math-pedagogy

[turbid zenith]

Okay, but was that the very first time you ever heard of i?

[spiral elbow]

yeah, I guess you need some rug shoving to begin with. But is it a good idea to start with infinitesimals when in real analysis you're just going to be using limits and epsilon delta?

[turbid zenith]

Depends how you're using them I suppose.

Personally, my way of doing it is that I don't actually teach arithmetic with infinitesimals. I use them as a visualization tool to conceptualize the derivative and integral, but when it comes to showing why those tools work, I introduce limits, although I tell students that more recently people have figured out how to make infinitesimals rigorous

[spiral elbow]

I'm genuinely not sure how rigorous an intro calculus course should be, but personally I like it when it's not too handwavy, so that I don't feel "misled" so to speak when I learn the more rigorous approach later

[turbid zenith]

But I've found that whenever I have more explicitly taught about infinitesimals in a class for one reason or another, students are very comfortable with conceptualizing how they "should" behave

[turbid zenith]

Hot take: judicious handwaving is a skill.

I'm a fan of handwave first while we're playing around, clean up later (within the same course). To give students an idea of what the point of rigor is.

[spiral elbow]

yep, when you can introduce the rigor in the same course it's a lot more defensible to handwave to begin with imo

[tight star]

how do they do this if they don't know infinitesimal arithmetic

[turbid zenith]

What do you mean?

[tight star]

hm well you say you don't teach arithmetic with infinitesimals

but i would think you'd need that to use infinitesimals for limits

[turbid zenith]

So you're talking about in the calculus course still?

[tight star]

yeah

[turbid zenith]

Well, okay, say take derivatives to start with... my way of doing it is kind of a blend of the two.

Take a function, start looking at closer-and-closer slopes, notice that they're approaching some fixed amount. Then suggest that if you were to pick two points that were "infinitely close together" you'd have the slope of the tangent line. Then we use that derivative for almost the whole rest of the semester, same with the integral, but it's all built on a temporarily vague idea of infinitely close.

Toward the end, the last unit is the "why did this work?" unit. We talk briefly about the history, how the people who invented calculus played fast and loose with infinitesimals, and we define the limit instead to clean things up and do everything based on real numbers. And that's when I introduce the limit definition of the derivative, and we do some proofs using it.

[tardy ember]

honestly i think a lot of the basic behaviour of infinitesimals is kind of just, fairly guessable?

they just act lIke they're really small real numbers

[turbid zenith]

From what I've seen, the usual way of starting with limits has a lot of students wondering why they're doing the things they're doing ... then they learn the limit definition of the derivative and struggle a lot, but once they learn the Power Rule they feel cheated. "Why didn't you just teach us that to begin with?"

By introducing limits at the end, they already are comfortable with the Power Rule and other derivative ideas. They have a good feel for how derivatives work. So the limit definition actually adds value because now it opens up the hood and adds a more solid foundation (and gives an idea of what rigor brings to the table).

[tight star]

i'm fine with these, it's infinitely small real numbers that confuse me

[tawny slate]

ive been traumatized by #foundations , at this point im not convinced the "behavior" of anything nontrivial should be taken for granted

how does the power set work? at this point, who can really say?

[turbid zenith]

Rigor is relative!

[tardy ember]

i mostly just mean things like "if you multiply an infinitesimal by a real number the result is still infinitesimal"

[tawny slate]

my view is that in the case of complex numbers, yes we can arbitrarily introduce the imaginary unit and then try to discover its properties, but it is precisely these properties that we care about and is part of the important ideas we are trying to learn, it feels like it is the point

but with these infinitesimals, it feels more like you're inventing a totally unhinged branch of math, albeit consistent, that has nothing to do with calculus, just to say that calculus works, in an abstract way that is not clear, like attaching a supercomputer to a bicycle

[tardy ember]

because like, if you have an infinitesimal $\varepsilon > 0$ and two real numbers $r, s$ with $r > 0$ then $\varepsilon < \frac{s}{r}$ and therefore $\varepsilon r < s$, so $\varepsilon r$ is positive and less than every real number (because $s$ was arbitrary) so it's a positive infinitesimal

[burnt vessel]

bee [it/its]

[tardy ember]

(you can also do this with negative numbers, i'm just restricting to positives so i don't have to deal with thinking about signs)

[turbid zenith]

Starting with limits (especially if including epsilon-delta) feels the exact same way to me.

Especially when you just throw out the limit definitions as soon as you learn the calculation tricks

Introducing Riemann sums and then getting to FTC as quickly as you can so that you never have to do Riemann sums again

[tight star]

riemann sums are important for recognising when a problem is an integral problem

[tawny slate]

personal opinion, at least limits start with something tangible and visual, an approximation, that gradually approaches the "ideal", so if you accept the rough approximation that is explicit and concrete, it is not too much of a stretch to try to improve the accuracy of that approximation

infinitesimals dont feel "real", they feel like an arbitrary abstract construction

[tardy ember]

this is a rigorous valid argument under NSA, but i think it's also just, intuitively reasonable given the concept that "infinitesimals" are a thing at all, even if you don't know how to make them precise

[tight star]

you can't put infinitesimals into a calculator, but you can put complex numbers (on some calculators)

[turbid zenith]

I would argue that an infinitesimal is just as visualizable and tangible as 1/1,000,000 😛

[tight star]

well you can put that fraction into a calculator

you can't put an infinitesimal into one, though

[turbid zenith]

And this is a limitation of calculators not a limitation of the concept 😛

[tardy ember]

as it happens there is a meaningful mathematical sense in which NSA infinitesimals are fundamentally less concrete than most things

i think the best way to get at the problem at this level is to ask, for an infinitesimal $\varepsilon$, what's $\sin(\frac{1}{\varepsilon})$?

[burnt vessel]

bee [it/its]

[tight star]

i'm illustrating how they're not as tangible

[tardy ember]

obviously when infinitesimals are involved it can be tricky to give exact values but you can just ask for the first five decimal digits and this "shouldn't" be a problem

[tight star]

for that i think calculators are relevant

[tardy ember]

and in the NSA formalism there is an answer, in the sense that each particular infinitesimal will give you something

you just get different answers from different infinitesimals

so infinitesimals contain more information than you might be expecting

we can take this further: from any infinitesimal $\varepsilon$ we can get an infinite "natural number" from $\left\lfloor \frac{1}{\varepsilon} \right\rfloor$, let's call this $\omega$

[burnt vessel]

bee [it/its]

[tardy ember]

then for \textit{every} $X \subseteq \mathbb{N}$ we can ask whether $\omega \in X$

[burnt vessel]

bee [it/its]

[tardy ember]

and this gives a subset of $\mathcal{P}(\mathbb{N})$ with these properties: \

1. for any $X$, either $X$ or $\mathbb{N} \setminus X$ is in the set, because either $\omega$ is an element of $X$ or it isn't \
2. for any $X$ and $Y$ in the set, $X \cap Y$ is also in the set, because if $\omega \in X$ and $\omega \in Y$ then we also have $\omega \in X \cap Y$ \
3. for any $X$ in the set and $X \subseteq Y$, $Y$ is also in the set, because obviously if $X \subseteq Y$ and $\omega \in X$ then $\omega \in Y$ \
4. there is no natural number $n$ for which every $X$ is in the set iff $n \in X$, because for any $n$, the set ${x \in \mathbb{N} : x > n}$ contains $\omega$ but doesn't contain $n$

[burnt vessel]

bee [it/its]

[tight star]

wait do we know $\omega \in \mathbb{N}$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tardy ember]

(to be clear i'm being a little loose with notation here, $\omega$ is not actually a natural number and so if $X \subseteq \mathbb{N}$ then strictly speaking we should never have $\omega \in X$)

[burnt vessel]

bee [it/its]

[tight star]

oh ok

[tight star]

ultrafilter

[tardy ember]

exactly

so any infinite "natural number" gives you an ultrafilter on N

and... these things are always a bit delicate to state in a way that isn't technically false but there are various results to the effect of "it is very hard to explicitly write down an ultrafilter on N"

[tight star]

usually it requires choice or smth

though apparently ultrafilter lemma is weaker than choice

[tight star]

-# non-principal ofc

[tardy ember]

oh right yeah

obviously

[tight star]

otherwise it's v easy

[tight star]

do you have one of these results that come to mind

[tardy ember]

...ok now that i think about it i have a vague sense that they should exist but i'm not sure i've actually encountered any of them

[tight star]

yeah i've never encountered an explicit free ultrafilter

[tardy ember]

well there's "it's consistent with ZF that there are no non-principal ultrafilters on N"

although thinking about it i also don't know if that requires any large cardinals or if it's just Con(ZF)

[tight star]

mhm mhm

i think there's also the Q about whether the hyperreal fields you get from different ultrafilters are isomorphic

i read somewhere this is equivalent to CH

[tardy ember]

...yeah that sounds right

[tight star]

https://en.wikipedia.org/wiki/Hyperreal_number#Ultrapower_construction

[tardy ember]

ok i did some googling, it's consistent with ZFC (with no large cardinals needed) that no free ultrafilter on N is definable even with ordinals and real numbers as parameters

[tight star]

interesting

[tardy ember]

https://www.jstor.org/stable/2272118

[tight star]

so it's probably quite hard to come across these in the wild

[tardy ember]

well, in some models, yes

the problem with statements like these is that, to take the simplest example, V = L implies that there is a definable free ultrafilter on N, or indeed on any set

"the first one, in the definable global well-order of L"

[tight star]

right right

[tardy ember]

as far as i know this characterisation is essentially useless for most purposes, but it is technically a very explicit definition

[tight star]

i remember being really surprised in my forcing course when we covered L

because having a global well-order seems like an extremely strong property

[tardy ember]

well you can add a global well-order to any model of ZFC without adding any new sets, it just won't be definable

if you mean specifically a definable global well-order then yeah, the existence of that is equivalent to V = HOD (every set is definable from ordinal parameters) and a lot of models of ZFC don't satisfy that

including a lot of the simplest models you can construct by forcing

[tardy ember]

this is part of how you prove that NBG with global choice is conservative over ZFC, in the sense that any statement it proves about sets is already provable in ZFC

[halcyon glade]

The rule as I see it is that you take the two parallelograms, match them up so that one side of the parallelogram lines up with the other, and then delete that side so that you get a bigger parallelogram formed by the remaining sides (this probably makes more sense if you draw a picture...). The caveat is that you can't always do this matching up procedure, in which case, you can't simplify any further, which I don't think is necessarily a problem, just something for students to get used to in the same way that you can't simplify 3 + i any further.

Like this is how I'd imagine adding up two unit bivectors in the same plane (red + blue = green, assuming they're oriented the same way)

Also my two cents on infinitesimals: it's a lot easier to rigorously define complex numbers (it's just an ordered pair of real numbers with a unique way to do multiplication) than to rigorously hammer down the properties of infinitesimals in NSA (need a whole tutorial on first-order logic and its limitations), so I think the cognitive overhead between the two is rather different. Whereas the standard definition of limits, while confusing when you first see it in formal notation, is actually a fairly simple and natural idea that people that work in quantitative fields have to think about all the time anyways (if I only want this much error in my final value, how precise do my initial measurements need to be). So I think it's not worth it to present infinitesimals except as a fun aside to explore for those that are interested.

[turbid zenith]

Simple and natural… to people who already understand it

I sure didn’t find it simple and natural when I first learned calculus

[halcyon glade]

I think it's often presented in an abstruse way which doesn't help

[tight star]

How would you prefer it to be presented?

