math-pedagogy

[zealous kettle]

Maybe the more accurate categorization is "Wants to learn", "Will learn if they think it's their best option", "Want to not learn"

[halcyon glade]

What went wrong?

[turbid zenith]

Of the 9 students that showed up to class (out of 21), none could explain why 0.5 = 1/2

i.e. it’s 5/10

They were so just-accept-it-pilled that I had to really push them to even question 0.999… equaling 1

[halcyon glade]

Did they understand the question? I feel like if you asked me on a normal day to explain why 0.5 = 1/2 I wouldn't really be sure what there was to explain

[turbid zenith]

Or their reason they accepted it was because “just round it”

I asked “where does the 5 come from”

“What does 5 have to do with one half”

It’s not like 0.5 is a magical incantation that makes the number 1/2 materialize… it should MEAN something

[halcyon glade]

How much time did you give them to think about it

[turbid zenith]

Enough for the silence to be awkward

There are a lot of awkward silences in that section that don’t happen in the one before it where everyone is excited and engaged etc

Where they’re all looking around seeing who’s going to be the one to answer

I do as much “turn to your neighbor” as I can but I can only milk that for so long before we need to do something as a class

[marsh thistle]

I like the question but I would definitely make sure to give two clearly distinct definitions of the LHS and RHS of that equation before asking it
Otherwise it feels like asking students to prove 1+1=2, from which they're probably going to respond "from what?"

[turbid zenith]

And I try to design the lessons to have as much of them doing or discussing something as possible

[turbid zenith]

Makes sense yeah… this all came from wanting to take two “everybody knows” statements of 1/3 = 0.333… and 2/3 = 0.666…

But those were getting blanks

So I was trying to reactivate what should have been prior knowledge and coming up dry

[marsh thistle]

Yeah makes sense that's a good warmup

[turbid zenith]

The intent was to frame all this in terms of Aristotle’s potential infinity vs actual infinity, point out that there’s nothing stopping you from asking “what if” questions about infinitesimals, ask further “what if” questions about decimals going to the left… like play with it and see what happens

Keeping it in the realm of familiar computations from back in elementary and middle school but exploring their consequences

But oh well. Test is next week, then we get to do taxicab geometry. That’ll be a good reset. I just needed to vent and commiserate I guess.

[tawny slate]

if life gives you rocks, you're gonna struggle to make lemonade

you do great work, so try not to let this weigh you down too much. you did your best, and that's pretty close to the best you can do

happens

[tall bolt]

Yeah I have to agree with eric on this im not sure id really know how to answer this question

[clever girder]

tbh i wouldn't know what to say for this. "they're two representations of the same value"?

[turbid zenith]

I wonder how to ask it then

Like the answer I wanted was “because it’s 5/10”

[quasi maple]

tbh, the other day I did have a student struggle to determine ½ - ¼, while another student suggested converting them into decimals to "make it easier"

[finite ravine]

I’ve always wondered about the 0.99999 one and I know they probably taught us that at least in real analysis but still… seemed odd

[turbid zenith]

Decimals do seem easier for some students just because place value I guess

[austere delta]

Maybe it would be easier to ask about 0.1. Once they realize that's 1/10 it might become more clear that 0.5 is 5/10

[torn apex]

I'm surprised they didn't enjoy this; could it be that your misreading they're confusion at "what is the right answer supposed to be?" for "disinterest"?

My personal experience/anecdata is quite the opposite. I just have to frame it out the outset that "there is no right answer" and then guide them through a historically motivated first-principles re-discovery process.

My go to in this line of questions is more along the lines of "Hey, what do you guys think life was like before humans discovered the number zero?"

The ultimate goal is to hopefully induce an epipheny in them that:

1. at some point, someone had to invent this really complicated thing
2. after enough time, that groundbreaking thing can be understood by 5 year olds
3. ergo, you too have the potential to invent something!

I find I can only drive the last point home only if they themselves end up "rediscovering" some foundational concept

[turbid zenith]

That’s sort of what this whole course is about actually!

[quasi musk]

I had a conversation with a student, trying to explain what a limit was in pre-calc this past week

After giving multiple examples he said "None of that is realistic -- what can discontinuities model"

And my answer was an off/on switch, but this, he said was not instantaneous. It seemed at every stage of the conversation, the student was just insisting on refuting any possible explanation

Instead of trying to learn...I felt like I wasted 10 minutes; I feel this way because there was a group of students I didn't get to talk to as we ran out of time

[halcyon glade]

Isn't he right though, nothing irl is discontinuous

[tall bolt]

Isn’t that sort of a matter of physics and philosophy

But it’s also a moot point because maths isn’t physical, or rather it doesn’t need to be

[halcyon glade]

Yeah maybe I could rephrase to nothing irl requires discontinuous jumps to be explained

It's just a matter of mathematical simplicity/convenience

[quasi musk]

I guess the point I was trying to make is that the person wasn't wanting to learn how to do the math to pass the class

But just challenge any possible reasonable explanation, and argue over it

Even saying things like "The function value at x = a"

"Why do you keep saying that? What does that mean?"

[austere delta]

I'd argue nothing irl is continuous.

[tall bolt]

Zeno2808

[turbid zenith]

Hey everyone, meet Connie. She's gonna help me teach my demo lesson on differential equations for my tenure track interview Tuesday. She likes bungee jumping. :3

(The lesson is on higher order linear ordinary differential equations.)

[quasi musk]

Yes this is the issue; the refusal to work with the math on its own terms, because I don't understand how to apply it. Then, when I give potential applications, they all get rejected because nothing is instantaneous or truly continuous/discontinuous

Moreover, when giving multiple examples on how to use it, rejecting the math terminology of "f evaluated at x"

It felt like I was talking to a wall, with no progress being made

[turbid zenith]

...tbh this is why I personally don't think it makes sense to talk about limits before you've done derivatives and integrals

At least one reason

There isn't an intellectual need for it otherwise

That's really annoying that the student was fighting you like that though ... what do you think their motivation even was?

[quasi musk]

I think they were just frustrated with not understanding the math. This is a freshman remedial course, for students that scored a C-, C, or C+ in the first quarter pre-calc sequence

We have 3 quarters of pre-calc/intro-calc before they go to calc. This is a lab discussion to support them through active discussions at the whiteboard with their TA

Believe me, I have my own ontological misgivings of the real numbers, validity of limits from a logical point of view, and these infinite sequence stuff

[vestal tangle]

maybe explain to him what the word model means

[lethal hornet]

yea, was thinking this

how else are you going to approximate discontinuous stuff

[vestal tangle]

just about every model involves some idealizations/approximations. maybe the flipping of a switch isn't truly discontinuous, but we can model it that way for simplicity

[turbid zenith]

Yeah that sounds about right ... fighting for fighting's sake

[tall bolt]

Even this I’m a bit iffy about. Like true, every model makes simplifying assumptions and we’d pretty much always assume this to just be actually discontinuous, however I still don’t think that’s fundamentally the issue. The argument that “well this isn’t physical so I’m just not going to engage” is the issue. We’re doing maths, physicality has no bearing on what we’re doing, it can be nice for intuition but it’s a guiding principle only. We talk about points and what not all the time that are infinitely thin, it’s just a construction

[tall bolt]

I am interested to hear about your thoughts on limits though

[dim blade]

I would ask them "if a transition happens in a nanosecond, but we are measuring the behavior of the system over minutes or seconds or milliseconds, does that nanosecond matter?"

[dim blade]

There is some pretty solid evidence that electrons moving between energy levels happens discontinuosly

[midnight scarab]

I don't think discontinuity is unphysical. For a visual example, something like shockwaves has a discontinuity (in velocity), at least at a macroscopic scale. Less visual but more convincing, the dependence on a coupling can be discontinuous at zero (where one switches from stable for positive coupling to unstable to negative coupling). For example, this is why the perturbative expansion in qft cannot be convergent (Dyson's argument)

And also just phase transitions

Though the discontinuity only exists in infinite volume

So yes, in a sense, the discontinuity only arises because you've taken a limit, an approximation

But it's a very useful one, and is physically very meaningful

[tall bolt]

Oh of course, to be clear I think the whole argument is nonsense but I guess they’re arguing for physicality from an engineering/modelling perspective. I think if we told this person how useful the Dirac delta is they’d have a stroke

[midnight scarab]

I mean even in terms of modelling or engineering, you often don't care about the fine resolution of the discontinuity. E.g. when designing a billiard, you just want the ball to bounce back on the edges following the laws of reflection, you don't care about the finer details of how the angle is actually a curve (and the edge of the billiard slightly deforms during the shock)

To quote Kapitsa's intro lecture to 1st year students, if a ship engineer computes stuff to twelve decimal places, you fire him because that's just a massive waste of time (well, now we have calculators but you get the idea), and dumb because he's not taking into account a bunch of effects that would matter at this level of precision

[tight star]

Absolutely

[torn apex]

One "trick" i've found really useful is to redirect the line of questioning:

"Why do you keep saying that? What does that mean?"

to

Why do you think other people defined it this way? What should the foundations of math be? Etc?

I found reorienting their frustration/anger is much easier than trying to abate it. And usually a "hmm, I haven't thought about that. what do you think?" puts them enough of a "teacher" role that it'll perturb theri behavior pattern.

ymmv

[vestal tangle]

what i got from it was that the student didn't think that the flipping of the switch (or anything really) could be modeled by a discontinuous function because it's not truly instantaneous. so that suggests to me that he doesn't understand what a model is. maybe if he knew that models basically always involve simplifying assumptions he wouldn't have objected to it. it didn't sound like he had a problem with the concept of discontinuity per se, just that he thought it was useless

but he definitely could have been fighting just for the sake of fighting

[shadow swift]

Is there really a market out there for PhD Qual prep?

[quasi musk]

No, not really. Usually the institutions handle it

[tawny slate]

the weakness of this tactic is that some students are already disinterested in math because "when is this ever going to be useful?"

if you then tell them "don't worry about it being useful, it has nothing to do with physical reality" it will only corroborate those initial criticisms, not assuage them

obv the real issue is much more nuanced but the point is that many of them will shut down by this point, need to directly address what they take issue with

and tbh, does anyone really know whether reality is continuous or not? i literally don't think it's possible to know for sure. there's a limit to what we can measure but it doesn't mean there's nothing beyond that

my reaction to the student claiming reality is discontinuous would be a simple "prove it"

[tight star]

there's always the nielsen-ninomiya theorem

[tawny slate]

youre gonna have to eli5 this because i know far too little background to gain anything from this reference lol

[tight star]

https://en.wikipedia.org/wiki/Nielsen–Ninomiya_theorem

no lattice can support chiral fermions

in particular, you cannot put the standard model on a lattice because the of the weak force

so it's not exactly clear how you'd model reality as a lattice

[tawny slate]

what does it mean to "put the standard model on a lattice"? i thought the standard model was a theory

[tall bolt]

That’s not at all what I was saying though, it being useful and maths not being constrained by being physical are not mutually exclusive statements

[tawny slate]

hmm i guess i just didn't see how the statement in context fit then, my bad

[tight star]

Describe the QFT as a lattice theory

[tawny slate]

ill jump to the chase here. would it be accurate to simply say the theorem tells us "physics as we know it cannot be modeled by a discrete grid"?

where physics as we know it is "the standard model"?

[tight star]

Yes

And hence, is some evidence for reality being continuous

[midnight scarab]

The Standard Model a priori does not make sense by itself. It's only defined "with a cut-off", what we call an effective field theory
There are continuous ways to deal with this issue in perturbation theory like dimensional regularisation, but if you want the full thing (which you need for strong interactions, and also to get effects like instantons (though you can see some of it in perturbation theory through resurgence), you need some cut-off definition of a path integral
And the main way to do that, which is also good for numerics, is replacing space-time by a lattice

[tawny slate]

this is alien language to me, sorry

no clue what any of these words mean, other than "lattice" and "path integral"

[midnight scarab]

The tl;dr is that in qft one has to integrate over functions on space-time, which generally we only understand as integtating over functions on a lattice (and taking a scaling limit where the lattice goes to zero, if the limit is well defined (which we don't believe to be the case for the standard model))

So anyway you "need" the lattice

Except we don't really know how to define a "chiral gauge theory" on the lattice

In facf, Nielsen-Ninomiya says it can't be done, at least the way we'd expect

So maybe we actually have to learn to do without the lattice

Working directly in the continuum

Which ig is Pseduo's point

[white fulcrum]

I'd like to motivate bijections for things by an example of a bijection between two sets whose size we don't know. It'd show that we get something 'for free', in that once we find the size of one set, we know the size of the other

[vestal tangle]

yea but you say two sets have the same size if there exists a bijection between them

by definition

[white fulcrum]

yes, but for finite sets we can just count the elements. an example from chatgpt is the bijection between even perfect numbers and mesernne (sp?) primes

we don't know if either is infinite but we know that both have the same "number" of numbers or are the same size

[lethal hornet]

if you don't know a priori that for a finite set X, its powerset is of size 2^|X|, you can prove that P(X) and 2^X are in bijection, and so counting the size of one of them gets you the other "for free"

[white fulcrum]

how is 2^X defined if not as the powerset?

