math-pedagogy

[tall bolt]

I think the requirement that it be cultural is already baked into it being relevant. I think the point here is to consider that many students cultural could be different from your own. Certainly this makes a hard task harder, but it doesn’t mean it should be ignored

[dire harbor]

I do think if youre clever enough you can find some way to make math useful to almost any situation but it takes a lot of creativity

[empty gull]

Math as a field is highly westernized
I wouldn't go as far as saying it's a priori "white supremacy" but this is a subtle point people don't think about enough. If you try and educate people on the history of math as was taught in different parts of the world (in my experience) 9 times out of 10 you'll be laughed out of the room because "it wasn't general enough until (mostly white men) came and made everything rigorous"
This is especially common with math as it was being developed in india but there is a subtle point that many of the people who try to talk about this are hindi nationalists but that's another can of worms
But math pedagogy as a whole acts as if it is some "neutral third party with how rigorous and serious everything is" but that approach is exactly what is cultural about it
That's the european enlightement way of thinking and that needs to at least be pointed out instead of being ignored
I'm not saying the way we're currently teaching math is "bad" however it is not as neutral as people seem to think

re what I responded to tropo with but I think the language here is just bad however the ideas that are baked into it are somewhat relevant
Math as is constructed these days is based primarly on european philosophical ideas and ideals and historically when credit is given in math is it horribly skewed towards white men

[tawny slate]

what no, no one in that field thinks this way jesus

those "swatika-toting" supremacists are absolutely the worst of the worst, but they are individuals who are so obviously racist it borders on cartoonish

when we actually want to solve nuanced issues prevalent in society, that are not the result of individuals being racist, but a game theoretic reason why even non-racists are incentivized to perpetuate structures that have racist roots, this is what we label as "systemic racism" in an academic sense

by not acknowledging this difference it is actually playing into the hands of the other side, because it gives them an excuse to continue to perpetuate those structures while pointing the fingers at extremist individuals saying that they are what you should be focused on

just because we have to focus on immediate issues like famine and war doesn't mean we can't have other people researching things like mathematics

[cosmic ibex]

what no, no one in that field thinks this way jesus
I'm saying that is the consequence of shaming working math teachers out for being "white supremacists", no matter those who are doing the shaming intends it to have that consequence.

[tawny slate]

well they shouldn't be shamed, not necessarily at least, that part i agree with

[empty gull]

idt they were being shamed I think it was just badly phrased

[tawny slate]

but the correct reading shouldn't be "teachers are white supremacist", but "teachers are, often unknowingly, perpetuating white supremacist structures"

[cosmic ibex]

If you insist of having no word that can be used to speak about the worst of the worst that doesn't also refer to your local math teacher, then you've lost the ability to distinguish.

[tawny slate]

unfortunately this is just how language works, i dont like it either but it isnt a proper reading of the literature

i also dont like that "hypothesis" and "theory" mean something different colloquially than they do in science

[cosmic ibex]

I simply cannot agree with the view that ordinary (even bad) teaching has a "structure" that should be referred to with the only word we have to speak about the swastika-toting worst of the worst with.

[tawny slate]

i also dont like how "argument" has a quarreling connotation rather than a term to separate it from its rhetorical definition

[empty gull]

I think you're conflating neo-nazism with white supremacy in general

[tawny slate]

but thats how language do be

[cosmic ibex]

Meaning what?

[empty gull]

meaning that white supremacy means any structure or belief that furthers the privlege and advantages you have if you're born white

it doesn't mean "swastika-toting maniacs"

[tawny slate]

if we can abstract the word "topology" or "space" in math to extremes, I don't see why we have to adhere to such rigid terms for other things

[spiral elbow]

What is neonazism if not white supremacy? Isn't that their main thing?

[tawny slate]

yes its tied to things in the real world and that language is often complex, imposes a reality, etc

[empty gull]

it's not a two way implication

[tawny slate]

i get all of that

[cosmic ibex]

White supremacy means the deliberate belief that white people should be oppressing all the other people.

[tawny slate]

its not a fair reading of what the academics are saying

[empty gull]

In a colloquial term yes definitely
But I don't think this was a document meant for the public and I very seriously doubt the people who wrote it meant it in this context

[cosmic ibex]

And if you instist on using it about ordinary nice (if perhaps struggling or not terribly competent) math teachers, then the message you're sending is that the swastika-toting maniacs are not significantly more of a threat to sociecty than a bad match teacher is.

[tawny slate]

its a claim about the system not the individuals

this is now reaching levels of "how can scientists be sure of evolution when its only a theory"

[spiral elbow]

I don't think it's about privilege, I think it's about superiority, as defined in the paper we're discussing:

[cosmic ibex]

It's reaching levels of "let's think we're making the world a better place by shouting to teachers they are bad people and should feel ashamed of themself for their innate racism".

[tawny slate]

bruh no one is thinking that

[cosmic ibex]

Someone wrote that long linked document without thinking, then.

[tawny slate]

i didnt read whatever document that was, im just letting you know this is absolutely not the consensus in this area of academic study

so either someone slightly unhinged wrote it or you just misunderstood what they were saying

[empty gull]

This has the same vibes as saying "why are we shaming boys for locker room talk it's totally natural and not bad at all it's just boys being boys"
Nobody is shaming anyone to start out with it's just pointing out biases and trying to make corrections

[cosmic ibex]

So you're saying theachers deserve to be shamed?

[empty gull]

well I mean this kinda still is about privilege but I get it

[empty gull]

I'm saying nobody is being shamed and you're making a problem out of nothing

[tawny slate]

the not-so-cartoonishly-evil racists justify their racist views by assuming that their cultural interpretation of values is the only correct one, by dismissing the values of other cultures as beneath them

this is how real conservatives hide and justify their racism

[spiral elbow]

You should read it then, because we are not talking about privilege in math education, we are talking about a paper claiming that objectivity and teaching math linearly is white supremacy

[dire harbor]

I do think its the responsibility of the authors of documents like the one from earlier to use language that best fits with their target audience. If its intended to be a guide for teachers, thats a target audience where you gotta use non academic language cause teachers are not academics

They are far from it usually

[tawny slate]

yeah i didnt contest this lol

[cosmic ibex]

Did you READ the document we're talking about?

[tawny slate]

i have no opinions on it either way until i read the doc

im saying that if that is what the document is talking about

[empty gull]

I have read what was posted in screenshot form

[tawny slate]

shaming individual teachers as white supremacist is a long shot

and i doubt any serious academic paper would do that

[spiral elbow]

And you agree that teaching math linearly is white supremacy? Can you expand on your view, because I genuinely can't understand the reasoning

[cosmic ibex]

It's not shaming "individual" theachers in the sense of making a distinction between some teachers and others. It's declaring everyone to be uniformly guilty.