[turbid zenith]

Probably closer to the thing in parentheses

[halcyon glade]

Yeah

[tight star]

I guess with limits it’s like

[halcyon glade]

I feel like a lot of ppl just slap the statement in formal symbols and Greek letters on the board and call it a day

[tight star]

You can have a more “quantitative” approach to them, or a more “qualitative” approach

In the former you think of delta as an actual function of epsilon

In the latter you don’t really

Like there’s a difference between “how precise do I need my inputs to be” and “in theory, would precise enough inputs ever give me a small enough error”

[halcyon glade]

Yeah there definitely is a conceptual difficulty that comes from the nested quantifiers

[tight star]

There are a few ways to address this

In the more “quantitative” approach, you can ask for delta to explicitly be a function of epsilon

I.e. your goal is to find a tolerance function $t(\epsilon)$ such that $0 < |x - a| < t(\epsilon) \implies |f(x) - L| < \epsilon$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

This removes a few of the nested quantifiers

In the more “qualitative” approach, you can define a different notion of truth

You say that a statement p(x) is true “for x sufficiently close to a” iff $\exists \delta > 0, 0 < |x - a| < \delta \implies p(x)$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

The advantage of this is that you can use the language of “true sufficiently close to a” without invoking a specific delta

[halcyon glade]

(Also I think a surprisingly small number of math students are taught explicitly that epsilon stands for "error" and delta stands for "distance" which really helps clarify things)

[tight star]

The main properties you use are:

• If p implies q, and p is true sufficiently close to a, then q is true sufficiently close to a
• If p is true sufficiently close to a, and q is true sufficiently close to a, then “p and q” is true sufficiently close to a

Using these, you can actually avoid the use of delta entirely

[turbid zenith]

I agree there are ways to make the epsilon delta definition of a limit more palatable and intuitive

[tight star]

And just work with epsilons

[turbid zenith]

But I fundamentally disagree with the notion that introductory calculus, before a student touches a derivative, is the appropriate time to teach that definition

[tight star]

(There’s also a way to get rid of the epsilons and make it fully qualitative, though im not sure how useful it is)

[tight star]

Why do you think this

[turbid zenith]

It’s asking for too high a level of abstraction before students have any idea why they would care about it, all just to be thrown out the moment they learn they don’t ever need to use it.

[tight star]

Hm…

Does it really get thrown out

[turbid zenith]

Hence the deer-in-headlights reaction from many students.

[tight star]

Personally I enjoy using $\approx$ for pre-formal reasoning about limits

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[halcyon glade]

What's wrong with the conventional way of blackboxing limits as "the value the function gets close to as x approaches ..." and then using that to define derivatives and integrals? That's the way I was taught personally

[turbid zenith]

This is fine I think

Starting with limits informally isn’t nearly as much of a problem

[halcyon glade]

Ah okay I thought you were arguing against that, my mistake

[turbid zenith]

I’m arguing against starting limits with epsilon delta

At the beginning of a calculus class

[halcyon glade]

Yeah I don't think it's worth teaching the full epsilon delta definition before starting calculus, I just think there's value in working with a system where the stuff you're blackboxing is a small technical tidbit as opposed to the large amount of background to blackbox if you work with infinitesimals

[turbid zenith]

I accept that my not-anti-infinitesimal stance puts me in the minority and I’m okay with that 😛

I did decide against using infinitesimals as an alternative formulation instead of limits, that did seem like too much

But I’m going to continue to draw infinitesimal pictures at the beginning and clean them up with limits at the end

While mentioning without much detail that we’ve since found ways to make even the infinitesimals rigorous, but it takes a lot more machinery to fully develop them to the satisfaction of modern standards

But I dunno, maybe students at other schools would complain, in year 1 calculus, if the methods used to solve lots of real world problems are at first relying on something that hasn’t been fully and meticulously laid out beforehand?

[halcyon glade]

Haha I don't think anyone's complaining there, I think it is a bit unsatisfying to students that want to ask follow-up questions though, but if you're presenting limits at the end anyways, maybe you can hold it as a piece of suspense

[turbid zenith]

Yeah, it’s part of the narrative

[tight star]

So long as you show empirically that it’s good, you’ll be fine

[turbid zenith]

I’d say it’s okay to do it if you say that’s what you’re doing

[tight star]

If you don’t show it’s good either empirically or logically then you’re screwed

[turbid zenith]

Empirically how?

[tight star]

Stuff like showing how the derivative acts as a good linear approximation

Or that it actually gives the slope of a tangent line

[turbid zenith]

Oh! Yeah absolutely

[turbid zenith]

how do you define the tangent line

😂

[tight star]

Well it’s like

For a lot of curves, “touches the curve only once” does a decent job

So long as it’s not vertical

[turbid zenith]

The line y = -x touches the curve y = x^3 only once

[tight star]

Yeah cubics are where this starts to get trickier

[turbid zenith]

My point there is, it’s hard to give a definition where it doesn’t get circular unless you define the tangent in terms of the derivative

[tight star]

Well this isn’t circular

It’s also how tangents to circles (and conic sections more generally) get defined

[spiral elbow]

Maybe touches the graph only once, plus you can nudge it so that it doesn't touch at all? But then again we're back to epsilons

No, that definition doesn't work either

[turbid zenith]

And that fails for the tangent to y = x^3 at the origin

It’s surprisingly harder than you’d expect at first I think

[swift hatch]

"a line such that no other straight line could fall between it and the curve" is what Euclid went with apparently which still doesn't help with x^3 much

[halcyon glade]

"If you graph the errors between the tangent line and the curve, then the error graph looks flat"
Personally this is how I might think about it, but you can get in some handwringing about what "flat" means

Maybe I'm too linear regression brained

[turbid zenith]

Is handwringing the dual to handwaving?

[tight star]

Yeah for general curves the linear approximation pov works better

[halcyon glade]

I have no idea, I couldn't figure out what the right word was so I made one up lol

[turbid zenith]

And by the IVT does that mean in the middle there’s something that you can do with your hands as an instructor that’s just right?

[turbid zenith]

But really I might have to steal this lol

[halcyon glade]

I see you're instituting the "if you laugh at my jokes, you get extra credit" policy

[turbid zenith]

Don’t we all?

I think it’s an unspoken rule of teaching math

[halcyon glade]

Speaking of, let me say, I still remember what the mean value theorem is because my calculus teacher brought in a speeding ticket he got and had us debate how the speed cameras could know how fast he was traveling instantaneously if they only recorded his position at two far-away time points

It's still engrained in my head as the "Mr. Benzing speeding ticket"

[turbid zenith]

Love it!

I would love to hear people in here share a lesson they’re proud of like that

[spiral elbow]

This isn't that funny of an anecdote, but last semester I took a diff top course that was pretty fun. There's one proof I remember particularly well, because the lecturer didn't really remember it, so he stood at the blackboard just thinking for a couple of minutes. And there were just 2 or 3 students, so we sat there thinking too. Until suddenly he shouted "AHA! oops, sorry" and wrote down the proof. I love lectures like that, where it's kind of informal and it feels like you're exploring the subject together with the teacher

[bronze heart]

Suggestions on what textbook to use to teach my friend linear algebra and multivariable calculus? He knows up to calc 2 and how to multiply matrices. He says he needs it to understand the mathematical techniques used in biology research, which come up quite often when he reads papers

[rapid tusk]

try asking in #book-recommendations, you'll prob get answers faster there

[bronze heart]

Ahh ok 👍

[feral swan]

I always think of continuity meaning that f(x+h) = f(x) + O(1)

I still haven't come to term with this epsilon delta malarkey although I'm fully proficient in its usage.

[tight star]

You mean little o right

And yeah this is a great way to think about continuity

[compact isle]

and then differentiability is f(x+h) = f(x) + f'(x)h + o(h)

[tight star]

Mhm, though this doesn’t continue to higher dimensions

[compact isle]

it does actually

[tight star]

What do you mean?

[compact isle]

With the same formula although you replace f'(x) with Dx(f)

[tight star]

Oh sorry I meant higher derivatives

[compact isle]

and then x,h are vectors

[tight star]

Higher dimensions have the same formula

[compact isle]

higher derivatives can be somewhat put into the same paradigm but I don't think there's a general one

Or at least it's complicated

Involves a lot of tensors n shit

[tight star]

That’s not quite the issue

Twice-differentiability is strictly stronger than having a second-order Taylor expansion

[compact isle]

Oh I wasn't aware that's annoying

[tight star]

Yeah it’s because twice-differentiability is defined as “derivative is differentiable”

[compact isle]

Ah yeah

[tight star]

So in particular you need a derivative in some neighbourhood of a point to talk about twice-differentiability

The standard example I use is e^(-1/x^2) times the indicator of the rationals

This has a Taylor expansion of all orders at 0

But is only once-differentiable at 0

[compact isle]

Yeah

[rapid tusk]

bombed my interview

[turbid zenith]

What happened, if you wanna talk about it?

[rapid tusk]

seems that my conception of the target audience for the mock lesson diverged considerably from what they had in mind

and i paced poorly and got cut off before i was able to finish

also the interviewers didn't seem all that enthused afterward

[tight star]

What were you trying to teach?

[rapid tusk]

was supposed to build a lesson around a problem selected from a list given to me

the one i chose involved triangle inequality

[tight star]

mhm?

[rapid tusk]

basically I paced it poorly given their time constraints

• some introductory thingies to get them thinking about it (walk the shortest path to a corner, what’s it look like? draw buncha triangles, verify it for those)
• proving the inequality
• addressing the degenerate equality case
• going to the problem (on minimizing distance between points on two disjoint circles), proving our answer with triangle inequality

[ashen hearth]

Hi everyone,

I'm an undergraduate student in Applied Mathematics from Brazil, and I've been thinking a lot about the way pure math is taught at the university level—specifically, the very formal and linear (Axiom -> Theorem -> Proof) structure that's standard.

I'm looking for recommendations for books or papers that discuss this. I'm not just looking for "teaching tips," but for deeper works on the pedagogy, philosophy, or even the history of why we teach pure math this way (and its impact on the learning process and on building context).

I'm also very curious about your cultural perspective: in your country or university, is this "philosophy of math teaching" a relevant topic of discussion among mathematicians themselves, or is it mostly seen as a separate field, just for educators?

Any insights or recommendations would be awesome. Thanks!

[turbid zenith]

Good quote I was just reminded of!

[cursive plover]

this is exactly what I meant with my message above yours! though in retrospect i guess that wasn't clear enough. or err.. most questions the smallest period of time? nevermind i'm completely misinterpreting you. i like this idea though.

However, I think the idea is flawed because high school students are not gonna be motivated by a prize like this.
I don't even think I mentioned what the prize was! unless you mean the journey to the prize as opposed to the prize itself..?

The rest will decide "It's not worthwhile to spend my energy to try to earn this prize, I'd rather read my book or scroll social media or brush up on the material for my next class."
thats fine! i can only do so much to try and motivate them. i wanna see how this goes.

[twin locust]

Hello.

[tall bolt]

I think you might be lost

[sonic kayak]

Ive been tutoring for a month now just a few middle/high school kids and im still so stuck, i feel like i dont know everything, and i also feel like im not good enough or that im bad at explaining does anyone have tips on how to just like explain, but also how to teach better

for reference im not in school i just teach thru someone locally

[lethal hornet]

before you start explaining, make sure you can see the answer all the way through.
ask the student for some time for you to figure it out yourself.
if that isn't the issue, and the issue is like, you don't know how to get through to the student, then try asking probing questions to pinpoint exactly where the issue in their understanding is. this is the main tactic i use when tutoring, especially students in this age range

[turbid zenith]

This isn't QUITE pedagogy but I don't know where else it fits. It's related to academic job searching, so it's sorta related.

Anyone have experience writing a research statement? Applying for tenure track and this is the first time I've had to write one.

[halcyon glade]

I would maybe ask in #advanced-lounge or #「graduate-lounge」 (or perhaps #graduate-applications )

[turbid zenith]

Hmmm, I can do that

It's a very new thing for me to do ... my focus has always been on teaching but I'm starting to try to get into research

[rapid tusk]

hopefully this doesn’t drag me down too much given my strong earlier rounds but I’m not holding my breath

[quasi musk]

https://arxiv.org/abs/1409.7660

I couldn't find any research articles on how mathematicians teach math right away, although I do recall reading some mainly aimed at lower division

Here's one where our sister field, Physics, is looking at how students use math

Which could be interesting in its own right

Here's one I found, but haven't dug intohttps://www.tandfonline.com/doi/full/10.1080/0020739X.2024.2309281#d1e163

[twin locust]

I am. Could you clarify please?

[twin locust]

can you clarify please?

[cosmic ibex]

You posted a weird question which had nothing apparent to do with this channel -- it appeared to be about registering on some website. (You later edited that post to just say "Hello."). Nope charitably assumed that you had intended to post it somewhere else, most likely in a different server.

[twin locust]

Thank you for clarification. I am going to post my job issue somewhere else. Best regard!!!

[ashen hearth]

I wasn't expecting physics to be brought up. Really cool

[ashen hearth]

This article on analysis touches on a lot of things I've been wanting to understand. I read the introduction, and the critiques seem to fit my personal experience with analysis. I will definitely give it a read

Thanks

[quasi musk]

I think physics is interesting because they deal with math, from a viewpoint that isn't Definition Theorem, Proof

Although most of what they communicate is intuition through mathematical symbols

[tepid cypress]

Hello guys! what's up?
I wanted to see if any of you have had this experience

I've found myself only getting excited/engaged with math concepts that are "at my level" or that are "non-trivial" to me. Of course that this is at odds with teaching, since I will always find myself teaching something I understand quite well.