[tight star]

set of functions from X to a 2-element set, such as {0, 1}

[halcyon glade]

There are a ton of these in combinatorics

The Catalan numbers are one fun example of something that has a ton of different representations you can convert between

But if you flip through an enumerative combinatorics book, you'll get plenty of other examples too

[patent flax]

Any last minute tips for teaching word problems to high school students?

[white fulcrum]

not really. I teach highschoolers and I just recommend the basics: patience, consistency, empathy, and so on

[tawny slate]

make sure they are disciplined in showing their work

example:
Ben has 2 apples, Steve has 3 apples. How many apples do they have total?

let x = number of total apples

2+3 = x

make sure they actually show the middle step where they declare their variables "let var = descriptive name". these habits will both help your teaching/grading as well as get them into good habits

the main resistance to this is its a lot of annoying writing, which students will push back on, so perhaps allow them to abbreviate, without sacrificing descriptive quality

[turbid zenith]

I have a hard time seeing an intellectual need for variables in a problem like this

I think it depends on what the class is though

[dim blade]

In my experience high school kids who aren't good at math often get confused by which words mean which operations

Per, the difference, increased by, quotient, etc

You need to always make sure they're on the same page as you. Ask questions and let them speak instead of lecturing

[tawny slate]

primitive example improvised, but yes

[turbid zenith]

Fair point lol

It's a pet peeve with some word problems though

The ones where the algebraic notation doesn't actually help

[turbid zenith]

Also key words are something to be careful of

[dim blade]

Yes, that's why I always make my students explain what the problem wants them to do

It's also very important to stop every once in a while and ask your class whether anyone has questions so far

[turbid zenith]

Yeah definitely

There's a piece of advice that I've heard a number of times that goes like

Don't ask "DOES anyone have any questions?" Instead ask "WHAT questions do you have?" See that normalizes the idea that having questions is expected and it means they'll feel more comfortable asking!

(In practice, I always use the second phrasing and often get just as much awkward silence)

So I default to, if they have questions, they're going to come up when they have to DO some math

Which means having them solve problems in class is what'll get those questions to finally bubble up to the top

[dim blade]

Ye, kids usually struggle formulating a question even if they are confused

[zinc dove]

Less lecture, more problem solving

[upper spire]

I’m not a formal teacher or tutor, but one thing that I've seen help is getting into the habit of finding a ballpark answer beforehand (e.g., should it be closer to −10 or 10,000? should it be bigger or smaller than some value?). Usually catches major errors and forces them to think about the situation, rather than simply manipulating symbols.

[obtuse ember]

Hi, I always have difficulty conducting in class review sessions for the exams. What’s the best way to do it for Calculus class? I would appreciate any suggestions

[turbid zenith]

Or also "What's an answer that's too large? What's an answer that's too small?"

[turbid zenith]

How do you currently do them?

[obtuse ember]

first i ask them if they got any questions from quiz or homework or book. If they have I answer them (usually they don’t ask). After that I give them handouts containing a set of questions based on important concepts from each section in the chapters. I have them work on the problems and encourage them to do it in groups. I walk around and answer their questions while they are working. In the last few minutes I go over selective problems from the handouts where most of them were stuck.

[zinc dove]

Do all of them work in class? How do you motivate them to work in class?

[obtuse ember]

That’s the problem. Some of them don’t. I feel this technique doesn’t work quite well. Are there any better ways to do reviews before exam?

[reef pulsar]

Hi, does anyone have experience in teaching an autistic student? I’d like some advice in how to design the course for them. Appreciate in advance.

My student is at the level of learning LCM and GCF, and I’m trying to find an interactive way to deliver the concepts to them.

[turbid zenith]

This is pretty much what I do :/

And yeah last time I had students who would just wait for me to do the problems

I mean one thing you can SOMETIMES do is a Kahoot (because games = automatic engagement right?) but that only works for certain kinds of problems.

But psychology is, like, a thing...

[obtuse ember]

Thanks. I can try Kahoot next time

[turbid zenith]

It’s worth a shot

My students really liked it when I taught Elementary Statistics

And multiple choice questions can be surprisingly good at identifying misconceptions

[white fulcrum]

https://en.wikipedia.org/wiki/Chomp is another great game

[zinc dove]

Ah yes, one where player 1 needs to be nerfed

[white fulcrum]

I let the kids go first

[obtuse ember]

Thanks. Just thought to share.
The review session didn’t go smoothly. I did the same technique i shared before: they didn’t have any questions so i gave them a review packet with bunch of questions and had them tried selected problems. Then I went over them on board. But it feels like this method backfired. Students didn’t seem to be happy during the review session. Just a few more things- when i did problems on board, sometimes I did addition type algebra wrong in two or three problems towards the end of a problem, so students caught and corrected me.

I am teaching a full lecture based course for the first time. Sometimes I don’t know what kind of class setup they like.
Also, sometimes I feel like I al not able to anticipate the steps or concepts where they can easily get confused or need more explanation.
I have seen them little frustrated in class sometimes.

Not sure if this is normal for a beginner teacher.

But all this while it isn’t giving me a good feeling. It feels like I am not teaching in the right way maybe, but I don’t know exactly where am I going wrong. I want to improve myself.

[quasi musk]

The more I've tutored the AP Calc BC exam, the more of a fan I've become of MCQ with diagrams, really measuring if they understand the concepts

Limit, continuity, differentiability, and integral as a model for area

[turbid zenith]

That’s a great game but not so much for a review game

…….I think

[turbid zenith]

What’s the setting?

High school, college? What course? What kind of students?

[obtuse ember]

College,
Calculus,
Undergrads

[tawny slate]

ive also been a fan of MCQ for another reason: you can often cheese MCQ by process of elimination

that would normally be a negative, not a positive, but 1) if you design the test knowing this then you just design around it or embrace it 2) assuming you are using the standard "scoring" of like A=90+, B=80+, etc, this lines up the raw score more closely with common associations of mastery level

the reason i say you can embrace this cheesing is because i genuinely believe we should be teaching students to cheese MCQ as a skill, because eliminating obviously wrong answers is effectively sanity checking, and that's a foundational critical thinking skill that should be used in real life

[rain hawk]

https://en.wikipedia.org/wiki/Homology_(mathematics)

I was talking to a bunch of people about this article, and they said that intro section is poorly written. I will talk about more about my thoughts and whatnot, but I'm first curious: does anyone agree or disagree?

[turbid zenith]

It does take a while to get an idea of what it's supposed to be about

[tall bolt]

Yeah it’s a lot of waffle and most of it I don’t think people really need to know, at least not in the introduction.

I would also probably flip the order of things, start with the idea of holes in a topological space, generalise to chain complex’s, mention this has become very useful for ideas in algebraic geometry etc

[tight star]

I’ve been thinking a little about what people find difficult about early math

A big sticking point is when letters get introduced, and I’m trying to understand why

Is it that you can use the same letter to mean potentially many different things?

[quasi maple]

Personally I think it's the sheer disconnect

"dafuq you mean that this letter is a number" is a surprisingly persistent thought

[tight star]

could you explain that a little more?

[tall bolt]

I think there is quite a bit of research on this and I do believe this is one of the suspected issues as well. I guess it is also just quite a big transition in maths. Until that point your vision of maths is basically just arithmetic, and varibles add in a whole other layer of complexity that you now need to reincorporoate into your understanding. You need to "reconstruct your schema" as they may say in the literature

Im blanking on all of the relevant reading but I know we looked at this in my maths education course

[quasi maple]

/s

A lot of the maths that students learn before algebra tends to focus heavily on numeracy and arithmetic, which are important concepts to learn but IMO ultimately create a false assumption that maths necessarily requires only numbers

Geometry might look astoundingly different in early maths compared to arithmetic, but the association with maths at that stage is still often defined using numbers (e.g. you define a triangle as a 2D shape with 3 sides)

[quasi maple]

Essentially, yh, IMO it's more or less because of the conceptual leap a student has to make compared with what they've just learnt

[tight star]

i'm trying to drill down precisely on what that conceptual leap is

as you say, geometry is quite different from arithmetic, but it doesn't appear that it causes the same reactions in students

[turbid zenith]

This is always weird to me

I remember learning about variables in 6th grade

And my teacher pointed out it’s just like what we had already done in earlier grades with like 6 + [box] = 9, what number goes in the box?

[quasi maple]

Yeah, it's the letters themselves

Because you have those box-exercises much earlier

[tight star]

Interesting - so do students find the box exercises a lot easier than letters?

[quasi maple]

I recall when I did some TA work in a primary school how I basically managed to teach the basics of algebra in this manner

[turbid zenith]

I think it depends in part on what they think the = sign means

A lot of students think it means “and the answer is”

So they’ll say that 7 = 2 + 5 is incorrect because it should be 2 + 5 = 7

[quasi maple]

oh that hurts

[turbid zenith]

Then I get them years later in college and they will write both sqrt(x) = x^(1/2) but also x^3 = 3x^2 in the same problem

[quasi maple]

I've had to correct that on occasion too

[turbid zenith]

When asked to find a derivative of a single polynomial like expression

[quasi maple]

In more subtle ways than that, but the underlying thought process persists

[turbid zenith]

So their conception of math is “when you see this, do that”

[quasi maple]

And this is whee I see it more frequently

I also find it common to have to explain that some answers require writing... like, words and sentences

[turbid zenith]

Another symptom of that issue is writing 2 + 2 = 4 - 1 = 3 ||quick maths||

[quasi maple]

I reckon stemming from this mindset still existing ^

[tight star]

Interesting

How do you resolve this?

[turbid zenith]

What I try to do is just re emphasize what the equals sign means with lots of examples

It doesn’t always work though

The habits are SO ingrained by that point

[tight star]

such as?

[turbid zenith]

The ones I gave so far

Like in Calc I’ll put up x^3 = 3x^2 and ask, are these literally equal? For all x?

And model better ways to write it more carefully, with d/dx or separately writing y = vs y’ =

But it still too often doesn’t stick :/ I haven’t found a surefire method to dispel that habit

[turbid zenith]

Okay so I suppose I could ask this here... I'm about to teach the calculus sequence at my university again, and I'm planning on having a "capstone" lesson at the very end of each course, not something I would test them on but something that would either tie what they've learned in a nice bow or give them an interesting new topic that uses many things we've learned. Here's what I've got for my other classes:

• Calculus I — Proving the Fundamental Theorem of Calculus
• Calculus II — The Riemann hypothesis
• Calculus IV — Differential forms and exterior calculus
  I'm stuck coming up with a good capstone for Calculus III. For reference the topics we go over in that course fit in five major themes:
• Thinking in higher dimensions
• Partial derivatives
• Vector-valued functions
• The gradient
• Multiple integrals
  For example one topic that I'd considered was calculus of variations, since you're really rethinking "dimensions" by optimizing over an infinite-dimensional space of functions, and the questions are so interesting. (Hanging chains! The brachistochrone!) But I want something that can also tie into multiple integrals. Any suggestions?

For reference, here's the justification for the other three capstones:

• Calculus I — We start by learning derivatives and integrals, so they already know the FTC by halfway through the course. But in the last part of the course we do limits, showing how to more precisely define derivatives and integrals in terms of limits. We also look at limit theorems: Squeeze, IVT, EVT, and MVT. So the idea of FTC is to bring together all those tools and prove FTC as a culminating experience.
• Calculus II — Since our last unit is about series (Taylor series and a tiny bit of Fourier series), plus we'll have talked some about Euler's formula, it seemed that the Riemann zeta function touches on many of the topics they've been talking about, like p-series, convergence, complex numbers, functions like Γ and Li that need to be defined by an integral, and so on.
• Calculus IV — Since we're ending with Green's theorems, Divergence theorem, and Stokes' theorem, I want to look at how all the various versions of FTC can be unified using the generalized Stokes' theorem.

[midnight scarab]

In principle minimal surfaces have multiple integrals, but obviously you'd want to first do all the other examples you mentioned so I'm not sure you'd have time to say much

[turbid zenith]

Hmm, that makes sense

With minimal surfaces though, are there any that are (1) set up with multiple integrals and (2) are actually solvable?

