take an intuitive-limit stance
that an infinite sum of infinitely small things is not terribly meaningful but you can take increasingly accurate approximations using finite sums
i also do think of integrals as continuous summation/generalized addition, which i don't think is mainstream at high-level analysis, but i also don't think it's wrong for an intuition

[ornate otter]

Huh, That's an interesting approach! I'll think on that!

[twin shell]

i don't think infinitesimals are inherently bad for an intuitive approach because that's how i learned differentiation/integration for physics but that also isn't how i think of it now

[long pelican]

https://arxiv.org/pdf/0909.4178.pdf
Hot-cha, I have serendipitously managed to find a paper that explains what I just said in more detail

[twin shell]

d e r i v e d

[ornate otter]

Physicists often rationalize analysis rules in this way:
"df/dx = df/dg dg/dx", basically treating the chain rule as a fraction extension. Or the substitution rule:
"dx = dt dx/dt"

[ornate otter]

Nice I'll take a look 🙂

[cosmic ibex]

A counterpoint could be that such a definition doesnt make it clear that a limit we can see on a graph is in fact the limit.

Hmm, perhaps that's covered by your "least upper bound property" in practical examples?

[long pelican]

I probably missed something, because I just whipped up the 3 items out of the top of my head

I should say this is inspired by the axiomatic definition of characteristic classes in Milnor and Stasheff, where they define Stiefel-Whitney classes and Chern classes by the very properties that people most use in practice, then construct it later

This isn't just restricted to Milnor and Stasheff; this kind of pattern pops up everywhere in math

[cosmic ibex]

Are you proposing to stil present a handwavy "if x gets close to this, f(x) gets close to that" in addition to the kind of axioms you listed? Just without mentioning the dangerous words "epsilon" and "delta"?

Otherwise, my misgiving would be: Yes, these axioms might well make it easier for students to find out what a given limit is, but it sounds like they're left completely without guidance as to why they should care what the limit is in the first pace.

[long pelican]

Yeah, I guess. Even graduate classes present handwavy things for motivation all the time

[wide ice]

All math starts from doing handwavy when you're not supposed to, then later try to make sense why you can

So I guess it's fair

[cosmic ibex]

My other less precise worry is that when you say declare certain functions to be continuous by axioms, then it sounds like the official procedure for finding a limit will be something like "find a continuous function that agrees with your function in an erased neighborhood and then evaluate that" -- as opposed to the current situation, where (particularly for derivatives) we do the same thing but can present it as a calculational trick rather than something fundamental. I'd worry that this would encourage the misconception that a limit is just a function value with fancier symbols, which students would need to unlearn later.

[long pelican]

How do you propose to prove sin(x) is continuous in a classroom setting? The optimal way in theory is of course to use epsilon delta and the addition property of sines, but realistically it will be guaranteed that no one follows this in a typical high school calculus class. Another way is to be handwavy, but you are supposed to be handwavy for motivation but not for demonstrating something.

[cosmic ibex]

Hmmm.

[wide ice]

you can of course cheat

[cosmic ibex]

I think I'm fine with just asserting that it is continuous. But preferably in a situation where that claim has meaning before I make it.

If we just start out with "there's some mystery class of functions that I call 'continuous', and I'm not going to reveal what they have in common other than the word, but the sine is one of them". -- Then I think I'd feel cheated as a student.

[wide ice]

You can bound |sin(x) - sin(y)| by some function of |x-y|, then you argue that as |x-y| tends to 0, so does |sin(x)-sin(y)|. This is of course delta-epsilon behind the scene, but you can get away without any handwaving

[long pelican]

This is more understandable than an epsilon delta proof but alas, the percent of calculus students in a US high school that will understand that is also approximately 0

[wide ice]

A standard exercise in HS is to compute the derivative of sin using definition, so it's not impossible. I would expect a HS student to be able to do that

Unless now you will proceed to destroy my faith in humanity US education system

[long pelican]

😛

[cosmic ibex]

My impression is that in the US, ordinary high school students never even hear about derivatives, unless they're in special accelerated "you're learning university stuff now" classes.

[wide ice]

in what world are derivatives for university???

[cosmic ibex]

The new world, apparently.

[wide ice]

You should either change the world, or make it great again, cus it does suck to me

[long pelican]

Here's the rough outline of how calculus might go:
Motivation: instantaneous slope of a function at a point
--> hard to define
--> Let's approximate with secant lines
--> take width to be 0, get 0/0, motivates limits
--> Characterize limits by some fundamental properties
--> State the fundamental theorem of limits: there is exactly one way to have a notion of limit for real-valued functions satisfying the above characterizations
--> say the proof of that is outside the scope of the course

[cosmic ibex]

But let's not get sidetracked into discussing when this or that concept is introduced.

[twin shell]

||my uni doesn't even teach formal derivatives in our one real analysis course lol||

[wide ice]

Isn't this how it's taught usually?

[long pelican]

It's supposed to be similar in content, on purpose, but the execution makes it so students have a clear idea of what follows from what logically and what can be used to prove things

[cosmic ibex]

The novelty in Icy's program is, IIUC, that he won't present any definition of "limit".

[long pelican]

I'd say

[wide ice]

the way math is done in HS is that you never care about defintion, only properties. So long as students can manipulate it well, it's considered done.

[long pelican]

It's still defined by the properties and the uniqueness theorem

[wide ice]

I do hope for a more thorough approach. But then again, not everyone is Terence Tao...

[long pelican]

What's not being presented is the set-theoretic Cauchy-esque definition

Like, there's nothing against doing this... except for the fact that 0% with rounding of students have the equipment to understand it

[wide ice]

Fyi, when Cauchy introduced delta-epsilon, even students at Polytechnique didn't understand it. They hated it with passion.

[cosmic ibex]

Still not convinced. I can buy that it is possible -- perhaps even simpler -- to find a limit with just such a characterization. However, I'm not sure how I'd go about drawing useful conclusions from an assumption "the limit of this function is such-and-such" without having something very close to an epsilon-delta formulation of that assumption available.

[wide ice]

I think it's safe to assume that HS students won't do any better

[long pelican]

What you are asking for is kind of impossible without requiring students to understand epsilon delta

[long pelican]

Quite interesting!

[cosmic ibex]

What I'm asking for is just to have some way to use the new concept being taught. If that requires epsilon-delta (which for all I know it might), then I think it's hopeless to teach the concept at all to students who won't understand epsilon-delta. Otherwise you'd just be teaching them parlor tricks and test passing, but nothing they actually have any hope of using for anything.

[turbid zenith]

Wow how did I miss this discussion 😛

[wide ice]

I mean, isn't that basically HS math? You need to teach integrals anyway, which involves Riemann sum, which then involves limit, delta and epsilon. It's not that hard to convince someone that "looks, the area tends to that under the curve when the width gets very, very tiny", but it's hopeless to use any Greek letter to make that rigorous.

Unless y'all don't even teach integrals in US, then idk what to say anymore

[long pelican]

So yes, I stated that what I wrote a compromise between the current status quo and the ideal. To reach the ideal, students need to be actually taught everything, especially algebra, from a deductive reasoning standpoint, with explicit quantification, so that they are ready to interpret nested quantifiers without too much trouble.

[turbid zenith]

I haven't taught using hyperreals (yet... I'd love to) but it's possible to motivate lots of calculus using infinitesimals first and then bring the limits in later

Like how they were historically

[wide ice]

You mean with Newton's fluxon? 😄

[long pelican]

This is better than the status quo because it is more similar to how math is done, with way less pretending

[turbid zenith]

I mean I still called it a derivative

But I deliberately handwaved over the "ratio of infinitesimal quantities" concept

And then brought it up again later as "okay, we handwaved that, how can you make that actually a thing?"

[cosmic ibex]

The status quo being that the students get told about the definition of limits, but only a few of them understand it.
I think that is still better than a compromise where NO students get a chance to learn anything useful.

[turbid zenith]

And gave a bit of the history

But we spent most of our time on applications of calculus before we got there

[wide ice]

I mean, you have to start from somewhere. If you teach First-order logic to 6yo kids to make math rigorous, then that's gonna be a big oof

[long pelican]

Oh no, the status quo is students get told the definition of limits in 2 minutes and then told "This was in the textbook but no one, me included, understands this, and we will never talk about this again"

Followed by the school math treatment of calculus we all know and love

[wide ice]

to be fair, the majority of them will never need delta-epsilon

[cosmic ibex]

It sure doesn't feel like I know whatever you're decribing as "we all know and love".

[wide ice]

Not that they will remember it either

[cosmic ibex]

In that case I don't see how they will need limits either.

[wide ice]

oh, the engineers will still find them useful. "Looks, when this gets tiny, that should also get tiny"

But who cares if you admit infinitesimals or not for that?

[cosmic ibex]

"Looks, when this gets tiny, that should also get tiny"
How is that not the epsilon-delta definition that you want to keep a secret from students?

As a "compromised"???

[long pelican]

Computation, test prep, exact steps to perform for each type of problem that will appear on the test, lip service to conceptual understanding, students being told to stop thinking logically, the usual

[twin shell]

calculus is generally somewhere between 4th year HS and 1st year college
we normally don't use epsilons tho
we learn results (F'(x) = f(x), int f'(x) dx = F(b) - F(a), sin' = cos) and we learn to calculate or approximate stuff like the volume of certain shapes

[wide ice]

It is! Just that you cheat and not mention how to make it rigorous

[wide ice]

hence my point earlier: you don't care about definitions in HS, only properties

[cosmic ibex]

Are we discussing whether we should be using the words "this" and "that" instead of the Greek letters "epsilon" and "delta"?

[turbid zenith]

To some extent yeah.

Because symbols make students' eyes glaze over.

[long pelican]

The main issue for the students is the doubly nested quantifiers and the precise logical thinking needed

[cosmic ibex]

But the quantifier nesting is present just the same if you paraphrase it with "this" and "that".

[turbid zenith]

Sure. WE see that.

That doesn't mean THEY see that.

For another example, you show them Bayes' Theorem in its full symbolic presentation, students have an aneurysm. You show them how to all of the exact same calculations with a tree diagram, things start to make sense.

[cosmic ibex]

They'll need to be able to avoid the pitfalls that we would describe as mixing around the quantifiers. No matter whether they get to know the word "quantifier" in the first place.

[turbid zenith]

Depends who "they" are. Are we talking about math majors? Then yeah.

[long pelican]

When I taught sequence limits last year to university freshmen, I spent a good chunk of time teaching them the principles of logic and what quantifiers are. And even then, about 1/3 to 1/2 of the class understood it (which is the same as a typical real analysis class, I'm told). It really has to be taught from the beginning

[wide ice]

If someone pulls off Yoneda's lemma to argue about Universal property and completing the basis, my eyes will also roll.

[cosmic ibex]

I'm talkin about anyone who has any reason to learn what a limit is in the first place.

[turbid zenith]

If we're talking about, say, my AP Calculus students from years ago? Probably not.

But I would argue most students who take a calculus class do not need to know epsilon-delta definitions of anything.

At least not in an introductory calculus class.

[long pelican]

My position is closer to Troposphere's, actually: properly understanding logical implication and how quantiifers work is a necessary condition to understand the definition of limits sufficiently to use it in a mathematical setting

[cosmic ibex]

If they don't get taught anything they can use to argue for an inference of the form "the limit is such-and-such, THEREFORE we now know this-or-that useful or interesting fact", then it's a pure waste of time to make them learn how to argue "bla bla bla and THEREFORE the limit is such and such".

[turbid zenith]

That doesn't mean "never teach them limits" though.

[wide ice]

here during the first year, there's one specific class about how to write proofs, where these symbols and their meanings are introduced. No one expects a HS student to understand them.

[long pelican]

You don't mean the symbols, hopefully? It is equivalent to write out "for all" and "there exists" everywhere

[twin shell]

we have a first year proofs class for math majors
i think it's nice

[cosmic ibex]

Right, but if you do teach them limits, then don't hide from them what it is they're learning.

[wide ice]

some ppl never knew what the difference between them is before uni

[turbid zenith]

A lot of the time when you teach something, there are often things you're hiding.

We teach elementary school students addition of whole numbers while "hiding" the Peano axioms, and they do just fine.

[cosmic ibex]

Yes, but in this case what is being proposed hidden is the entirety of the reasons one would ever want to learn the stuff in the first place.

[long pelican]

Yes. We don't need to touch formal logic with a 10 foot pole in math classes. Sufficient understanding of how to read statements containing "for all," "there exists," and "implies" and some other stuff is enough

[wide ice]

So now when you teach integrals, you'll teach measure theory too? You have to start from somewhere

[cosmic ibex]

I have not made any claim that integrals are useless to know without measure theory.

[turbid zenith]

I still suggest teaching limits at the end of first-semester calculus. That way you can be more honest about what their role is.