So I fall into this kind of gatkeepey sort of mentality where I think that people learning more "elementary" maths have to get it right away because it's "easy".

It's the same attitude as looking down upon engineers doing math, or people in finance. I wanna be able to put myself in these people's shoes to get better at teaching.

Y'all ever felt like this? How do you overcome it?

[austere delta]

I haven't necessarily felt that way, but there is joy to be had in coming up with different ways to understand something you understand well, and trying to anticipate ways in which someone could misunderstand it.

And it might be too late now, but to empathize you can try to look at how long you actually struggled with something yourself if you have notes / can recall that time.

Accepting that people are different and not looking down on people for those differences is also a good idea I guess

[zealous kettle]

Instead of focusing on the investment (or lack thereof) in the topic itself, try getting invested in the students' skills and knowledge. Get excited that the student understands the thing

[feral swan]

You know I thought I was pretty good at it but yesterday at this recitation one student was plodding along with some question going really slowly and a different student who asks exciting questions wanted to ask as well and I got really pissed at the slow student although im usually patient.

I didn't show it thankfully. I just focused on knowing that if I acted out I would feel very dissapointed with myself and reminded myself it's not the student's fault im reacting badly today and in the end it passed.

[feral swan]

I do have a few more tricks i use but they're more personal so i'll send it to you privately

[tepid cypress]

Please do so!

[turbid zenith]

Anyone else find the notation $M_x=\sum m_i y_i$ and $M_y=\sum m_i x_i$ annoying in Calculus II? And the related $M_{xy} = \sum m_i z_i$? Am half considering using different notation when I teach it again.

[burnt vessel]

Solid Angles

[turbid zenith]

I'd much rather it be $M_x = \sum m_i x_i$, $M_y = \sum m_i y_i$, $M_z = \sum m_i z_i$.

[burnt vessel]

Solid Angles

[turbid zenith]

Is there anything sacred about the other notation?

[tall bolt]

What is this notation?

[turbid zenith]

If I asked a student, "Quick — how far is (4,5) from the y-axis?" I bet 9/10 would immediately want to say 5 and would have to really think about it before they said "oh wait it's 4"

[cosmic ibex]

Weird.

[coral crest]

Probably only makes more sense if students have to know about the inertia tensor

[tall bolt]

I know that i covered this stuff in calculus but i dont have the slightest idea what the notation was, so im going to guess using whatever is fine

Maybe there’s some area where this matters more, but at least for my maths it hasn’t

[turbid zenith]

Yeah I might just end up not introducing the intermediate notation and just going with $\bar{x} = \frac{\sum m_i x_i}{\sum m_i}$ etc

[burnt vessel]

Solid Angles

[halcyon glade]

Wait, is this a multivariable or single-variable class? I don't see why one would discuss Mx, My, Mz at all if it's a single-variable class.

[turbid zenith]

Multivariable is when I introduce them

Though I've seen them done in some Calculus II courses as yet another application of integration

[pure light]

the M_x notation is mostly there to be analogous to the rotational inertia notation which might be introduced later

although it's a formula that only makes sense in 2D anyway and the moment of inertia notation is mostly useful for 3D

[quasi maple]

If you think the M stuff is in a similar vein to this example, it's possibly then because a definition - or at least some useful property - collapses under the proposed alternative

(4, 5) being "5 units away from the y-axis" for instance (and I acknowledge this is a common answer) fails when you consider (4, 0), which is not on the y-axis but whereby the same logic would conclude otherwise

[midnight scarab]

I am a physicist and I am apalled by the notation

I agree with the people saying it's to be analogous to the notation for the moment inertia, but it's stupid nevertheless

[rapid tusk]

ok update: they gave me a summer intern offer? i was worried my bad final interview was gonna lock me out completely

still bummed tho

[midnight scarab]

Also I dislike the fact that students are taught to think in terms of rotation vectors etc, because the fundamental notion is the antisymmetric matrix

Rotations happen in a plane

Packaging things using this duality (which is specific to 3D) leads to quite some confusion down the line

TL;DR I'd also prefer writing I_xy = sum m_i x_i y_i

But using I_z here is less absurd than this M_y thing

[halcyon glade]

You mean like pseudovectors?

Yeah they're weird

[turbid zenith]

I'd love if we used bivectors instead 😛

But I don't know how the computations would go

[halcyon glade]

As far as I understand it, computing a cross product involves exactly the same computations as a wedge product, so I think it'd be basically identical, just with different notation

[rapid tusk]

stats flashbacks (eric's nickname)

the homeworks i had in that class kicked my ASS

[turbid zenith]

Yeah, but it's harder for me to draw 😛

[warm valley]

that's correct, the advantage of the wedge product can really only be felt in other contexts

which I think is one reason differential forms can be hard to motivate at the level of multivariable calculus

[rapid tusk]

the calc3 course I took had a random section on differential forms + generalized stokes right at the end

it felt uh

quite tacked on

[warm valley]

yeah, like differential forms hit the level of "obviously useful" in manifold theory, but are certainly teachable before that. I've seen calc 3/4 try to teach them with mixed success. I know some people who think they should just be taught directly and ditch the classical vector calculus, but this makes working with the classical stuff more difficult

[compact isle]

formula for the cross product might be easier to remember using wedge products since the submatrix you use will correspond to the indices of the basis vectors in the wedge product I think

though remembering the signs still take some effort I suppose

[warm valley]

I've not seen students have a particularly hard time remembering the cross product via the determinant in the traditional way

[compact isle]

yeah its not that hard

either way

but perhaps its a bit more intutiive using wedge products

[warm valley]

it definitely is more transparent where the formula comes from

[compact isle]

sicne each component is going to be a wedge product times a 2 by 2 determinant which is the area of some parallelogram

when you project onto that plane

which more naturally maps onto the cross products visual interpretation as the bijector spanned by the two vectors with magnitude equal to the area

[turbid zenith]

See I did that when I taught Calc IV

I wasn’t expecting everybody to perfectly grok every bit but I wanted them to see the running thread behind everything

What made it feel tacked on in your case?

[rapid tusk]

"oh it won't be tested!"

when everything else in the cursed text known as st*wart was fair game on the final exam

[tall bolt]

I learned about them separately to vector calc and multivarible, and I honestly think I prefer that. Like I think so much of the reason why I appreciate them is because you introduce them to do geometry in higher dimensions and whatever, but then after you spend a little time building up the theory of these “new objects” you can very easily just recover everything you saw in vector calc with pretty minimal effort, and that was so much of what convinced me they’re useful

And I don’t even mean with generalised stokes, I had some question to recover all the big identities just using differential forms and iirc the hodge star? But also just shout out to generalised stokes, its just sick, probably my favourite equation

[midnight scarab]

Yeah. Really the fact that for a matrix $M_{ij}$ in so(3) (so antisymmetric), we define the rotation vector $\omega_i = \frac12 \epsilon_{ijk} M_{jk}$ etc

[burnt vessel]

afqt

[turbid zenith]

It's sad when whether something is on an exam or not is the main metric of something's worth in a class 😛

[rapid tusk]

i was a crashed out high school senior at that point lmao

[turbid zenith]

Fair point lol

[rapid tusk]

i found the few exercises they made us do with it mildly interesting but

poorly motivated

[turbid zenith]

Oh you got to take multivariable in high school

Lucky

[rapid tusk]

"here's a firehose of new notation that we're going to vaguely show you how to manipulate without telling you how it works"

aaaaaaaand then my uni didn't take it for credit

so i had to waste a semester redoing calc3 😭

[turbid zenith]

Okay yeah true, that part of multivariable is pretty damn rushed

Trying to do all of multivariable + vector calculus in one semester just seems like a terrible idea to me

[rapid tusk]

the prof paced it so badly that

line/surface integrals were THE WEEK AFTER THANKSGIVING

and it was ONE THIRD OF THE FINAL

<@&268886789983436800>

thanks

[turbid zenith]

This is why my uni made Calc IV and honestly I like it a lot better that way

Though it's still not perfect

[rapid tusk]

issue becomes when that pushes out being able to gun for the big upper divs early

i should've done analysis my sophomore year 😔

[turbid zenith]

LOL imagine having enough math students at your school to offer analysis to sophomores

[rapid tusk]

big private uni

[turbid zenith]

We're a small private uni, 1500 students total

[rapid tusk]

total

damn

[turbid zenith]

Single digit math majors

[rapid tusk]

i think we're like in the 5 digits?

[turbid zenith]

We can only offer analysis once every other year

Same with abstract

But okay I appreciate the insight on the Stokes thing ... I want to keep offering it at the end of my classes and I do kinda like the last day to be something that isn't gonna be tested, some kind of culmination

So they can just kinda sit back and watch the pieces come together, soak it in without worrying about having to reproduce it

[rapid tusk]

unrelated but i like to think of the not-generalized stokes with a "butterfly net" analogy

[turbid zenith]

That does seem to be a good way to visualize the surfaces and the rim

[rapid tusk]

i can move the actual netting wherever i want but as long as the frame stays in the same place (up to orientation issues) the surface integral of the curl will be the same

yea

one of our profs that helps out with our math contest was showing me a buncha surface models he uses to teach gradients

[turbid zenith]

Oooo, like 3D printed?

[rapid tusk]

dont think so?

idk how they were made

would've been fun to play around with those when i was taking calc3

but the one i did in hs was some shitty online self paced course

the "videos" they had were bad quality summaries of the same info i could find just by reading the corresponding section in stewart

etc

[turbid zenith]

Sounds about right

Stewart is so weird

[long pelican]

In a 4D space, torque truly lives in the space $\Lambda^2 \mathbb R^4$ which is 6-dimensional and there is no trick akin to $\Lambda^2\mathbb R^3\simeq\mathbb R^3$ as in 3D

[burnt vessel]

icy

[quasi maple]

Honestly, I feel this is the biggest problem with formal education as a whole tbh

Exams are so ingrained into educations systems that for some people it even becomes a literal defining part of them

I could swear in the UK this was to the extent that someone'd made a play about this, which then became something that was taught in Drama classes

(The name of the play evades me however )

[quasi musk]

Yeah, but to be fair to our students, many of them have a lot on their plate from classes. It'd be great if everyone had an interest to learn everything in a class, but at some point one must be pragmatic about learning only the things to get through the class with the desired grade.

There was a multivariable calculus instructor at my college whose response to any question of 'will this be on the exam?' was "It wasn't, but it is now!"

[tall bolt]

I absolutely agree with this. I would love to say I read every “non examinable” section of every set of notes I’ve read, but that would be a lie. I do care, but you’ve just got to be pragmatic at a certain point, there’s only so much time in the day.

I do think there’s levels to that though, but like at the end of the day, what can you do

[plain pebble]

For me my courses mostly get mentally divided into “for exams”, “for general interest” and “for specific interest”
If it’s non-examinable, then the first category it’s not getting read, the second it may or may not (depending how much I like the course) and the third I’ll read it
I suspect in general non-examinable stuff will be treated like “read/listen if you’re particularly interested”

[quasi musk]

Clearly if a student is having the attitude of "I'm going to do the bare minimum work to just get by", then this isn't a great attitude to learn; but likewise having the attitude that you're going to chase down every detail until it's beaten to death is just not feasible

[turbid zenith]

If you were teaching vector calculus and wanted to give a name to the operator that takes in a vector function F = P i + Q j and gives back ∂Q/∂x - ∂P/∂y, what would you call it?

Specifically I mean giving a specific name to that operator itself, not just calling it "the z-component of the curl of P i + Q j + 0 k"

Would you still call it "curl F"? Give it its own name like "2d-curl F" or "rot F" or "swirl F" or something? Or what?

[quasi musk]

I work with this, and just call it the curl or 2D curl

It comes up in fluids all over the place

[turbid zenith]

My gut is to just call it the curl and be done with it

(And to say that ad - bc is the cross product of <a,b> and <c,d>)

[quasi musk]

The only misnomer is that students might think that curl gives you a vector

but 2D curl gives you a scalar

[turbid zenith]

Yeah

...I know you can get around it using the wedge instead but I'm finding that a bit more annoying to deal with than I'd like

Especially if we're talking about people's first exposure

[quasi musk]

Yeah, just avoid the $\times$ notation and use $curl(F)$

[burnt vessel]

MoonBears-C-

[turbid zenith]

I mean I like the × notation because I think of × as a "perpendicular product"

[quasi musk]

yeah, I like it as well, I just only use it for the proper curl in 3D

[turbid zenith]

And · as a "parallel product"

Another definition I've seen is <a,b> ⟂ <c,d> = ad - bc

"Perp product"

Would it be too confusing to say that in 2D we have curl F = ∇ ⟂ F but in 3D we have curl F = ∇ × F? Or would that actually help?