[tawny slate]

for one, believing that math is something "objective" (the philosophical definition of the word here is doing a lot of heavy lifting) is something that arose from western philosophy

it is not one that is objective or universal across the world

so i can see a plausible argument to say "objectivity is white supremacist"

[empty gull]

I think it's sort of echoes towards a larger argument about education and meritocracy as a whole and how it basically is built as a white supremacy thing in the sense that the modern education system is sorta built on a foundation that was established during a time when racism was very very prevalent

[empty gull]

I think it is to the extent of math being taught linearly is a tactic that was established a long time ago when people had interesting views about people of other races and how such a system alienated marginalized groups that now might require a different style of education/pedagogy to succeed

[cosmic ibex]

Being taught linearly is not a "tactic". It's a natural consequence of the fact that time flows linearly and you can only say one thing at a time.

[spiral elbow]

It's also often very necessary in math, because it's so cumulative. You can't teach pre calc and algebraic geometry at the same time

[empty gull]

Idk if you're taking the piss or not with this comment but that is not what was meant lmfao

[cosmic ibex]

Then what was meant?

I don't see how one could propose to teach math or any other topic in a where where there's not something that comes before something else.

[empty gull]

sometimes it is necessary but not always
I think what was meant isn't teach one thing after another it's more like
You have a rigid linear way to progressing through math in school
You take algebra I, then algebra II, then geometry then blah blah instead of trying to recombine things to make them more open to exploration
Ofc you can't teach alg geo and pre calc at the same time but the way the current system is built doesn't require things to always be linear
Why can't you learn how algebraic manipulations work in the context of euclidean geometry for example etc.

[cosmic ibex]

There can be a choice of what to take first, but in the end each student is going to end up experiencing things in some linear order.

[empty gull]

I think this is a very narrow minded view of education

[cosmic ibex]

Is just a fact of living in a world with a single time dimension.

[empty gull]

no it's a fact of living in a world where you sit down and listen to lectures 8 hours a day for 12 years with barely any input from yourself to try and discover things on your own

[spiral elbow]

Okay, so it's about choices of the subjects you can take? Why is the lack of choices white supremacy?

[cosmic ibex]

No matter whether you sit down or stand up or whatever you do, everything you experience in your life will be linearly ordered by whether one event happens before another.
This includes events of learning and understanding mathematics.

[cosmic ibex]

You never answered what it is you say is meant.

[empty gull]

I mean the way I see it that you're sorta focusing on the wrong tagline here
The system as a whole is built on white supremacist foundations
Back in the good old days when your skin color determined your rights the education system (and math pedagogy in particular) was literally constructed in a way as to exclude marginalized groups as they were often lower class
This is why math is considered "hard"
And these systems over time alienated these groups from being able to be on a fair playing field with their white peers and it's a system that's still being perpetuated today involuntarily or not

[empty gull]

it just means "you have to take X class or understand X thing before you're you're ready to dare try learning Y thing that's meant to come after X"

[cosmic ibex]

Are you claiming that having white skin somehow biologically confers a better ability to work in that way than everybody else has?

[molten void]

you all seem busy but - what do you guys think about the use of humor in math pedagogy - like one of the professors I’ve had was genuinely funny (and kinda weird tbh) - does it have the capacity to help with retention and focus? like with challenging subjects like analysis and PDEs. like one of my profs legit used weed jokes in his class which is crazy but idk he was still popular

[empty gull]

I'm not saying it's a biological thing but a social one

[molten void]

this is a very interesting discussion btw

[cosmic ibex]

Then why are you claiming that it is motivated by supremacy for a particular skin hue?

[atomic pulsar]

i think humor has a brilliant place in pedagogy

[spiral elbow]

It doesn't matter what the system is built on, I'm asking whether teaching math linearly is white supremacy, ie. do we do it because white ideas and beliefs are superior to those of people of color?

[molten void]

can you all send me the paper? i would like the share with my stem ed club at school?

[cosmic ibex]

Here.

[atomic pulsar]

sometimes when very large numbers come up my teacher will crack a "thats more than i make" etc which i think helps maintain focus and contributes to a nice atmosphere

[molten void]

ohhh ok

what is meant by mathematics being taught linearly like focusing more on solutions than just general progress?

[spiral elbow]

It seems like you're arguing that people of color are disadvantaged in the educational system, and I'm not disputing this at all

[empty gull]

because that's how society works?????
Do you just live in a perfect world where racism never existed????

[cosmic ibex]

You seem to be perpetuating racism by wanting to label things by skin color.

[molten void]

yeaaa it’s nice to see professors more as people i suppose than people who just straight up spew information

[empty gull]

again I would personally interpret it as "the style of teaching in math we use today doesn't suit everyone's cultural background for various reasons" and this makes sense to me at least
The way math is contructed is based on european philosophical ideals of how knowledge should be transferred and understood

[cosmic ibex]

"Because that's how society works" is a cop-out where you essentially say you don't want to argue for why that is how society should work.

[cosmic ibex]

The thing is that most things in the document we are discusssing are things that would make teaching better for everyone, including "white" students.

[empty gull]

It's important to understand how systems that were built at a time when people were being labeled by their skin color still affect those same people today like this isn't mental gymnastics
This is like saying affirmative action was racist lol

[empty gull]

well it would yes and that's the point it should be better for everyone
But we should be allowed to point out that the current system is bad because it doesn't help either students of other skin colors and in many cases it even makes white students less likely to succeed even tho it's literally built to give them a head start

[cosmic ibex]

If we agree that "the current system" is bad for everyone (with the caveat that I don't have any firsthand experience with American education, so I can only rely on secondary sources saying it is), then why not make it better because it would be better for everyone?
Why bury the improvements in accusations of white supremacy among the teachers if the bad teaching does not in fact benefit white students either?

[molten void]

I mean perhaps you can look at it this way

• In the western world, many people of color have been put in disadvantaged positions (like in the case of class and social position) historically at the very least
• Not only have they been disadvantaged, they have been abused by their white counterparts intentionally and unintentionally (consider segregation and micro aggression like stereotypes and bully etc.)
• Consider this leads to an environment that is not conducive to learning because for most people basic needs must be met
• On average in Western society, individuals with privilege are wealthier white individuals have better access to education and are more likely to have their basic needs met and perhaps have more resources in general to excel
• I think linear progression in math hurts most people because everyone has different capacities for education - POC or not and idk my general opinion that a linear approach to education is pretty ableist and treats the schools as if most students live in a vacuum

[spiral elbow]

This doesn't really answer the question. I know this is the case, but does it boil down to the idea that white people are better than black people? Like, the way we must take alg 1, alg 2, calc 1, etc. in high school seems to me to be a product of lots of reasons - lack of resources, logistics etc. But I can't understand how the color of our skin has anything to do with it? Do we teach math that way to oppress people of color?