[twin shell]

what is the issue here again
teaching limits but not the limit definition?

[cosmic ibex]

I HAVE made the claim that "how to compute limits" is a useless skill unless you also learn "how to use the knowledge of a limit to conclude something at all".

[turbid zenith]

You don't need limits to calculate derivatives. You do need them (as one way) to make derivatives precise.

[turbid zenith]

I showed it on a slide in my video on the subject after going through the basic idea with numerical examples, but I didn't emphasize it. 😂 I went back to "if you understand 'as x gets arbitrarily close to this, y gets arbitrarily close to that', you're in good shape at this point".

[long pelican]

I'm not sure why you think this? It's going to be presented as "Here's how we will define 'this approaches/gets close to this' in this class.
Definition: The limit of a function f(x) as x approaches a is the unique linear functional (except not using this word) from partial functions to real numbers satisfying properties 1, 2, 3, 4.
Theorem: There is a unique linear functional satisfying these."
I think some boundedness thing will be one of the properties

Lots of revision needed 😆 for example the codomain can't strictly be the real numbers

[cosmic ibex]

What I don't see is how an argument of the form

1. bla bla bla
2. therefore the limit is 5
3. because the limit is 5, bla bla bla
4. therefore (something interesting)
   can happen in your setting. It sounds like your axioms will make step 1 more straightforward, but I can't see how step 3 is possible at all without knowing something like the epsilon-delta property is what the claim in (2) tells us. It doesn't have to be worded with the exact symbols "epsilon" and "delta", but I don't see how to make any productive use of the claim in (2) without having the meaning of the usual definition present in some shape.

[long pelican]

What's something in calculus class currently that makes use of the fact the limit of a function is 5?

Maybe the limit of another related function

Maybe that |f(x) - 5| < 2 in some interval (is this ever talked about though?)

Limits of other related functions are covered by the properties

[tardy ember]

i don't know what people currently do in calculus classes but if the only thing you could do with limits is compute other limits then why are you computing limits

[long pelican]

Derivatives are limits 😛

Computing derivatives has independent interest

[tardy ember]

is it? under this definition the only thing you can do with them is compute other limits

[cosmic ibex]

The intuitive meaning of a derivative? By saying that the limit of (f(x)-f(0))/x is 5, I can argue that the slope of the chord gets closer to 5 the closer x gets to 0. I can't do that if I only had rules that told me how to calculuate the limit, but I don't have any rules that tell me any consequence of that.

[twin shell]

back to infinitesimals! [x^2 - (x - h)^2]/h = (2xh - h^2)/h = 2x - h = 2x

[long pelican]

" I can argue that the slope of the chord gets closer to 5 the closer x gets to 0" is not a precise statement. If this is all you wanted to get across, it's covered by the handwavy explanation at the beginning

if you were to state the precise statement, you will lose everybody

[cosmic ibex]

Heck.

What do you want from me here?

[long pelican]

I don't know, but I will remind you that all I'm saying is
[students coming in with proper understanding of logic and teaching them limits rigorously] > this > the status quo

[cosmic ibex]

You asked for something that can be done in calculus, and I suppose you will reject anything precise with "that sort of precision doesn't happen in calculus", so I downplayed the precision for that reason.

[turbid zenith]

Yeah I'm now confused as to what side anyone is on here lol

[cosmic ibex]

I am disputing [this] > the status quo.
When status quo means that at least some useful definitions get presented to students and perhaps 5% or 2% get something out of it, and [this] means that NO students get to learn something that will allow them conclude anything from the mystery concept they're they're being taught.

[cosmic ibex]

Perhaps my response should be: if YOU don't imagine teaching the students how to conclude anything from "the limit is 5", then why would you bother with teaching them to find out whether the limit is 5 or not?

[long pelican]

Tbh, thinking back to the uniqueness theorem

You still get the power to conclude anything concludable

You will just need to prove it

It's conceivable epsilon delta can be avoided for what's used in calculus class, but I haven't thought it through

[cosmic ibex]

I would like to see an example of how such a proof goes if it DOESN'T go through something like the epsilon-delta property.
(Where "something like" means I don't fucking care whether the Greek letters epsilon or delta appears, or whether symbolic quantifiers do).

[long pelican]

One way limits tie back to properties of functions that are defined before limits is: f' > 0 implies f is nondecreasing. Maybe we could tackle that

How does the status quo do this?

I think the comparison property axiom of limits (f(x) < g(x) on some interval implies limit f(x) <= limit f(x)) can handle this

[cosmic ibex]

I'm also pretty skeptical that you can get a uniqueness theorem from your plan at all, but since you just sketched something like "constant sequences converge to their common value" and "certain functions are declared to be continuous by definition" I don't really know how the rest of it goes.

Those two proposed axioms, at least, would be satisfied by a concept that says lim_(x->a) f(x) just means f(a).

[long pelican]

An "equality on deleted neighborhoods" axiom would deal with that

https://arxiv.org/pdf/0909.4178.pdf
I linked this earlier

[cosmic ibex]

I didn't get that that was YOUR plan.

It starts off with a forall-exists-forall statement as axiom 2. It doesn't use Greek letters; I suppose that is something.

[long pelican]

They're pretty much the same as the properties students are told to use on problems, so you can't really complain that this is too advanced

[cosmic ibex]

No, you're the one who is complaining it's too advanced.

[long pelican]

Yes, this wording would not be used. How is it worded in textbooks?

[cosmic ibex]

That's the whole reason you won't let students see the epsilon-delta definition.

[long pelican]

Hah! This is where you misunderstand

I will gladly let them use it, if they are ready to use it

This is a proposal only for classes where it's known ahead of time the students are way behind on many things

which is most high school classes

This by no means solves anything fundamental with the system

That needs a larger solution

[cosmic ibex]

If they're too behind to learn what limits can be used for, then I don't think time should be spent on teaching them to calculate limits they won't be able to use anyway.

[long pelican]

Yes, that's something I can get behind as well as a possibility

[elfin sleet]

I think that what is proposed in there is more complicated and less intuitive than the epsilon delta definition, and therefore it seems very unlikely that it is possible to teach it in a way that is simultaneously more rigorous and more easy to learn than using an explanation that is somewhere on the spectrum of "actual epsilon delta definition" to "handwavey intuitive explanation that hints at the epsilon delta definition"

[long pelican]

Yeah, the notation is scary and it would defeat the point for it to be delivered as is

But all that's being done

is to upgrade the properties that are drilled into students into axioms

It's the "inversion of definition and property" pattern that I've seen throughout math

[cosmic ibex]

The Arxiv paper looks very different than what I understood from "declare certain functions are continuous by definition", so possibly not many of my criticisms apply as-written to it.

[long pelican]

Inversion of definition and property is generally considered when it turns out that 90% of your proofs go through the properties rather than the underlying definition

As it is with characteristic classes

also done when the construction (which used to be the definition) is complicated

Now, epsilon-delta is not complicated by any mathematician's standard, so this proposal would not be fit for an ideal world

[turbid zenith]

I'm pretty sure this will lose most high school students

[elfin sleet]

yeah but I don't just mean the notation, like the whole idea just seems to make less sense. Like to me the general idea of epsilon delta is pretty intuitive whereas if you use these axioms I feel like the natural response is going to be like, "so what? What does this actually mean?" The axioms would feel totally arbitrary and meaningless to any student I think

[long pelican]

Ah, but If "general idea of epsilon delta is pretty intuitive" then we don't need this, I agree. The reality is "general idea of epsilon delta is pretty intuitive" is vehemently not true for the target audience in calculus classes in 2023

[turbid zenith]

But this would not be better

[elfin sleet]

exactly

[turbid zenith]

If anything it would be worse.

Honestly high school calculus is not the time for axiomatic proof in general.

[elfin sleet]

We can argue all day about how intuitive epsilon delta is on an absolute scale, but on a relative basis, I think it's more intuitive than these axioms, so what advantage do they offer

[long pelican]

These axioms are merely what students use to solve problems

How are we claiming they are not intuitive?

From the students' point of view, the properties listed here are what's used to solve problems while the definition is a red herring. Either make the definition not a red herring or invert the definition and properties

There is a point that it does not help intuitive understanding of the concepts, which is a valid one, but that wasn't the point

[elfin sleet]

I'm not confident that the inequality one is used as-is to solve problems but maybe I'm wrong

[elfin sleet]

I was arguing that they're not intuitive as a definition of a limit and give no insight into what a limit is

It seems like you want to make the definition of a limit more rigorous than what students usually are taught, and my point is that if you introduce something based on this with any amount of rigour, it's going to be just as "difficult" as something based on the regular definition with an equal amount of rigour

[elfin sleet]

maybe I'm not clear on what "the point" is

[long pelican]

This is a good understanding of what I was doing

So my thinking behind all this is that in both methods, we are presenting students "rigor". But in one of them it's just "pretend" rigor because we show them an impenetrable definition, and properties are generally not proved. So what students are using to solve problems is basically rules given by the teacher

In the other, it's less of a pretend, because the definitions (which used to be properties) are now all used directly to solve problems.

An underlying assumption I have: A logical experience of math is much easier to understand than an illogical memorization-and-handwaving-based experience of math

(To be clear: handwaving is important in both experiences, but only as motivation, and not as technique)

[cosmic ibex]

There's a type of homework problems that basically go

here's the graph of a function f. Read off the graph what lim_(x->2) f(2) is.
Would such problems be relevant exercises in your "compromise" course? I'm genuinely unsure whether your answer will be
a) yes, of course!
b) no, the purpose of those problem is just to drill the student's understanding of the epsilon-delta definition, so they won't be necessary in my course.

[long pelican]

Neither! Such a question is testing the handwavy understanding. We still give them the handwavy understanding but we don't pretend it is mathematics

[elfin sleet]

Then it seems like the only difference is whether you tell students that the properties are the definition, or that they’re properties. I don’t think most students would even notice this difference, and will just perceive them as rules either way. It’ll only feel more rigorous to the instructor

[cosmic ibex]

Do you think students who take (and pass) your course should nevertheless be able to identify the right answer to such an exercise?

[long pelican]

Hm, I suppose. I guess my proposal is not an improvement or de-provement for this kind of exercise. Epsilon-delta is definitely not being used to justify how to answer this question in practice as it is

[long pelican]

Mmm, if students are poisoned and think math is divorced from logic, it might well be the case that one calculus course cannot make much of a difference. So treat this proposal as part of a large overhaul of the system

[cosmic ibex]

To me, epsilon-delta is a fairly direct encoding of the fuzzy idea in my head I would approach the exercise with. I'm trying to imagine which fuzzy idea will be in your ideal student's head for it.

[long pelican]

Yeah, we all think epsilon delta is fairly direct and intuitive, me included. The point is it could not be further from the truth from the students' point of view

And say, me at the beginning of when I took calculus class

I had to do work to make it intuitive

the other students had no chance

[elfin sleet]

Ok but my point remains that if epsilon delta is unintuitive then any similarly rigorous thing is equally unintuitive

[long pelican]

Yeah, it won't be an improvement in the intuitiveness. I already addressed that

[cosmic ibex]

And therefore you're trying to construct something else the students will think is direct and intuitive instead. I just don't see how it will apply to "here's a graph (or a table of function values), identify the (likely) limit".

[long pelican]

I think the hold up here is that traditionally in calculus, a big question is "How do we make limits more intuitive?" I'm not answering that question. I'm attacking a deeper issue

[cosmic ibex]

That sounds a lot like your answer to "how do we make limits more intuitive" is "replace limits with something that doesn't have an intuition attached to it that the students can miss".

[elfin sleet]

yeah lol

[long pelican]

Ehh, the way it's made intuitive is that we teach how to interpret limits on graphs as a separate thing completely from how to solve problems with limits

If you want, you can continue to teach how to interpret limits on graphs completely separately from my proposal

[cosmic ibex]

Can you give some ideas of what you mean when you say "solve problems with limits"?

[long pelican]

Calculate limit of (x^2-3)/(x-sqrt(3)) as x approaches sqrt(3)

[cosmic ibex]

Except that your proposal has eliminated the definition that the intuition I think should be taught is inution ABOUT.

[long pelican]

So again, this is for the scenario in which the actual definition is completely useless to the class

[cosmic ibex]

Do you have any problems that draw any consequences from knowing such a limit?

[long pelican]

In this scenario, including it is just lip service and pretending

[elfin sleet]

ok this seems to contradict what you said earlier. I thought you had said "epsilon delta is too unintuitive; we can't introduce it". But now you're saying that this is no more intuitive. So why wouldn't you just do epsilon delta?

[cosmic ibex]

Why would you care about what the limit of (x^2-3)/(x-sqrt(3)) as x approaches sqrt(3)?