I'm fine with giving a high-level hand-wavy look if needed

[turbid zenith]

I'll also say, it may help that when I teach Lagrange multipliers, I actually include the Lagrangian function $$\mathcal{L}(x,y,\lambda,c)=f(x,y)+\lambda(c-g(x,y))$$

[burnt vessel]

Solid Angles

[turbid zenith]

So that's something they would have already seen

[midnight scarab]

I believe the most systematic way is to just prove that minimality is equivalent to zero mean curvature and then just compute the mean curvature and show it vanishes.
However, if you consider the soap bubble hanging on two parallel rings, you can take revolution symmetry as an Ansatz and explicitly reduce the surface integral to a 1d integral and then do Euler-Lagrange on r(z) to find a catenary (and hence a catenoid)

[turbid zenith]

Yeah I believe I've seen that case — it just becomes a surface of revolution

But I could see, say, a ring created by intersecting x^2 + y^2 = 1 with z = xy

That would be an interesting shape that you could imagine a minimal surface for without the rotational symmetry, and you'd be maximizing a double integral

But solving it? No idea XD

[midnight scarab]

So if you did the catenary, you can just quickly talk about minimal surfaces and soap bubbles and just have a nice quick exercise on surfaces of revolution to reduce to the catenary problem

[midnight scarab]

Yeah, if you're truly in two variables than I don't think things are very nice

What you can do in two variables is deriving the PDE, perhaps show it implies zero mean curvature, but solving it is a whole other beast

Though Idk if you discussed curvature

[turbid zenith]

I'm okay with at some point saying "actually solving this by hand is beyond the scope of this course, but it could be done numerically"

[turbid zenith]

We do talk about curvature and torsion, but not mean curvature. But this would actually be a great time to give a nod to mean curvature.

It's mostly just so they can see something that unifies and/or extends a bunch of the things we've talked about, even if they wouldn't be able to do any problems themselves

I think the more I think of it, Calculus of Variations is the perfect fit. 🙂

[midnight scarab]

Yeah, then you can just derive the Euler-Lagrange equation, say it's proportional to the mean curvature, and then show some nice pictures ig 😅

[midnight scarab]

Yeah, I think CoV is very cool

(Definitely not biased as a physicist)

[turbid zenith]

Of course 🙂

IS there a good way to solve a CoV problem numerically?

[midnight scarab]

I'd imagine they just solve the E-L PDE (or Hamilton-Jacobi) but idk

Or maybe some kind of gradient descent

I suppose the question is whether you want a "smooth" solution or a triangulation

[turbid zenith]

Triangulation would probably be fine in practice I'd imagine

Like getting a picture

[turbid zenith]

So I’m watching Trefor Bazett’s video on CoV

To derive how you do things, you need to differentiate under the integral sign

How do you prove that?

Fubini’s theorem!! 😄

So even if I don’t do a full on surface area functional, double integrals explain one of the most important steps!

[midnight scarab]

Oh, you prove Leibniz in calculus? Unexpected but nice!

[turbid zenith]

I mean not normally but this capstone would be a great time to do it

Sort of a "just in time" approach

[midnight scarab]

I see

[turbid zenith]

They already have Fubini from doing multiple integrals

[midnight scarab]

How much stuff do you actually prove in calculus

[turbid zenith]

Depends on your standards of proof

[midnight scarab]

Or I guess how much time do you spend on explaining limits

[turbid zenith]

In my current plan, I spend 7/25 days on them at the end of the course

[midnight scarab]

Of Calc 1?

[turbid zenith]

Yes

1. Limits and continuity
2. Analytic techniques for limits
3. Limits and derivatives
4. Limits and integrals (includes limits at infinity as part of the lesson)
5. L'Hôpital's Rule
6. Valuable theorems (Squeeze, IVT, EVT, MVT)
7. Capstone: Proving the FTC

[midnight scarab]

Sounds very nice and makes a lot of sense

As a non-American I assumed most "calc" courses were mostly just about computation, and the real stuff about limits, IVT etc only came in RA

[turbid zenith]

Most of the time they are

But they do start with limits

[plain valve]

Hm isn't L'Hôpital usually proven after things like IVT

The way I am used to proving it is like cauchy MVT

Or maybe you mean merely to apply it

[turbid zenith]

You spend a bunch of time learning algebraic techniques for limits, then learn the difference quotient definition of the derivative and trudge through, say, the derivative of x²

Then you learn the Power Rule and promptly forget all of it

[plain valve]

I guess you can also initially assume your function is C^1 or smth

[turbid zenith]

I wasn't planning on rigorously proving L'H

[plain valve]

Sure ye

[turbid zenith]

Some pictures can be plenty convincing

But in RA yeah sure prove it

[midnight scarab]

Do you do Taylor before or after l'Hopital?

[turbid zenith]

I do Taylor in Calc II

[white fulcrum]

I'm looking to refresh myself on the motivation for the turing machine as the definition of a computer program. when we think of a turing machine as a function fron nats to nats, why must it be the case that some of them don't halt? I know that in order to have the functionality we expect a program should have, some programs won't terminate, but I'd like an example. I wanna teach my students the halting problem.

[halcyon glade]

Even high school calculus (AP calculus) includes IVT, MVT, etc

Even though it doesn't involve proofs on the exam

[halcyon glade]

It seems like you have two questions here, one about the motivation and one about the halting problem. The halting problem proof is classic, you can find it here (and it applies not just to Turing machines but any computation process which can be communicated over text): https://www.youtube.com/watch?v=macM_MtS_w4
As for why the Turing machine is a compelling model, Turing argues at length for it in his original paper on the nature of computation, but if you think about the processes of computation you do on a piece of paper and are okay with reducing the 2D paper to a 1D tape (just go down the paper in lines) and sticking to a finite set of symbols (e.g. the symbols on a keyboard), it becomes clear after some playing around that Turing machines can capture anything you can do on the paper. Maybe it would help to demonstrate some sample Turing machines to illustrate this point.

There are of course other models of computation (most famously lambda calculus), but they all turn out to be equivalent in power to Turing machines, a statement that goes by the name "Church–Turing thesis"

[white fulcrum]

I'm aware of all of that, my hangup is on why tms can sometimes never halt. for example, in turing's paper, he puts forth a machine that writes the sequence 0010110111011110111110...

we could make a program that takes a number n and prints the nth digit of this sequence instead, which would always halt. same thing for computable numbers like pi and e. printing a result might go on forever, but it seems like figuring out the result doesn't have to. I feel like I'm missing something easy here, I used to know this really well

[halcyon glade]

Okay tell me if this program halts for all integer inputs n > 0:
Input a number n. Set a counter to 0. Repeat the following until n=1: (If n is odd, calculate 3n+1 and then add 1 to the counter. If n is even, calculate n/2 and then add 1 to the counter.) Now output counter.

[white fulcrum]

oh yeah I feel really stupid now

ty comrade

[halcyon glade]

You're welcome, halting is very confusing

[white fulcrum]

what if we insist that programs that don't terminate instead output -1 or smth like that?

we could say the collatz program exists we just don't know what it is

but we could still program the naive collatz machine couldn't we

so a program would have to know if it would halt which seems too meta

[halcyon glade]

If you want everything to always halt, you'd have to restrict the capabilities of the programming language, but then there will necessarily be programs that Turing machines can run but you can't

[halcyon glade]

If you don't know what the program is though, how do you execute it

It kind of defeats the point of having a program in the first place

One obvious thing you can do though is write a Turing machine as a "limit" of programs that always halt

Just run the Turing machine for n steps and output -1 if it hasn't halted yet; that gives you a sequence of programs for each n that always halt and get closer and closer to the output of your original program

That sort of method gets used a lot in computability research

[white fulcrum]

I think i'm asking for "what do we lose if we insist on halting? what sorts of problems can't we express or compute?"

actually I'm gonna think this thru for a bit and then come bacck

[halcyon glade]

It's a good question

Basically you lose the ability to describe in general sets of the form "{y: there exists x such that (some property of x and y)}"

Those are called Sigma1 sets

A lot of common things you might want to do fall under this category

For example checking whether a number is composite (y is composite iff there exists x1 and x2 such that x1 < y and x2 < y and x1 x2 = y)

In many cases there may be clever things you can do to get rid of the need for the "there exists" quantifier (for example, checking if a number is composite is possinle because you always only need to check a finite number of values for x1 and x2), but in general you can't always do that

"Arithmetical hierarchy" is a good search term here, it's basically a hierarchy of sets that get harder and harder to compute (also "recursive" vs "recursively enumerable" sets)

[white fulcrum]

my approach with my students is to start from "why don't we just make a supercomputer that can solve all math problems so we never have to learn math anymore?"

I think that's the best motivator for approaching halting and set theory madness

it's basically the irl motivation too

[halcyon glade]

Yea I think that is a very good question

[white fulcrum]

so it fits in nicely with historical narratives, and I love the chance to spruce up lessons with things like that

[halcyon glade]

That sounds like a great throughline for a lesson

Oh here's a good example of something you need halting to be able to compute

You can write a program that, given an input program, will halt and return "YES" if the input program does and will not halt (interpret running forever as a "NO") if the input program doesn't. (Just run the input program.) So in this sense, a program that's allowed to run forever can "solve the halting problem". But one that is guaranteed to always halt can't. So the set of programs that halt is recursively enumerable but not recursive. There's a good number of these types of sets.

[drowsy otter]

I have a question. I am doing a side hustle, where I am tutoring an aspiring Data Science student who struggles with mathematics. They are studying degree-prep level Mathematics, but they struggle with a GCSE topic related to factorizing quadratic equations.

So far, I have taught them to use the quadratic formula to find the values of $x_i$ and then express the original function as products of $(x - x_i)$, but this only applies when the coefficient of $x^2$ (which I will call $a$) is equal to 1.

However, when $a$ is not equal to 1 (denoted $a \neq 1$), it doesn't work. I have attached three slides in the form of images (since attaching a Google Slides presentation is technically against the rules) explaining this concept.

Please could you advise me on what I should do to get the student to understand. The student is in their 40's (so much older than me), and this is my first time teaching someone of that age despite having 3 years' experience teaching students younger than me.

[burnt vessel]

Kardashiana7

[drowsy otter]

Another thing. You may have noticed that the third slide isn't complete. This is because I am struggling to explain why the technique on the second slide works.

[drowsy otter]

Also, I get my words jumbled up sometimes due to having a communication disorder (hence my use of these slides during lessons). I mean screenshots, as opposed to images.

Reference: "I have attached three slides in the form of screenshots"

[turbid zenith]

There are a few things you might do in such a case

One thing you might want to look for is the “box method” or “area model”

[drowsy otter]

I'll try the former technique. Thank you very much.

The Box Method (which is the same as the Area model) is actually quite simple, but I'll read up on it anyway to make sure I'm teaching that correctly.

[midnight scarab]

Isn't it more natural to say that a polynomial is determined by its roots up to an overall prefactor

And it's clear, looking at the x^2 term when you expand, that the correct expression is a(x-x1)(x-x2)

[marsh compass]

Even if the coefficient of x^2 is not 1, you can still get non-integer solutions

[drowsy otter]

I mean, I am able to factor out the value of $a$, to form $a(x^2 + (b/a)x + c/a)$, but we're talking about someone who isn't a school leaver whom I have only just started tutoring, so it's best I stick to how to get to integer solutions, otherwise they will get confused.

[burnt vessel]

Kardashiana7

[drowsy otter]

In summary, while I acknowledge that what you say is mathematically correct, I don't think my student will quite get it yet.

[midnight scarab]

I don't understand how telling your student that "when a neq 1..." instead of just telling them that solutions don't have to be integers is helpful or even less confusing

[drowsy otter]

It's OK to not understand. I too am beginning to question it. Perhaps I am going about it the wrong way. I'm going to try to change my mindset that "simple first is always good" and not be afraid to delve into more complicated stuff so that it's easier in the long term. After all, my student is, as I said this morning, in their 40's.

[marsh compass]

I just worry that it might be misleading, attributing "fractional" solutions to non-unit leading coefficients

[drowsy otter]

To be fair, I'm happy that my way of thinking is being challenged, because that's a sign of me becoming not a good, but a great tutor in the long term.

I've figured it out. Thank you. I'm just glad that the solution to my issue was simple.

[midnight scarab]

Yeah, I think it's very important not to be misleading. To quote Einstein (😛), "Make it as simple as possible, but not simpler."

[drowsy otter]

Thanks.

[white fulcrum]

I'm trying to understand graham's number better to show it to my students, because it's one of the easier big numbers to explain. are there visual examples of various colorings of low dimensions that do and don't contain a single-coloured complete subgraph on four coplanar vertices?

alternatively, are there other big numbers that are easy to explain how big they are?