[cosmic ibex]

None of the badnesses listed in the document seem to be ones that there's any good reason to assume that having white skin will make a student less harmed by them.

[empty gull]

why is it bad to say the current system has X issue why does it have to be framed in a neutral way like nobody is at fault
This is quite literally what those "all lives matter" morons were doing during the black lives matter protests
You're burying the voices of the people that are being most impacted and literally doing the thing they're pushing against

[cloud zealot]

incorrect:
https://youtu.be/P3yfGQivroE

[cosmic ibex]

I think it is bad to say that a system has a particular problem for non-white students, if the problem is actually one that affects white students in the same way and for the same reasons.

[warm valley]

I haven't read the whole conversation (but I have read most of it), but I can assure you that those academics involved in discussions of systematic racism (within education and otherwise) do want to make it better for everyone. They are in discussions of white supremacy in education (they are not accusing the average teacher of being a white supremacist, they are arguing they exist within a white supremacist system) because white supremacy in this academic sense is at fault for many structural problems facing students

[empty gull]

again this is the same argument people had during the black lives matter protests to drown out black voices without realizing it

[woeful jackal]

A system which favors richer people will lead to a dichotomy between white and poc people if the initial conditions are that of "white people will start rich, poc people will start poor"

[molten void]

Omg I have a personal anecdote a weirdo professor said that people in my proofs class weren’t talented but hard working in a kinda condescending tone that was so disappointing to me (this professor is a boomer old white man tho)

(I’m not white and …)

[lost raptor]

no bro look it's not supremacy if they also have their schools that are separate so long as they're equal

the supreme court said so

[woeful jackal]

ill say tho as a brown person studying in europe ive seen an insane amount of racist remarks, today a professor told my classmate "how did you make it to france if you are so bad" because she made a simple mistake, but when white students make such mistakes you dont hear any such remarks

[molten void]

Like don’t you need to have the capacity to put enough effort into learning to appear “talented” or just get “good” at something

[woeful jackal]

yesterday*

[molten void]

He was also weird about me having disability accommodations

[cosmic ibex]

Does the style of teaching in the US system actually differ depending on the skin color of the student?
If it does, then I agree there's a problem that needs to be fixed.

[tawny slate]

there are plenty of ways to teach concepts in a nonlinear fashion, such as through the use of abstract puzzles

[lost raptor]

are you fr

[tawny slate]

yeah im seriously wondering if youre taking the piss

[molten void]

REAL. My prof was legit favoring this other white kid it was the weirdest stuff like and it was just bc he was getting all the answers right he didn’t seem to care that much if other students were putting effort into and asking questions

[empty gull]

tropo I am saying this in like the nicest way possible
But you need to take a good look in the mirror and realize that you're kinda being very racist rn

[cosmic ibex]

What do you think is racist about what I'm saying?

[tall bolt]

It does. Racism didn’t just go away when segregation ended. This issue isn’t even specific to the US, though it is structurally worse there.

[lost raptor]

tropo are you deadass like im ngl you have to have opened a history book before

[tawny slate]

ok this truly cannot be a serious comment

you cannot be serious

[lost raptor]

I was referencing the "Separate but Equal" decision in particular

[cosmic ibex]

If you have a point to make, it would be more productive to make it instead of just spewing generalized insults.

[lost raptor]

also theres a high degree of segregation in schools in NYC in particular even iirc

[molten void]

yea Chicago too

[lost raptor]

the point was all the things that went over you

[molten void]

Jim Crow is still going onnnn

[tawny slate]

tropo its very difficult because when we try to patiently explain the concepts to you you keep retreating into very reductive positions we already rejected in the comments we previously made

[cosmic ibex]

And if some of them teach in an effective ways and others hold students back, then definitely that is a problem worth solving.

[tawny slate]

its very frustrating, it doesn't feel like you are taking us seriously or listening

[lost raptor]

The points were made in multiple other formats too you just genuinely dont know anything about this

[molten void]

what happened to the guy that was lowkey making properly racist commentary lol

[tawny slate]

you are dismissing the experiences of everyone here outright without bothering to understand how they feel in detail

[lost raptor]

Talking down, even

[tawny slate]

i do not want to label you racist but this type of behavior is literally a tactic racists use to mask their racism

im js

[lost raptor]

you genuinely sound like a caricature of a commentator around Brown v Board of Education

[cosmic ibex]

Why?

[tawny slate]

i want to be constructive, so i really hope you dont take it personally, i simply want to point out that the discussion feels frustrating

[lost raptor]

these are the things they said

when resisting integration

[cosmic ibex]

Because I agree that it's a problem if kids get different educational opportunties depending on their race?

[tawny slate]

its not that, thats an easy position to hold

[cosmic ibex]

I'm trying to respond to you, sorry, but it feels like we've gotten to a point where there's very little concrete I can pick to respond to here.

[cosmic ibex]

Then what did I say that's wrong?

[lost raptor]

many things actually

[molten void]

Dam this is getting heavy

[tawny slate]

i think what is more important is not the individual things because you have demonstrated that no one feels like you are listening to them

[cosmic ibex]

Could you explain some of the many things? Otherwise I have no chance of learning.

[tawny slate]

i think what is more important, as someone else mentioned, is you take a step back

[warm valley]

(This is not just US specific.)

it's not that the style of teaching of a given teacher depends on the skin colour of a student. It's that the education system as a whole (and this often includes teaching styles taught to teachers and encouraged in curriculum) is heavily biased towards students who have resources (e.g. familial wealth or able to devote free time to studies), among many other biases (e.g. towards ablebodied students). Given the existence of racism historically (and today...), this leads to disadvantaged outcomes for students of colour. it is this structure that is, in the academic sense, white supremacist.

[woeful jackal]

you are pretending that the teaching style is the only things that factors into becoming a mathematician when in reality it is probably the least important thing when compared to networking, treatment of students after class, skin color factoring into acceptance, mental health, encouragement from the environment, and the financial abilities

[tawny slate]

reflect on why it is people feel like you are not listening to them, why they feel like you are dismissing what they have to say

[molten void]

REAL

[lost raptor]

also, ya know, different schools can afford (or convince to hire) differently qualified teachers

so some is literally teaching ability differences even

[cosmic ibex]

I'm not pretending that teaching style is the only thing that factors into thing. I had the impression that teaching style was the particular topic this conversation was about in the first place, though.