[long pelican]

If it's something actually non-trivial, it would be beyond the scope of typical calculus classes, generally

[long pelican]

I think I need to remind myself of the context in which this proposal was made

[cosmic ibex]

And then we get back to: What is the point of teaching students to compute limit if they don't get to use the limits they compute for anything other than pass tests?

[long pelican]

I feel like in practice, it's not actually a question of "derivatives via Cauchy definition of limit" vs "derivatives via infinitesimals" vs "derivatives via hyperreals", it's more a question of "derivatives via handway intuitive concept of limit + rules" vs "derivatives via any logical definition of limit"
Unfortunately, anything of the second type is inaccessible to most students due to the simple fact that math classes give them zero training in understanding logical definitions

Here's the context

[long pelican]

If you reject the usefulness in "finding derivatives" or "finding integrals", yeah, this is now a criticism of calculus class as a whole

[cosmic ibex]

If you claim finding derivatives is useful, it must be because you have an argument that draws a conclusion FROM the existence and/or value of a limit.

[long pelican]

Generally applications of finding derivatives that students most often see are something like calculating velocity in physics

There are a lot of assertions in that that students never see proved

[cosmic ibex]

They can do that quite well with symbolic differentiation rules, can't they?

[long pelican]

Yes

[cosmic ibex]

So the limit definition is part of a plan to let them see why the symbolic differentiation rules are the right ones.
But that plan won't be complete if you only tell students an opaque deliberatetly-not-connected-to-any-intuition definition of limit that will only allow them to calculate limits, but won't allow them to even begin to think about the possible significance of the limit being this rather than that.

[long pelican]

What you just wrote applies equally well to the epsilon delta definition of the limit in practice

[cosmic ibex]

No. I'm saying the epsilon-delta description of limits is what is MISSING here.

[long pelican]

In the context of mathematical theory I can't imagine anyone disagreeing with you

[cosmic ibex]

Calculating a limit is a meaningless parlor trick unless you've gotten to know what it is you have accomplished when there's a concrete string of digits after the final equals sign on your paper.

[long pelican]

You just described what students feel even after having seen epsilon delta 😛

[cosmic ibex]

No.

[long pelican]

that is, if they haven't already been jaded by their math experience

[elfin sleet]

even if we accept this as true, it doesn't show that your way is better, only that it's not worse

[cosmic ibex]

I describe the feeling of a student who has not been allowed to see epsilon delta, in the name of a "compromise" because were judged too stupid to deserve the privilege of being told what the computations they do are FOR.

[long pelican]

Oh I see. Epsilon delta will belong to "The existence theorem"

The existence theorem (which contains epsilon-delta) will be called "outside the scope of this course" but available to be seen by motivated students

Mmmmm

[cosmic ibex]

And unmotivated students will have their lack of motivation justified by being denied knowledge of what their computations are for.

[long pelican]

Do you agree that if the existence theorem was taught instead of being outside the scope of the course, then your current objection will disappear?

[cosmic ibex]

Partially.

I suppose it would still feel weird to students that the reason for doing what they've been taught is hidden away as an internal detail of a proof instead of front and center in the course.

[long pelican]

Let me turn this around actually. Why is the epsilon delta definition of any value (from a student's point of view), assuming they have already been taught the handwavy definition of limit?

Assume this student will not understand it

[cosmic ibex]

I'm not sure which "handwavy definition of limit" you're talking about if it's not equivalent to the epsilon-delta definition.

[long pelican]

It can be as simple as "The value the function approaches as x approaches a"

This (plus some training) is sufficient to answer graph type questions

[cosmic ibex]

Okay, so you'll tell them that, and then you'll hit them with a rigorous theory that is completely divorced from that intuition, and then perhaps at the end of the course in the middle of an existence proof that, ha-ha-ha, that thing about the function approaching something as x approaches a was the same thing we've been doing the whole semester.

[long pelican]

Rigorous theory divorced from that intuition is what the epsilon delta is to a student lacking in some prerequisites

You haven't answered my question, though

What is the value of epsilon-delta to such a student

[cosmic ibex]

Which question?

[long pelican]

here

[cosmic ibex]

The value of epsilon-delta is that it is the explanation for what AT ALL the formal manipulations they're being taught has to do with any intution.
At least as long as you don't provide any alternative intuition that your alternative axioms would encode.

[long pelican]

That is not from the student's point of view

This student thinks math is about computations and arbitrary rules and answering test problems

they would not think what you wrote

[elfin sleet]

what does your way bring to this student? Aren't they still going to see it like this?

[long pelican]

They will be doing something mathematical while solving problems, at least. They will know what a definition is and what it means to use a definition

[cosmic ibex]

If the student is just taught (a) a handwavy concept and (b) a formal theory that has no apparent connection with that handwavy concept, are you expecting them just to take your word for it that they're the same thing? If you are, then why show them proofs at all, if they're required just to take on faith that the proofs they're shown are about anything?

[cosmic ibex]

And teaching should be about disabusing them of that notion, not enabling it!

[long pelican]

Yes, in practice I aim to teach them a crash course on logic and how mathematics works. I have been doing that. This is a compromise for when this method is unavailable

Also, there are probably plenty of ideas better than it

Think of my compromise as setting a lower bound (or upper bound?)

at the very least it does not pretend to do something it doesn't

[cosmic ibex]

I don't see a single way in which your "compromise" is an improvement on the usual teaching, however bad that might be.

Unless you think it is an improvment that more students will be able to find the right answer to test questions that are meaningless to them.

[long pelican]

At the very least it makes it so that what the students use to solve problems can be traced to something rigorous, instead of behind a smokescreen and effectively a rote thing

[cosmic ibex]

Formal symbolic calculations that are not connected to intuition ARE a smokescreen of rote learning.

[long pelican]

Currently: -uses comparison property of limits- Why is comparison property true? Because the teacher said so, or because there's a proof of it I don't understand and I don't care about
Now: -uses comparison property of limits- Why is the comparison property true? It is defined to be true

[elfin sleet]

exactly

[long pelican]

Using the comparison property is now rigorous instead of rote

[elfin sleet]

how is "it is defined to be true" different (from the student's point of view) from "because the teacher said so"?

[long pelican]

It may not make a difference in the span of 1 year, but at least it involves no pretending, and it does not compound the problem

[cosmic ibex]

It definitely compounds a problem if you go about encouraging students to think math is about computations and arbitrary rules and answering test problems and that intuitive meanings is something they shouldn't worry their young little heads about.

[long pelican]

I've seen many students think theorems and definitions are the same thing

[cosmic ibex]

That. Is. Not. A. Reason. To. Deny. Students. An. Intutive. Understanding. Of. What. They're. Doing.

[long pelican]

Epsilon delta is not an intuitive understanding...

[cosmic ibex]

So you keep claiming.

Counterfactually.

It encodes exactly the intuition one needs about what limits are.

[long pelican]

Ask any college graduate who has taken calculus 1

if they think epsilon delta is intuitive

the answer will be no

They will probably not even be able to state it

[cosmic ibex]

We've been talking about this for hours, and you still have not proposed any OTHER intuition about limits than the one that the epsilon-delta definition embodies.

[long pelican]

If you disagree with that, then that is probably a fundamental disagreement about assumptions

I am NOT proposing a more intuitive definition

I am proposing a definition with less pretending

[cosmic ibex]

You're proposing a DELIBERATELY LESS INTUITIVE definition.

[long pelican]

No, epsilon delta is about as unintuitive as it gets for aformeentioned college graduate

I don't think I can convince you of this with words

[cosmic ibex]

And apparently have the cheek to claim that it will be easier for students to handle NOT getting any intution.l

WHAT THE HELL do you think a more intuitive concept would be?

[long pelican]

OK let me conclude

IF epsilon delta is able to be even partially successfully understood and thought of as intuitive by the average student, then my proposal is absolutely useless

I hope we agree on that front

[cosmic ibex]

You can't just come and say we must avoid epsilon-delta because it is insufficiently intuitive, and then refuse to propose something you think is BETTER.

Basically you're saying "this is unituitive, let's replace it with something even less intuitive".

[long pelican]

OK, maybe you have a problem with this axiom

This is probably not the only way to axiomatize it

[long pelican]

No, this is not what I said

There are 2 steps.
Epsilon delta has zero value to the current cohort of calculus students. So it makes no difference if we have it or don't have it

Instead of that, let's try axiomatizing what the students use about limits in practice

I mean you disagree with the first premise

[cosmic ibex]

No, I have a problem with the whole approach of not teaching students that formalizations correspond to intuition, to the point of deliberately eschewing formalizations that do correspond to intuition and instead choosing ones that don't. It's as if you want to PREVENT that any student could get the crazy idea that things make sense!

[long pelican]

And this is largely down to what you think the average US student has learned in their education

[long pelican]

You keep going on about intuition, so here

[cosmic ibex]

Apparently the US school system is doing its damnedest to prevent students from discovering math can make sense.

[long pelican]

Constants have limit equal to their value <-- about as intuitive as possible

[long pelican]

Yes, it is

[twin shell]

it is

[long pelican]

Limits of functions on deleted neighborhoods <-- also intuitive

Inequality axiom probably can be made intuitive if educators work on it enough

\forall epsilon \exists delta.... <-- not intuitive

[cosmic ibex]

And your problem with the epsilon-delta definition is that it might accidentally make a student suspect math can make sense. We can't have that! Don't show them the sensible epsilon-delta definition, find something meaningless but formally equivalent to drill them in instead!

[long pelican]

I already addressed this like 2 times

[elfin sleet]

You keep saying that epsilon delta has zero value for these students. I think what I disagree on is that epsilon delta can only be the full formal definition. You can still explain the concept of it without introducing greek letters if you think that's what's going to scare people off. To me, teaching with the epsilon delta definition is a sliding scale from "function approaches value as x approaches a" to the actual definition, and you should pick whatever point on that scale that students will be able to understand. Then the intuition is aligned with whatever level of rigour is possible for them

[cosmic ibex]

And yet you still champion NOT TELLING STUDENTS THE THING THAT MAKES SENSE.

[long pelican]

Is there a thing that makes sense to them?

[elfin sleet]

hey, I'm on your side here but this message is unfair. I'm getting frustrated too but obviously this is not what they're claiming. You shouldn't deliberately strawman people

[long pelican]

The only thing I can think of is the handwavy explanation

[cosmic ibex]

Okay, sorry.

I think I shoul just go to bed now.

[long pelican]

Oh you live in Europe right

[elfin sleet]

Yeah I think this conversation is not going to be productive. I'm going to head out too

[long pelican]

Yeah, we obviously are coming from very different contexts and not explaining the differences properly

[turbid zenith]

If we just did hyperreals we wouldn't have to worry about whether the epsilon delta definitions make sense because we wouldn't need them anyway

*ducks*

[zenith slate]

Wow these hyperreal shenanigans really took off huh

[elfin sleet]

lol it wasn’t even hyperreals we were discussing during that long conversation, it was another idea for teaching limits

[zenith slate]

Yeah, I saw

I'm wary in general of attempts to completely overhaul the way things are taught, especially when motivated by vague speculation that the way things are currently taught is the sole reason for any and all perceived failings in education

The biggest issues in education are systemic, anyone claiming to have some kind of panacea for mathematics education by proposing we define or present things in a non-canonical way is really missing the point

Even if this axiomatic approach to limits resulted in demonstrably better outcomes for students, you'd still have the enormous logistical challenge of overhauling curricula on a national scale both in HS and in university

Teach this to some poor highschool student and they're gonna be completely unequipped to deal with a first course in real analysis and have a lot of ideas about how the discipline is actually structured that would need to be unlearned

Just teach limits as limits good lord

[long pelican]

You lost your blue @zenith slate
I think a context Europeans are missing is that epsilon delta is completely omitted in AP calculus in US curricula, unless the teacher thinks the class is strong enough. I don't know why I didn't think to mention that

[zenith slate]

This is also true in the UK curriculum, where there is no formal treatment of limits whatsoever

[long pelican]

I was not planning to be pinned against the wall for several hours on an idea I threw out off the cuff as well, nor strawmanned to the extent I was

[zenith slate]

Students will see the limit definition of the derivative and the actual question of convergence being a well-defined concept is omitted

Yes I saw that the conversation above got quite heated

[long pelican]

This is easy to say, but at the same time it is not healthy to discourage improvements nor is it healthy to strawman them as somehow the person proposing a panacea for all of math education

[zenith slate]

I do not discourage improvements - far from it - but good pedagogy is evidence-based and an approach that looks good on paper is meaningless unless there is evidence to back up the efficacy of said idea

Regarding my "panacea" comment, you're right - this is a more general gripe I have with particular types of people and it was unfair to direct toward you

[elfin sleet]

Yeah sorry if I contributed to that. I do think I mostly understand what you were getting at even if I’m not convinced that your idea would accomplish it. I think it was hard to effectively communicate, especially with multiple people saying many things at once - I felt like it might have gone better in person or something - and I think that caused everyone to get frustrated

[long pelican]

Even though evidence-based methodology is superior to opinion-based, I still feel like there is something very wrong with what people are doing scientifically in this field.