. . . . oh hey, that actually works just fine for curl F if you extend the exterior calculus version!

$\mathrm{curl},\mathbf F=\left(\star\mathrm d(\mathbf F^\flat)\right)^\sharp$

[burnt vessel]

Solid Angles

[turbid zenith]

That actually makes the 2D curl work just fine

[quasi musk]

lol

[quasi musk]

This runs the risk of getting into $\nabla^{\perp}$

[burnt vessel]

MoonBears-C-

[quasi musk]

In 2D, $(x,y)^{\perp} = (-y, x)$ so just take the perpendicular vector to the gradient for grad perp

[burnt vessel]

MoonBears-C-

[turbid zenith]

Yeah I'm playing around with grad perp as well

[turbid zenith]

I was kinda hoping that if you took the curl of a 2D scalar function using this definition you'd get the grad perp

. . . instead you get its negative D:

Which is going back once again to trying to find a way to explain why the signs for div-free and curl-free vector fields are uncannily like the Cauchy-Riemann equations but with the wrong signs

Now I've got two places where the signs don't match, which obviously means one must be able to explain the other :V

So I think I have no other choice at this point than to read Needham's section on it, apparently there's a nice geometric reason that the conjugate lets you visualize it

[quasi musk]

Yeah, my complex professor had a whole list of exercises on this matter

[coral crest]

I like the x-notation since it’s like you’re taking a cross product of your del-vector

[quasi musk]

Sorry, I only avoid it in 2D

in 3D its fantastic

[green siren]

Does anyone here teach at high school or first university years?

Or has taught at that levels?

[austere delta]

I have been involved in the teaching of some first year university courses. I'm sure others here have aswell.

Why?

[abstract grove]

I've taught some high school

[cloud zealot]

https://www.rxddit.com/r/math/comments/1or2gn4/is_there_any_optimal_way_to_teach_kids_mathematics/

[turbid zenith]

I've done both!

[feral swan]

do you reduce points for proofs that are too long?

we're thinking of starting to reduce because some of these proofs are ridicolous, 10+ pages for a proof that should take 2 pages if you belabour yourself.
Admittedly it gets skewed a bit because sometimes they both write long proofs and write in big handwriting and it honestly kinda ticked me off so my memory is not the most objective I guess.

[rapid tusk]

i'd penalize for obviously superfluous fluff

or if the method is obviously overly inefficient

[midnight scarab]

My 1st year prof would go "too long ; didn't read" so I think just penalising is fair 😂

[tall bolt]

Yes. If it’s waffly I’d 100% take marks off. I think:

1. It shows a lack of understanding, both because if you really knew the material you could probably find a better argument, but also because you could just me trying to cover all your bases when you’re unsure of your argument (I did this in combinatorics a lot )

2. I think it’s important to teach good writing, and if your argument is harder to follow than it needs to be, that’s not good writing.

I had a friend in UG who was terrible for writing solutions which were so much longer than they needed to be. He for sure fell into camp 2) because he knew his stuff, and he only stopped after our group theory prof told him he’s not reading anything over 2 pages lol

[ashen hearth]

Is the argument correct? If the reasoning is sound, sometimes you just have to give full marks. If you think solving it another way would be more interesting, it might be worth adding a comment showing how to do it. Don't you think being this restrictive just makes it harder to develop skills that will be essential for future research?

[quasi maple]

Another thing if the proof is stupidly long - could they have at least broken it down?

Like, suppose they'd written a proof that's 4 pages long and boiled down to "A -> B -> C -> D"

I'd hope at least to see that breakdown somewhere at the top, so I can at least see what their goal here was

[feral swan]

There were some proofs that were obsecenly long.

I'm talking 10 handwritten pages for a routine definition checking argument for which I would call a page long

This type of proof I cannot physically check in my 4 minutes I have per sheet, so I gave it as much marks as I could see were true from super fast skimming, but If I had full control over the grading I would give very little

right now I don't think I can remove points over it because it will cause a lot of student drama.

for more reasonable arguments I leave a note, it's ok to stretch a proof 3x or 4x as a second year student.

[feral swan]

I think it shows a lack of understanding what a proof is, the part where it's a device to communicate. I also think it shows a lack of understanding which parts are important and which not, and the physical reality of living with constrained time.

specifically I remember improving immensely as a mathematician when a professor in my second year took me aside and just told me to condense my proofs.

thanks everyone else for their input as well.

I think I feel a bit guilty because I myself don't write really good proofs, (and I/am a lazy writer of sheets), so I feel like a hypocrite, and i'm a bit apprehensive of the confrontations it's going to cause with students if we dock points for seemingly arbitrary reasons like proof length.

Also I have to admit it's just ticking me off for no good reason.

[tepid smelt]

I was under the impression everyone here teaches in some way. I have noticed many are grad students or tutors though. I teach at an urban highschool and teach gifted middle school kids in the summer.

[green siren]

I teach at one too

But my students are far from being gifted

They drive me mad

Here in Uruguay 12th students can barely do a derivative

[green siren]

The students here are awful

[tardy ember]

...honestly i feel like you should be able to say something more specific about a proof like this than "it's long"

whatever it is that they're wasting all of those words on, you could point out that to them instead of just the length

like if they prove something they never actually use (in a way that doesn't add any clarity), or prove the same thing multiple times, or if parts of the "proof" are just rambling that doesn't contribute anything at all

...i guess trying to do that does get back to the problem of you not having the time to read through such a long proof in detail

[tepid smelt]

No that's typical in the states also. There are huge differences between schools though.

My hope is we will see changes in how we teach younger kids. The move away from standard algorithms and direct instruction has really hurt many kids. I have seen a shit in how reading is taught though and the hope is something similar will happen with math

[turbid zenith]

How has that move hurt many kids?

The edu-bigwigs that post all over social media are all about how moving away from standard algorithms and direct instruction is THE way to go

(My hunch is the right thing to do is somewhere in the middle)

[tepid smelt]

Young kids lack arithmetic foundations which makes foundations in algebra difficult.

Yeah I feel strongly direct instruction is the key. There was a strong push for awhile for discovery based teaching and doing multiple approaches to arithmetic instead of dialing in tried and true algorithms

[turbid zenith]

What about all the students for whom the "tried and true" didn't work?

In at least some cases I can see why not going with the standard algorithm would be a good way to build number sense

Example

[rapid tusk]

it seems like it could be rather difficult to teach those "new" methods

particularly the one on the left

how do you properly motivate it to a young student who likely doesn't have the number sense for it yet?

intuitively i think i picked a lot of this up by just doing a lot of computations (benchmark by "friendly" multiples of 5, 10, etc) but im not sure how to communicate that well to a primary schooler

[turbid zenith]

You give them problems that are designed to show where it's helpful

[green siren]

Thats sad.My 12 th grade students dont know what the plot of the function 1/x looks like

[abstract grove]

pretty sure that's the case just about everywhere

with the possible exception of small, private, elite schools

[green siren]

I always wonder that

But I agree

As a student I wasn't like that though

[abstract grove]

there are usually a few kids in each class that aren't terrible

I'm being a bit cynical, there are usually a few who excel

[quasi musk]

My calc 3 students don't know either

[quasi maple]

I mean, imo, number sense should be a concept you should ideally have learnt by age 10 imo

Perhaps it's a combination of "parents having learnt this younger than they can actually recall, so they don't remember doing these" and "teachers setting exercises that are only clear to anyone who's sat in their classes, at the cost of being unclear to parents from whom kids will ask help" that results in these sorts of question being clowned on

Another point about the "new" way in teaching number sense is that it's supposed to improve mental arithmetic

Like, if I had to compute 429 - 245 in my head, then if I can see the line "245 to 300, then to 400, then to 429", and then go "well that's 55 + 100 + 29 = 184" (there're other approaches) in my head

[green siren]

Really? How is that possible?

[cosmic ibex]

I think it may be that some students actively resist learning what the graphs of standard functions look like, because they've gotten the impression that visual intuition is cheating and they're supposed to think only in terms of symbolic manipulation.

This syndrome can coexist with doing a lot of "graph this function" exercises, once the students have convinced themselves that the graphs are supposed to be only output of their work.

[tight star]

That’s interesting and really strange to me

[turbid zenith]

There does seem to be a lot of “pictures bad, symbols good” in how math is taught.

I watched a video series from The Great Courses about proofs recently. The prof was Ron Larson. He did a whole lecture on “proofs without words” and ended it by saying he didn’t really consider them proofs unless they were using some kind of algebraic manipulation.

I blame Bourbaki. 😛

[cosmic ibex]

Always a safe bet.

[turbid zenith]

His Calculus I series is actually terrible imo

Lots of “please memorize this formula”

When it introduced the derivatives of sine and cosine, he basically said “yeah it’s unfortunate there’s a minus sign, but you just have to know it”

Know what he didn’t do? Look at a graph of sin x or cos x

The Calc II and Calc III ones are better but ugh

[cosmic ibex]

"His" = Larson or Bourbaki? 😛

[turbid zenith]

Larson

Sorry should’ve been more clear

[tight star]

as someone who grew up on 3blue1brown this is quite foreign to me

[cosmic ibex]

Even where the teacher doesn't consciously subscribe to that philosophy, it's something students can easily end up internalizing as a result of repeatedly getting homework back with "not symbolic enough" because the point of the exercise was to train the ability to convert intuitive understanding into symbolic reasoning.

[midnight scarab]

Or literally draw a tangent vector on the unit circle. How can you not have geometric intuition with trig??

[quasi musk]

Damn, that's pretty bad. You can make the comment "Yeah, there's a minus sign and at the end of the day you have to remember that; but here's why" would be way better

[quasi musk]

Most students don't really know the pre-requisites as their knowledge, it's more like a vague mist of understanding. Many students are just trying to get by the class, and each class they have comes with a different set of rules how to succeed. I try not to blame my students too much, because it is dizzying to keep track of "This professor grades like this but now this new professor grades like this"

[turbid zenith]

Is 3blue1brown part of any actual class though?

[turbid zenith]

Yeah, this makes sense. The symbolic reasoning has to serve a purpose, and if a student can solve a problem just fine without it, I don't think it's a good problem to teach its utility

[cosmic ibex]

It may not be a good problem to teach the utility, but problems that are within the reach of intuition can be good to practice proficiency in translating intuition to symbols ... if it's made clear enough that's the point of them.

[turbid zenith]

Fair enough

[quasi musk]

Interesting, I've found 3blue1brown to be detrimental to a lot of students. Especially his essence of Calculus or Linear Algebra videos. They're beautiful explanations, ones that I've had with my colleagues and professors, but I think that it really gets students into spectator mode a little too much; almost as if they're lulled into a false reality that Grant Sanderson's understanding is their own. That doing these worksheet calculations isn't doing math, the real math is in this beautiful, carefully sorted explanations

[turbid zenith]

But I think it's even better if the symbolic reasoning actually helps

One example I've given in the past is a real-world problem about finding a quadratic maximum, lemme find it

[tight star]

Not usually

[tight star]

I’ve heard this sentiment a lot from people on mathcord but never seen this irl

[quasi musk]

I was made to read Lockhart's: A mathematician's lament, and G.H. Hardy's A mathematician's apology as part of my Differential Geometry

[turbid zenith]

Here we go

[quasi musk]

I've seen it with at least a dozen students so far in real life

[cosmic ibex]

I think both kinds are needed; neither stands well on its own.

[tight star]

I don’t really understand the people here who worry about students thinking they understand more than they do

1. Why would that even be a problem

[tardy ember]

i think the problem with trying to do this is that a problem that's too hard to get through on pure intuition might also be too hard to get through with symbolic reasoning if you have no practice with symbolic reasoning

[tight star]

2. Is the alternative explicitly not showing people the geometric intuition behind these concepts, for fear that they might understand it too well?