[tawny slate]

pedagogy relating to big numbers is an area i'm actively working with right now in my drafts. could you provide some more background as to what your goals are and what you're trying to accomplish with this?

because the main issue with introducing large numbers is 1. it's mostly not useful or practical 2. it leads to misconceptions about mathematics 3. students with low math competency don't know how to appreciate just how massive some of these numbers are, so just telling them this number is huge is underselling it. you don't need to go that far just to have them appreciate big things

and the issue with graham's number specifically is that it is an exceedingly loose upper bound on a problem that has long since been improved on. its appearance in a math paper is socially derived rather than being something inherent about math itself, which makes it harder to justify it being meaningful

[white fulcrum]

my students are gradeschoolers and highschoolers. I usually save 5-10 min at the end of a session for exploring fascinating but unrelated math concepts. past things include collatz, other open number theory conjectures, https://en.wikipedia.org/wiki/Chomp (and the proof that player 1 wins), the mandelbrot set, and so on

[tawny slate]

so idk how much you can cover in 5-10 min, this kind of thing is difficult to squeeze in a reasonable amount of time, because of the meta-issue of students not having the intuition framework for understanding large numbers

if you can do a multi-part sort of thing, there are two approaches I think you can take here. you can do a from core principles approach by explaining growth rates of functions, how they apply to the real world, so they can build some intuition and appreciation before going to bigger stuff, or you can just throw them the wild stuff one by one and tease them

so if say you're doing the tease approach, perhaps goodstein sequences are a little bit more accessible, because rather than throwing the students large numbers immediately, you can show them where they may come up in a natural sounding math question

first walk them through how this sequence actually works, using some basic small examples. in fact, i would say probably don't even use hereditary-base notation, just use normal base notation first. then show the smallest example that appears to explode. ask "do you think this sequence goes on forever or stop at some point? what about if started with [some large number]?" most students should reasonably assume the sequence goes on forever here, or at least for some large enough n

then hit them with the "actually all of these sequences are finite, no matter how big a finite n you start with". then you can show them the hereditary-base notation and ask them if now the sequence goes on forever, show them just how much more aggressive the growth rate potentially is, then ask the same question, give the same answer

then to summarize, tell them that in order to prove that no matter what number you start with, that the seqeunce is always finite, you need to use concepts involving infinity. so understanding infinity can sometimes be necessary to better understand finite structures. this gives them an example of an unexpected use case for large numbers, while priming them for potentially going into like infinite ordinals or something later on

[white fulcrum]

I also taught the hydra as an alternative to goodstein

Same idea, counting with ordinals

[tawny slate]

if you're doing the core fundamentals approach, i would say they should first learn how growth rates of functions are dominated, that n^2 always outgrows n, what that means

then one layer at a time, explain examples of things that grow linearly, things that grow quadratically, things that grow cubically, things that grow exponentially, and then actually explicitly compare them to hone that intuition. for instance, the rice grains on the chessboard problem demonstrates how exponential > cubic growth (because physical space is cubic), the fact that a well-shuffled deck of playing cards will never see the same order twice, ever (factorial > polynomial > quadratic, because number of seen orderings is time * num of people). this can extend to real world applications, like showing why multi-level marketing is doomed to be unsustainable

once they have a core understanding of growth rates, they are ready to tackle bigger things. challenge the students to write down the largest number they can think of on a small sheet of paper in a short amount of time. now if you introduce to them, say, the slow-growing, fast-growing, or Hardy hierarchy of functions, you can point to where each of these functions live on the hierarchy. for instance, if you show that on the fast-growing hierarchy, f2, is already exponential, then f3 is stupidly massive already. compare some of their answers to values on this hierarchy, and almost certainly their answers are going to be on the "order" of f3

obviously you can go into omega and beyond here, but i think just giving them this much is already enough to get them excited, they'll start doing recusive nestings of the fast-growing hierarchy or something, thinking it's clever, but then you just show them that just ticking the base number/input by 1 already achieves this clever thing they're doing, and they'll realize just how powerful the hierarchy is

[tawny slate]

in my opinion and experience i find goodstein is cleaner than hydra when it comes to connecting to ordinals and the math behind it. the hydra makes for a better story obviously, but more difficult to work out the math, it's more obtuse. so i usually start with goodstein

but ultimately i just think this kind of thing is not plausible to do in a 5-10 min interval, probably needs to be multi-parted at least

even defining the big numbers take like 5-10 min

[white fulcrum]

maybe I'll do a short intro to encryption. since today is valentine's, I'll make it about texting your secret crush to your bestie

[halcyon glade]

That's such a good idea

[sacred skiff]

Hiii everyone!! I'm new to this chat, so please forgive me if this kind of question has been asked many times before :((.

I'm running a 2-hour uni tutorial (for me, this looks like a classroom of students, maybe 10/15 or so, and a whiteboard for the tutor) this semester. I'm given a worksheet of problems to go through as well! I'm wondering what the best way to get through the worksheet is

I'm new to teaching, so I'd appreciate any suggestions!! My biggest fear is it being a little dry if I'm the one talking most of the time. Is it a good idea to ask students to work on the problems first?

My experience so far has been that I then walk around to help anyone that needs it. But often, everyone is busy trying and no one asks for help

There are lots of times I ask for students to recall definitions, or results, but I get lots of silence in return

[halcyon glade]

If you're having difficulty getting engagement, sometimes it helps to prompt them to talk about the question with their neighbor

[tight star]

idk if this is the best advice but i always try to call the students' bluff with this

[halcyon glade]

Even something as simple as bringing desks together into groups if possible can help

[tight star]

like if i ask a Q, i will not continue until someone gives an answer

[halcyon glade]

Yeah use silence to your advantage

They are more scared of awkward silence than you

[sacred skiff]

Thank you!! Is there something I should do in the meantime??

[sacred skiff]

I'm so curious what you mean by that haha!

[tight star]

the thing is, if you ask a Q, wait a bit, and then continue without anyone giving an answer

then the students know that they can get away with not answering

[halcyon glade]

You can go around groups and see if they have questions, you don't have to be active 100% of the time though, often it's good to give students space to think and talk

It's really about their learning after all

[sacred skiff]

I don't quite understand, because you'd have to continue at some point right??

[halcyon glade]

You don't have to do anything

[tight star]

yes, when someone answers the Q

[halcyon glade]

If the students don't say anything, it's their time that they're taking up, you still get paid the same haha

Obviously make sure that they understand what you're asking though, sometimes a question can be unclear

[sacred skiff]

Do you go through the problem in detail after they have a think?? I'm not sure how you'd get the feel for if the students will get bored from it

[halcyon glade]

If you're having trouble getting engagement, invite one of the students up to the board to explain the problem, or if that's too much commitment, have the students take turns explaining the steps

That way your role is minimized to just facilitating and supporting the students, and the students are forced to demonstrate their mastery

[sacred skiff]

I like that so much!!

[halcyon glade]

These are all just different strategies you can try, not rules; what works best will differ from classroom to classroom

[sacred skiff]

For the first class, is it typical to do a self introduction, or an intro to the course?

And for the last class, doing a "where to go from here" kind of thing?

[halcyon glade]

Yes to both

I also have students make nametags and introduce themselves for the first class

[sacred skiff]

I've had tutors in my earlier courses do this, but in later courses, they stopped!

[halcyon glade]

The further you go in math, the more the teaching quality drops tbh

Persistent weird problem with mathematical culture

[sacred skiff]

Thank you so much for your suggestions @halcyon glade @tight star . I'll give it a try!

[halcyon glade]

Yw

[tall bolt]

I need to get better at this. What I’m giving this semester is supposed to be supervisions, not a support class like last sem, but I often end up basically lecturing, especially with one of my groups, because they just refuse to talk

[halcyon glade]

Lecturing can be good, it just depends on the situation

[tall bolt]

Yeah it definitely works well sometimes, like I spoke about how to actually come up with and then write an epsilon delta proof the other week which seemed to work very well. But sometimes I just want to go through some problems, and it ends up with me just saying “ok cool give it a go then we can talk about it in 5-10 minutes” and no one seems to do anything and I just give them the answer

Which works but like, it’s not a productive use of anyone’s time, and I have made that point lol

[white fulcrum]

^ From smbc ofc

[turbid zenith]

I like having the students introduce each other!

[turbid zenith]

I feel this

[white fulcrum]

explaining why a neg minus a neg becomes add a pos is still tricky for me

[halcyon glade]

First the first neg doesn't matter so I would abstract to "why is minus a negative the same as plus a positive?" And then I would try to rephrase the minus into a plus and give some concrete numbers. So for example why does 7 - 4 = 3? It's because 7 = 3 + 4. So why does 7 - (-4) = 11? It's because 7 = 11 - 4. So subtracting a minus gives us the same thing as adding a positive. The same applies if the first number is negative too, since it doesn't matter in our argument.

[turbid zenith]

I really like doing it using tokens or "piles and holes"

If you're trying to show that 7 - (-4) = 11, you can start off with 7 positive tokens:

(+) (+) (+) (+) (+) (+) (+)

Then you want to take away 4 negative tokens ... but you don't have any! But you can always add zero pairs without changing the total value:

(+) (+) (+) (+) (+) (+) (+) (+) (+) (+) (+)
                            (-) (-) (-) (-)

Now you can take away 4 negative tokens:

(+) (+) (+) (+) (+) (+) (+) (+) (+) (+) (+)

And voilà, your result is 11. In order to take away 4 negatives, we had to introduce 4 zero pairs, so we end up with 7 + 4 positives.

[white fulcrum]

omg I like it

[tawny slate]

a weaker way to tackle that is to treat numbers as positions on a number line

you add by shifting right, you subtract by doing the opposite, the inverse, shifting left. in this way, the negative is treated as a "direction flip". so subtracting a negative is like taking a left shift and flipping it, shifting right

this is admittedly weaker than the other explanations but i think for completeness sake, i think it's good to mention, because while the other explanations are very good at explaining why the phenomenon is true, this visualization is imo more practical and easier to use when doing calculations

[tawny slate]

is it appropriate to start a conversation on this? because i have some of my own hypotheses as to why this might be the case and im curious what others think

[halcyon glade]

Sure

(I am not a mod though)

[tawny slate]

so some of my hunches are as follows:

• for early education, teachers don't need to be exceedingly well-versed in math, their main focus is general education, which makes them more informed teachers in the general sense (compared to some professors who are there not to teach but to do research)
• the easier the math concepts, the easier it is to find hands-on activities and practical applications. its easy to motivate why we care about 3+4, its harder to motivate the intermediate value theorem
• math is one of those subjects that requires and depends on prerequisites more than other subjects, so its harder for teachers to keep up when the students are at different levels
• for the individual, often times it is easier to just do math for the sake of math, without too much regard for practical use or purpose, but many students need motivation. and so the ones who are very good at math may be unable to adequately provide this motivation, but they are the ones who are most proficient at math and therefore teach it

wonder if anyone else agrees/disagrees or even has academic support for or against any of these

[white fulcrum]

a month or so ago I had someone ask why long division worked...

I work at a place that has premade worksheets/problems that guide you thru things so luckily that was handled

I know how to explain it just not how to explain it quickly

[turbid zenith]

I agree with pretty much all of it

With the caveat that I don't think everything needs a "practical" application

In the "it'll help you paint your fence next Tuesday" sense of practical

I also think it's worth pushing back against the narrative that in order for something to be worth learning it needs to be "practical" ... if you learn to play chess or Fortnite, it's not because you're going to find a "use" for it ... but of course the fact that you're not graded on either of those leads to some incentive problems

All this to say, there are lots of possible motivations, and practicality is oversold but it's only one of them

[rapid tusk]

I always remember that one line from lockhart’s lament about the beauty of math being how irrelevant to our everyday lives it can be

[halcyon glade]

I will say there is also a cultural/structural component—in academia, teaching is often seen as a chore rather than an interesting task, and even if you are interested in it, it's not required or even difficult to find proper training on how to teach in the first place

On the contrary, more introductory subjects like calculus are more likely to be assigned to permanent lecturers instead of research staff

[tawny slate]

i dont disagree, but the point is that it doesn't help the students. they will then just ask "so why should we even waste our time learning this?" and then it makes answering this question harder, not because you cant answer it but because now the explanation is long-winded, esoteric, tedious, confusing

[vestal tangle]

i like to say that the positive integers represent piles of marbles (or whatever) made out of matter and that the negative integers represent piles of marbles made out of antimatter.
7 + (-4) means 7 matter marbles plus 4 antimatter marbles. the 4 antimatter marbles annihilate 4 of the matter marbles leaving you with 3 matter marbles.

so adding 4 antimatter marbles to the pile is functionally equivalent to taking away 4 matter marbles. with that as motivation we can define a - b to be a shorthand for a + (-b)

once you know that subtraction isn't a fundamental operation and is really just a shorthand for a certain kind of addition i don't think it gives students much trouble

i like @turbid zenith's approach, i think it fits in nicely with the matter antimatter picture. we can add as many antimatter marbles to the pile as we want as long as they're balanced by an equal number of matter marbles, because this is equivalent to adding nothing, and then we can think of "taking away" the antimatter marbles

[turbid zenith]

I disagree that it doesn’t help the students

One big example being that sometimes you learn math because it ties the other math you’ve learned together

I don’t think dividing polynomials is particularly useful on its own, although some of the things it eventually leads to can be probably