Can't really put my finger on it right now though. Metrics are a part of it

[zenith slate]

I don't think people are doing things wrong - it is that it is extraordinarily difficult to collect high-quality data

[long pelican]

Yeah it's not necessarily "doing the science wrong". I think there's some lack of understanding of what you want to measure, among other things

[zenith slate]

Like even a basic comparison study where you teach half of a group of students with method A and half with method B is borderline unethical

It is a tough problem and I do not know the answer to it

[winged urchin]

I haven't read everything but thank you to those that put energy into this discussion

It is enlightening

[tardy ember]

tbh from what i can tell it's somewhat ambiguous what the point is actually supposed to be

like if you ignore all the words that get thrown around and just look at how the system is actually structured, the point of "teaching students maths" is apparently so that they can pass various tests, that's the direction the incentives point in

there isn't systematically anyone who checks whether the students have any clue what maths is and then does anything about it

[cosmic ibex]

After sleeping on it, I now notice that these two first axioms directly imply the epsilon-delta property (namely, compare f to the constant functions L-epsilon and L+epsilon one at a time).
I think my reaction would have been a lot less negative if the plan had been framed as "teach the epsilon-delta property split into these two components" rather than "don't teach the epsilon-delta property".

[cosmic ibex]

On the other hand this also underscores that the logical structure of the second axiom there is not really any simpler than the epsilon-delta definition:

Suppose you're looking at a function f with (left) limit L. Then for every epsilon>0 there exists a delta>0 such that for every x with x0-delta < x < x0 it holds that |f(x)-L|<epsilon.
versus
Suppose you're looking at a function f with (left) limit L. Then for every function g with limit less than L there exists a < x0 such that for every x with a < x < x0 it holds that g(x) < f(x).
If students find the latter easier to internalize, it can't be because of the quantifier nesting. Perhaps one could argue that one-sided inequalities are less of a cognitive load than also needing to grok the behavior of |this-that| in the "it holds that" parts.

[wispy slate]

asjdasjklhdajhddsa why didn't I find this channel earlier

I wish I could read 10s of thousands of messages in a short amount of time. Unsurprisingly, that is not the case :(

[long pelican]

I didn’t expect that the translation was that simple! On an unrelated note, do you have any thoughts on the fact that both US and UK curricula leave out teaching epsilon delta completely in any form

[pallid night]

Epsilon Delta is literally just :

"If you are at a point $x$ sufficiently close to $x_0$ , then the value of your function $f$ at this point is very close to $f(x_0)$

A good diagram is surely more than sufficient for making this intuition.

That's a lot of sufficiency lol

[burnt vessel]

Ama Dablam

[pallid night]

The lack of teaching of this makes limits much more awkward in my experience.

Limits are nice maths, but without an actual definition of what they are they're just not very nice to work with

Not the person you were asking but thought I'd chip in anyway

It would take a matter of one lesson to teach at most.

The definition that is

[long pelican]

Oh yeah, but more on why it happened. For AP it was around the 2010s, because they found it was too hard for students

[twin shell]

I remember my AP calc BC class mentioned the definition of an integral as a limit of riemann sums once and never touched it again

[pallid night]

I mean ... yea you have to keep the questions very straightforward at HS level.

Can't realistically expect them to prove polynomial continuity with $\varepsilon - \delta$ at HS level

[twin shell]

putting an empty line in between two paragraphs produces a newline

[burnt vessel]

Ama Dablam

[pallid night]

In any case Uni questions with $\varepsilon - \delta$ could be stuff like

Use $\varepsilon - \delta$ to prove that $f(x)=x^2$ is continuous for all $x \in \mathbb R$

At HS level you can't realistically expect them to do any tricks so you'd probably avoid anything that requires spotting bounds or stuff like that.

[burnt vessel]

Ama Dablam

[pallid night]

I think part of the issue is that even seemingly simple looking functions can become hellish to apply epsilon delta arguments to directly

[zenith slate]

Yup - this the reason I think teaching epsilon delta would be largely a waste of time for a high-school class. While the definition may be straightforward (though I contest this for several reasons), actually constructing proofs using this definition can be extremely tricky for students

[pastel sundial]

Any advice for a first time TA? (calc 1)

[long pelican]

One thing that happens often for first timers is students not following because you assume too much mathematical maturity. So lower your expectations, then lower them some more

tbh you probably already know this since you've seen what I shared 2 years ago

[pastel sundial]

Yeah I definitely have your stories in mind

Though my expectations will probably still end up being too high tbh

I remember you had a bunch of good exercises you wrote, Icy. Mind sharing some of your favorites again?

[long pelican]

None for calc 1 sadly

[pastel sundial]

Oh were they for calc 2?

[long pelican]

yea

[pastel sundial]

It's probably gonna be the Prof writing most of the exercises anyways so no big deal

[long pelican]

One thing very relevant for calc 1: Your students likely don't know what a mathematical theorem is or what the difference between a theorem and a definition is -- you could make that an exercise and see for yourself what they think. For example, they have seen properties of limits. Ask, "Are the properties of limits theorems?"

Also ask "Is the fact that the derivative of x^2 is 2x a theorem?"

[pastel horizon]

Not understanding what exactly "definition" means in general.

Context: mechanics problem where they need to provide a bearing.

By definition it's the clockwise angle measured from north, student asks why I need to put the north line on the diagram

Like it should be clear that because of the definition you need something to do with the north direction or at least I would have thought that's the case

[pastel horizon]

Yup will add to this no matter how far you lower them, it still won't be enough

[twin shell]

your average engineering student just won't care what the difference is

[long pelican]

My point was not to later assess them on what a definition or a theorem is (it's sad that students always ask "do we need to know this for the exam"), it was to help them have a chance at understanding the tangled web of mathematical statements for once

[ivory ember]

Re: limits: I recall seeing one textbook that I thought handled them well. Not rigorously, but well enough that a student would later be able to connect it to a delta-epsilon definition.

Anyway, I'm popping in to ask: does anyone have recommendations for software with which I can make geometric diagrams that would allow me to indicate congruences, parallel lines, etc?

I can't seem to find a way to do this on Desmos or GeoGebra, but I'm also new to this so maybe I'm missing something

[pallid night]

Geogebra had a geometry mode last time I checked.

Can't recall if it allowed indicating those though

[ivory ember]

I guess I could make up something with Tikz, but I rather use something with good in-built functions.

[ivory ember]

I tried that earlier and didn't have such a good time when I was trying to make a drawing that wasn't to scale. I wonder if scripting (but currently I don't have time to learn that) would help.

It's driving me slightly crazy though, because you'd think that this would exist, and I really would like to avoid drawing diagrams on MS Paint like it's 2000.

[ivory ember]

Turns out there is a GeoGebra solution to this, but there's a lot of clicking around to get to it: https://www.youtube.com/watch?app=desktop&v=8C2qH-qSCP4

[cosmic ibex]

On an unrelated note, do you have any thoughts on the fact that both US and UK curricula leave out teaching epsilon delta completely in any form
I have a feeling there's a disconnect here in that we seem to have quite different understandings of what a "form of epsilon-delta" means.
If the teacher says, "to say that the limit of f at some point is some number means that we can force the value of the function to be as close to the limit as we want, just by requiring the function's input to be close enough to the given point", then I consider that a form of teaching epsilon-delta . (Several additional sentences will be necessary to explain how the function value at the given point is ignored and so forth; I'm ignoring that). My understanding is that you don't consider that to be epsilon-delta in any form, and rather it is exactly that "handwaving" that you contrast to epsilon-delta. But I'm honestly unsure where you draw the line.
In my understanding there's a sliding scale of explanations between the prose and the fully symbolic rendering with Greek letters and upside-down A's and E's, but as long as the semantics of the explanation is the same, that's what I'm asking for.

[long pelican]

"Remove epsilon-delta because it's too hard" was always about the formal definition itself, or any rewording of it thereof. Softer things are how it's taught in practice, and frankly how it's always been taught, even when epsilon-delta was part of the curriculum.

I am not aware of a teacher using "to say that the limit of f at some point is some number means that we can force the value of the function to be as close to the limit as we want" as the de facto formal definition. In any case, it's unusable as a definition that is used to prove the first limit properties anyway, until we define precisely "as close to x as we want." So there's going to be incomplete logical development of limits no matter how you cut it

[cosmic ibex]

That doesn't help me understand where your line is, because an explanation starting along the lines I sketched (plus all the subsequent discussion to explain and illuminate what it means, which I didn't care to type out) does in my understanding qualify as "any rewording thereof".

[long pelican]

OK, can you outline how a proof of the quotient property of limits might go in a classroom with you as the teacher?

[cosmic ibex]

No, I can't do that.

[long pelican]

Right, because to demonstrate such a proof you would need to use the doubly quantified version of the epsilon-delta definition. So there is a difference between your rewording and the actual definition!

[cosmic ibex]

I don't understand why you seem to think that the explanation I typed is not "doubly quantified".

[long pelican]

I just mean that it's not a faithful rewording of the epsilon-delta definition

or fully faithful perhaps

[cosmic ibex]

We might have to agree (or not) to disagree about that.

[long pelican]

Huh, ok

https://tenor.com/view/lets-agree-to-disagree-no-eyes-gif-16165896

Couldn't resist

[cosmic ibex]

In that I don't understand what it is you feel is missing, and it appears to be so obvious to you that your explanations don't seem to make it any clearer to me.

[long pelican]

I'll try. Imagine the logical development of limits and derivatives as a graph, where there is an edge between two nodes iff the student understands the proof of the more distant node using the closer node. As it stands, the graph is not connected, because the definition of limit is an island

[cosmic ibex]

I ... sort of get the impression that you're answering "what you feel is missing in mathematics education" there, whereas I was trying to understand what you feel is is missing in the style of exposition I tried to describe, such that it cannot be a "faithful rewording of the epsilon-delta definition".

[long pelican]

So your goal is to have something that someone can walk away and say "this is what it means," my goal is that plus the ability to connect it logically to the rest of the stuff in calculus

And if I had to choose one or the other I'd slightly favor the latter but not lightly

[cosmic ibex]

Are you saying that a statement of the definition in words rather than symbols cannot possibly be "connected logically to the rest of the stuff in calculus"?

[long pelican]

I asked you to try with the quotient property, as an example, and you couldn't :^)

while I'm quite sure you could do it with the epsilon delta definition

[cosmic ibex]

Then we get back to the fact that I genuinely don't understand how you distinguish between the explanation I sketched and something you would agre to call "the epsilon delta definition".

[long pelican]

How is this not a big difference??

One is precise enough that you can begin to use it in a proof, the other isn't

[cosmic ibex]

And no, I wouldn't be able to extemporate a proof of the quotient rule starting from a symbolic rendering of the definition either.

[long pelican]

Wait, so you answered the wrong question

I just asked you to outline it, without details

[cosmic ibex]

And I said I won't be able to do that, and my inability to do that is completely unreleated and independent of the style the definition has been presented in.

[long pelican]

You will have to come to grips with using the hypothesis "close enough blahblah" and you'll find yourself having to make it more precise

You traded inequalities for vaguer words

[cosmic ibex]

I don't understand why you think an inequality becomes "vague" by being stated in words rather than in symbols.

[long pelican]

I do not see an inequality in your rewording...

[cosmic ibex]

Huh.

I wrote:

to say that the limit of f at some point is some number means that we can force the value of the function to be as close to the limit as we want, just by requiring the function's input to be close enough to the given point
The words "as close to the limit as we want" speak about a comparison between how close the function value is to the limit versus how close to the limit we want it to be.
The words "close enough to the given point" speak about a comparison between how close the input is to the given point, and how close we're requiring it to be.

[long pelican]

No, I do not agree with those readings

[cosmic ibex]

I think we're back to the agree-to-disagree point then.

[long pelican]

Just to name one thing, "we want" is terribly unclear as far as mathematical terms go

It would be nice for you to demonstrate a proof of something using this definition

You'll face the same challenges I can imagine facing if I were to sit down and do it myself

From Wikipedia

Thank goodness I am not hallucinating

[cosmic ibex]

All I can say is in my world it would go exactly the same as anyone elses proof from the epsilon-delta definition becuase what I'm describing IS the epsilon-delta definition, simply rendered with different surface syntax. But there's no semantic difference between what I'm saying and the symbolic rendering, and therefore the proofs would go in the same way.

[long pelican]

Since you're familiar with syntax and semantics, I guess what I'm saying is that it's not clear a priori how the syntax of what you said translates to the semantics of epsilon-delta

[cosmic ibex]

I mean, shrug. You're the native speaker, so it won't work well for me to try to convince you about what the words in your language mean...