[quasi musk]

Yeah, it's because when you have an "understanding" in your mind, you tend to turn away from things that don't match that "understanding" or begin to rationalize against the professor's assignments, grades, etc. I've seen it happen a lot of times, even amonst grad students that I'm mentoring

[tardy ember]

not a perfect analogy but imagine someone suggested that, in order to teach people to swim, you should put them straight into the deep end of the pool, so that they can't get around by just standing on the floor and walking, in order to demonstrate the utility of being able to swim

[turbid zenith]

Yeah of course, it has to be gradual

You need the intuition for a hook, but the precision gets you the rest of the way there

[tight star]

I can see this, but this also feels like an easier problem to solve than not understanding anything in the first place

[turbid zenith]

You don't want them to stay in the shallow end of the pool either if we're going with that analogy

[tight star]

If the choice is between a student seeing the geometric intuition and getting a bit too far ahead of themselves

Or not seeing it and just suffering through the calculations

I’d take the former any day

[quasi musk]

I think undoing misconceptions is actually harder than dealing with a blank slate, of course depending how deep that misconceived belief is

[compact isle]

tbh mostly my hope is that videos like his can inspire students to actually care more about the math, but yeah they do still have to put in the work

[turbid zenith]

This greatly depends on which misconceptions we're talking about

[compact isle]

or at the least can give them the impression that they actually can understand the material

[tight star]

I’m not sure this is the case for the kinds of misconceptions in math

[quasi musk]

I don't see it as much in undergrads to be fair, but I do see it in first year grad students a lot

[tight star]

I don’t see the need to gatekeep understanding for fear of students not being able to handle it

[turbid zenith]

If we're talking about the idea that not all continuous functions are differentiable or something, that's pretty easy I think

If we're talking about thinking that (x + y)^2 = x^2 + y^2 ...

. . . well, if you figure out how to cure students of that, let me know

[tight star]

Hm what do you usually try

[turbid zenith]

Because once it's entrenched it seems like nothing, not even all the geometric intuition in the world, helps 😛

[tardy ember]

i mean if you think about the misconception "maths is just about shuffling symbols around, none of it means anything"...

[turbid zenith]

I've tried a bunch of things

[quasi musk]

Just draw a square of side length x+y, chop it up, count the pieces

[turbid zenith]

I've tried "well why don't you plug in 3 and 4 and see what happens"

[quasi musk]

That's the fastest way to get them to "see it"

[tardy ember]

my instinct would be to ask them to just plug in x = y = 1

[tight star]

Isn’t this exactly the kind of thing geometric intuition helps with

[turbid zenith]

I've tried "that's not what the distributive property is actually about"

I've tried "a puppy died"

[tight star]

Also by misconception I mean a misunderstanding of a particular mathematical concept, not maths as a whole

[turbid zenith]

But it goes in one ear and out the other no matter what it seems because it is THAT deeply entrenched 😛

[turbid zenith]

They may "see it" in the moment but then on the next assignment they do it again

[quasi musk]

This is 100% TRUE

I can't even get my students to write limit on all steps of their calculation

[turbid zenith]

The next thing I'm going to try is to just give that mistake a catchy name and point it out whenever I see it

[quasi musk]

I'm not sure why writing limit is very difficult for students

[turbid zenith]

So that that gets stuck in their heads XD

[green siren]

They do that?

Uni students?

[turbid zenith]

I've started calling it the "ABBA principle"

[quasi musk]

lmao the name and shame

[turbid zenith]

(The A of the B's is not necessarily the B of the A's)

[turbid zenith]

Yes.

[tardy ember]

i've seen this kind of mistake called "the law of universal linearity"

[turbid zenith]

My gut is it happens because the students think the distributive property is about "here's what you do when there's parentheses"

Instead of "here's how multiplication plays with addition"

[green siren]

Maybe because it is annoying to write it several times

[tardy ember]

i imagine it doesn't help that we often write multiplication without any actual symbol and just indicate it by putting things next to each other

[turbid zenith]

This is precisely it. Students view writing the limit as a "Simon says" exercise and don't see what the actual reason is for writing it.

"You didn't write lim every time" = "Simon didn't say"

[green siren]

Ohhh I shouldn't be so mean with my 12th grade students

[quasi musk]

Sure, it's annoying to write every time, you can do the algebra work without it, then say by the algebra work we get this equals this

[quasi musk]

No, no no, keep being a stickler. We need you to drill it into them before they get to Uni

[tall bolt]

You may have been my analysis tutor, guy docked 10% of one of my homework’s because I forgot to type one single limsup half way through a calculation

[turbid zenith]

I've started phrasing the limit thing as "we can cancel out the (x-3)s because we're in the safety of the limit" and "now we can plug in 3 safely so we don't need lim anymore"

[tall bolt]

Still pissed about that

[green siren]

Fair

[tall bolt]

If it was a consistent thing, sure, but like one in a long computation is clearly just a typo

[quasi musk]

If 10% is one point out of ten, then I can see that. But if it's out of 100, and they dock 10, then it's silly

Proportional reasoning is hard for graders "it's just one point!"

[green siren]

I would have docked 2 %

[tall bolt]

It was out of 10, but like cmon. I’m pretty strict about writing standards in my marking but like clear typos I’m happy to let slide

[quasi musk]

I'm not sure I'd dock anything if it's clear all the other steps had limits

I would've probably circled and said "Hey, this should be there" and moved on

[tall bolt]

Yeah that’s how I approach my marking

[green siren]

0.02 %

[tall bolt]

If it’s consistent then it’s a lack of understanding and should be penalised, otherwise it’s whatever, typos happen

[quasi musk]

In my real qual, I had "State Fatou's and Monotone Convergence, prove that Fatou implies MCT". In my statement of MCT, I literally forgot to write the integral sign. Just got a circle saying "you forgot the integral" but no pts were taken off. Without that mistake, I got a perfect on my real qual

Felt pretty good, and kinda funny

[turbid zenith]

Unless you're teaching mathematical typesetting/communication as part of your class

But even then it should be minor

[green siren]

But you must check your work before handing it in

[compact isle]

I try to stress communication to students because I do think it matters a lot

if a student makes something that could be a typo but could also be them not understanding, sometimes I have to take off marks

but if they are writing english to explain what they are doing and they make a typo, then I know they actually understand

[green siren]

Here some students cant solve quickly
(x-2)(x-3)=0
They use the distributive property and then use bhaskara's formula

I knew a uni student that used to do that too

[turbid zenith]

My coworker calls that "see math, do math"

Launching straight into a procedure without taking a few seconds to think

I was tutoring a student for the GRE subject test, let me find the problem they were doing

Here we go

The student solved it by rewriting |x+1| as a piecewise function, splitting the integral into adjacent intervals, finding antiderivatives of each, and plugging in endpoints

[green siren]

Pretty common

[tall bolt]

This is true, but also like mistakes make it into books with professional editors. If something is riddled with typos that’s one thing, but the odd slip up is whatever

[turbid zenith]

Man I'm grading essays for my liberal arts math class and this question should've been a gimme but it's amazing how much people just did not listen

What Counts as Proof?
Scientists and mathematicians both rely on evidence and reasoning, but they mean very different things by "proof." Write a short essay comparing and contrasting how proof functions in science versus mathematics. What are each discipline's standards of evidence, and why are they different? How do these differences shape the kinds of knowledge each one produces?

And I've got a bunch of students talking about how both disciplines base their reasoning on evidence

When we specifically discussed that in mathematics all the evidence in the world isn't enough

[green siren]

Actually they are very different sciences

[turbid zenith]

Yes, agreed, and we've talked about that in class

[green siren]

Empiral sciences a whole different thing
Your students are weird

In which year are they?

[turbid zenith]

3rd and 4th year university

This is the one math course that's required for all majors, "Mathematics and Human Nature"

[green siren]

Ohh ,they are not math or science majors

[turbid zenith]

Even some science majors are saying this!

But no, I actually have no math majors in either of my sections :/

Because we have like single digit math majors in the whole school (of 1500 students total)

[green siren]

I math major would never say that

I hope

[turbid zenith]

Meanwhile there are other students, not STEM majors at all, who clearly understood the assignment

I have English and Theatre majors etc just absolutely nailing it

[quasi musk]

Trying to get students to draw two triangles and add? That's too difficult

[rapid tusk]

ah the whole class of "don't overthink it" type questions

[turbid zenith]

RIGHT?

[feral swan]

that's a good point.

i don't really have a good other reason.

[neon current]

Yeah

[neon current]

I see that

I've stopped watching 3b1b for that reason and I'm focusing on the texts for now

And problems

[quasi maple]

ffs I was about to this message as though I didn't right it

[zinc dove]

You can always indicate an important aspect of what you're writing, as it draws the attention of the reader to that part.

[quasi musk]

It's totally fine to watch as you're reading. It can inspire you to do more, but I just think that students use it as a substitute for work. As long as you're working on it you're fine

[rapid tusk]

it’s a supplement not a replacement

many students fail to realize that unfortunately

[turbid zenith]

Everyone wants a shortcut

[spiral elbow]

Everybody wants to be a mathematician, but nobody wants to read no heavy-ass book

[turbid zenith]

How do y’all feel about the idea of using $f^{(-1)}$ for an antiderivative of $f$?

[burnt vessel]

Solid Angles

[compact isle]

Don't like it

could be easily confused, and also I think it should be more emphasized that there's not a unique antiderivative

tbh I dislike the antiderivative notation with integrals as well

[charred solstice]

That's inverse

Topic change: When my calc BC teacher taught finding the error bound in a alternate series everyone was so confused there was barely any explanation for how the a_n+1 term is the upper bound of the series.

So I suggested that she should write the first 8 terms down of the infinite sum of ((-1)^n)/n with n=1. And label which is the partial sum and the rest is the remainder which the first term being the error bound because the rest of the terms will always just make a_n+1 smaller because of how it's a decreasing sequence with changing signs from + to - .

(I used khan academy video to understand it better took me like 2-3 days to get everything straight for me to understand the error bound shannigans; with me constantly asking why this happened in my head and finding out why.)

[spiral elbow]

I like it, sin^(-1) could then mean either arcsin, 1/sin or -cos

[turbid zenith]

I’m specifically talking about having it in parentheses btw

Just as $f^{(n)}$ is the nth derivative

[burnt vessel]

Solid Angles

[turbid zenith]

I mainly want an “annotated” notation for antiderivatives like prime notation does for derivatives, rather than an operator notation. I’m not worried about nonuniqueness since that happens with traditional notation as well.

[midnight scarab]

Idk, I feel like forgetting that anti-derivatives are non-unique is such a common mistake that it's dangerous to "encourage" it through such notation

[arctic coral]

i am having an issue with a constructive proof, students really don't like that construction proof isn't intuitive all the times

[quasi musk]

I suppose it's fine, but man I don't like it

I mean there's H^{-1} space for the dual of H_0^1. If that can fly, then why not this?

[turbid zenith]

Same is true for square roots but we have the notation $\sqrt{x}$.

[burnt vessel]

Solid Angles

[turbid zenith]

I wonder if the idea of a “principal antiderivative” makes any sense

(The one where C = 0)

[cosmic ibex]

When you're just given the function it's not clear what C is.

The same antiderivative could be either sin²(x) + C or -cos²(x) + C for different values of C.

If 0 is in the domain, it would make sense to speak of the antiderivative that satisfies F(0)=0.

[arctic coral]

i guess it is too much effort to make a notation convenient

which makes it more of bending the nature of antiderevatives to suite a notation

while it should be the other way around

[turbid zenith]

Good point!

I'll tell you the two reasons I've wanted such a notation

(1) Being able to introduce antiderivatives to students before they do integration and the FTC. Saying that ∫ f(x) dx is "a symbol for the family of antiderivatives" is really unsatisfying to me.

[cosmic ibex]

I can sympathize with the desire -- if one was available without breaking too much stuff, everybody would be using it!

[turbid zenith]

I want to be able to say that the ∫ and the dx mean something (smooth sums and little widths) rather than "that's just the notation"

(2) I guess it feels clunky to start with a function f(x) and introduce a function F(x) that is an antiderivative of f(x) whenever you want to use it. What if the function I'm integrating is already F(x)? I can't make F any more capital.

Oh, apparently it's already a thing... I probably should have looked it up to begin with 😛

But I don't know if I've ever seen it in practice, I wonder if there's an example somewhere

[midnight scarab]

Not surprised that's the notation because it's the most logical one, but I've indeed never seen it

But still, I feel that something like I_0[f] for the antiderative of f which vanishes at 0 would do a better job at getting the non-uniqueness ingrained

[turbid zenith]

But not all functions would be defined at 0

[midnight scarab]

Or A_0 if you want to emphasise the distinction between integral and antiderivative

[midnight scarab]

I just took 0 as an example

[turbid zenith]

Oh I see

[midnight scarab]

Should've written I_a for some a in the domain

[turbid zenith]

That's interesting

I'd have to see if that notation would be useful though

[midnight scarab]

Yeah I've never tried it out with students so idk how well it'd work in practice

[turbid zenith]

With discontinuous functions it could get even worse

(Discontinuous in the calculus sense, not the topology sense)

So yeah, if there were such a notation, it would need to solve more problems than it creates.