[lost raptor]

it was never

[woeful jackal]

i thought it was clear that it wasnt but like sharp pointed out even if it were theres still an issue

[lost raptor]

(it was clear)

[cosmic ibex]

Okay, all I thought I was discussing was the claim in the linked document that there's a particular style of teaching that is "white supremacist".

[woeful jackal]

plus ngl we learn most of the math outside the class and most professors dont put the same amount of effort into teaching you stuff outside of the class when you are brown

ive seen it first hand

[tall bolt]

As to this point about it not being specific to the US, this is documented quite well in the UK with regards to class (because typically this is a class issue, there’s just also the related compounding factor of race). I avoided arguing with someone here about this a few days ago. Your social class matters massively in as far as the quality of your education and what you can get out of it

Hell even i e experienced this first hand. In my teaching placement as part of my education class last year I did a week at two different schools, one in one of the countries most affluent areas and the other in one of the most deprived. In the former case we managed to teach them modular arithmetic (they were like 10) and in the other case it was essentially just baby sitting, and the difference in the quality and knowledge of their respective teachers was staggering. I do not in any way think that these students are somehow dumber or less capable because they’re from poorer backgrounds but there is a clear difference in attainment and ability. The whole discussion here is to dissect why and what we can do about it.

[woeful jackal]

also dont forget that the usual curriculum is indeed white supremacist, it is too eurocentered, but ngl i think this is a minor thing when compared to the other things we talked about

[tall bolt]

It’s not even really an issue of does this thing exist because it just like clearly does

[cosmic ibex]

You have spent many comments to tell me off for not listening, but I'm finding it difficult to hear what it is you think I should be listening to.

[molten void]

real

[lost raptor]

Particularly around historical things, but that's less of an issue for math in particular at least

[woeful jackal]

yeah

[lost raptor]

though US history taught in the US has its moments some places

[molten void]

i kinda have trauma related to this 😭 like

it was with a popular professor too

[woeful jackal]

🫂 same bestie i understand

[tawny slate]

for what it's worth, i do want to mention again that while I haven't read the document that details these things, everything in this list are things that rub me as "conservative" and "white supremacist", even in a mathematical context. i think people who value the sum total of these values are absolutely questionable people, these are red flags when i try to get a feel for how much i want to trust someone

[woeful jackal]

its tough out there

[molten void]

nah like when you’re a young 18-19 yo like this stuff hits ofc cuz you’re young and vulnerable and you’re getting mistreated like consider that … and now that im older none of it makes sense to just treat someone different based on race and ability over academics

[empty gull]

I think this means you should take some time and actually reflect on what was said here today and try to understand and empathize
And I'm being 100% genuine when I say this please for the love of god try and think about what people told you

[turbid zenith]

Okay just now seeing this...
No. Claiming that 2+2=4 is not racist now.
But 2+2=4 has become a dog whistle for the sorts of people who think that math is "woke" now.
It became a "thing" because James Lindsay (aka Binomial Jimmy) posted a thing about woke math and someone unfortunately took it too seriously.

[cosmic ibex]

Again, could you point out some of that? It's a very long discussion.

[empty gull]

everything
Just reread everything if need be
Try and understand the counterpoints people gave you to your ideas

[tawny slate]

#math-pedagogy message

other people?

[turbid zenith]

Past that ... oh my god, I missed a lot

[tawny slate]

i just didnt want to get into specifics because everyone else said everything already

[turbid zenith]

Is there a tl;dr or is there no hope lol

Do I just have to read to catch up?

[cosmic ibex]

Sorry, I don't think I can use "just reread everything" for very much.

[turbid zenith]

Fair enough lol

[tall bolt]

It’s kinda just been going in circles tbf, you get the gist from what you’ve seen already lol

[turbid zenith]

This was the tweet that set everything off btw

[tall bolt]

Maybe the Americans were right and we should have gen eds. Sociology for maths.

[empty gull]

someone said math was racist? Then somebody posted the papers where that was "apparently" claimed and certain individuals sorta were making fun of the papers for making "ridiculous" claims and then there insued a big argument where some other and myself tried to convince said individuals there is nuance here and what the papers were saying wasn't without substance

[turbid zenith]

Which like ... come on. I teach liberal arts mathematics. And I teach PLENTY about the overrepresentation of European mathematicians. And yes there are examples where 2 + 2 doesn't equal 4. But that tweet took it a bit too far.

It ended up with a lot of mathematicians coming up with ways that 2 + 2 comes out to something other than 4

And non-mathematicians largely not getting it and hence missing the point

And pretty much no progress being made.

[empty gull]

I think you should try to be more open towards what was said

[woeful jackal]

ngl not much naunce cuz the arguments are very clear people trying to deny them are just white supremacists

[cosmic ibex]

Again, if you want me to take something away, I'm afraid I'm gonna have to ask you to say it once again.

[tall bolt]

This has been much of the discussion summed up.

A lot of “that’s a ridiculous claim, so silly haha” to which the response was, it’s likely just a provocative title for a nuanced point, that is likely being vastly overstated. And the ensuing debate has vaguely been on the validity of these positions and sometimes restarting the whole thing

[lost raptor]

Very true

[turbid zenith]

To be 100% fair, both sides have used 2 + 2 = 4 as the bastion of objectivity

[spiral elbow]

I'm not sure that is the most unbiased summary. It started with this paper: https://equitablemath.org/wp-content/uploads/sites/2/2020/11/1_STRIDE1.pdf
And it turned into a discussion on whether objectivity and teaching math linearly is white supremacy. Then I think it just turned into an unproductive discussion on whether people of color are disadvantaged in the school system (which again, I'm not disputing)

[turbid zenith]

As something along the lines of "only absolute idiots would say that 2 + 2 is anything other than 4"

It's just that the one in most recent memory has been largely conservatives using it as their rallying cry

[empty gull]

math education and the education system as a whole are inherently racist institutions that were built in a way to disadvantage non white students and we are still seeing the effects of this today and it's important to point this fact out and try to build a better system that benefits everyone especially those that were hurt the most

[tall bolt]

The point here is that you made the claim that the paper said “objectivity and linearity are white supremacy” and the response has been, well I don’t think that’s a correct reading, and no one seems to want to budge