[long pelican]

I mean your syntax includes words like "enough", "desired", "required", "want" which could someone could theoretically incorporate into a formal translation into semantics

but it has not been done (well really, what I mean is it is not in common enough usage in formal mathematics for it to have been done), and that's why they are words that belong to intuitive explanations for now

Apparently you have a clear translation in mind

[zenith slate]

I'm inclined to agree that your "handwavy" definition maps precisely onto the algebraic statement of a limit in terms of epsilons and deltas

[long pelican]

Look at that, two native speakers disagree with each other 😳

[zenith slate]

Indeed

It certainly takes some effort to translate the worded definition into algebraic notation if you're not already familiar with the definition - but I'd say the converse is equally true. The epsilon-delta statement in its algebraic form is very intuitive and, unfortunately, I feel that students are often lacking this intuition precisely because it's stated in a language that students at that level are not fluent in.

I'm speaking from experience here, as a student who was taught the algebraic definition but didn't grok the intuition needed to actually construct proofs with the definition until a TA sat me down and explained it to me in a wordy form.

In any case, to return to the original point, a formalisation of the limit doesn't appear in the A-level curriculum in either of these forms

I go over it with my students and most mathematics teachers with the requisite background in mathematics will do the same - though this highlights another reason why certain things are often excluded from mathematics curricula. A question arguably more important than whether students can learn any given content is whether the average highschool mathematics teacher can adequately teach it.

[pastel horizon]

I wasn't even assessing them on it that's the scary thing

It was just a case of "this is what a bearing is so therefore we have to draw north on the diagram"

[halcyon mortar]

hi! i'm making a website for teaching math, but im teaching things that i learned quite recently, so if anybody's down to help fact-check/give general thoughts some of the things im writing then lmk in dms!

[pastel sundial]

Limits are weird

I was reminded of this fact today when thinking about how best to explain them.

I've internalized them for so long that I had forgotten what a weird concept they must seem when they're first being introduced.

Like how do you even teach them to a new calc student? The epsilon-delta definition feels like it would fry most students' brains, but without it, what option do you have besides vague analogy...

[storm kettle]

how about explain epsilon-delta using some sort of game semantics?

[topaz current]

my chemistry teacher touched on the intuitive concepts of calculus when teaching the topic of rate of reaction, so if students take chemistry, that could be another example, since first order reactions are naturally continuous, exponentially decaying curves

[mild shuttle]

Can I get some feedback by those of you in this disc on my situation? im debating swithcing majors

I am enrolled in an online math degree program at public uni
Pros: prof does a good job w lectures, and textbook is somewhat comprehensible, prof is kind and answers emails pretty quickly.
cons:
-quizzes, which is 25% of my grade, are typed out, you dont submit work with pens and paper or draw a graph, you TYPE out your answers using only words.. so I have to work the problems out, but then have to type them out with words like "the limits of integration are x = 0 and x = 2, when we integrate our function 2x we get x squared which evaluates to 4 - 0 ..." this is INCREDIBLY time consuming and I wonder if prof is doing this so that they can just use an AI to grade our work... in addition, we are to type out a worded description of a graph if we draw one.. there is also a large degree of implied things that we are supposed to say on these quizzes but are never given any explicit instructions to mention, for example if I am working with a probability density function, I have to define it using the definition, and state where it has probability and where it does not, this cannot be left implied that it is "zero elsewhere". Nothing wrong with this inherently mind you, but the fact that this is expected of students and NOT explicitly stated is really frustrating, I only know this from AFTER getting my quizzes graded..

homework: prof does not give any solutions manual or information on how to verify your answers to the homework (thus making me spam reddit and discord to verify my work) and wants us to "work together" to do the homework. Problem is its online and since the prof gives brutal quizzes if I know the answer to a question, I will not benefit from helping my other classmates, in fact the poorer my classmates do, the better off I will be if i understand how to do something (obviously this isnt me bc I dont understand anything but Im just saying there is incentive NOT to help classmates out).

Grading and feedback: prof does not have office hours or any way to get your questions answered by them. They also do not grade quizzes and homework in any reasonable amount of time. For example: there are assignments I submitted on August 28th, almost a month ago, that still have not been graded. I very well could have learned that entire unit incorrectly and will not know about it until much much later...

I have two options here:

1. switch majors and graduate early
2. stay in this program, and continue to take more courses with this prof. (they are the department chair). I had originally planned to take multiple courses with another professor but he and another prof are retiring this year, so I am not even sure what courses I will be able to take as I finish out my degree...

what I would like to know from you all is if this is something that I can make work/ get something out of? I feel like I have been learning some of the content and the material but it seems like the effort is wasted. There have been circumstances in this course where I haven't gotten anywhere after >1 hour of working on homework problems, and it becomes counterproductive to still keep on going. In theory to keep on going seems like the better choice and I would learn much from it, but it will take a huge amount of time which doesnt feel justified by the added understanding.

[pastel sundial]

Wrong channel

This is a channel for teachers/TAs, not a channel for students to ask questions

[pallid night]

wrong channel but you should complain

Also maybe worth learning LaTeX

[mild shuttle]

ah sorry

is there a place to post questions for students not on specific topics in math but rather math learning as a whole

[twin shell]

try #math-discussion

[deep kindle]

How many of you guys grew up learning alternatives to the PEMDAS mnemonic for order of operations?

[zenith slate]

I was taught BIDMAS

UK is most often BODMAS iirc

[deep kindle]

I've never liked ones that have MD and AS because people wrongly misremember as it meaning that multiplication strictly comes before division regardless of operand order

See figure 1.1: every order of operations fight on TikTok ever

[zenith slate]

Yeah I mean I think the whole premise falls apart under inspection

PEDMAS/PEMDAS/whatever are so simplistic as to be borderline incorrect

Nevertheless, students still struggle to remember and understand even this simplified version of things

So I see why it exists

Like eventually you get used to parsing equations etc like a mathematician and can subconsciously construct the abstract syntax tree representing any given mathematical expression

But you can't really teach this to middleschoolers

[cosmic ibex]

I'm pretty sure I was taught syntax trees for arithmetic expressions in middle school, though of course without calling them that. Our textbook had cute little drawings of boxes with input funnels and output slots that represented addition and multiplication machines, and conveyor belts (or was it slides?) to connect the output slot of one machine to the input funnel in the next one.

[pastel sundial]

Any suggestions on what to say on day 1 of TAing?

Like I obviously I'll say my name, pronouns, and office hours. Thoughts on having the class do introductions? (I'll be TAing a smaller recitation) Thoughts on other things to say?

Is it smart to mention that I'm a first time TA? I'm a little concerned it would make students lose respect for me. Oke suggestion I heard was to maybe just say I've never TAed this class before, or that I've never TAed at the university before (which is true but a weaker statement)

[zenith slate]

Yes, I think I know exactly what you mean

"Function machines" are what they are called here

This kind of thing is what I'm talking about

When I said syntax tree earlier, I meant to point out that mathematical expressions are often a lot more tree-like than conveyor-belt-like

Operators have precedence over one another and this can be modified with parentheses

But also one can do things in different orders if one knows what they are doing

For example, (3 + 4) x 2 can equally be computed via
(3 + 4) x 2 = 7 x 2 = 14
(3 + 4) x 2 = 3 x 2 + 4 x 2 = 6 + 8 = 14

As mathematicians we can do this fluently with complex expressions because we understand the, like, recursive nature of how they are constructed and how these syntax trees can be "refactored"

I've always thought it's honestly remarkable that we can do it as well as we can

(Something something chomsky something something recursively enumerable)

[wispy slate]

is this the channel where i can ask for tips for teaching

[zenith slate]

Yep

[wispy slate]

cool

im a ta for cs (not math ig technically but still a ta) and i feel like i struggle at explaining things but idk how to get better at that

do you have any tips?

the problem isnt my understanding but conveying my understanding ig and idk how to explain stuff

[zenith slate]

I feel this is more of an art than a science

[wispy slate]

tbh yeah

[zenith slate]

My general approach is to try and make an educated guess as to the level at which a student understands something

[wispy slate]

im just really bad at art :>

[zenith slate]

And then lower this assumption by a fair amount

[wispy slate]

hm

[zenith slate]

It is better to underestimate than overestimate and it is much easier to fix in the former instance

[wispy slate]

thats true

i generally just ask if they understand a concept before i assume they understand it or overexplain something they already know

[zenith slate]

There is often some hesitation doing this with new educators out of a desire not to patronise people

Obviously this is totally reasonable

[wispy slate]

yeah

i dont wanna assume they are less knowledgeable than they are,,,

[zenith slate]

But in my experience students are quite happy to demonstrate when they already know something

[wispy slate]

true

[zenith slate]

It gives them an opportunity to exceed your initial expectations right from the get-go

[wispy slate]

also the point you made about "its better to correct assuming they dont know than assuming they know" is something i never thought about

thats true :>

[zenith slate]

Yeah

It also gives them a bit of control over things

[wispy slate]

tru

[zenith slate]

Like, they can set the level appropriate for them as you explain something

[wispy slate]

me omw to go "well recursion is just induction due to curry howard isomorphism" in an intro to cs class /j

(i wont do that dont worry)

[zenith slate]

The worst feeling is needing to say "ok actually i'm not understanding this and I need you to simplify it for me"

This makes students feel stupid

[wispy slate]

yeah :(

i feel like its harder to explain some concepts to others for me at least since sometimes they just become so intuitive that idk how to break it down

[zenith slate]

Yeah this is a distinct issue I think and is really something that comes with time

[wispy slate]

like i didnt know how to explain recursion until someone else said "work on base case, then you assume it works and work on the recursive case assuming the function works" and suddenly it clicked how to explain it

[zenith slate]

A good exercise is to sit down and write down the thing you're going to be teaching

Then write down what students have already been taught

And try to justify what you're going to be teaching using only facts from what they have already learned

[wispy slate]

a hm

[zenith slate]

With something like recursion I find it is helpful to write out explicit examples and go through them in what feels like excrutiating detail

Write out each step

[wispy slate]

yeah tracing seems to help people understand what is happening

[zenith slate]

I taught discrete mathematics and introductory programming for math students for about 6 years

Did this all the time

Recursion and iteration are big stumbling blocks for a lot of students regardless of how well you teach it so don't feel too bad if you can't get it first time

[wispy slate]

yeah thats fair

mm

[zenith slate]

It is also important to recognise the unfortunate but inevitable reality that some students will simply never come to understand these concepts

[wispy slate]

what do i do if a student is behind on understanding former concepts so they cant understand future concepts,,

o

[zenith slate]

The only thing you can do here is revisit those former concepts

[wispy slate]

thats fair

the thing is though that since its a lab period with like 20 students and theres only two tas trying to help everyone i cant focus on one person reteaching them what functions are so i can explain recursion

[zenith slate]

It is better for them to have solid fundamentals than to try and grasp concepts that they don't have the ability to access due to lack of said fundamentals

[wispy slate]

thats true

[zenith slate]

Yes this is a very similar environment to when I taught this

Another unfortunate reality of teaching

[wispy slate]

do i just go "id suggest office hours"

[zenith slate]

There is simply not enough time

[wispy slate]

because that feels mean :(

but also i gotta help the other like five people raising their hands at a given time

[zenith slate]

Yeah you do need to be confident giving a student something to think about and moving on

At university students need to take some responsibility for their own learning and your time is limited

I'm in the UK where we don't have office hours so I don't have experience with this in particular

[wispy slate]

yeah thats fair

office hours we just sit in an office and wait for students who have questions to come in

no one comes to any of our office hours :(

oh wait speaking of office hours and lab periods

i gotta go to a lab period

i said id help a ta with their lab period since shes the only ta in that lab period but all others have two and this lab is harder than the previous ones

thanks for the tips ill try implementing them :>

[zenith slate]

No problem 🙂

This channel is fairly active so do drop in whenever

[wispy slate]

thanks :>

[wispy slate]

update

i think i did a lil better at explaining stuff today

:>

[cosmic ibex]

What I (sort of dimly) recall would have looked more like this.

(but with better art)

[lethal leaf]

I haven't seen this in so long

elementary school flashbacks

[wispy slate]

damn

thats like how we explain functions to people whove never coded before

"so basically a function is a funny lil machine in a box that takes in inputs and spits out an output"

"when using a function the details of how the function works doesnt matter, only what its inputs and outputs should be"

[cosmic ibex]

So my point here was that all the machines together with how they're connected is basically an abstract syntax tree for "x·3+2".