[neon current]

I know, but I find it better for me to read through first so I have a good understanding. I also deleted YouTube to eliminate distractions anyways. The courseload is HEAVY this year

[midnight scarab]

You mean piece-wise defined? Arguably the default would be to pick constants on each interval to ensure continuity of the antiderivative

[arctic coral]

change your name to solid lagrange

[cloud zealot]

https://www.rxddit.com/r/math/comments/1ouk3wt/what_criteria_do_you_weigh_most_heavily_when/

[upper nacelle]

What do u guys think about teaching mathematics through very good geometry foundation?

[austere inlet]

Arnold is a master ragebaiter

[turbid zenith]

This hits though

Always annoyed me that precalculus textbooks would say that the determinant is ✨ a number associated with a matrix ✨, and here's how you can use it to compute the inverse, etc

But would never say what a determinant is

[turbid zenith]

Question if anyone happens to know… does writing a textbook count as “scholarship” in academia, as in for evaluation of scholarship during a tenure track?

[vocal phoenix]

Yep, although there's always the niggle of explaining why that volume is sometimes negative and what this depends on

[turbid zenith]

Yeah but I feel like that’s easy enough to explain

[quasi musk]

I think Arnold was just deeply frustrated with the French system. I believe in the US that there's a strong emphasis on "applying" what we learn, and I largely agree with Arnold's approach to mathematics. I think professors need to think much further beyond definition-theorem-proof, and I have great admiration for how Physicists teach their students to think about math

I also think that the way "applied math" courses use math to go solve something, in a way that isn't penalized if you're not 100% rigorous is very refreshing

[zinc dove]

This can be applied to in CS, where, e.g. in crypto, there're a lot of definition-theorem-proof going on.

[quasi maple]

Ironically the geometric definition is the one I use if I'm asked what a determinant of a matrix is

Though I only do that when ik the person asking me knows how to apply a matrix onto a vector

[arctic coral]

I am teaching advanced algebra, and i have an issue with constructive proof, my students doesn't quite understand them and get stuck just thinking about where the construction came from instead of why it works

any useful ideas dealing with this kind of issue

[tall bolt]

Could you be any more specific?

I often find with constructive proofs it’s best when they’re first presented as an “easy” computation for a specific space/object and then show how that process generalises to be a generic proof

[arctic coral]

i see

interesting approach

in my case the question is to find a bijection between two sets
and when i provided the bijection they got confused

my answer to that was just it is not something trivial , and it is based on careful remarks and handling of some elements of each sets

and that can take ages or it can take 5 mins

so probably i should have just changed the question to prove said aplication is a bijection instead of asking them to find one

interesting appreach it raelly desolves the issue

ty so much @tall bolt

[upper nacelle]

Geometry and imagination book by Hilbert starts with ellipse construction + conics, then moves toward cone and more about conics and surfaces of revolution. Honestly we have never analysed these objects in a such depth at school nor in my math bachelor. We had a pure algebra course in the first semester, and the same Prof. Was giving linear algebra. I don’t know, but it always felt like he was in another dimension, and basically we had to memorise techniques without really understanding any of the concepts. I really wish we had much more geometry back then

[quasi musk]

Neat

[feral swan]

I don't know i quite like thinking about a determinant as a number attached to a matrix.

sometimes it's nice to not have meaning attached to the symbols. It's like having another angle of approach i guess.

For example I am taking algebraic topology and it's very hard. It's nice that I can calculate some homology without knowing what it means because I'm new to the topic and everything is very hard. Should I want a better understanding I can translate it to geometry.

But im just a master's student so i don't really have an opinion everything seems good.

[tawny slate]

soft question, sanity checking

there's no reason to assume sin is more "natural" than cosine or vice versa right? i know the obvious usual arguments like that cis is the real trig function

and also obviously they are slightly different things entirely, for instance one is an odd function, the other is an even function

but do any of those differences give rise to any relatively general context for which we should consider one of them primary over the other?

[compact isle]

They're both two sides of the same coin (the complex exponential) so I wouldn't say one is more natural

[vocal isle]

Hi. Any recommendations to teach a linear algebra student that matrix multiplication is function composition of linear maps? I convinced myself the long way but I want to know if there's a quicker way.

I was preparing for the lesson but like was "nah, I can't eat up an entire class doing this".

But this is what I did to teach myself

My student knows matrix multiplication but not why we do it like that

[cosmic ibex]

If you've shown that multiplication is associative, then I think behaving like composition should only be a few lines of handwaving away.

[vocal isle]

Yeah, I have an idea thanks

[feral swan]

I like looking at the sigmas and seeing that indeed they line up on both definitions.

[turbid zenith]

Do it with a 2x2 example.

[austere delta]

Step 1: multiplicative matrix by basis vectors gives the corresponding column
Step 2: matrix multiplication AB is the same as multiplying A by each column of B.
Step 3: a linear map is determined by its value on basis vectors

[slim path]

Its a pretty simple proof to give as an assignment imo so long as they know how to construct the matrix representation of a linear map.

You could do a visual representation of it in class.

And ask the students to show that two composable linear maps T and S represented by matrices A and B satisfy upto a change of basis (T○S)(x) = (AB)x.

[slim path]

A very important starting point would be to give a general definition for what a matrix is if they already have the intuitive picture and then restrict that by adding a linear structure. Then go in reverse to show how to obtain said matrix from the linear transformation in a given basis.

[slim path]

I think the natural-ness is mostly due to convenience of usage in well-known cases which have now been formalised so much that most ppl do not care.

Things I can think of have to do with local realism in physics prior to the advent of quantum theory when it came to dealing with oscillatory and wavelike behaviour and even after in some cases.

Plus there is a weird pedagogical bias towards the horizontal axis of ℝ² and ℝ³ perhaps stemming from the whole Euclidean vector addition, inner products, projections and stuff.

[tawny slate]

thanks for the response, i wish i could understand any of it 🥹

[slim path]

Wait a second. Is this your first introduction to trigonometry? Or have you just not come across vector spaces?

[tawny slate]

i understand those, I think I have a good grasp of undergrad math stuff, but that's all I am, I'm a math dilettante with an EE bachelors

[slim path]

Cool. So the last bit should be easy enough to grasp if that is the case. As far as the whole local realism before qm goes is that physics pedagogy often wanted its wave-like and oscillatory behaviour to be formulated using real functions like cosines and sines but it became obvious that the complex exponential was nice to use so ppl started taking the real part instead. Its only after quantum mechanics came along that most modern treatments stopped talking about the imaginary part as an actual figment of our imagination lol.

[tawny slate]

oh i see, that's a pretty good layman explanation

i think i got the very vague gist, thanks

[bronze heart]

So last night I tried teaching my Uzbek cousin (doesn't speak English) mathematics at the 7th grade level. However, I noticed he is not well acquainted with basic mathematics, such as addition and division, and he can't really solve equations like 7x+4=41. What kind of steps should I go through to make him be able to: solve a system of quadratic equations for example?

[slim path]

Train with examples like you would a neural net. First have to drive in the point that the variable is a number. Stop thinking about general systems of higher order equations before your bro understands how to solve a linear equation in one variable. Before all of that, use geometry, teach what a polynomial is and how a linear one should look and why. That intuition tends to help a lot.

[slim path]

You cannot just "provide" a bijection and check if it works while teaching. You need to work out a bunch of examples where you use the elements of the two sets to deduce a relation between them. You can then proceed to show that the relation is indeed a function and a bijective one at that but the deducing part is important.

You need to start with simple and obvious ones first. Then move on to slightly tricky ones and give them assignments on them and then take the intuitive leap of faiths on the harder ones and the students will now be able to follow you having spent time on trickier constructions.

[cloud zealot]

https://www.rxddit.com/r/math/comments/1ox4btq/growth_of_remedial_math_at_uc_san_diego/

[arctic coral]

thank you so much i will work on that, seems like that was the issue, me taking a leap of faith before a preliminary introduction of what i was dealing with.

[turbid zenith]

What do y’all think of grade inflation? Leaving this vague to get different responses.

[rapid tusk]

hate that it’s made grades meaningless

[native iron]

My personal experience was that it collapsed the difference between classes where I did pretty well and classes where I did very well.

[slim path]

Depends on what you mean by it. In thr country where I presently teach, schools and certified educational boards compete for a higher pass rates leading to decline in quality in proportion to inflated grades. AI has made things even worse.

[turbid zenith]

How so?

[rapid tusk]

an A no longer carries any weight when you can pull one with frankly minimal effort

[turbid zenith]

What does that look like in your experience? Like, what class, and what does it seem to take to get an A?

[rapid tusk]

in intro analysis

i completely bsed my final and somehow walked out with an A

prof also graded way too leniently

[turbid zenith]

Okay that does sound like "too easy to get an A"

[rapid tusk]

also hw grading was like “oh solve like one of these problems to get full credit!!!”

just a totally unserious class all around

oh and to add insult to injury the pace was so fucking slow

we didn’t get past rudin CHAPTER TWO 😭

[tall bolt]

I think this question would get a different answer from different unis in different countries.

I guess my perspective is that there are certainly just degree mills that churn out first classes for anyone willing to show up, but these places are typically given appropriately less respect.

As for inflation at bigger name places, because I have seen that the percentage of As etc is going up, I don’t know that I’d say it’s due to things getting easier as I’ve seen some people say. At least in my very limited experience from my UG and now where I’m doing my MSc (both of which are good unis), the work hasn’t really gotten any easier. I would actually say in the case of my UG the exams I sat were harder than in the past. But these days there’s so many more routes into uni, so more talented people who couldn’t make it before are participating, and there’s more resources available to learn the material well.

I guess there’s also an element that you have to do better now, there’s just so many more people going to uni now, so while a 2:1 may have been impressive 20 years ago, there’s simply just so many people going to uni now that you need a first to compete. So I guess there is an element of inflation, but I’m not sure it’s due to things getting easier, not across the board anyway. But this is just my extremely anecdotal and limited opinion

[warm valley]

My experience at my current institution is that it feels mostly like an issue at the high school level, because of how high school teachers are directly compared against how many of their students get to good universities, so that students need approximately 100s to get into competitive programs.

At the university level, I don't think it matters as much, in terms of both severity and the number of students it affects. For the former, I have not seen instructors pressured to give higher grades than they felt students deserved; if students are getting higher grades (which they are at our institution, but not drastically so. Excluding the two COVID years of online classes, they have gone up by 1-2% on average), most instructors I talk to genuinely feel the students have gotten stronger and are succeeding at the level the course hopes for them to succeed at.

For the latter, for students not applying to graduate school (which is the VAST majority of students), grades simply don't matter, no employer will look at them or ask for them, so many students try at the level to comfortably pass and not more (so ~high 60s low 70s, where I am from everyone uses percentage grades) and then use the rest of their time on things that actually affect their employment prospects. In this sense, grades have failed at one of their original motivations, which is to incentivize students to try harder to learn the material (many classes at my institution have grades that look bimodal, split into the students that aren't trying at all except to pass, and students that are actually trying to learn the material)

For what it's worth, my institution certainly counts as one of the bigger places, at least in Canada

[empty gull]

lol somebody posted this and added a little arrow like "tiktok was founded" lmfao

but yeah I remember being at UCSD and it was rough for a lot of people

[turbid zenith]

So continuing with the grade related discussion

Which of these should count as an "A"?

[native iron]

I would prefer 4, but since an A is expected in so many settings, it's kind of unfair to make 4 an A. I think 3 is fine.

[midnight scarab]

Depends on whether this is a course in basic bicycle riding or a course in bicycle tricks

Anyway imo an A means you satisfied the standards. Not that you went above and beyond

Moreover there's no limit to above and beyond

[tall bolt]

I feel like I’d prefer a “can do basic tricks like a bunny hop!”

I feel like an A should mean, you have the basics on lock, and know how to extend that knowledge to simple enough new ideas

That’s generally what was required for me in UG to get an A. If you wanted that, you were going to have to prove some genuinely new and unseen stuff in the exam

But I guess given those options I’d had to go with 3, I don’t think As should only be given to those who go significantly above and beyond, that gets you into the silly corporate thing of “you have to exceed our expectations or you’re failing”

[plain pebble]

I'd say "Can solve basic unseen problems on the material" is required for sufficient mastery in anything proof-based

[tall bolt]

Yeah I agree, and I guess I could see that being included in “student can ride the bike themselves” but on my first reading of it I guess I felt that should be something a bit more

From what I’ve seen though, the US seems to be far more bookwork heavy, and it’s why like a 90 is an A, but over here we’re looking at 70-75ish because we tend to have significant amounts of truly new stuff on exams (also the fact we tend to just have one massive exam worth all of our grade etc)

[plain pebble]

(and tbh it's also 70-75 because of tradition, and scaling exams that fall outside the desired distribution to match that tradition)

[cosmic ibex]

I find it weird when people talk about percentage completion of a randomly constructed exam as it if was a more well-defined measure of performance than a letter grade (or grade on another few-steps scale).