But pointing out that it’s just like what you already did with fractions can be a motivation for why you’re learning it

You can probably also throw in something about it being like weight training for your brain… I’ve told my students before that I don’t actually care if ten years from now they’ve forgotten some random formula, but if they remember how to think through a problem, break it down, look for patterns, etc, then I’ve done my job

[tall bolt]

Why do anything

[turbid zenith]

https://youtu.be/-OEjRnsAqjk?si=dbF1XUaM0xUg-sRx

[tawny slate]

no i mean it doesn't help some of them get motivated

why should we learn fractions?
so that you can learn algebra!
why should we learn algebra?
so you can do trig!
why should we learn trig?
so you can do calculus?
why should we learn calculus?
so you can do multidimensional calculus!

for many students, this will just frustrate them, and you lose trust rather than gain trust. again, i understand the "beauty" of not needing to be practical, doing math for the sake of math, im simply saying that this wont satisfy some students

you can say "it helps problem solving in the abstract", and this is an infinitely better answer you can give from the start that is independent of the point that math can be "detached from reality"

ah whoops this response is blending in someone else's comment too so im taking your point a bit out of context, sorry

anyways the point is typically when students ask why we do math at all, the three biggest answers i give are:

• less likely to be scammed/exploited as you get older
• helps keep your mind active and brain healthy, which also keeps physical body healthy
• helps with problem solving

my answer would never be anything remotely of the form or close to "dont worry about that, just do math for the sake of math, devoid of practicality [in the sense we described it]" unless we were specifically addressing this idea in context

[tall bolt]

I dont think doing it for the sake of doing it bad though, and I think with context its perfectly ok to say that sometimes we do maths just to do maths. Like youre points are very valid and where id start, but like, we dont need to make up reasons for doing art. We study art because its beautiful, and important as part of a broad education, its part of being human. I dont really understand why maths is the one subject that seems to need to constantly justify its existence with practical application. Its not a complaint I think ive ever really seen directed at any other subject in school (beyond people broadly complaining about needing to go to school at all)

[tawny slate]

again, fully agreed. but i have had multiple students argue very passionately with me that math is not art

to some extent i can understand why they feel that way, but if i have to explain multiple ideas all in one go:

• math is art
• math doesn't have to be practical
  this just makes me look crazy. if i explain one at a time, not convincing unless i can isolate one idea at a time

[turbid zenith]

I don’t think that’s what I was saying

I was thinking of it the other way around. Why learn algebra? It explains why fractions work. Among other reasons sure. But what I’m saying is that this is ONE motivation.

[turbid zenith]

I agree with all of this too

But what I’m saying is that there are some lessons where I’m not going to be able to give a clean practical application of it

And absent those clean practical applications, one possible thing that may help depending on the lesson is, well it ties together some of what you’ve previously learned

I’m not going to claim that polynomial multiplication will help you build a better bridge or whatever, no matter what contrived examples the textbooks use

(I guess I could throw in some hand wavy stuff about Bézier curves)

But I would love my students to walk away thinking polynomials sure are a whole lot like integers, with a lot of uncanny parallels

Any time I CAN give a practical application I’m going to obviously, but sometimes trying to come up with one becomes really ham-fisted 😛

[midnight scarab]

Also, giving a contrived practical application is likely to backfire

[severe moon]

assume i ever have the chance to teach a linear algebra class in the future, a typical first course.

should i change the order of how stuff is taught generally?

[tall bolt]

That’s a pretty vague question

There’s some things that probably could be quite fluid and others that aren’t

[turbid zenith]

This, 100%

Dan Meyer has given some great examples of "pseudocontext"

[turbid zenith]

It's up to what your goals are for the course

Shuffling the order of topics is certainly a nice freedom to have, but you should have a clear reason for doing so, and you should be ready to defend why it'll help the students learn it better

[severe moon]

we want some computation ofcourse, i would assume the ones taking this class are stems majors that aren't math majors.

but at the same time, having matrices be introduced without at least spending a class going over linear transformations is downright criminal.

also change of basis is a MUST, it is insane that i went through a linear algebra class that did not cover change of basis at all, it would make diagonalization so much simpler to understand.

i should at the very start discuss vectors and their properties

for the most part, this class would focus on R^n, i can't assume i would be able to cover vector spaces

and honestly, going over guassien elemenation at the very start should still be okey, but it will not be introduced with Ax = b.

determinants should be included, but the usual presentation seen in conventional linear algebra classes should be scraped, it doesn't make sense.

also stuff like the coffactor formula for inverse matrices and Cramer rule should be removed.

so maybe something like this

1-guassien elemenation
2-vectors and 3d geometry 3-linear transformations
4-matrices
5-inverse
6-determinant
7-spanning sets
8-subspace
9-basis and dimension
10-change of basis 11-eigenvectors and eigenvalues
12-inner product in R^n and orthogonality
13-gram schmidt process and orthonormal basis
14- least squares

[tawny shard]

I've been a math tutor for almost nine years. I have over 4000 billed hours on Wyzant. I am always trying to improve what I do to better serve math and physics students. I don't want to overthink teaching the times table, but I definitely don't want to underthink it. What kinds of counter-intuitive thinking do you find has helped you working with a student one on one?

[quasi maple]

I've forgotten who it was who suggested this, it was some speed-calculator person who also apparently coaches; but in some video he'd suggested that the times-tables up to 10x can in practice boil down to having to memorise 6x7 = 42, 6x8 = 48 and 7x8 = 56

Found the name -> Arthur Benjamin

[tawny shard]

I do tend to focus on the multiples of seven. I didn't have research saying do more drills on 7's.

[tawny shard]

What was your most effective undergraduate course for teaching you to write rigorous proofs? For me, it was MTH320 at Michigan State, Real Analysis I. As I tutor proof writing techniques at different levels, I find myself repeatedly asking myself what unspoken rules of proof I am not bringing up.

[rapid tusk]

anecdotally it was always the 7s that gave me the most trouble when i was memorizing times tables way back when

in terms of rapid recall

[halcyon glade]

Sorry I feel like I'm missing something, aren't there tons of linear algebra books that discuss linear algebra in R^n, introduce the idea of a linear transformation later on, etc.? This sounds more or less how my first linear algebra class went

Why not follow a linear algebra textbook geared toward applied math

[white fulcrum]

I'd like examples of pi showing up in unexpected places, but for kids who don't yet know about euler's formula bc they don't know e or i

the leibniz formula for pi/4 converges too slowly to illustrate the point, same with the pi squared over six series

[white fulcrum]

I also know about buffon's needle but I don't have time to try that out experimentally

[tardy lynx]

How would you guys go about teaching an intro to complex numbers? The lesson is supposed to be ca 1 hour and meant for high school students

[rapid tusk]

whats their assumed knowledge and what context

id cover first the simple case of quadratics with negative discriminant and then some basic complex plane stuff (vector analogue)

and then at least state fundamental theorem of algebra?

depending on what background the students have you could also touch on multiplication/division and how to interpret geometrically

but that requires at least some trig background

and at that point de moivre/exponential form might be a bit out of scope

[turbid zenith]

You can teach that multiplying by i is rotation by 90 degrees without going full trig

That one little fact carries so much intuition that it opens up complex numbers

I also like to frame it in terms of alternating current for a real world application

Yeah you can’t have “i apples” in your hand, but that’s not what complex numbers are good at — they’re good at describing rotation and fluctuation

I think students would find that take more compelling than “we took a useless problem with no solution and made up an answer anyway” lol

[tight star]

I like teaching them via yoneda

[plain valve]

Lol

[tawny slate]

one of the major issues with this topic ive found is students tend to think that real numbers are real, imaginary numbers are fake, and complex numbers are complicated, due to the names

to dispel these myths, i like to demonstrate how all numbers are kind of made up to begin with. we don't say that fractional values or negative values dont exist just because we cant have that amount of people. numbers are useful as long as any kind of useful structure can map onto them, and are equally made up in this way

in this way, i think another utility point to mention, although too complicated to go into full details, is that complex numbers are closed under exponentiation

probably also important to explain to students that 0 is the only number that is real, imaginary, and complex, and visualize this on the complex plane

[brave wave]

yeah showing 0 as the intersection on the plane makes it click for kids

[tawny slate]

additionally, once you show the analogy between the real line and the complex plane, it helps to mention how multiplication and division work visually, as someone here mentioned, but if you want to fit it into an hour with students who may have varying skill levels and backgrounds, here are some ways i recommend framing it:

• to explain addition and multiplication visually, it helps to first demonstrate how it works on the real number line, because students need to get comfortable with the idea of treating a number as a transformation as well as something geometric and visual. show that addition is translation and multiplication is scaling, THEN use this: https://m.youtube.com/watch?v=-dhHrg-KbJ0&pp=ygUabWF0aG9sb2dlciBjb21wbGV4IG11bWJlcnM%3D (at 9:20)

this will explain how rotation works, and this derivation doesn't involve trig at all

make sure to emphasize that the "stretching focal point" is always at 1

• once students understand that complex numbers are essentially "2D" numbers, now you can demonstrate the usefulness of them, a few practical applications. show that the geometric interpretations make calculations involving sliding, rotating, and stretching images almost trivialized. show that the two attributes of polar form, mag and arg, can be mapped onto amplitude and phase of a wave (both arg and phase have a cyclically repeating property)

and honestly, i think if stretched on time, you can handwave a lot of things, i dont think students need a rigorous explanation of everything at this point, they just need to orient themselves in the generally right direction

[tardy lynx]

Thank you guys so much for the tips! I'll keep all the above in mind when teaching

[tawny slate]

oh couple more points i just remembered

i literally use a nonstandard notation for polar form complex numbers:

$$r \angle \theta$$

[burnt vessel]

Cozmogrgdfschkipkhrshtensi

[tawny slate]

i find this is just more convenient than using e^it or cis, both being shorter and carrying less baggage

so now it becomes very easy to state the product rule:

$$(r_1 \angle \theta_1 ) (r_2 \angle \theta_2 ) = r_1 r_2 \angle (\theta_1 + \theta_2 )$$

[burnt vessel]

Cozmogrgdfschkipkhrshtensi

[tawny slate]

if they know trig, they may appreciate that translating this equation to rectangular gives the trig sum identities, one of the nastiest identities to memorize and intuit, for free

[rigid coral]

Hey guys, I would love some feedback on this video, it's a geometric visualization of the Euclidean algorithm
https://youtu.be/Zi6aTbVN1sM

[tawny slate]

cadence is slightly weird in a few places, where there is actually less pause where there is a period than where there are commas. when taking splices of your recordings and throwing them into the video editor, its very easy to just jam them together, but often this results in cramped pacing, feels unnatural, a bit harder to follow. give the audience a little space to just digest what they just saw and heard, even if its half a second

i think from the viewer's perspective, it may not be entirely obvious why the 3x3's must always tile the original 9x9's, if we were to extend this arbitrarily. its not so easy to visualize this as cleanly for complex examples, and the logic for why this property must hold is not that self-evident

comparing it to the algorithmic view of the euclidean algorithm, this may lose some viewers, because while some viewers might recognize the euclidean algorithm, many might not, and that table just looks obtuse and confusing

additionally, the target audience, someone who struggles to solve the teaser problem in the beginning of the video, is likely to be someone who doesn't know what the euclidean algorithm is, imo

i would be careful of saying "this is the answer" before the actual answer of the original question is solved, because this is a common mistake students will make, solving a major step and assuming the problem is solved, and writing a wrong answer. doing it this way causes a moment of cognitive dissonance; "wait what? i thought that was just gcd"

also in these videos, it helps to go the extra mile to visualize something if you can do it. for example, the last example shows the visualization for the two big squares but then does not show the animations for the smaller squares, even though you had this in the 9/30 example. anywhere where you can just put in a bit of extra work to show something more clearly goes a long way

suddenly introducing finding the gcd as a problem about rectangles might be surprising, yes, but the viewer may be feeling more overwhelmed rather than less. they already found finding the gcd to be intimidating, and now they are being thrown geometry where they dont immediately see any connection.

i think the core point here is that 1) there is an easy way to do this, but 2) its not that easy to understand why this easy way works, and 3) most importantly, this geometric intuition makes this complicated obtuse thing simpler

im not sure I got a feel for any of these three main points in sequence before being introduced to this seemingly irrelevant geometry thing i now have to concern myself with. laying this out more clearly i think would help immensely

and i think my final point might be that if you have a single tiny thing on the screen and nothing else, make it a bit bigger and fill the screen a bit, both for visibility and accessibility. this isnt a whiteboard, people sometimes watch these on their tiny phone screens, use that space

good concept though, and if this is your first attempt at a video its a good one, good luck!

[rigid coral]

Thank you so much for the detailed feedback, how was the audio btw?