[wispy slate]

i mean this is just that but you curried an argument

well not currying exactly but

you know

instead of (x * 3) + 2 its like (+2) ((*3) x)

except ordered in a more intuitive way

also i just realised calculus multivar and linear algebra are under uni instead of pre uni

do people just,,, not learn those in high school,,,,,

[cosmic ibex]

Such questions generally vary between countries and systems. In my personal experience both of these were only taught at university. We had some 2D and 3D vector algebra in high school but no abstract vector spaces.

[wispy slate]

my high school had abstract vector spaces,,,

wha

[twin shell]

that would definitely be unusual in 🇺🇸

[wispy slate]

o shit i still need to do some grading i forgot lol

damn wtf

i went to an american high school

[twin shell]

my HS capped out at AP calc

no one knew what a proof was

[wispy slate]

my ap calc experience started with defining the real numbers and lead into epsilon delta and did funny epsilon delta proofs

also did funny proofs about the reals obviously

[elfin sleet]

In Canada it depends on your specific region but they’re usually re-covered from the beginning in university. Some regions cover a bit of calculus in high school, in mine there’s a bit more calculus but it’s optional so you can also do none; in some regions they do a bit of vectors but no abstract vector spaces, in mine there wasn’t any vectors at all

[wispy slate]

do i remember much? not really, but it was really fun and cool

i should relearn

:>

my high school math courses were more rigorous than the uni courses ive taken so far

:(

to be fair im only a second year in uni rn but still

[zenith slate]

Ohh interesting

Yeah no this is neat

Very neat, actually

Might try this approach out with my Y7/8 students when we come around to algebra

[tawny slate]

i make time for myself to do this because job sort of calls for it but perhaps it's not a viable method for a lot of people who don't have as much time

but i sometimes do research by checking out how other people teach this same concept

and i aggregate any good ideas i come up with from the content that I find, whether it's youtube videos, interactive applets, textbooks, etc

when there's a very particular thing that you're looking for, might be better to just directly ask on a public forum like this one, you get a direct answer

but when you're looking for general direction, just seeing what's out there helps me personally at least

like someone else said, it's more of an art than a skill

you'd be surprised how many comically simple math concepts have comically simple methods of explanation that go completely unnoticed

[long pelican]

What metrics do you use to judge how good an explanation is?

[tawny slate]

i generally use the principle of pleasure

how effortless was it to construct the reasoning

how well do you remember the reasoning

how well can this reasoning be abstracted

what prerequisites are needed to understand this reasoning

is the reasoning aesthetic/beautiful

i think the principle of pleasure is best used to teach a concept, the principle of "pain" is most appropriate when you're practicing with exercises

and then beyond that there's like the meta considerations

for instance, you might want to explain one concept in multiple different ways because the connection between them teaches something

or there is a particular habit that you want to reinforce, whether it's a methodical habit or a psychological pattern

[zenith slate]

I'm not really sure what you mean by "pleasure" or "pain" here

Nor do I see why "pleasure" is appropriate for concepts and "pain" is appropriate for exercises

Indeed I think educators should do all they can to avoid inducing "pain" in learners

Also I generally think that ideas like "beauty" are useless at best and exclusionary at worst in a mathematics classroom environment though this is particular bugbear of mine

[stable verge]

are you speaking for all the "math levels"?

[wispy slate]

me when category theory doesnt cause pain 😳

[tawny slate]

So what I mean by pleasure is that the concept comes together more effortlessly, in a way that minimizes cognitive load and stress, requires less working memory to put together

"pain", on the other side, im more referring to the work that you actually have to put in to practice applying the concepts in problems

In the sense that, if you didn't struggle to apply the concepts you learned, they were just trivial consequences of the lesson, then you didn't learn anything new because the problems were too easy

Or it's a concept that didnt require any practice to apply to anything, in which case I doubt you're doing math

Applying new ideas to problems builds up some of your experience and helps internalize the patterns, concepts, and intuitions, and that takes cognitive load and something you actually need to work at

There might be some savant out there who can just watch 10 YouTube videos and achieve perfect scores on the IMO but I suspect most mortals need to actually practice

"pain" here is not saying you want to leave students so far in the dust that they are totally lost and break into a puddle of tears

[zenith slate]

I see

[long pelican]

The reason I asked my question is because I sense that explanations generally tend to fall into three categories: either an explanation intending to tie a new concept to a real life intuition (e.g. fractions as slices of pies), or a step-by-step explanation of what to do for the current type of problem, or an explanation of why a rule works or why something is true.

The first type of explanation has motivational utility but limited utility in using the concept, the second type of explanation simply has limited utility, and the third type of explanation is the most useful but is generally the hardest to pull off successfully, especially to a large group of students who have different backgrounds in logic and reasoning

I guess I can elaborate with the fractions example I started. From what I've read and my own experience, the concept of fractions are explained using various tools such as pie slices and manipulatives, but then rules for how to add, multiply, and divide fractions are presented as a separate concept, but there is relatively little attention paid to getting students to connect the two explanations. And I think logic is what you need to successfully do that

By the way, it's unfortunate that the term "conceptual understanding" has been hijacked by the education community to mean that you know the concept as in how fractions relate to pie slices and manipulatives, while the knowledge of the rules and why they are true are considered a separate entity

Part of me says (in connection to what CosmoVibe said) that even the best explanations in the world cannot do everything to bridge the gap between intuitive concept and problem solving; you need (well-chosen) problems and practice with the right mindset (especially a logical one, rather than a memorization-based one) to achieve it

[zenith slate]

Teaching fractions with slices of pies is kinda old-school

We tend to prefer box-/grid-like diagrams these days, for precisely the reason you mention

Using grids you can represent fractions graphically in a way that extends readily to arithmetic operations

[elfin sleet]

Interesting, how long ago would you consider old school? I remember being taught with circles when I was an elementary student and I’m only 20. Different county but still. It does make sense more sense to do it with grids now that you point it out

[zenith slate]

Yeah to be fair this is a fairly recent development and it's not universal

The grid method is great because you can represent the addition of fractions entirely graphically and without any arithmetic required

In my experience this helps kids to understand the mechanics of what is going on under the hood, after which they generally prefer to just move on to the much less cumbersome methods of, well, arithmetic with fractions

But it helps weaker students who may struggle to remember how to do something like 5/7 - 1/4 consistently

If they so choose they can just stick with the grid method

[zenith slate]

In general I think you have a tendency to underestimate how important learning how to follow and execute mathematical procedures is to many students who are not mathematically apt

If you're talking about teaching fractions you're talking about teaching, like, 8-11 year olds

I do not see how this "logic-based" approach you suggest is supposed to actually work

Like what do you propose should be taught, specifically, to explain why the procedure of adding fractions is "true"

You suggest that "logic" is required to mesh together the visual intuition of slices of pie with the rules for manipulating fractions but don't say what precisely this logic is

[elfin sleet]

yeah that’s great

now that I think about it I may have been taught with that as well as circles

[long pelican]

So grid is okay for adding fractions. What about multiplying? dividing?

My UK non-math friend has told me he was really good at math in school but he forgot how to divide fractions. In fact he avoids fractions in favor of decimals, at least until he re-taught himself how to divide fractions

Logic is important in dividing fractions: p/q is the rational number r such that rq = p. Simple definition. That's what division means

[long pelican]

"Not mathematically apt" can either mean "low potential for math" or "poor grasp of background material". Which one is it you mean? Do you believe both exist?

[zenith slate]

Yes you can do multiplication also

And, by extension, division

[zenith slate]

This definition is not appropriate for kids who are learning to divide fractions

[long pelican]

Do they get to see any definition of division of fractions?

[zenith slate]

You are introducing three arbitrary variables that satisfy an abstract relationship

[zenith slate]

As for this I was referring primarily to the former and yes, I believe both exist

[zenith slate]

No I do not teach my Y7 students definitions of fractions in algebraic terms because they have not even encountered the idea of a simple variable substitution yet

[long pelican]

Well obviously not teaching a definition of division of fractions is consistent with the traditional way to teach fractions

But we can do better than that

[zenith slate]

How

[long pelican]

Well, what I proposed is a starting point. The variables criticism is one about language and their unfamiliarity with the language of expressing a relationship using variables and quantifiers.

So you can either work around that by teaching them the language or working around it as a temporary transitional measure

Working around it would consist of rewording it in the pre-18th century way, using a wordier language

[zenith slate]

What?

[long pelican]

Example: "for all a, .... P (a)" you would just say "for all numbers, P of that number"

that was how pre-introduction of variables into math prose worked

[zenith slate]

Alright

[long pelican]

The one with variables is a strict advantage of course, but with some learning curve

One point of view I've read about and like is that (a) education is stuck with teaching the pre-1920s way of doing math; 1920 was the decade that marked a math revolution in the mathematical community. The revolution mainly dealt with the issues of set theory and the need for logical precision to deal with that problem. But I'm not talking about introducing set theory, just introducing the logical precision that it sparked. Also, before 1920, very few people were good at math. Nowadays there are way more, but virtually always in spite of school math classes, not because of it.

[zenith slate]

virtually always in spite of school math classes, not because of it
do you have any evidence to suggest this is the case?

[long pelican]

Off the top of my head my experience and what I read very much point to it. That's all I have without digging into stuff

[zenith slate]

Uh-huh

[long pelican]

Do you think it's the other way around?

[zenith slate]

Well do point me toward any peer-reviewed literature you happen across that supports this statement because it is not a conclusion I have seen before

Do I think that kids having mathematics education produces more mathematicians than it would if they did not have mathematics education? Yes

[long pelican]

Ah no I'm not talking about mathematicians. Programmers included, in fact there's a lot of casuals out there who have a strong grasp of math

[zenith slate]

The way you have phrased it suggests that mathematics education is outright deterimental

[long pelican]

Actually when I was in high school and interacted with classmates this is exactly how I felt

Nowadays my view is more nuanced

[zenith slate]

I asked you for evidence of the claim, not anecdote

I don't see much nuance in the statement "in spite of school math classes, not because of it"

The conclusion of this statement is that doing away with school math class would produce better outcomes for students

[long pelican]

No, they got the proper math thinking from outside sources, which was not provided by school

Is what "in spite of" means

This isn't a statement that supports doing away with math education

[zenith slate]

I do not follow your line of reasoning here

If students do not receive any "proper" mathematical thinking capabilities within school (a 'fact' you have still not sourced), then what is your argument for not doing away with math education?

Let me rephrase this last part

Since "reform" is not an adequate answer to the question here

What I am attempting to get out of you is where or not you think math education, as it is presently, is a net negative or a net positive on students' mathematical capabilities

[long pelican]

Ah, I can answer that

So it depends on what the goal is.

In terms of the current goals which is to "prepare the world for the 21st century" and expressly in terms of skills and specific content, well, it is probably at least partially accomplishing those goals. It would be horrible it didn't accomplish that.

In terms of the goal I would like it to have which is to empower or at least expose everyone with powerful thinking tools, one negative is that it causes people, particularly the unsuccessful people, to associate math with calculation and memorization (something my aforementioned friend used to believe). This is a negative because in the absence of math education, such an association would probably not be made. A positive though is the great teachers who do inspire people in math. So hard to say

[long pelican]

The reason I'm not sourcing it right now is because I'm mainly using this channel as a place for discussion, not to demonstrate that anything in particular has been proven, nor to claim anything is a best practice. (Coincidentally, I just received an email for a colloquium whose title is "Against 'Best Practices' in Mathematics Teaching", but I don't know any details.)

[river jewel]

Hi all - my name is Rebecca, and I am an 8th grade math teacher in South Carolina.

I have used MAP NWEA to find my kids knowledge and I am working on a project to use that data to fill gaps before their end of year test.

The issue: that’s over 400 topics and I havnt toughed geometry or statistics. I really need help.

I need to narrow down what my kids actually need to learn, and be successful in future classes…

The mission: find the skills they “need” but don’t really need and eliminate them from my project.

• telling time on an analog clock
• using a protractor to measure angles.

I need math educators that know high level standards to help me sort through this. I only have ever taught 7-alg 1 and I need these kids to grow.

[long pelican]

I'm guessing you mean 400 skills, listed in fine-grained format like

Write equations for proportional relationships from tables (8-X.2)
Write equations for proportional relationships from graphs (8-X.5)
Find the slope from a graph (8-Z.1)
Find the slope from two points (8-Z.2)
Find the slope from a table (8-Z.3)
Graph a line using slope (8-Z.5)
Slope-intercept form: find the slope and y-intercept (8-AA.3)
Write a linear equation from a slope and y-intercept (8-AA.6)
Write a linear equation from a graph (8-AA.7)
Write a linear equation from a slope and a point (8-AA.8)
Write a linear equation from two points (8-AA.9)
Rate of change of a linear function: graphs (8-CC.5)
Interpret the slope and y-intercept of a linear function (8-CC.6)
Write a linear function from a table (8-CC.7)
Write linear functions: word problems (8-CC.10)
You can condense these to a single category, to reduce the overwhelmingness of the number of things involved. Listing skills like this seems to satisfy some psychometricians' dreams but are (in my opinion) the wrong direction for what math standards should look like.