[warm vortex]

If a 12-year-old can arrive at the answer in his head and write it down but does not write all the steps in a advance math class
Will
Tutoring help him

[warm vortex]

How is AI affecting curious

[tawny slate]

yes, i had nearly the exact same scenario for several of my students

easy habit to fix in theory, takes a little bit of time because it is a habit

so you just have to be persistent and firm

relatively speaking, outside of special cases, one of the easier problems to fix

the only thing that might be mildly tricky is figuring out how to convince the student this habit is worth fixing, which you may need some prep beforehand, with examples

[cosmic ibex]

It's not clear from the question whether the kid can explain his reasoning and just doesn't want to bother to, or he wants to but doesn't know how to be more specific.
In the first case, I'm not sure tutoring in particular is what would fix the problem (if it actually is a problem: "the steps" is a bit of a false idol at some educational levels).

[turbid zenith]

Points and percentages are objectivity theater.

[tribal crest]

Does anyone here teach PK-2nd grade math?

I have the Singapore Math textboooks for those grades

Based on the textbooks, I'm trying to figure out what kids learn in those grades and in what sequence the learning builds on on the layers below

I spent a lot of my time one this – like 2 hours every day for 2 months. Still, it's very challenging to reduce down what they learned

It looks like the main topics I really care about are counting, comparing, addition and subtraction. There are others like geometery, time, money, ordinal numbers, etc. I'm not too interested in these.

But, if anyone can help me understand the progression for counting, comparing, addition and subtraction, that would be very helpful!

[slim path]

Kids using it to do assignments and projects have become a headache. The fact that they have to resort to it more often than not I have observed is correlated to not having any grasp of the material or care to grasp for that matter.

There are of course exceptions where students use AI only to make life easier and do their due diligence but this is pretty rare in my experience. Only just now a student said google is her source for some jargon in her project. Since when did a search engine become a source? Since it started providing those largely hallucinated AI overviews.

[warm vortex]

How can my sister and brother in law best help my nephew 12 years old in an advanced math class in 7 th grade, already doing geometry and algebra. He is very smart, and arives at the correct answers quickly and write it down. But his teacher want him to show all the work. And that part slows him down the writing all the work. Would Math tourting like Koumon or matheism help. They won't give him in the advanced math class more time. But this is the first time he see point off. he always had over 90 to 100 grades A all semster of his school life.

and he did have advanced math last year and passed with A and passed state test in math in top of state

But he slow writer and needs to show all his work

[tawny slate]

those tutoring companies are unlikely to be of much help, because the help your nephew needs doesn't quite align with what they do

the quickest fix to this is to simply have him coordinate with the teacher about shorthands and abbreviations for things

if not writing fast enough is the only issue, that is

[midnight scarab]

If writing speed is the issue, wouldn't that show up (even more strongly) in language/humanities

[lethal leaf]

Showing your work is an important part of not just math but life

Often when showing people that you did something, they'll want to see how you did it

So that they can do it themselves if needed

[quasi musk]

A private tutor can help him develop the executive function skills to take notes or to write down their thought processes. I think it needs to be a gradual change to taking notes and writing work. You can't just go from no writing at all, to full solutions with perfectly legible work overnight

It will be a gradual change. I could get 5th graders to take notes decently well, but 6-8th grade, it was very difficult to get them to write things down

Part of it is that the problems are much more abstract for them, and multi-step equations become difficult to do. So they find tricks to multiply and divide in their hand to find the answer

[warm vortex]

He gets to use chrome book and text to speech

So a private tutor would be better off

[warm vortex]

Would an old school teacher approve of this ?

[tawny slate]

not sure, but that's what the teacher is there for, discuss with them

[turbid zenith]

To be fair, when I was in school it wasn't "Show your work so you can demonstrate your thinking"

It was "Show your work so we can see you didn't just use a calculator"

So a lot of this depends on whether the student sees showing their work as purposeful or just an exercise in compliance

[warm vortex]

Oh i not sure the reason

But it reminds of the movie about infinity when littlewold tell the students we need your proofs

Now how to find and select a good tutor

[quasi maple]

"Littlewood" and "student" - this was directed at Ramanujan, i.e. the 1 person

But I'd argue (and this is coming from me who for most of his school life was also a slow writer) that it's still in his best interests to write his working out down

If for whatever reason your answer is wrong, in pretty much all exams you then don't get any marks

[quasi maple]

In so far as addressing this point is concerned, I wouldn't argue that mathematical tutoring is needed so much as literacy tutoring, i.e. to either write faster or more efficiently

As for which is more the priority, I'd rather see what his working out would be, had he had ample time to write it down, before making that judgement call

[warm vortex]

Yea

They said he wlike get
Ore time but in not in advanced math class

[quasi maple]

Now this'll depend on the country, but depending on what "advanced math class" means, the working out itself will often be the answer

Any question that begins "Prove that..." does definitionally not have an answer of the form "The answer is..."

[native iron]

Advanced math class for a 12 year old likely includes a lot of computational work still.

Or all computational work.

It's not clear to me from the initial message what precisely the problem is. Is it specifically that he runs out of time on tests?

Or is it more that he's impatient and does not enjoy showing his work?

Or some other thing?

[quasi maple]

Yeah, again hence the "depend[ing] on the country" bit - it's unclear what "advanced math class" entails here, especially in the context of this being a 12-year-old

It could be anything from "next grade maths" (which would be 8th grade maths) to high-school maths, to yet other things, hence the need for some clarification

[cosmic ibex]

!nogpt

[lethal hornet]

switch to ecosia!

[warm grail]

ecosia my beloved

[quasi maple]

fair, though my point still stands

[lament basalt]

True now it’s my turn tho

[midnight scarab]

@turbid zenith I remember you were asking about (historical) motivation for envelopes etc.
Don't know whether it matches the history, but pretty neat example I came across while scrolling through this math methods book (Mathews & Walker)

[turbid zenith]

Oooo thank you!

[midnight scarab]

There's also this physics (kinematics) classic

Idea 28

For a neat challenge problem using this idea (EuPhO 2019 pbm 3)

https://eupho.ee/wp-content/uploads/2020/07/EuPhO_2019_theory_solutions.pdf

[cosmic ibex]

(I filled a sheet of paper deriving the shape of the envelope from scratch, but then got stuck figuring out how to find the point of tangency from the graph without being reduced to very tedious trial and error.
Then discovered that the model solution essentially just says to accept that tedium! 🤣)

[cloud zealot]

https://blogs.ams.org/inclusionexclusion/2021/12/30/failing-on-its-own-terms/

[cosmic ibex]

That sort of polemic always sounds slightly unhinged when one is not privy to whatever it is they're arguing against.

[dire harbor]

I was eager for the punchline but then it was just accusations of racism

Or i guess technically more allusions than accusations

[cosmic ibex]

Apparently claiming that 2+2=4 is racist now?

[dire harbor]

I think theyre mad about some decision that was made and jumped to "mathematicians are so stuck in their ways also theyre racist"

Which is a shame cause the way it was set up i thought it was gonna be about new ways of thinking about education research

But the only new way i found was dont be racist lol

[midnight scarab]

My vague memory is that it's actually relatively nice if you take y(x) and rewrite it as a quadratic in tan(initial angle)

[midnight scarab]

So the max value of y is given by the discriminant vanishing

[cosmic ibex]

But max value of y does not happen on the envelope.

[swift hatch]

-# jumpscare; I feel like I'll be seeing or remembering this question the rest of my life

[swift hatch]

I think the point was that a whole sheet of paper is a lot to derive just the shape - that writing it as a quadratic in tan(alpha) does it in just a view lines (either way it didn't count for marks 😩 )

[swift hatch]

It's like 10 lines of actual algebra starting from newton's second law

[abstract grove]

Some criticisms of math classes from a CRT/DEI angle have made a bit of a splash in media in the past few years:

https://www.thetimes.com/world/article/focusing-on-the-correct-answer-in-maths-is-racist-96gcztfs2

https://www.reddit.com/r/math/s/uJZLAaJjOy

https://katu.com/news/local/debate-emerges-over-racism-and-white-supremacy-in-math-instruction

at the high school I taught at for a few years, all teachers underwent mandatory DEI training annually over the summer, and I saw it get rather uncomfortable more than once

[dire harbor]

People love slapping the racism label on stuff to make their argument better man 😔

❌ "conceptual learning is more engaging and meaningful"
✅ "anything that isnt conceptual learning is racist and homophobic and you will go to hell before you die"

Tho tbh the journalists that rile people up over this are equally as bad

[tall bolt]

Yes I am willing to bet a significant amount of money that most of this is things taken massively out of context and blown majorly out of proportion by shitty “journalists”

[abstract grove]

this is the document that seems to have caused the main stir: https://equitablemath.org/wp-content/uploads/sites/2/2020/11/1_STRIDE1.pdf

[spiral elbow]

I'm sure there are some good points in this document, but I'm really struggling to understand how "math is taught in a linear fashion" is white supremacy

[dire harbor]

I dont see a single thing on here that has anything to do with white supremacy 😂 it almost seems like somebody wrote a book on good teaching practices and then told an ai to make it white supremacy themed

[quasi maple]

"Latinx", dear lord if that doesn't cause sparks to fly

[cosmic ibex]

If you keep telling people they are white supremacists, perhaps they will eventually believe you.

[dire harbor]

I always read latinx as latinks and it cracks me up every time

[spiral elbow]

yeah, I don't think the term "latino" was a problem until a bunch of white people suddenly decided it was

[tight star]

this is kind of a silly argument

[dire harbor]

Idk if this is how they meant it but i think its pretty valid to say that there are a lot of people who have been tricked into thinking theyre inherently evil

[tight star]

there’s a big difference between that and actually becoming a white supremacist

idk it’s an excuse I commonly hear from these people and I have 0 patience for it

[austere delta]

All the points seem like good criticism of practices in math education, but I don't see what any of them has to do with white supremacy.

Other than I guess math education being considered an antidote to race equity problems.

[tight star]

but regardless this is very off-topic for #math-pedagogy so I will refrain from saying anything more on the subject

[tall bolt]

I haven’t read it, but I am quite sure the point will be along the lines of “Because maths compounds so much, if you ever miss any amount of maths it can be really hard to close that gap. Non white students (certainly in the US) typically have worse schools, less stable backgrounds etc so this affects them more acutely.”

I’d also be very surprised if the point here is that maths shouldn’t be taught so linearly, I think it kinda has to be. But like my point is it’s very easy to take like a right wing “Look how crazy this is!!!!!” Approach to these things, because it’s probably actually very nuanced

It could also just be a ridiculous document, as I said I haven’t read it either

[dire harbor]

I hate how much nothing is said in this

[tall bolt]

But yeah, TLDR, don’t be a reactionary, read stuff

[dire harbor]

Ive only ever heard tales of these "needs of Black, Latinx, and multilingual or migrant students" ive never seen any real research

[tall bolt]

I do think this could be far better written and substantiated, but I think their key point is in:

“…without addressing the underlying causes of why certain groups of students are underperforming…”

[dire harbor]

Actually thats a lie in non math education classes ive seen vague mentions of how to help ESL students but that was with like reading passages for a literature class

[spiral elbow]

but they specifically state that the table lists examples of white supremacy, so I don't see how you can interpret it differently? They also have a list of characteristics of white supremacy. I don't get the impression that they go into any nuance at all, it seems to be more that anything negative is a sign of white supremacy

[tall bolt]

I find it incredibly hard to believe that there isn’t writing on this. It’s not something I’ve personally looked at because I’m not in the US, but I would bet there is a lot written on it.

[abstract grove]

I mean this controversy is literally over whether standard practices of math pedagogy are inherently “white supremacist”

[tight star]

I’m not going into this

[dire harbor]

I would think so too but i havent seen anything yet, i gotta go looking

[tall bolt]

Without either of us having read the paper by Jones and Okun I don’t really feel like we can meaningfully say anything here

This fully could be a poorly written document with flawed arguments, I’m not taking a side here, I’m just saying be careful not to fall into the reactionary trap

[spiral elbow]

I mean, you can definitely talk about how black and latino students are disadvantaged in the school system, I agree on that, but as Manifold says, this document talks about the actual way of teaching math is white supremacist. It literally says that teaching math in a linear way is white supremacy. How can you defend that statement?