[turbid zenith]

https://www.youtube.com/watch?v=Zh3Yz3PiXZw
For an example of the progressive side using it as their rallying cry

[tall bolt]

I think the discussion started vaguely civil and productive and sort of just unraveled

[spiral elbow]

I just haven't seen anyone proposing an alternate reading? You keep saying that it's more nuanced, but I haven't seen anyone precisely explain why teacing math linearly is white supremacy. The only answer I get is an explanation on why people of color are disadvantaged in the school system, which doesn't really answer the question

[empty gull]

I think you're coming at this from a bad faith approach where you're attaching yourself to a single sentence that isn't even literally stated anywhere in the document to make yourself seem like some pariah

your point was talked about

multiple times in fact

and reasons were given

but you never listened or thought to listen

[cosmic ibex]

I may have to accept that's just how it is in the US. It matches badly with what I know from the education system I know. When it was built, there were no non-white students (or non-white population in the catchment area at all, in any significant number) to disadvantage in the first place, so I find it difficult to internalize how one would go about constructing an education system with that purpose.

[turbid zenith]

So for this, I'm not sure how much I currently agree with this assessment in terms of math assessment in particular, at least not in terms of it being actively built to disadvantage non-white students

Other aspects of the education system, absolutely, 100%

But I haven't seen evidence of that with mathematics in particular, again, talking about actively making it anti-non-white

What I have seen is plenty of examples of it passively being such because of not addressing inequities

To put it another way, for example, I don't think the people who teach only about white (and Greek, but that's a WHOLE other rabbit hole) mathematicians are doing so to nefariously twiddle their handlebar mustaches and make sure that non-whites stay out of mathematics

But that can certainly be the effect it has, and we've discussed that plenty in the classes I teach

[spiral elbow]

The paper literally says "The table below identifies the ways in which white supremacy shows up in math classrooms" and then "Math is taught in a linear fashion and skills are taught sequentially without true understanding of prerequisite knowledge". I think you're being unfair when you say I don't listen - I do, it's just that you seem to answer a different question. I feel like you're trying very hard to underline how bad it is for people of color in the educational system, like I don't believe that. We are most likely on the same side of the political spectrum, we have a lot of the same opinions. Like, this isn't a discussion about how to structure the school system, or what political system is best, because we most likely agree on that! It's just a discussion about the merits of that particular paper, which I feel is making claims that are unfounded

[turbid zenith]

There's a LOT to unpack here, wow

[warm valley]

I think it's important to emphasize that the broader critical theory argues (and I fully believe) that this phenomenon is not US specific, and reads dangerously close (from my POV) to arguing systemic racism and other systemic bigotry doesn't exist in Europe (?, unsure where you are from)

[turbid zenith]

There are a lot of issues here that I agree are relevant issues, but I'm not clear offhand on how they're connected to white supremacy

There are others that I can absolutely see the connection, such as tracking

[lethal hornet]

with the story that somebody shared here about their professor being disrespectful to non-white students when they made a mistake, that is an example of the first bullet

[rapid tusk]

yea I don’t think it’s at all correct to attribute all of them to white aupremacy

[lethal hornet]

i guess they are thinking independent practice is more valuable if you are predisposed to better education

[rapid tusk]

those seem more a byproduct of the Industrial Revolution kicking off in Europe than anything else?

[lethal hornet]

which ones?

[rapid tusk]

18th cent

[turbid zenith]

Also I think there are some other big ways that white supremacy does show up that aren't mentioned here

[lethal hornet]

i’m confused as to what you mean by this

[lethal hornet]

i’m confused. what do they want the classroom to look like?

[rapid tusk]

the sort of prescriptive teaching methods that resulted from them wanting to train “good” obedient factory workers

[turbid zenith]

There are too many things mentioned in that screenshot to give a clean answer to this

But it makes some good points

When also taken with said nuance

Each one is a link btw with more info

[warm valley]

I think the linked image is probably overstating its case, but I think there is a plausible reading along the lines of "the severity of these issues and their effects on outcomes are worsened by white supremacist structural issues"

[turbid zenith]

Here's one example

[turbid zenith]

This is a good way to put it I think

[cosmic ibex]

Denmark.
No, I'm saying that to the extent there is a systemic problem with exclusion of ethnic minorities here (which I accept there probably is), it cannot have been built into the system from the beginning because the ethnic minorities that are here now were not yet here when the education system was built. They started arriving around mid 1960s to 1970s, and by the time their second generation reached education age in significant numbers, the whole school system was already in place after an explosive growth in the first postwar decades.

[turbid zenith]

Like, these things don't seem like evidence of white supremacy on their own, but white supremacy certainly exacerbates them

Meanwhile I'm thinking about these kinds of problems that showed up in materials I used to teach from :V

[lethal hornet]

so they want non-standard teaching catered to every student while also accommodating to their background?

i’m not trying to argue for the white supremicists

but like

sounds like a great use case for AI if it were good at teaching math

[turbid zenith]

"I do, we do, you do" is seen as very traditional, and a lot of progressive education has been pushing back against that

[cosmic ibex]

What does "I do we do, you do" actually mean?

[turbid zenith]

In favor of more inquiry-based methods

[tardy ember]

it cannot have been built into
i think there's some nuance here depending on what exactly you mean by "built into" - it's not immediately obvious how it could have been intentional, but it's entirely possible that, by coincidence, properties of the educational system that were built at that time would later turn out to be exclusionary to ethnic minorities when they showed up

[rapid tusk]

how do you implement that in a typical classroom where 95% of students don’t gaf

[turbid zenith]

It's also called "gradual release of responsibility"

1. The instructor models how to do a problem up on the board
2. The students work on a problem guided by the instructor though some kind of back-and-forth
3. The students work on problems on their own, with the instructor going around and helping where needed

[rapid tusk]

and/or simply don’t have the wherewithal to make inquiry effective

[cosmic ibex]

This was in reference and comparison to a claim that (as far as I understood the claim at least) school systems in America were deliberately designed to disadvantage non-white students.

[turbid zenith]

That's the hard part 😛 a lot of those recommendations seem to come from affluent schools where the (privileged) students act like "model" students

Then you take it into a real-world classroom and there are other issues to deal with

[rapid tusk]

which makes that interview I did last month even more nonsensical

“☝️🤓 uhm ackshually you were being too prescriptive”

bruh how else am I gonna do it if the typical student is not very receptive to this inquiry-based whatnot

[turbid zenith]

Meanwhile I'm like, what about using Method A when Method A is appropriate, and Method B when Method B is appropriate?

Why does it always have to be all one way or all another way?