[river jewel]

So not standards - but skills like, determine unknown divisors, determine unknown factors

But some are “within” 100 and some are “greater than 100” so I could combine those for sure

[long pelican]

Yes, but even different skills for the same topic can be unified. Right now skills tend to become problem types, exam questions become in one-to-one correspondence with skills, and students learn that studying for math is about practicing identifying which problem type a problem is and executing the correct procedure on it -- a very common belief among high school graduates.

[river jewel]

Hmm. Right. They need to understand how angles work, not well if it’s interior vs exterior vs parallel lines vs xyz

[wispy slate]

what is the root of 3

[twin shell]

i dunno

[pallid night]

Idk ... is there a root of 3 ?
What does it mean to be a root of 3 ?

[wispy slate]

I meant what is the roof of a tree

Root*

[timid topaz]

what is root(tree(3))?

https://tenor.com/view/very-big-big-bri-bri-brian-may-queen-gif-16622798

[wispy slate]

$\sqrt[3]{3}$

[burnt vessel]

aurora

[twin shell]

<@&268886789983436800>

[wispy slate]

wait what did i do

[winged oar]

I would assume something else happened

[winged urchin]

And no one will ever know

@.@

[tardy ember]

i just remembered this, and,
yes you can and it's just the normal definition of a vector space over a field because *R is a field

[deep kindle]

pedagogy is my favorite help channel

[deep kindle]

If your state does have a solid list of standards then that might be a good place to start. Standards-based curriculum is where a lot of the schools I work with are moving towards (or at least trying to LOL)

[river jewel]

Yea, that would be awesome. I do my standards pretty well and have been working with the coherence map to try to find previously taught skills as well. The biggest issue is district wants this textbook and it’s just too advanced for them but at the same time if I baby them they’ll never grow. It’s just hard

[wide ice]

... can you? what would be the neutral element here tho?

[tardy ember]

...uh
zero? the same as it normally is?

or am i misunderstanding what you mean by "neutral element"

[austere delta]

R* is the hyperreals, not the nonzero reals if that's your confusion

[wide ice]

Y'all should come up with a new notation, lol

[austere delta]

$\mathbb R^\times$ is nonzero reals

There you go

[burnt vessel]

jagr2808

[tardy ember]

or use *R for the hyperreals like i did

[wispy slate]

wait

whats the difference between hyperreal numbers and surreal numbers

am i stupid or are they the same thing but with different wiki pages??

[cosmic ibex]

Both of them are ordered fields that strictly extend the reals (and contain elements that can be interpreted as "infinitely small or large"), but beyond that they're quite different constructions.

"Hyperreals" are not even a single thing -- IIRC there are several different constructions that give something with the right properties to be used for nonstandard analysis but which are not automatically isomorphic.

"Surreals" are one particular thing, but that thing won't qualify as hyperreals. For one thing, the surreals form a proper class. And I don't think they satisfy a transfer principle.

[wispy slate]

huh

neat

[deep kindle]

So every degree of polynomial equation has a standard form

ax + b, ax^2 + bx + c, ax^3 + bx^2 + cx + d, so on

And every degree has this other form
point slope: y = a(x-h) + k
vertex: y = a(x-h)^2 + k
inflection: y = a(x-h)^3 + k

Is there a name for the latter form of equation?

They all indicate a reference point, but are also named after their reference points so they’re all named differently. Point-slope has a point of reference, vertex has a point of reference always being the vertex

Cubics have a point of reference being the inflection point

But I’ve never heard this called reference form or point of reference form

[pallid night]

Not all cubics have that form

A general cubic does not

[deep kindle]

I guess what I’m asking is if there’s a more general name that point-slope and vertex fall under

[elfin sleet]

I'm not sure there is a more general form to name - like I'm not convinced that point-slope and vertex are two instances of the same general thing. For one thing, point-slope doesn't seem super distinct from the standard form, since the standard form is point-slope with the point being the y intercept. The vertex form of a quadratic, on the other hand, displays a specific predetermined point that is an inherent property of the function, as opposed to an arbitrary one that can be adjusted to recover the standard form. Also, as you just discovered with the cubic example, it's not obvious how this reference form would be generalized to polynomials of other degrees

[pastel horizon]

I guess you would say an nth degree polynomial has a maximum of n-1 turning points (proof: turning points are calculated by finding the roots of the first derivative)

Higher order polynomials could theoretically have multiple inflection points

This isn't the best place to ask those kinds of questions btw this is more to do with the teaching and learning of mathematics

[elfin sleet]

I was assuming they were asking it in the context of presenting these forms to students but if not then yeah

[cosmic ibex]

Max n-2 turning points, I think you mean.

[elfin sleet]

no because quadratics have degree 2 and have 1 turning point

[cosmic ibex]

Hmm, what exactly do you take "turning point" to mean here? For me, e.g. y=x² keeps turning to the left as y increases, there's never any plance where it starts turning to the right instead.

[elfin sleet]

I've heard the term turning point as referring to a point where the first derivative changes sign

ie the graph of the function "turns around"

sounds like you're thinking about the second derivative which would also be a reasonable interpretation of the meaning but I think Gaunter was talking about the one I've heard

[deep kindle]

I was asking in the context of presenting this to students

I know exactly what this channel is for

[deep kindle]

What cubics can’t be represented that way?

[cosmic ibex]

x³-x, for example.

[pastel horizon]

I guess I should be more clear, like global / local optima

You could also interpret them as changes of signs within the gradient function

[deep kindle]

Could be represented with y = a(x-h)^3 + bx + k

Where (h, k) is still the point of reference and represent transformations

bx would represent something funny going on around the inflection. Like a vertical stretch but in the opposite direction of the sign of each x component

But not really the point of my question; point-slope and vertex still fit under a similar umbrella and I can’t figure out if there’s a name for it

[pastel horizon]

To get something in that form you would be doing the inverse of a binomial expansion I guess

With the linear case being (ax+b)^1

[long pelican]

This is almost the form of the depressed cubic which is the first step to solving cubic equations in general 😁

[pastel horizon]

For cubics I have heard of the term "completing the cube" which takes on a different form

Than the one shown

[long pelican]

On topic, the general phenomenon that's happening here is that if you are able to express an equation $y=g(x)$ of the form $y-k=f(x-h)$ then this expresses that the graphs of $f$ and $g$ are translations of each other in the plane. Is there a name in use in math education for this? I don't know.

[burnt vessel]

Icy0

[long pelican]

I didn't mention the $a$ factor but that corresponds to dilations. And you see cubics don't have this form because cubics are the first class of polynomials whose graphs aren't all similar to each other

[burnt vessel]

Icy0

[long pelican]

Maybe a name for this is "Encoding precompositions by certain Euclidean transformations of the plane into equations" (edit: I had GL(2) but actions by GL(2) preserve the origin)

[elfin sleet]

yeah ok if that's your cubic form then you could generalize that to all polynomials. You're taking advantage of the fact that all cubics have exactly one inflection point, just as all quadratics have exactly one max or min, which are instances of the fact that a polynomial of degree n has one point where the (n-1)th derivative changes sign (since that derivative is linear). I'm not sure how useful it is in general to express polynomials in a way that highlights this point but it's a thing you can do

[cosmic ibex]

If there's a general phenomenon at play here, it would be that a polynomial can always be expressed in "depressed form" as $$p(x) = a_n(x-h)^n + a_{n-2}(x-h)^{n-2} + a_{n-3}(x-h)^{n-3} + \cdots + a_1(x-h) + a_0$$
where there is no $(n-1)$th power. For linear functions, this is effectively point-slope centered on the x-intercept. For quadratics, getting rid of the 1st power gives us vertex form. For cubics it's only slightly nicer than the general cubic, and the relative niceness diminishes as the degree increases.

[burnt vessel]

Troposphere

[cosmic ibex]

(Unless we're working over a field whose characteristic divides the degree, bla bla bla).

[long pelican]

Quadratics over F_2 are fun

[elfin sleet]

yeah that's what I was getting at

[deep kindle]

So in the end, there’s not really an umbrella name for the similar form that point-slope and vertex assume?

Besides “depressed form” I guess lol

[elfin sleet]

Well I still don't think point-slope is a "similar form" to vertex form. If you just take the instance of point-slope where the point is the x-intercept then it is. Or you could extend the vertex form to take any point like this: $f(x) = a(x - x_0)^2 + m(x - x_0) + y_0$ where $(x_0, y_0)$ is a point the parabola passes through, $m$ is the slope at that point, and $a$ is the same $a$ used in the standard and vertex forms. If $(x_0, y_0)$ is the vertex then you recover the vertex form. It's kind of neat that you can do this so nicely - by choosing familiar variable names like I did here, you could fairly easily present this to students and explain what each of the parameters do (to be be clear, I'm not suggesting you do this - there isn't much reason to - but it's neat that you could)

[burnt vessel]

pigeon

[tawny slate]

function transformations

https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:transformations

personally, i think math educators need to very comfortable with the pedagogy surrounding function transformations

i think a big jump in difficulty from algebra 1/2 to precalc is being able to start thinking about functions rather than expressions and numbers

functions are the key to understanding trig and calculus

function transformations are one of those things that is used pretty ubiquitously but often taken for granted and is present in many of the foundational principles

[vagrant meadow]

any suggestions for how to get students to actually talk or work together for group work? usually they just sit silently and work on their own in relatively close proximity to other students.

[pastel horizon]

What do they need to work together for

[vagrant meadow]

obviously they don't need to. but it would be good if they did

[pastel horizon]

Maybe you need to structure an activity that necessitates some form of communication

Also another thing you should avoid is having a group task that effectively consists of four individual tasks. Have some sort of overlap e.g. a solution to one problem is needed in another problem

I don't know for certain but I would guess they aren't talking either because they're too shy to look stupid or they don't see a point as they can do the tasks independently

[vagrant meadow]

I get a list of textbook problems from the primary and she wants me to give them to the students to do in groups. I suppose I could modify the questions in some way but atm it's just like basic row reduction/elementary row operations so I'm not sure how I could do that

[turbid zenith]

https://twitter.com/solidangles/status/1709723409188303097

Bill Shillito - now at @solidangles.bsky.social (@solidangles)

Trying out ChatGPT Vision for math.

This has the potential to be a HUGE game changer for students.

Twitter · Oct 5, 2023, 12:13 AM

[pastel horizon]

Yeah that's too bad. As I said it's probably falling under the too easy to warrant group work category in that case

[tawny slate]

im more concerned about how to handle edge cases. in the rare instance that gpt gets something wrong, how do we correct it or what do we do about it?

i think it could be a good teaching reference, but i hope students aren't using it like they use calculators, relying on them for fact-checking

[turbid zenith]

That's why I said "potential"

It still needs some work but I can see it being quite useful once the kinks are worked out

[deep kindle]

Thoughts on using angry birds to show how parabolas model projectile motion in frictionless environments?

[viral parcel]

sounds fun, don't see why not

[turbid zenith]

It's neat

But it also reminds me of a talk that's worth watching

https://vimeo.com/113714091

Dan Meyer has things to say about how everyone's first thoughts with Angry Birds were "Oh my god we can use this to teach parabolas"

[deep kindle]

Definitely will watch that in a little bit

Kids are coming back from lunch in a couple minutes lol

This is for students who have been learning vertex and standard form but have thus far mostly done abstract problems

Like figuring out vertex form from a graph with no other context, just a regular graph

This could also be used in precalc or whatnot where you might introduce vectors

[turbid zenith]

My go-to is actually profit

[deep kindle]

Which courses do you teach again?

I feel like I’ve watched some of your videos but coulda been someone else

[pastel horizon]

Profit is nice

What about using the "dumb" despicable me villain? It does actually give an accurate description

I also like to use golf as well for displacement vs distance.

You might hit a golf ball a total distance of 300 yards but sadly it can roll backwards and you only have a displacement of 200 yards from the tee

[deep kindle]

elaborate on the despicable me one

Idr which villain was dumb

[pastel horizon]

https://youtu.be/A05n32Bl0aY?feature=shared

It's not really dumb but it's cheesy af

Which I guess is the point of the character but it does help explain it

[analog trail]

when I was in school this clip was everywhere during year 11/y12 physics and advanced maths

people loved it, and they were also very nostalgiac about it

[turbid zenith]

Oops sorry for lack of response

Right now I'm teaching Calculus III, Elementary Statistics, and liberal arts math

[deep kindle]

What’s in liberal arts math?