[tawny slate]

audio is mostly fine i think, not much to say there personally

[white fulcrum]

I am not sure the motivation works very well as I think that's a strange seemingly-out-of-nowhere quesiton even tho it appears on the exam. the visuals are quite nice

[white fulcrum]

re-upping this one cuz I didn't get responses

[halcyon glade]

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

https://www.youtube.com/watch?v=cy8r7WSuT1I

[turbid zenith]

I use this too!

[lime turret]

to this day i don't know what the number pi really means

[turbid zenith]

Exact same notation!

[lime turret]

i don't believe it's just about circles anymore

[halcyon glade]

It's half of tau

[halcyon glade]

I think it is all about circles, circles are just really important

[lime turret]

what about the root pi in the gamma function?

[halcyon glade]

Oh yeah here's another 3b1b video about pi and finding the secret circle lol he does a lot of these https://www.youtube.com/watch?v=8GPy_UMV-08

[halcyon glade]

it's because you get a gaussian integral and gaussians are spherically symmetric (see the 3b1b video above about the normal distribution formula)

[lime turret]

D:

i was hoping it'd be this deep analytic constant and circles just happen to be one way of expressing it

[halcyon glade]

hahaha well I think it is still a very deep constant

[lime turret]

there are number theoretic angles to it too

[turbid zenith]

Well, for Euclidean circles at least

If you use taxicab circles then π = 4

:3

[lime turret]

and instead of "find the hidden circle", why can't you just say "find the hidden number theory" or "find the hidden factorial/gamma function"?

[brave wave]

taxicab geometry is so trippy, basically just a tilted square

[halcyon glade]

I think that's fine too, but circles are more elementary than any of those and visually intuitive, so it's the easiest to relate everything to

[turbid zenith]

I’m currently teaching taxicab and other kinds of distance in one of my classes

So that’s fresh on the brain

[lime turret]

it's essentially our favourite clothing we like to dress pi in

not the essence of pi!

[halcyon glade]

For Euclidean circles too https://www.youtube.com/watch?v=VYQVlVoWoPY

:^)

[brave wave]

lol that staircase paradox is such a classic trap for calc students

[lime turret]

the fact that the gamma function shows up in hypersphere surfaces and volumes makes me think gamma is the root

[halcyon glade]

Eh sure, essence is subjective, I think finding the circle gives powerful visual intuition for any problem where pi shows up though

[lime turret]

nvm spoke too soon, it also nakedly contains pi by itself

[halcyon glade]

It's nice that a number relating to the surface areas of 1D spheres also shows up for surface areas of nD spheres

[lime turret]

what e does is pretty clear, it provides exponentials. you never really encounter a formula where e isn't being exponentiated by something.

pi usually shows up via multiplication, and curiously, it shows up with a bunch of rational powers (pi^1/2, pi^2...). like e, you never add it to anything though

the constant that likes to be added to things is euler-mascheroni. dunno, just thought this was interesting.

[brave wave]

lol that's a sick observation, what other constants did you have in mind

[lime turret]

none really, maybe the feigenbaum ones but i'm neither particularly familiar with them, nor are they arithmetic in any way

pi really likes integrals, harmonic stuff and number theoretic stuff

[lime turret]

this is what chatgpt has to say

so clearly we should call pi the "square constant" :clue:

[white fulcrum]

restating that I work with grade/high schoolers and anything I can show them in 10 min tops is greatly appreciated

last time I talked about encryption bc you want to text about your secret valentine

[tawny slate]

why would that be a square constant though? this makes no sense, this has nothing to do with squares

[white fulcrum]

chatgpt is dumb

[pure light]

it means it has to do with "squaring". although that just comes back to circles when you start adding things which are squared

[brave wave]

yeah basel problem is a classic for that, sum of 1/n^2 giving pi^2/6 is wild

[lime turret]

since it was mentioned, maybe the basel problem can be condensed to 10 minutes? that would be cool

[lime turret]

not just that, but also all of the zeta(2n) yielding funky monomials of pi is funny and absurd

[white fulcrum]

but how do I explain it without trig? most of my students don't know that yet

[brave wave]

maybe the light intensity analogy? 3b1b has a sick video on it

[lime turret]

ah if it's too technical and their prerequisites are too low, then it's not possible

[tawny slate]

that was i think the initial difficulty, why i think few people responded. if even just presenting the series doesn't qualify, buffon's needle takes too long, then i dont know how many examples there are left that can go any simpler. plenty of examples, just not many that basic

[turbid zenith]

Just gave this to my students

[brave wave]

this is perfect for thederkus, it always takes exactly mn-1 breaks no matter what

[turbid zenith]

Correct!

It’s a game of no strategy 😂

[tawny slate]

i feel like there should be a super clean elegant direct proof

but i cant see it

[turbid zenith]

Well we used strong induction to get practice with it

But you can also just count the number of unbroken components

It goes up by 1 after each move

[tawny slate]

omg im so stupid

now my stupid is on discord permanently for the entire federal government to see gdi

never been so humiliated in all my life

[tardy ember]

to be fair i also didn't see that

[tawny slate]

genuinely feel better now lmfao

[lethal hornet]

this reminds me of another game with very little strategy.

alice and bob have a list of numbers on a chalk board. they take turns erasing one number each. a player loses if they erase a number and the xor of the remaining list is 0.

the xor of the singleton list is the only number of that list, the xor of the empty list is 0. if a player starts off with a list whose xor is 0, they win automatically.

[lethal hornet]

||alice wins if the length of the list is even or the initial list xors to 0, bob wins if the length of the list is odd and the initial list doesn’t xor to 0||

[white fulcrum]

one of my fave stories is that one where gauss is told to sum 1 to 100 but uses the trick to solve it quickly

[turbid zenith]

Bitwise xor, like in Nim?

[lethal hornet]

yea, bitwise

[turbid zenith]

Gotcha

[turbid zenith]

So uh

Remember how I said I was working on my own calculus book?

https://bill-j-shillito.github.io/calculus-scbi/online/frontmatter-4.html

It finally feels like a thing!

Very very very much a WIP (more of an outline right now than anything) but I got the whole thing converted to PreTeXt!

OER here we come

[azure dove]

just took a peek at it

is chapter 21-26 for students that are interested in pursuing real analysis?

[turbid zenith]

Yup yup

Or students in one of the earlier classes who want to look ahead and see on a deeper level why some of the results are true

But formalization coming at the end instead of the beginning is a big part of the overall philosophy

[tame tulip]

is there any study comparing how well students would grasp a rings first vs a group first approach for abstract alg

[white fulcrum]

groups are simpler right? just one operation

[analog oar]

though one could argue that people have seen some rings before, e.g. from elementary number theory

[turbid zenith]

Yeah that's usually the tradeoff ... groups have one operation but are more abstract, rings have two operations but are more familiar

I'm asking my advisor if she knows of any studies

If anybody would know it would be her

(Plus gives me somethign to do other than grading, which I'm getting really tired of)

I had some students multiplying quaternions, and they were given $i\cdot j\cdot i\cdot k\cdot i$

[burnt vessel]

Solid Angles

[turbid zenith]

Here's something a student wrote:
$$i\cdot j=k\cdot i=j\cdot k=i\cdot i=\boxed{-1}$$

[burnt vessel]

Solid Angles

[turbid zenith]

(They still got most of the credit, but uuuuugggggghhhhhhhhhh)

[lime turret]

loll

[dim blade]

I believe there isn't a comparative study, only pedagogical literature arguing for a certain method

[tame tulip]

LOL

[tame tulip]

Why does # of ops matter

[dim blade]

More axioms

Idk, it's really hard to tell where the confusion is inside your student's head if you were never confused yourself to begin with

[turbid zenith]

The Curse of Knowledge.

[tame tulip]

Yeah lol

Groups and rings don't seem any diff from vector spaces in the barrier to figure them out

[halcyon glade]

Groups are def more abstract, in terms of the examples usually given

Rings you can start with asking what holds in common between Z, Q, and R

[turbid zenith]

Why are textbooks like this :V

[halcyon glade]

Okay this is hilarious

If I were a student I would be very amused doing this problem

[turbid zenith]

Maybe so, but I wouldn't call this an application 😛

[quasi maple]

"If Linda leaves the squirrels be, they gon fuuuuu"

[turbid zenith]

Honestly is this kind of curve fitting ever used?

Like using a rational function like this?

As in, you've got the current value f(0) = a, and the intended asymptote y = b, so you'd set up f(t) = (a + bkt)/(1 + kt) for some k

[tawny slate]

i know for a fact that ive used this before i just cannot remember where or when or why

[turbid zenith]

See that process would ACTUALLY be interesting

Not just "here's a function handed down from upon high"

[tawny slate]

oh holy crap i think i remember

so in vertical scrolling rhythm games, like ddr, guitar hero, pop'n music, rift of the necrodancer, etc

there is a pattern where you have to press a single button repeatedly in quick succession. the community defines this pattern as a "jack", short for "jackhammer", so ill call it accordingly call it a jack here

[marsh thistle]

I mean this sounds more like you have boundary conditions for some ode, and the squirrel problem is a perfect example of that because the solution of the logistic equation is expressible in terms of y_0 and y_inf. the rational function looks a bit weird so I doubt there's any significance to it

[tawny slate]

the question is, if you wanted to define a difficulty metric, how would you do so? it turns out this is an absurdly hard question with so much subjectivity and data analysis required that i will simply metricize one highly specific aspect of this: how would you assign a difficulty to jack speed?

it turns out there is a human limit to how quickly you can press a button, so that is the asymptote. you also have a baseline scaling difficulty, where the speed of the jack is so slow that it functionally shouldn't change anything, so depending on how you utilize this metric, you can set a f(c) = 0 or 1

for the scaling you would basically do some kind of curve fitting. given levels of a particular difficulty, you would expect that speed within a particular range is reasonable, but anything beyond that should impact the difficulty in a significant way, so that gives a specific point

not sure how helpful that is to you, perhaps you can generalize this in some way to, say for instance, assigning difficulty to some other hardcapped feature, like human reaction time

[turbid zenith]

. . . wow I did NOT expect a connection to rhythm games

I'm very very very familiar with jacks

I remember the early days when Can't Stop Fallin' in Love Speed Mix was notorious for them 😛

[turbid zenith]

Logistic does make more sense here

I figure the rational function was just because they wanted to test understanding of limits of rational functions

And so they felt the need to shoehorn an "application" into it

[halcyon glade]

I think rational functions are natural in any case where you know the asymptotic behaviors

[turbid zenith]

A lot of the applied rational functions I've seen tend to just be f(x) = k/x

[tawny slate]

ah i totally forgot you are part of that, i remember now haha

[halcyon glade]

although if you allow translations horizontally and vertically though and dilations, that's just the same thing as smth like f(x) = (ax+b)/(cx+d)—maybe a good point to bring up

[quaint basalt]

was thinking of trying to make some lecture-style videos for some niche topics in the future but my handwriting is not very good. https://youtu.be/aS6R3epK3ms?t=484, any ideas what software he is using to write and present in this video or is he just writing very small and neatly?

[tall bolt]

Appears to just be hand written using a writing tablet of some sort

[dense lichen]

Google notebook...

[vestal tangle]

I wouldn't call that neat tbh. I find it hard to read

[hybrid river]

as a TA i have to check answer sheets , students are not doing good, how to maximize the marks so that most of them pass

even that is difficult

no we dont get complaint, but failing them isnt an option

okay will try to add hints in paper

[halcyon glade]

why?

[hybrid river]

students commiting suicides here

[slim path]

Then this is not really an evaluation problem for math educators but for psychological counselors. Doing poorly in math shouldn't equate to suicide unless you are mentally affected by it in a way that merits the help of a professional.

Also, as a TA it might be in your best interest to discuss this with the professor teaching the course.

[quasi maple]

Yeah, I would strongly suggest this be sought out as a board discussion, especially if it's commonplace enough to be able to drive that sort of fear into a teacher

It's not that beneficial to give people minimum passes if they're still far below that requirement, nor is it beneficial to induce... such a thing from a test paper

[turbid zenith]

...I hate this SO much.

[halcyon glade]

People are gonna start bringing AI holograms to oral exams

[turbid zenith]

lol welp

[midnight scarab]

Feynman teaching Einstein is a bit anachronistic (yes, this is obviously the only thing wrong with this)

[midnight scarab]

"teaching busywork", "what actually matters", "reclaim your time" 💀

[turbid zenith]

Yeah at first I was like "okay maybe a good use of AI might be identifying patterns in student work that you might not have noticed"

Then I saw it talking about learning to imitate you and teach for you

Uhhhhh no that's what I LIKE doing

[tawny slate]

hey i have a great idea, why dont we replace teachers with AI, then replace students with AI, win-win

[turbid zenith]

Okay enough about this

It's making me angry

Something more practical ... has anyone ever found a good way to explain why we have both $\lim\limits_{x\to a}\dfrac{f(x)-f(a)}{x-a}$ and $\lim\limits_{h\to 0}\dfrac{f(x+h)-f(x)}{h}$ for defining the derivative? As an answer to ``why do we need to learn both''?