[tall bolt]

Yeah when I did my placement we had initially planned to go mostly inquiry based at both schools, and while we found great success in the affluent school with this method (like seriously a couple of the kids essentially came up with modular arithmetic on their own and were using it for numbers around 100), it just didn’t even come close to working in the less affluent school.

Among the small subset of kids who actually tried to engage with the tasks, I don’t think any of the got anything out of it and we had to pivot it a much more structured approach with them.

I feel like it’s very intuitively obvious that “different strokes for different folks” is true, but seeing just quite how true first hand was cool

It was also very upsetting because it clearly displayed how lower income kids were being set up for failure but ya know…

(I mean as a lower income kid myself that wasn’t a secret but still)

[turbid zenith]

Yeah, at some point, instead of picking a philosophy and sticking with it no matter what, you have to meet your students where they're at

You can move them toward being able to do that inquiry, but you may not be able to start right out with that

[tall bolt]

Yeah for sure. There were a few kids who were clearly capable and passionate and could’ve been taken there, but their teacher…

Eve ignoring that though it still shows a systemic bias (and not even just blaming the schools, this also comes from home and society at large) at play, because I don’t think for a second that there just so happens to be only 3/35 capable kids in the lower income school but some how basically 30/30 in the higher

[turbid zenith]

And part of it is the later grade you teach, the more you have to potentially "undo"

Which is also systemic

[rapid tusk]

I remember there was a paper my brothers had to read for high school English

“the ‘banking’ concept of education” or smth?

and like yeah that method has its shortcomings but sometimes it’s the only one that works given the students’ level

[turbid zenith]

That's the one where you view teaching as just "depositing" information into students

[rapid tusk]

yeah

[turbid zenith]

I think there's always a way to have at leaset SOME active learning on the part of the student

But how that's accomplished and to what extent depends on the students

[tall bolt]

I’ve been finding I’m doing this a lot with my TAing which I don’t love but I’m struggling to come up with much that’s better. Last week I experimented with getting someone to come up to the board and solve a problem (if I’m being honest this is largely because it gave me a way to kill 10 minutes and I didn’t have enough stuff prepared), but I’m not sure how effective this was. People weren’t super keen to come up to the board, despite them being reasonably happy to answer questions, and when a girl finally did come up, she wasn’t really speaking up and explaining and I didn’t want to push her to do so too much

I think the issue is that, the course is pretty easy, and the students are pretty competent so I struggle to know what to say. (they also have 3 lectures a week with someone who actually knows the course and differential equations so there’s that).

Like when I can I give some intuitive discussions of stuff. This week I spoke a good bit about how and why eulers method works and basically did a manual implementation of the algorithm (unbelievably difficult to free hand graph something well) which I think was pretty helpful, but there’s been a good few weeks where I just genuinely don’t know what to add and so I spend 50 minutes just writing down the solutions to some problems

[turbid zenith]

I almost never have students come up to the board

I know that used to be a thing when I was in school, but I remember my first year teaching any time I did it they would just do it wrong and it would just be embarrassing

[tall bolt]

I try to ask as many questions as I can, rather than just writing the solution. Simple stuff like, what kind of ODE is this, how do we solve one’s of this form etc, but I’m not sure that’s amazing either

They do seem pretty switched on though, it’s at 9am on a Friday so I’m usually still half asleep and a calculation mistake or two always slips in and I’m always quickly corrected which is nice

[tall bolt]

Yeah this is a valid concern.

[turbid zenith]

Also I found that when I did that the other students would just shut off

So I'm much more a fan of having students work on a problem in small groups

...but then you run into other issues

That I'm still trying to figure out how to deal with

Nothing is more frustrating than four students working on a problem silently in parallel with three of them spinning their wheels and doing every symbol-pushing mistake under the sun while the fourth one does it perfectly, when you specifically asked them to talk through it together

[tall bolt]

That’s fair, possibly that’s something I’ll suggest in future though, I notice most people sit in groups of 2-3 anyway so it could be a good way to get them to do some work that isn’t just be talking at them

[turbid zenith]

Yeah, if they talk it can be very effective

I really need to figure something out for the next time I teach the calculus sequence

I really want to give another serious shot to group roles now that I've seen them work in another class I'm coteaching

[tall bolt]

But I also do think all the people who show up are perfectly capable of doing most of this stuff on their own. The content is pretty easy and they’re all quite competent. In my marking basically the only things I ever have to say is like writing standards stuff, the actual mechanical maths is fine for probably 95% of them

[turbid zenith]

That's good!

[tall bolt]

It is! I’m glad they get it but it also makes my life harder in a way lol

If there was some common issue I could at least plan to deep dive on that, otherwise I’m just kinda like uhhhh

I’ve been considering pulling up some of the stuff from my introductory dynamics class at UG, because that was a lot of the same material but significantly harder, I could maybe run through some derivations from there at some point but then I worry if that’s just too much of an aside that would possibly only confuse them

[turbid zenith]

Calc I? II?

[tall bolt]

Intro to ODEs and modelling

It’s pretty basic stuff, the usual integrating factors, separation of variables, second order linear ODEs, difference EQs and basic numerical methods, systems of DEs and like dimensional analysis stuff

My intro to dynamics course did all of that in about 4 weeks and used like horrifically complicated systems, shit was awful, I failed, but it does mean it’s a nice source of harder examples using the same material

[turbid zenith]

Has anyone here ever actually taught the tangent half-angle substitution in a Calc II course? If so ... do you happen to know any real-world applications of it?

[turbid zenith]

All I can find is that apparently that substitution can be useful for robotics, but no idea about integrating such a thing

[tardy ember]

somewhat tangential but i feel like a lot of the arguments over this are just based on conflating two different things that "2 + 2 = 4 is objective" could mean, and more generally two different notions of the objectivity of maths - the idea that notation is objective, and the idea that something else which there doesn't seem to be any clear word for in this context is objective

there are contexts where you could write down the string of symbols "2 + 2 = 4" and, under the notational conventions in use in that context, it would denote a false statement, or fail to mean anything at all

but this doesn't really refute the claim that "2 + 2 = 4 is objective", at least in its strongest form - there might be some people who believe this strawman that notation is objective, but like, for any view that's sufficiently well-known you can find people who believe a ridiculous version of it, so proving that they're wrong doesn't really do anything

the more sensible reading is that there's an abstract idea, that we're denoting by "2 + 2 = 4" just because we aren't telepathic and can't just directly send someone an idea without using any sort of words or notation

and that idea is objectively true, in the sense that, if someone else is thinking about the same thing but thinks it's false, they are actually just wrong

obviously there are a few places that the hypotheses of this statement could break down:

1. they might not be thinking about mathematics at all - i think by my standards a lot of people just do not have any concept inside their mind that is "the same idea" as what i mean by "mathematics" (to be clear, this goes in the other direction too - there are all sorts of concepts from other philosophies and cultures that i don't have), and even if they do have the idea of "mathematics" they might just not be considering it at that time
2. they might be thinking about some other mathematical statement
3. they might not be thinking about truth - they might instead be considering some kind of social or cultural property of the statement, or trying to create art that represents it, or something, which again means they're just doing something that isn't specifically asking the question "is 2 + 2 = 4 true?". also this is again a case where i mean something very specific by "true"; if they have some other concept that they call "truth" but which is actually something different, then that's still a different thing

with all of these reasons put together, if someone says that "2 + 2 = 4" is false, i would consider it very likely that they just mean something different than what it sounds like - maybe they're using it as a rallying cry for the opposite side to whichever one is using "2 + 2 = 4 is true" as its rallying cry, or maybe they're trying to express some sort of opposition to the culture that produces the idea that 2 + 2 = 4 is objective, or maybe they mean "true" in some sort of relativist sense instead of in a mathematical sense, or something else i haven't thought of

but all of that is just the harsh reality of human communication, and especially communication between people with very different political views - it says nothing about the abstract ideas that are not themselves about humans at all

if i could somehow be certain that when a person says "2 + 2 = 4 is false" they mean exactly the same thing as i do, and yet they go on saying it, then they are just wrong, and there isn't some clever way around that because any clever solution will fail at the step where the statement means exactly what i mean by it

[abstract grove]

good grief, I started a real shit show eh

look upon my works, ye mighty, and despair!

[quasi maple]

Nothing beside remains.

[quasi maple]

I mean, you don't need to go majorly philosophical to find a place where the assumption behind 2+2=4 breaks down

A: "I saw two people in the library yesterday."
B: "I saw two people earlier today in the cafe."

How many people were seen in total?

Otherwise, at this point, however, this is more a matter of metamathematics

[tardy ember]

and that's an example of either thinking about something non-mathematical or thinking about a different mathematical statement, depending on exactly how you're thinking about this question

a more maths-terminology-flavoured way to phrase it is "how large is the union of two sets that each contain two elements?", which is a different question from "what is 2 + 2?"

[abstract grove]

but are those two-element sets disjoint or…

[calm egret]

I'm convinced most folks understand "2 + 2 = 4" formally, given how often people get upset about pemdas being ambiguous.

One of my favorite thought experiments/graphics on this is:

I think the trouble is hiding in the use of equality here, but I genuinely find it hard to think of a framing to never make mistakes like this.

[abstract grove]

I don't understand what that's conveying

[calm egret]

The number of squares on the left before concatenation followed by after

[tardy ember]

four small squares and the one large (yellow) square is 5 in total

[abstract grove]

ah, gotcha

[tardy ember]

the problem here is just that that's not what + means

[calm egret]

Kareem mentioned it wasn't even an artificial example, they were confused by some combinatorial issue

[abstract grove]

not an articial example of what?

[calm egret]

They claimed they didn't cook it up to prove a point, they were genuinely confused by some other combinatorial question and framing this paradox that way helped them sort their confusion.

[abstract grove]

that's cool

[calm egret]

Ahh, I see this conversation was started by the cultural relativism/race controversy.

If that's the context, it's worth knowing the phrase was born out of Galton's scientific Racism.

From this article (free online but I don't want to post a link) "Savage numbers and the evolution of civilization in Victorian prehistory"
MICHAEL J. BARANY
The British Journal for the History of Science
DOI: 10.1017/S0007087413000356

Galton then presents his most direct illustration: handing a tribesman four sticks of tobacco in exchange for two sheep, the going rate, he watched as his counterpart matched two sticks to the first sheep and seemed surprised to find exactly two sticks remaining for the second sheep. Galton reports that ‘[the man’s mind] got hazy and confused ... and he broke off the transaction’ and required that the sheep be bartered one by one.

This story was retold in many places including:

• ‘Are twice two four?’, Troy [New York] Weekly Times, 6 May 1886;
• ‘Are twice two four?’, Daily Evening Bulletin (San Francisco, CA), 8 May 1886;
• ‘Twice two: the mathematical problem which puzzled a savage’, San Francisco Chronicle, 13 June 1886; ‘Young folks’ column’, Atchison [Kansas] Daily Globe, 29 December 1886.

(Now I'm going to back off this topic because I see in the rules political discussion is discouraged, I wanted to share the above paper though because it gives some context for how cursed this topic can be. There are much better angles to talk about objective reality in mathematics)

[proven cape]

https://www.researchgate.net/publication/355459894_Objectivity_culturelessness_and_apoliticism_how_cultural_beliefs_prevent_the_advancement_of_equity_in_astronomy_graduate_programs

i don't want to restart shit but this sort of notion of the structures of higher education as being apolitical or objective is discussed in this paper

[cosmic ibex]

That seems to be speaking about the social structures and context that surrounds the teaching, rather than problems with the teaching itself. The problems the paper describes are important, but it doesn't seem to make any link to pedagogy -- they look like they could appear with essentially the same description in an organization that doesn't even aim to teach anything to anyone.
I see it does include "Utilize research-based methods for equitable pedagogy" as a recommendation in the conclusion, but I find it difficult to see the link from the body of the paper.

[proven cape]

i think if you want an example of how Western cultural viewpoints affect people's experiences with the teaching basic maths this is a good article

but yeah the one I linked doesn't have stuff specifically on pedagogy

[cosmic ibex]

My thesis, such as it is, is that the document that started the whole brouhaha when ManifoldCuriosity linked to it looks like one of those "performative efforts" your link trounces. Instead of addressing the underlying social causes for inequity, it proposes, for example, that teachers can make things better by ceasing to put a greater focus on getting the "right" answer than understanding concepts and reasoning. All while completely ignoring the employment context that heavily incentivizes teachers to focus on right answers by making students' performance in standardized tests a key parameter for evaluating the teacher.

[turbid zenith]

Just came across this, and since we've had some discussions about infinitesimals in here, thought some people might find it interesting!

Talking about, for example, the idea of 0.000...1

[dire harbor]

This is cool, i wonder if this introduces any hiccups in calculus

Thinking about it even if a student insists 0.999... does not equal 1 im not sure that would negatively impact their conception of what a limit is

In fact it might help introduce limits if they can conceptualize it as "plugging in" 0.999... for instance

[midnight scarab]

-# This is about students, but what about high school teachers

[turbid zenith]

What do you mean?