[wispy slate]

Hi

I've got an idea for a publication, I think but I wonder if this can be done as an individual, and whether pedagogical proofs or solutions to general problems could be submitted to a journal

Basically it's just approximating the linear problem Ax=b in different ways, including some less common problems like non negative least squares. Would this be possible? I'd appreciate some comment

(I've also implemented existing computing algorithms from papers into computer code so I thought this could be an addition.)

So like, several sections, including the calculus method to solve it, the linear algebra method, the equivalence between those methods, extension to quadratic programming, and code implementations, all not as a new thing, but as a possible tool for teaching

[stray grail]

sounds like a blog post?

[winged oar]

Not to a top tier journal. A blog post sounds fitting.

[wispy slate]

I see, thanks

I didn't mention it, but I was thinking of an education or pedagogy journal, just in case @stray grail @winged oar

This is because it'd be quite long, there aren't any good blog posts about it, and you only get this info from combining some good introductory courses, for undergraduate level math, especially of those from engineering and sciences. (Plus I don't have a blog.)

[stray grail]

i have no experience with that kind of journals. might be a possibility

[wispy slate]

Yeah that's why I asked in this tab @stray grail

I would do it just for fun, and for doing something slightly more rigourous as well. Thanks though.

[stray grail]

put it on github?

[wispy slate]

I don't see the benefit, it's the same I do all the time, and it's in there already the code part

But that's not very educational, especially as ideas for teachers

But I guess what you mean is that it doesn't fit what your idea is for a journal, which may be true in general for other journals than the ones you're used to. I may just do a couple of flyers and distribute it...

[vagrant meadow]

it's not that hard to make a blog tbh. if i can do it, being the technologically challenged boomer-at-heart that i am, then anyone can. you can make a free blog using github pages and jekyll. just find a template and github gives you free hosting.

^a template that supports latex. i was able to find one very quickly when i started mine.

you don't need to know any javascript or html either. most of the templates are very easy to customize. just sort of a plug and play operation.

(uhhh this is not a plug or advertisement ||please don't look at my blog||)

[wispy slate]

i am actually a typescript developer, and don't have any inconvenient creating a blog, but that is not neither in my interest nor the target @vagrant meadow

by the way, github gists support all you talk about without having to bother with anything else.

[vagrant meadow]

not sure what you mean by "not the target". i don't see the difference between making flyers and sharing a link to a blog post besides a link being far easier to spread around and making it more accessible to others. is your target audience teachers who don't use the internet?
also not sure what you mean by "not in your interest", i don't see how it could be bad for you. unless you mean you're literally not interested. in which case, i would argue it seems like you are interested in sharing something that you've found and could be helpful to others, and a blog would be a perfect medium in which to do that. what you described sounds exactly like a blog post that one might share in this channel. i can't imagine it would be of particular interest to publishers. but if you can find some journal that would be interested, that's great. you just have to hope the content isn't then behind a paywall, which would make it far more inaccessible.

[wispy slate]

Thanks

[lethal leaf]

One day I will set up my blog

[umbral sable]

I have an interview at a local community college tomorrow to become an adjunct professor. I graduated May 2023 with my M.A. in math, so this will be my first classroom teaching position. The interview involves a normal question/answer interview portion and a mock lecture portion. I feel pretty prepared for the mock lecture but less prepared for the QnA. Is there anything I should know or prepare for? What materials should I print and bring with me? etc. The more info the better, thank you. I very much want to ace the interview and start building my teaching resume

[tame hemlock]

Does anyone have any experience teaching about video game microtransactions and their actual costs? In the K-12 range. I'm wonder if it's ever taught to students, brought up in the classroom, or if there are websites that kids may know about that help them understand the conversions that the developers put in place - to make it hard to equate to real money.

[pastel horizon]

The QnA will probably be your standard teacher interview questions.

What's in a good lecture, how can you ensure progress is made, can you add any extra value?

[pastel horizon]

What do you mean

[pastel horizon]

Oh they may ask what material you can deliver as well try and cover a broad range as possible

[umbral sable]

As in, what types of courses I can teach?

[pastel horizon]

Yeah basically

Idk about the US but there's a shortage of lecturers within FE/HE so they usually try to hire people with a wide range in their tool belt since they can use them for different units

Being able to teach applied math as well should pretty much guarantee a spot

[vagrant meadow]

so my TA section is only 50 minutes long, which is a lot shorter than i thought it'd be. so far i've been doing a few problems on the board in front of everyone, and then hoping to get them into groups to work on the others. but last time, i spent the whole time doing problems in front of everyone. so i'm wondering if it would be better to just have them start working in groups right off the bat and not do any problems for that day's worksheet in front of them.
i'm planning to try that tomorrow, but i'm curious if anyone has any suggestions on at what point should i stop the group work to do something on the board? so far i've stopped group work when i've heard a common mistake or misconception, but perhaps i should instead just address the group i hear that from.

this is my first time as a TA and it's been kind of an overwhelming experience since i feel like i don't know how to effectively structure things. which is weird, because it's not like i haven't been taking classes for the last 20 years of my life (though, i have never attended a TA discussion myself). but still like how did i not pick up on effective ways to conduct group work? but i'm more worried because i don't want my lack of experience||/general incompetence|| to negatively impact the performance of my students.

[strong hearth]

how many students are there? I always consider collaborative to be superior than top-down

[vagrant meadow]

~30 approximately

[tacit adder]

We had a flipped classroom style tutorial for my linear algebra and combinatorics classes and I found them to be incredibly helpful for the student's learning process. However, a flipped classroom setting tends to work better when the class sizes are small and students are of a similar background. So I'm not sure how effective it would be with a class size of 30 (please let me know if you plan to do it, because the algorithms classes I teach have 30 students roughly and I've been meaning to try that kind of model!).

If you have a tutorial worksheet each week, then probably scan through the worksheet briefly before class and pick out a few problems that would be of interest to students, if students don't have any questions of their own. At least, you'll have a general schedule of problems that you can work through when the class is silent

the algorithms classes I teach are also 50 minutes long so I also tend to go overtime 😦

[vagrant meadow]

i'm only the TA so i don't think i have the authority to do that haha. i just get problems from the instructor to put on a worksheet to give them.

[winged oar]

Don't worry, I don't think a lack of experience will be the biggest problem in being a TA.
For a size 30 class most you can do is do problems and move on. You can ask them to email you specific questions - for these I don't get a lot of emails, even in classes as large as 100, though that might not be your experience. You should also set reasonable boundaries in the questions (i.e. I usually would not answer textbook questions on a first email)

Anyway the best you want to do is somehow keep student feedback in mind while developing your own teaching style. There's a lot of different ways to do it

[lethal leaf]

Am undergrad TA in a flipped classroom with hundreds of students (have a large large amount of course staff ready to answer questions)

It doesn’t work at scale at all

Also many students not motivated

[vagrant meadow]

does flipped classroom ever work? i've only ever heard stories where it's a miserable failure, and it's mostly because of this

[lethal leaf]

I think it can if all the students are motivated to actually watch the lecture videos outside of class

And actually take the time to work on questions and ask questions in class

But without motivation to do so, flipped classroom is worse

Hell even my office hours are empty

I even was running review sessions last semester and now this semester they were so empty I cancelled them

As in in a course of 600 students only 3-4 came to the review sessions even though they went over the material needed for the biweekly quizzes

In general there's 0 effort from these students to try to get any form of help

[strong hearth]

I wonder if it’s a generational thing, I notice this too and been getting worse by the year

[lethal leaf]

It was fine last semester

This semester it's gone off the cliff

[long pelican]

3-4 seeking help in a class of 600 students is really really rough

[strong hearth]

Demotivating for the instructors too

[long pelican]

2 years ago when I taught calculus to non-math majors my office hours were similarly empty. 1 year ago when I taught calculus (again to non-math majors) I had packed office hours and the TA had even more packed office hours. The difference being that I used non-standard materials

There was a lot of enthusiasm in the packed office hours too

[twin shell]

might be hard to conclude that the materials made a difference (although I’m inclined to believe so) considering that by that point we’d only barely recovered from covid

[long pelican]

One thing that crossed my mind is that standard materials (Stewart) gives students standard ways to get help, such as tutors, Professor Leonard, and solutions to textbook problems

maybe also past exams

but the last one depends if the professor bases exams on Stewart exercises

If an unmotivated student knows about this stuff, there's zero reason to use office hours

After all, that is more than enough to pass the class with nothing more than a half-assed understanding of the mathematics

[lethal leaf]

Yea and like

Last semester I started doing the review sessions since students were asking for them

So from literally asking for them to not even showing up

I was making review sheets and answer keys and everything 💀

[long pelican]

Come to think of it, what is the difference in your case?

between the two semesters

[quasi musk]

I had this for my Grad Real Analysis in my MS institution, and when I took 3-manifold topology. Most of the students were extremely motivated, and tended to be less than 7 people

In total in the class. 3-manifold topology had 3 students, including myself

[lethal leaf]

The summer happened

jk uh

For some reason the prof made the discussion sections optional

So no more attendance grade

So even less motivation

But still flipped 💀

[vagrant meadow]

I can see flipped classroom working better for grad courses, especially since the number of students is so small. it would probably be my nightmare to be in one though lol

[lethal leaf]

But for grad courses it's so small

I like being able to just ask questions in lecture as they come up

I think it works better in classes with more of an emphasis on doing alot of problems

Calc, discrete math (I'm an undergrad TA for this), numerical lin alg

[tacit adder]

Yeah in my experience, flipped classroom models work best when you have a class of motivated students which is great for grad classes

[long pelican]

With motivated students, lots of things work. With unmotivated students, nothing works.

~~ still an active area of research, in other words ~~

[shadow flower]

we’re using flipped class in a group theory class at MS program and it’s not going too bad but most of us prefer a standard lecture. it’s like read the book and give feedback each week via a survey and then the professor comes in and answers what we said we didn’t understand via the survey, on MW. F is reserved for an in class worksheet

it’s not like awful, but i’m unsure if we feel this way bc we’re fairly comfortable with group theory already. so i guess it’s just not bad enough to the point we’ve asked him to change it to a more standard convention

[cosmic ibex]

Cute problem here #help-37 message: Come up with a plausible reasoning for an answer you know is wrong. Possibly designed to allow the student to shake off the "I must never even consider anything that could be wrong" paralysis, though it didn't seem to work so well for the helpee here.

[pastel horizon]

On a similar idea, I think this is a brilliant question

This time it's the paralysis that there are more solutions than the "obvious" solution

I managed to get 10×10-1×4 while attempting it myself

[cosmic ibex]

I hope students are issued with loupes, so they can read that.

[pastel horizon]

It's possible you'll stumble on a different proof for Lukas while solving this even

It's a cool exercise I think by Don Steward though

Even if by the end of it they still don't see beyond the "split into two rectangles" method

[cosmic ibex]

I can see how Lukas might have cut up the shape, but I'm not sure I fully grok why he did it that way.

[twin shell]

I have no idea what lukas did

[cosmic ibex]

He cut off a 2×7 cm strip and used it to fill up as much of the gap to the right as it was enough for.

[long pelican]

Minimal number of cuts needed to reassemble the pieces into a 10x10 square missing a 2x2 square = ?

[twin shell]

sounds dangerous

[long pelican]

This vaguely reminds me of algorithms problems

I think the exact type of question I asked is likely on codeforces somewhere

If not project euler itself

They will probably restrict the cuts to horizontal and vertical though (and I don’t)

[cosmic ibex]

... leading naturally to a discussion of Hilbert's 3rd problem. :-p

[long pelican]

Who will be the next victim to learn about tensor products and their applications?

[pastel horizon]

That's not the purpose of the exercise though

I think it's more to improve their geometric reasoning by throwing curve ball approaches

There must be an infinite number of ways you can rearrange the shapes to get the same answer with a different sum even

[twin shell]

<@&268886789983436800>

[tawny slate]

The literal question "are you convinced by the reasoning" I would personally have different responses for depending on context I think

In an absolute sense yes, because he got the correct answer and the means by which he got it could be construed as a proper method

But as an educator I would probably explain that while it could be a correct solution, it's not a great solution because it doesn't actually give any reasoning, it leaves out a lot of detail

We should pay more attention to our presentation, our communication, our design, when presenting reasoning and arguments to be persuasive and minimize cognitive load

As a layman, colloquially I would say I'm not convinced by the reasoning

Too much calculation I'm expected to do

[lofty garden]

Put this in my math lesson today. Kids thought it was funny.

[long pelican]

How many of them thought it was obvious from intuition and didn’t think a rigorous proof is needed?

[near oriole]

this seems... non-trivial...

only approach I have is the sine rule, and it doesnt fall out immediately

[twin shell]

the longer side is opposite the larger angle

[near oriole]

yes, proving this fact.

[twin shell]

wonder what class this is for

[near oriole]

does anyone follow this https://math.stackexchange.com/a/3776783/243059

nvm yh, think i got it. non-trivial