[burnt vessel]

Solid Angles

[turbid zenith]

Like, we know why they're the same

But I've always found it annoying to try to justify you'd need both when you could just use one the whole time if needed. If anything I often find the former easier, say if you're proving the power rule for integer exponents

You just split up $x^n-a^n$ instead of needing to use the binomial theorem

[burnt vessel]

Solid Angles

[turbid zenith]

So I'd be interested in hearing how y'all frame it

[plain pebble]

You never need both
But like you said, sometimes one form is slightly more convenient
Like almost everything in maths with multiple forms

[turbid zenith]

Have you ever found the latter form to be the easier one to use in certain situations that you can think of off the top of your head?

[midnight scarab]

More compact if you Taylor expand

More seriously, the 2nd version matches delta f / delta x, delta x -> 0

[plain pebble]

For exponentials it’s easier

[turbid zenith]

Oh interesting

[midnight scarab]

I think it's also a question of logic

My starting point is always Taylor expansion

[turbid zenith]

So yeah for that section of my book what I want to do is show both and, for each, give an example of where one is more convenient than the other

[midnight scarab]

(Call me a physicist)

So introducing an increment and just expanding f(x+h) is more natural

[plain pebble]

Because you factor out the b^x, and the remaining term is purely a constant, and that’s more obvious to think of in the f(x+h) - f(x) form

[turbid zenith]

So the funny thing is, the Taylor series I've learned is closer to using f(a)(x-a) instead of f(x+h)x

[midnight scarab]

My point is more like, I don't naturally start with the difference f(x) - f(a)

[turbid zenith]

I know they're equivalent but the version of Taylor series you see in most calculus books is $$\sum_{k=0}^\infty \frac{f^{(k)}(a)}{k!}(x-a)^k$$

[burnt vessel]

Solid Angles

[midnight scarab]

Whether I call the original point x or a is just a matter of if I think of it as fixed or variable

[turbid zenith]

Yeah, that's an interesting conversation

But actually talking about it now, I think I can see why certain functions would be more convenient with the x+h version

I would have a much harder time finding $\lim\limits_{x\to a}\dfrac{\sin x-\sin a}{x-a}$ than I would $\lim\limits_{h\to 0}\frac{\sin(x+h)-\sin x}{h}$

[burnt vessel]

Solid Angles

[turbid zenith]

The latter screams "use the sum identity"

[midnight scarab]

And here I was thinking of this as an example where the first form is more natural lol

[turbid zenith]

Okay I think that's the connection I need

[turbid zenith]

LOL really?

What would you do if you were trying to show that the former comes out to cos a?

[midnight scarab]

Just sin a - sin b identity. I know that's what I have, and I want to rewrite

Ideally as a product

[turbid zenith]

Huh, I don't think I know the sin a - sin b identity off hand

[midnight scarab]

With something that depends on x-a

[turbid zenith]

I would have to look it up

Whereas sin(x + h) = sin x cos h + cos x sin h comes to mind immediately

[midnight scarab]

Oh in that sense. Yeah that's true

[plain pebble]

exponentials in disguise /hj

[turbid zenith]

That doesn't come in until much later 😛

[midnight scarab]

I generally derive trig identities from sin(a+b) etc

Though tbf there was a point in life wherebI remembered both, and now I'm not sure of either so yk

[halcyon glade]

Why do we need both 6+7+8 and 8+7+6? Seems to me like they're just two ways of writing the same thing :P

[turbid zenith]

So you'd have $$\lim\limits_{x\to a}\dfrac{\sin x-\sin a}{x-a}=\lim\limits_{x\to a}\dfrac{2\cos(\frac{x+a}{2})\sin(\frac{x-a}{2})}{x-a}$$

. . . and again it's a dead end I feel like

[burnt vessel]

Solid Angles

[midnight scarab]

Yeah. I think it's just about which form is more "inspiring"

[turbid zenith]

Okay, this I can absolutely get behind

Pick the one that lets you do the thing.

If I've proven the Power Rule using the x-a version, I don't need to redo it using the h version for any kind of completeness's sake. They're not telling me two different things.

[midnight scarab]

Well, now you use sin h/h

Or sin h ~ h

[turbid zenith]

Yeah I guess you would need to substitute h = x-a anyway

[midnight scarab]

Yeah, you need to pass to a result about lim at 0 of sinc

[lime turret]

protip: for all the funky addition theorems, use the exponential description of sin and cos

and if your class is ready for that too, show them

[turbid zenith]

Again, love it, but that's Calc II or later. 😛

That exponential description depends on Taylor series, and I'm talking about a "we just learned limits" environment.

So it's elegant, yeah, but it's elegance in hindsight.

[turbid zenith]

Well see clearly 8+7+6 won't get students laughing

[midnight scarab]

Yeah, I think it's important to teach the students that

That you can change variables in limits

I think it's actually a very common point of confusion after a course in calc/analysis

[turbid zenith]

Hmm, that's a good point

I was actually thinking of using some different variable names when first teaching it:
$$f'(a)=\lim\limits_{b\to a}\dfrac{f(b)-f(a)}{b-a}\overset{h=b-a}{=\joinrel=}\lim\limits_{h\to 0}\dfrac{f(a+h)-f(a)}{h}$$

[burnt vessel]

Solid Angles

[turbid zenith]

To remove some of the baggage of "x" being variable in one definition but fixed in another

Good idea? Bad idea?

[white fulcrum]

some of my worksheets on funciton notation have variable names like a sun emoji to emphasize the principle that variables can represent whatever we want and we can call them whatever we want

[midnight scarab]

My year 1 prof be like

[turbid zenith]

For all n in R, for all ya > 0, there exists ka greater than 0 such that for all ai in R

Les variables sont ... quoi?

I can understand the rest of the French

[midnight scarab]

Muettes

(Mute)

[turbid zenith]

. . . huh

Je ne comprends pas

[midnight scarab]

It's the French expression to say the name is arbitrary

[turbid zenith]

C'est une expression idiomatique?

[midnight scarab]

Or more specifically the variables aren't bound

[midnight scarab]

Oui

[turbid zenith]

J'apprends encore le français 😛

[midnight scarab]

So you'd say the variables in a sum/integral are "mute"

I.e. you can rename them

[turbid zenith]

Interesting!

[midnight scarab]

Not sure where the expression came from exactly, though I guess it makes a bit of sense

[midnight scarab]

Nice

[halcyon glade]

I think that's sensible

[midnight scarab]

Idk how "natural" b is since it's more of a parameter name than a variable name. Though it's true it's the natural pair to a.

But in a sense, the definition is not symmetric in a and b, even though the ratio is

[marsh compass]

variables can represent whatever we want and we can call them whatever we want

[midnight scarab]

See my prof's example with n in R

Sacrilegious

[marsh compass]

i get what you're saying though

[midnight scarab]

But yeah, variable names are just names, but they carry context from conventions

Which is very important

If someone were to call their functions x,y,... and points f,g,..., that'd be very confusing

[marsh compass]

That reminded me of my Christmas submission during undergrad

[pure light]

i've seen textbooks use "a" when they want to define the derivative at a point, and "x" when they want to define the derivative as a function. i can agree that using "x" in the limit can be confusing for that context, another solution i've seen was in thomas where they used the limit as z approaches x

[white fulcrum]

I'm looking to find a paper on why we do math, but it's one that readily admits that most people don't need to know math

it was 3 pages max

it occurrs to me this might not narrow it down a lot but I'll also take anything similar

[quasi maple]

c'est comme en anglais "dummy variable"
-# It's like the English term "dummy variable"

Though ig in formal logic, "bound" is closer

Wikipédia at least contrasts this with "variable libre" / "free variable" (which would make the equiv. English term for "muet" "bound"

[marsh compass]

For me, looking at the generalization to the multivariable or complex case, the former is more in line with a total derivative, while the latter is closer to the definition of a directional derivative.

[tame tulip]

when i took calc bc, the f(x) - f(a)/(x - a) form was useful when i had to evaluate a limit that took that form

so like ummmm

i cant come up with an example on the spot that isnt extremely trivial with normal limit tricks but there were questions that would make you solve a limit of that form by recognizing it was f'(a)

also you probably strengthen the understanding of the derivative by introducing both forms

[marsh compass]

Of course, in the one-dimensional real case, one is only interested in the direction of v = 1, so

\[\frac{\del{f}}{\del{v}}(x) = \lim_{t \to 0} \frac{f(x + t v) - f(x)}{t}\]

for the directional derivative, where t v plays the role of h.

[tame tulip]

[burnt vessel]

Zan (鹿乃 #1 Fan) ❀

[turbid zenith]

Is it weird that I don’t think I’ve seen this form of the directional derivative?

Does v need to be a unit vector?

[tight star]

I thought this was the usual definition of the directional derivative

[marsh compass]

v does not have to be a vector of unit length

the term "directional derivative" does not quite fit semantically when you extend it to vectors with arbitrary length, but i can't think of any better name

[turbid zenith]

Somehow I missed this in our book

Looking in Stewart, it claims you do need a unit vector

Which I suppose makes sense

But I had always just taught it as gradient dot unit vector

Wow, I feel kind of dumb 😛

[marsh compass]

this would be the case if the function is (totally) differentiable

[marsh compass]

then it is also always possible to extend it linearly to non-unit vectors

[turbid zenith]

Also the $D_{\mathbf{\hat{u}}$ notation always annoyed me, I'd never seen $\displaystyle\pdv{f}{\mathbf{\hat{u}}}$ before. I'm going to have to steal that.

[burnt vessel]

Solid Angles
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[tight star]

Honestly I’m not sure if I really like the directional derivative

[marsh compass]

\[\frac{\del{f}}{\del{n}}\]

is, for example, a quite common notation in PDEs to denote the derivative in the direction of the normal at the boundary of the domain

[burnt vessel]

Zan (鹿乃 #1 Fan) ❀

[marsh compass]

but there are a lot of other notation like

\[\del_n{f}, \, D_n{f}, \, (D{f}) n, \, (n \cdot \nabla) f, \, \nabla{f} \cdot n, \, \nabla_n{f}, \, \text{etc.}\]

[burnt vessel]

Zan (鹿乃 #1 Fan) ❀

[marsh compass]

What don't you like about it?

[turbid zenith]

I"m wondering this too

[tight star]

I don’t like it as a way to think about the derivative of scalar-valued functions

You already have these issues of what exactly counts as a direction

[marsh compass]

Yeah, I have an issue with the naming, not with the definition

[tight star]

But I also think the formula looks a little too similar to the derivative of a vector-valued function

[marsh compass]

But again, I can't think of a better name

[tight star]

I actually remember being fairly confused about the difference between the derivative of a function R -> R^n versus R^n -> R

[marsh compass]

Oh, so you don't agree with the definition?

[tight star]

It’s not that, it’s just I think it’s overemphasised maybe

[marsh compass]

Hmm, I still don't quite get what you want to convey

[marsh compass]

I mean, it is the same

[turbid zenith]

Well, thank you for pointing out that definition that somehow I hadn't processed I didn't include

https://bill-j-shillito.github.io/calculus-scbi/online/sec-directional-derivatives-and-the-gradient.html
This will help with how I treat it in my book, as pointing out that if you write it in this kind of vector form you can generalize to higher dimensions

[marsh compass]

Oh, did you just include the notation

[turbid zenith]

Both the partials notation (more cosmetic than anything) and the vector definition of the limit at the end

[tight star]

Right but they’re quite different objects

[tight star]

This is a big part of it

[marsh compass]

What are we comparing again?

[tight star]

Directional derivative of a function R^n -> R versus ordinary derivative of function R -> R^n

[marsh compass]

I think I'm too deep in the realm of derivatives of functions between two differentiable manifolds

[marsh compass]

I would say that the "ordinary" derivative is a special case of the directional derivative in the direction of the chosen one basis vector of R

[halcyon glade]

I'd say this is a theorem

And as a corollary, you can calculate the gradient (direction of steepest ascent) directly from the partial derivatives

[tight star]

Hm I guess I wouldn’t say this

[turbid zenith]

Or a way to motivate the definition of the gradient

[marsh compass]

I prefer to introduce the Jacobian matrix first before the gradient

[marsh compass]

Why not?

[marsh compass]

What's your preferred way then?

[tight star]

i think in terms of the linear approximation point of view

[marsh compass]

Yeah, sure, but I was more talking about the value \lim_{x \to a} (f(x) - f(a)) / (x - a), which is a particular representation of the derivative as a linear mapping and given by the directional derivative in the direction of the standard basis vector

[tight star]

hm right

[turbid zenith]

That’s a little too abstract too fast for me

[tight star]

hm what is abstract about the jacobian matrix?