[austere delta]

Objectivity?!

[abstract grove]

I don’t blame you. In a lot of circles, expressing the slightest pushback against these ideas is liable to get you called a white supremacist yourself

[tight star]

Please stop replying to me

[dire harbor]

Tbh i think this is a big problem ive been having with a lot of my classes, its a whole lot of basically "teachers should be good and do good things" without a lot of practicality. Ive only just now gotten into a class where i found anything helpful and im almost graduated 😭

[tall bolt]

It doesn’t literally say that as far as I can tell. And as I said, I haven’t read it and I’m pretty sure you haven’t either so I don’t think there’s much to be gained from this. There may be a valid point made and there may not be

[spiral elbow]

Exactly, it's very hard for me to see how you can defend a document saying objectivity is a characteristic of white supremacy

[pure light]

going down it seems their point on linearity was more 'look at what they know rather than rigidly sticking to prerequisites course sequences' but as with the rest of the report it's framed in a very clickbaity way

[tall bolt]

I mean have you read the paper cited? There may be good points made. The only thing I’m arguing against here is this kinda reactionary response

[tall bolt]

Yeah that makes sense, I could believe that. Part of why I’m so strongly making the point of don’t just take a reactionary stance here is because I know a lot of things like this are intentionally provocative, for better or for worse

[spiral elbow]

I haven't, I just strongly doubt that there's any context that can make the statement "teaching math in a linear way is white supremacy" true

[tall bolt]

I doubt this is the claim.

[dire harbor]

Has anyone else had this experience? Are there any useful resources people have come across

[abstract grove]

Pseudo, I didn’t mean to antagonize. I apologize, and wont push further on this

[dire harbor]

Cause im so tired of "meet the needs of students" what are the freakin needs

[spiral elbow]

it literally says "The table below identifies the ways in which white supremacy shows up in math classrooms", what other interpretation can there be? And also it literally says objectivity is a characteristic of white supremacy

[tall bolt]

I am a white Scottish man I have no horse in this race, but I’m just kinda urging you to read beyond the headlines here. It very very well could be nonsense but like you’re coming to that conclusion before engaging with it.

[tall bolt]

This is a headline, not the actual content, not where there is a substantiated claim. I also haven’t read the article about the characteristics of white supremacy so I don’t feel qualified to critique their claim. It could be nonsense, I agree that the claim “objectivity is a characteristic of white supremacy” is very bold and would require a lot of context and justification, but perhaps that is provided

[cosmic ibex]

Well, the document is right there for you to see the claim in ...

[tall bolt]

Headlines aren’t content, things can be true and stated provocatively. I could read it but it’s late and as a white Scottish man, I don’t really care that much

[cosmic ibex]

Yes, headlines are part of the content.

[tall bolt]

Again, it could also not be true. My only point here is to read before coming to any conclusions

[tall bolt]

Do you think a good argument is invalidated by a provocative or clickbaity title?

[spiral elbow]

I think this is the paper linked: https://edibleschoolyard.org/sites/default/files/resource library/WhiteSupermacyCulture.pdf
I skimmed it, and there doesn't seem to be much context that clarifies it for me. If you can figure out what they say beyond what I've pointed out already I'd be happy to listen. I feel like I've definitely "gone beyond the headlines" at this point, and I can't figure out a deeper meaning of their claim other than what is written in the table I linked

[cosmic ibex]

And when one reads it and points out to you where it makes outrageous unsupportable claims, you jump to its defense by saying that particular claim is not part of the "content"?

[spiral elbow]

I mean, I definitely agree with not being reactionary and only looking at the headlines, but I don't think I have done that. I've actually looked through the paper to try to figure out what they're saying

[tall bolt]

Then yes, it’s entirely possible it’s just flimsy bad arguments and can be disregarded. I’m not like defending a paper I haven’t read lol

[cosmic ibex]

It enrages me to see "white supremacy" brandied about in such a cavalier fashion -- it cheapens and normalizes actual white supremacy.

[tall bolt]

I’m not sure that is what I’m saying at all

[spiral elbow]

Agree, particularly in the political climate in the US today - there are actual racists in government, people who want to expel everyone non-white person from the country. I think it's kinda preposterous to then point a finger at math teachers and say that what they're doing is racist

[cosmic ibex]

What are you saying? You were pointed to a concrete page in the document where it claims in black on white that "objecitvity" is an indicator of "white supremacy", and your only answer is to redefine the document so it doesn't include that page?

[dire harbor]

"Damaging to both people of color and to white people" then why title it white supremacy 😭

Yet another decent points, horrendous branding moment

[cosmic ibex]

(Also what the heck is "right to comfort", and how on earth is it racist for students (I guess) to be comfortable? And/or comforted?)

[tall bolt]

My point is that a claim like objectivity is a characteristic of white supremacy is is clearly a very bold statement, and at first sight, completely ridiculous. However, without understanding the context in which the author uses it or argues about it, I wouldn’t simply write it off.

I completely agree that it sounds ridiculous, and it was take an incredibly strong nuanced argument to convince me that was a reasonable thing to say, but perhaps that exists. There is a history with these kind of things to be incredibly provocative to a fault, and I just don’t know without reading it that this isn’t the case

[cosmic ibex]

The context in which the author uses it is the list you saw.

[tall bolt]

I’ve seen a lot of complaints about “critical race theory” and things like that making “white students uncomfortable” as a reason for schools in the US to not discuss slavery or racism. I have to assume it’s in relation to this

[dire harbor]

Seems like their definition of objectivity is basically logical deductive reasoning and disregarding emotional thinking

[tall bolt]

They say that they follow the other paper, which I have not read, so I don’t think I would condemn it without having read that.

To be clear I agree completely that on the face of it it’s completely ridiculous, and any argument in favour of it would have to be incredibly strong and nuanced for me to be even a little convinced by it, but perhaps they make that argument. They likely don’t but I’ve not read it

[cosmic ibex]

It seems extremely racist to assume as given that "white" students are more comfortable with local deductive reasoning as a result of their skin color.

[dire harbor]

For right to comfort its feeling entitled to not have to think about uncomfortable things

[abstract grove]

the irony is crazy

[dire harbor]

And yeah i do think its kinda hilarious how these kinds of papers end up making non white people out to be this weird other

"Heres how we talk to black people guys... we gotta stop thinking logically and start being more comfortable with conflict" like bro what are we doing lol

[tall bolt]

And for what it’s worth I do think, at least in the recent past, words like decolonisation were thrown around too much, especially in education.

I very much suspect much of what they say has at best flimsy justification, and a lot of it is vastly overstated.

[cosmic ibex]

I don't understand why you think it is okay to make a completely ridiculous claim in instruction material for teachers, just on the off chance that somewhere in an academic publication that is not shown to the teachers in the audience for the document there might be a "strong and nuanced" argument that defends some variant of it.

[tall bolt]

My position here has never been defence of the paper. I also think the assumption that interested teachers are somehow incapable of reading said academic essay (which is clearly citied) is just weird. If they care to they can

I do strongly suspect most of what is said is at best flimsily supported and vastly overstated.

[dire harbor]

Its not weird, its reality :(

[cosmic ibex]

Your position comes across as a defence of the linked document stating as truth that being "objective" is a characteristic of white supremacists.

[pure light]

reading through the report, it seems the "right to comfort" is applied in the report to mean teachers prioritizing their own comfort [with a particular pedagogy] over students' needs

[dire harbor]

I cant speak for everywhere but ik where i live competent teachers are rare

[cosmic ibex]

That may well be true, but did they choose to be incompetent because they are racist?

[tall bolt]

I have repeatedly stated is not my stance. I don’t really have much else to say about this though so I’m going to go make dinner, this feels far more toxic than it is productive.

[dire harbor]

A lot of them are the type to fall into this exact trap of seeing something telling them theyre racist and then freaking out

[boreal agate]

no shot the pedagogy channel is now the politics channel bruh

[dire harbor]

I just say that to make the point that education draws in people who are very empathetic and these kinds of "handbooks" prey on that

The fact that talking about how to best service non white students is considered political is so sad

[frigid edge]

as if racism hasnt been a huge problem in teaching since like forever

[boreal agate]

hey i never said it wasnt

but i feel like this channel was meant for more meaningful conversations than "is objectivity racist"

if yall want to talk about how to make teaching better for marginalized groups, im all for it

but please frame the conversation correctly

[frigid edge]

That's fair, but its also not really what you said, so surely you understand why I responded the way I did. Politics is a huge and necessary consideration in pedagogy

[cosmic ibex]

The point is that most of the advice in the document would make teachning better for everyone, marginalized or not, and that bringing "white supremacy" into it is insulting to teachers and harmful for the larger discourse in society since it erodes the ability to recognize and reject actual white supremacists.

[boreal agate]

yes i understand, i apologize for not conveying my thoughts adequately the first time :3

[frigid edge]

It happens!

[dire harbor]

Ig to actually ask a legit question to yall, does instruction differ from white students to non white? Cause i really cant think of anything besides language barriers

[tawny slate]

got tired of reading somewhere in the middle but im starting to think that maybe math people should actually talk to experts before assuming what terms like white supremacy mean

its like "systemic racism"

just because you perpetuate a racially unfair system doesn't mean you're racist

thats not what the term means

[frigid edge]

I think class differences can be very prevalent between white and non white students, and many teachers do not take this into account when designing their curricula.

[tall bolt]

I would bet a lot of it is much more implicit. My best reference for this (being in the UK) is about how it’s been seen time and time again in multiple studies that teachers give less attention and support to students they deem to be from a lower class background

I fully belive that most of them are not explicitly classist, but this is the world we’re raised in, and these biases make their way into us weather we like it or not

[boreal agate]

then a good question in response to this is "what can be done?"

[cosmic ibex]

And the consequence of that opinion is to imply that actual swastika-toting supremacists are no worse than your local math teacher.

[frigid edge]

Including more marginalized people in the conversation is a good first step

[tall bolt]

That’s a very complex problem and I think requires a much larger discussion than just maths. A good first step is to get more people into teaching from a wider array of backgrounds, but this is hugely complex and I couldn’t give an appropriate at all answer just now

[frigid edge]

This is something that is on a case-by-case basis. Systemic racism and oppression in different countries looks different, and nobody knows marginalization more than people who are actively marginalized

But yeah tl;dr there's not one answer to "what can be done?"

Not that you were implying that either lol

[dire harbor]

Is it good to intentionally call on/check up on students from traditionally marginalized groups (more than others)? I think i would find it hard to do that without seeming patronizing to those students or ending up excluding others

[frigid edge]

I think a good way to not come off as patronizing would be to check up on all students when possible

The reason it can end up being patronizing is because you as (assumedly) a non marginalized person will then be deciding for yourself who is marginalized

There's also not a hard line between "marginalized" and "non-marginalized." These things are really nuanced

[frigid edge]

By assumedly I mean that this line of logic is with the assumption that the teacher in question is "not marginalized"

[dire harbor]

I think what really makes me cringe is all the stuff about trying to include "culturally relevant" content

Like i feel like thats just a recipe to make everybody uncomfortable

Like the adult is trying to be hip

[tall bolt]

I guess only if you do it badly

It’s just a wider point about meeting students where they are, and trying to consider that their background and experience may not be the same as your own

[dire harbor]

Definitely, i wanna be as respectful as i can

[tall bolt]

It’s for sure a hard thing to do though

As you said you can easily be patronising or like an adult who’s trying to hard

[frigid edge]

I think this is another reason that we need to include these marginalized communities in the conversation. I think a lot of the discomfort comes from people who aren't in these communities deciding what is "culturally relevant" for them

[dire harbor]

Thats the vibe i get, ive seen some pretty bad hypothetical (hopefully) scenarios in my classes

[cosmic ibex]

Isn't that the natural consequence for tasking teachers with coming up with "culturally relevant" disguises for problems in the first place?
Generations of teachers have failed comically to figure out ways to wrap problems that would make them "relevant for the students" or something such -- I don't see how sticking "culturally" in there is going to make the impossible task any less patronizing.

[dire harbor]

Tbh im pessimistic too but i dont wanna outright call it impossible to be culturally relevant

[frigid edge]

I don't think its impossible either, but it is undoubtedly hard, and also shouldn't be the responsibility of the teacher to just figure it out on their own

[cosmic ibex]

If it's impossible to find a relevant disguise for problem without the extra requirement that it should be "cultural", then I don't see how the chances of succeeding would be better by shoveling on more requirements.

[dire harbor]

I had the idea to try to make specific problems that have to do with things the students have actually expressed interest in