[marsh compass]

The issue I have with introducing the gradient as the vector of partial derivatives is that this is not true anymore if you work with more general coordinate systems and might be potentially confusing for advanced students later on, while the definition of the Jacobian matrix stays the same.

The Jacobian matrix is the more natural notion as it does not depend on the choice of inner product in your space.

But I can see how this might not be so easy to introduce to pre-university or first year students if they haven't dealt with linear mappings and their matrix representations before.

[tight star]

It does really feel like calculus and linear algebra should go hand-in-hand

They really seem to help each other

[marsh compass]

calculus: linearization, linear algebra: dealing with linear(ized) stuff 😎

[halcyon glade]

I think the definition is just the direction of steepest ascent

And then the theorem is that you can write it as the vector of partial derivatives

That way I think maximizes the suspense at wanting to figure out how to calculate this mysterious gradient object

[marsh compass]

Hmm, I would say that this is a theorem, too. The definition that I would use for the gradient (corresponding to an inner product) is that it is the Riesz representant (a vector as opposed to a linear functional) of the differential (derivative as a linear mapping) in the sense that <∇f(x), v> = df(x) v for all (tangent) vectors v.

[halcyon glade]

I think I would rather that as a theorem in a multivariable calculus because the definition doesn't seem very well-motivated

Like okay it's just the transpose of the total derivative, why should we study that

[marsh compass]

The total derivative is not an element of the original vector space, so it does not describe a direction in there, while the gradient does

[halcyon glade]

I understand, that's why I said the transpose (turning the covector to a vector)

I just think it makes sense to introduce something as a question, e.g. if we have a surface how can we figure out which direction will get us up the surface the fastest, so that you can get students engaged with a tangible image and have them start thinking about it themselves and wanting to know the answer

And in this case the answer is just remarkably simple

[marsh compass]

Okay, I can see your point of view. With the steepest ascent interpretation, you would also be able to generalize the intuitive notion of the "gradient" to normed vector spaces without an inner product.

In this case, how would you specify the magnitude of the gradient?

[marsh compass]

From this definition, you only get a direction at first. What if you, for example, want to talk about stopping the gradient descent method after the magnitude of the gradient becomes sufficiently small?

I think I would stick to the definition of the gradient that only works for inner product spaces, so that you can still talk about more general spaces that do not admit a gradient.

I would refer to directions of steepest ascent as just that: a direction of steepest ascent.

[halcyon glade]

How fast you ascend

[marsh compass]

Okay, so

\[u = \argmax_{\substack{v \in V \\ \norm{v} = 1}}{\norm{\del_v{f}(x)}}, \quad \alpha = \max_{\substack{v \in V \\ \norm{v} = 1}}{\norm{\del_v{f}(x)}}, \quad \nabla{f}(x) = \alpha u\]

[burnt vessel]

Zan (鹿乃 #1 Fan) ❀

[marsh compass]

I think I can live with that

[turbid zenith]

Okay back XD was teaching

[turbid zenith]

So first off for this, the idea of a general linear transformation is definitely on the abstract side

I wouldn't want to go into it until I've worked my way up to functions from R^m to R^n

And a big part of Calculus III for me is building student's intuition on what higher dimensions even are, so I want them to build intuition first with R -> R^n (curves) and R^m -> R (surfaces) before getting into R^m -> R^n in Calculus IV

Then I'd rather introduce the Jacobian as the generalization of all the derivatives we've seen so far

[turbid zenith]

This approach I can ABSOLUTELY get behind, it has enough for students to sink their teeth into

And if I understand it right, if $\mathbf{u}$ is the unit vector in the direction of $\nabla f$, then $$\nabla f\cdot \mathbf{u}=\nabla f\cdot\frac{\nabla f}{\Vert\nabla f\Vert}=\Vert\nabla f\Vert$$

[burnt vessel]

Solid Angles

[turbid zenith]

So you could define the gradient as the vector whose direction gives the maximum directional derivative, and whose magnitude is that maximum value

[turbid zenith]

Okay yeah this!

I just had to work through it on my own lol

[marsh compass]

I'm trying to figure out how to derive the characterization <∇f(x), v> = df(x) v for all v from this definition, but this doesn't seem too trivial (the converse is clear to me)

[tight star]

Right, but there are lots of specific linear transformations that are easy

[vestal tangle]

if linear algebra came before calculus in the curriculum multivariable calculus could be taught in a much nicer way

[marsh compass]

fortunately, that's how it's done in the programme i took

[halcyon glade]

yea!

[halcyon glade]

I took calculus -> lin alg -> multivar calc, but unfortunately multivar calc is needed for physics e+m so it's hard to do it in this order if you have to support physicists and mathematicians in the same class

my undergrad teaches calculus -> multivar calc -> lin alg + odes, but starts the multivar calc with a good amount of introductory lin alg, I think it's helpful

[turbid zenith]

I think what I want to do is drop hints of LA where I can in Calc III and IV

But I’m still opposed to doing the Jacobian first. In general I’m very wary of starting with the most general thing rather than building up to it.

Thinking of a function as transforming the entire line or plane or space is more abstract for students to understand than thinking of them in terms of inputs and outputs.

So starting with Jacobians seems kind of like that SMBC comic about teaching fractions.

[midnight scarab]

To add to everything that's already been said, I think starting with the definition of directional derivative as derivative of f(x+tv) is a good idea, because it encodes the idea much more directly than the dot product with the gradient, and more importantly reducing a multivar function to a single var in this way is a natural and very important idea that shows up in the proof of many multivar calc results, like the MVT

I also think that the fact that the directional derivative is actually a linear function of v for "nice" functions is non-obvious and "cool"

And I think it's a good idea to present this as a result, not a definition

(Proving continuous partial derivatives along coord axes => directional derivative = v . grad f is also a nice application of MVT)

[Also, a nice thing to show is that the directional derivative only depends on the velocity of the path at the point, i.e. you can replace the line x+tv by any other curve with velocity v at zero. It's a nice illustration of the locality of the derivative]

[tight star]

Ok I think I actually like this perspective

[white fulcrum]

I had the week off due to a combo of snow and sickness, I miss tutoring 💔

[still elm]

How do you guys grade proofs that are correct but have missing justification for a step

showing a chain complex is contractable, a short exact sequence arises and it splits

the chain complex is assumed acyclic and free (and bounded below)

however it splis because of an inductive argument at C_2 --> C_1 --> C_0

student writes it splits as C_N is free

rest is okay

[turbid zenith]

I would mark it as most of the credit, circle the issue, have them revise and resubmit it.

[still elm]

would you do that for a closed book exam

midterm

[turbid zenith]

Yes.

[still elm]

i see

thank you!

[turbid zenith]

Students should be able to revise work and get credit for it.

[still elm]

I agree!

[turbid zenith]

Fair warning though a lot of people don’t agree ;P

Because something something ✨ RIGOR ✨

[still elm]

idk i think if the goal is to train students for research (a grad school in pure math?) then assessments should be as "close" to research as possible

making mistakes (both stupid and not stupid) retracting them, reviewing and then making another one is part of it i guess

so i tend to agree haha

in fact, closed book timed exams shouldnt be athing imo

[turbid zenith]

I’ve had to go back to timed exams but I let students make a cheat sheet

Great way to study

Among other things of course such as working problems

[still elm]

yeah i see

[still elm]

any other opinions?

[midnight scarab]

I think it's my mom (or maybe my dad?) who told me the story of a classmate who spent so much time and effort writing a litetal cheat sheet, rewriting it to be tinier, that he ended up remembering the matetial

[turbid zenith]

Happens all the time!

[midnight scarab]

I think everyone agrees this would deserve partial credit. Traditionally, you'd have a gradibg rubric where you assign a certain amount of points for each idea, so the student would get all points except those corresponding to the steps they skipped because they stated something wrong

[still elm]

the problem is supposed to be out of 5

and i dont have a grading rubric

the original proof instrucotr had in mind was by inductoin

induction

like constructing the chain homotopy itself by induction

[midnight scarab]

Well ig I'd just go with a gut feeling of the relative importance of the part the student missed

[still elm]

i see

thank you

[turbid zenith]

Speaking of rubrics I should post my newest iteration of my grading system I’m using in my upcoming calculus courses if anyone wants to see it!

[lethal leaf]

this is half the reason people allow cheat sheets

My UG analysis class said "two of these 10 proofs from class will be on the exam" so that everyone would sit and memorize like 10 proofs

which made you really understand the material (easier to understand 10 proofs than memorize them word for word)

[midnight scarab]

Plus you can cover a large span of the material with just 10 thns

[tall bolt]

My UG allowed a sheet into a lot of the exams and they just explicitly said “the sheet probably won’t be useful, but the process of making it will be”

Really useful way to revise, and we just had very little bookwork or like known material in the exams

[lethal leaf]

yea I think it's a great policy pedagogically

especially when you can no longer feasibly do take home exams with LLM mumbo jumbo all about

[tall bolt]

If the PhD I’m starting decides to get rid of take home exams the year I start

[tall bolt]

For all cheat sheet exams are better, I’m just generally sick of sit down all or nothing exams

[lethal leaf]

my graduate real analysis 1 class got rid of take homes due to LLMs

basically the prof said "here are the notes, there are many exercises. A portion will be on the homeworks, and I will pick questions from the notes to be on the exams."

[rapid tusk]

i think the first time i did one of these was back in precalculus

our teacher had us write one up before each test; although we wouldn't be allowed to use it on the test it would be collected and graded along with said test

[empty gull]

In my entire UG we never got homework lol

And people wonder why so many ppl fail exams

[rapid tusk]

bruh

[empty gull]

And we never got cheat sheets either

Sometimes we'd get formula sheets that were pre made

[rapid tusk]

the only class where i had no actual hw was calc3

[empty gull]

It's cool that they tell you which proofs you have to learn

We have oral exams here where you have to know all the proofs and all the lemmas in the entire lecture notes lol

Really makes you understand everything

Tho the amount of stress I go through during exams isn't something I'd recommend

[halcyon glade]

Do ppl actually use LLMs to cheat in a graduate class, that's so sad

[halcyon glade]

I'm a big fan of oral exams but I see why people don't like high-stakes exams in general

[lethal leaf]

I'd wager people would if they were desperate to satisfy a qualifying requirement necessary to stay in the program

Such as real analysis 1

[quasi musk]

Yes, I've seen many do that

Many times they don't pass the qualifying exam

and complain that the quals are too hard

I wish my school had an oral examination based on your research topic

We just do a proposal for a thesis, and partial progress for our oral

[quasi musk]

I'm ok with revision and for credit, but I do question at what point can we expect students to answer basic questions about their major, and answer in a precise way

[still elm]

yes but i think basic is being worded like

being used in a malueable way

qualifying problems, for examples i woludnt expect them to be too basic

idk

[quasi musk]

I have many math majors in my real analysis that don't know how to structure an induction proof, or how to set-up epsilon-deltas, etc.

[still elm]

i see

[quasi musk]

They can only solve problems they've been explicitly shown before, that is the same. Any tweak to the problems, and it's out the window

I'm ok with revision, and for credit, and it can be a great learning moment. But I also wonder when they can just answer the basic examples or counter-examples without any help

[turbid zenith]

To some extent I think that’s what a final exam can be good for

That’s when you ask the “okay can you do the thing” questions I think

Oh okay so speaking of this, lemme tell y'all what I've got planned for next semester in calculus for a grading system

. . . . . . . uh, my message got blocked

Can any mods see my post and tell me what happened?

I'm gonna try posting little bits of it and see what's getting blocked...

So let's say we're talking about Calculus I. I've sorted the class into broad Topics, and you can earn Stars (⭐) toward those Topics:

• (0.) Basic skills ⭐
• 1. Differentiation ⭐⭐⭐⭐
• 2. Applications of derivatives ⭐⭐⭐⭐
• 3. Integration ⭐⭐⭐⭐
• 4. Differential equations ⭐⭐⭐⭐
• 5. Limits ⭐⭐⭐
     So there are 20⭐ total. If you get all 20, you get an A+, 19 = A, 18 = A-, 17 = B+, etc, down to 9 or below = F.

The way you earn those Stars is by earning XP. (Yes I like the gaming terminology. :P)

• For Topic 0 (Basic Skills), that's Precalculus stuff that you need to have mastered to do well in Calculus. Basic algebra/geometry stuff. There are some assignments for that online, you need 100% on all of them to get the one Star, but you get unlimited retries.
• For Topics 1-5, let's say there are 100 XP total you can earn; a maximum of 50 XP can be earned through things like classwork or homework (though there's a little bit of flexibility on how in some cases), and 50 XP come from the exams. Exam 1 covers Topics 1 & 2, Exam 2 covers Topics 3 & 4, and the Final Exam covers all Topics 1-5.
  So my scale is 50 XP = ⭐, 70 XP = ⭐⭐, 90 XP = ⭐⭐⭐.