[midnight scarab]

One of my hs teachers was trying to expain to a few of us after class that 0.999... is not equal to 1, because when you do x=0.999..., 10x - x = 9, actually 10x has 1 less 9 after the period so 10x - x is not actually 9, but 8.999... with a 1 after the infinity of nines...

I say hs but I mean something like year 8 or 9, for US people

[tall bolt]

I never really got how people struggle with this but accept that 1/3 = 0.333... and 3/3 = 1

[midnight scarab]

Me neither

[tall bolt]

Like its admitadley weird that deciminal expansions arent unique when you first see that, but to me, seeing the example with a third makes me go "Oh actually yeah fair enough, im used to that it just hadn't been pointed out before"

[midnight scarab]

-# (you edited your message and now the sentence is broken)

[tall bolt]

English is hard

(yes its my native language, im just dumb)

[midnight scarab]

Yeah I have no clue. Also very funny to me was that literally the year before, so last year of middle school (I kinda skipped a year but also changed school systems, anyway), we got gotcha on a test drawing like a Venn diagram of rationals etc because one of the points was 2.999..., and people like me who didn't turn on their brains missed the fact it was actually an integer

[midnight scarab]

I type broken sentences all the time too

[turbid zenith]

Oh! So speaking of this!!

The student in this case study was equally uneasy about 1/3!

[tall bolt]

Hey at least she is consistent, that I actually respect. Honestly the fact she says she feels uneasy about it probably makes her a prime candidate to be a good analyst lol

intro RA was a shock for me because, like I imagine many, I just didnt really question any of this stuff before

[cosmic ibex]

I've wondered myself why this isn't a more common response to the 3×0.333... argument.

The only reason to have 0.9 repeating is to show that it's not 1!
So I also wonder what Sarah would (hypothetically) respond to "okay, you're right -- we don't actually have 0.999...; it's an undefined notation that doesn't stand for any real number".
(Edit: And now that I've actually read the article, I see that such an approach probably isn't likely to convince if attempted point blank).

[tall bolt]

What is "this"?

[cosmic ibex]

"I never really believed 0.333... = 1/3 either"

[tall bolt]

Ahh I see

[calm egret]

That blurb to me reads like she has an intuition for the difference between what a type theorist might call a definitional equality, vs a propositional equality. I wish foundations were less fussy/controversial because it's a fun subject otherwise.

If you have a program that outputs "9" for every digit of the fractional part and 0 for the integer part, can you show this behaves "the same" as a program that only outputs 1 for the integer part and 0 otherwise?

This sort of noodlyness was useful for implementing android's calculator, which actually supports infinite precision!

[marsh thistle]

if you're bored there is an entire subreddit dedicated to trying to convince its admin that 0.99... = 1

you can learn many amazing facts such as 0.999.. / 2 = 0.4999...95

[elfin sleet]

I actually don't find it surprising that people find this hard to believe. Decimal representation is a sum so in order for an infinite decimal representation to be at all acceptable, you have to have some notion of series so that an infinite sum is not total nonsense, and you have to believe that these particular series converge. Of course you don't need full technical knowledge of those things, a handwavey explanation would totally suffice, but you need some kind of discussion of what we actually mean by "infinite decimals", and I don't think most people ever receive that.

Also, I think many people don't have any notion of a number beyond the decimal representation. They think that the number is its digits. If that's your definition, then non-uniqueness of decimal representations is impossible: 0.999... is definitionally not equal to 1

[cosmic ibex]

So, what should the pedagogic strategy towards "Sarah" in that article be?
The authors argue convincingly that she has some conception of number which deviates from the real numbers of standard mathematics, but which feels reasonably consistent to her -- and for all we know at this stage might even be internally consistent, just different from what mathematicians in general think about when we say "real number". Even if it is not consistent, poking holes in it at the detail level is more likely to make her try modify it to save her intuition, than to fold and accept the standard concept as objective reality.
To the extent the article argues for teaching her non-standard analysis, I don't think that's the answer either. Getting useful results out of NSA requires explicitly making a move between the hyperworld and the standard world at just the right place in the definitions -- which presupposes an understanding that the hyperworld is non-standard in the first place.
My best idea is to use it as a moment to preach the "we can definite anything we want" gospel -- that is, agree (at least for the sake of the argument) that her conception of number may not be objectively wrong, but the mathematics of the textbooks uses a more narrow concept of number and she ought to explicitly decide to be familiar with the details of that narrower concept so she can communicate, even if she doesn't privately buy into the notion that it's all that exists Platonically.

[quasi musk]

It starts with what the goal is: if you're trying to teach her standard real analysis, then you just lay down the law as usual. If you're trying to teacher her non-standard analysis, then you start by drawing pictures, and the history of calculus -- why people thought about these things this way.

If you're trying to get Sarah to develop her own conception of math, then you still teach the basics of standard real, non-standard analysis, and ask her to try to formulate the ideas of 'larger infinity' or 'smaller infinitesimals"

[dire harbor]

I agree, i think its good to encourage that kind of thinking but be clear that class is gonna continue with whats in the course description

[quasi musk]

Part of teaching someone necessitates them putting themselves out of the way, and allowing you to fill their mind with information that you're trying to teach

[dire harbor]

I think a lot of students dont even realize that math can be something they do outside of their school work and the sarah situation would be a good chance to communicate that to her and encourage that

[cosmic ibex]

I suppose the goal I have in mind is to enable her to learn standard analysis without wasting a lot of her energy with pushing back on 0.999...=1.

[quasi musk]

Ah! I see. I got into this argument in high school. My friend also gave the answer of 0.00...1 when asked what is 1-0.999....

I retorted "Oh so you get to the end of infinity and there's a magical one there"

But if you do the 1-0.9 = 0.01, 1-0.99, etc. as a sequence. Ask them to compute the limit. Or ask them to write down the geometric series and compute it

You can even get precise with epsilons. Just choose epsilon = 1/1000. Ask them to beat you by choosing a term far enough out

[cosmic ibex]

Ah, but Sarah believes in infinitesimals, so she already has an epsilon that this sequence doesn't reach. (p.130)

[quasi musk]

yeah, it's out of the standard way. I'd just say it fails the standard way of how we do things

[turbid zenith]

Honestly, I would point out what I'm seeing, and point out that there are two different number systems at play.

That she has good intuition about one number system, and that the one the teacher is talking about is another number system.

[cosmic ibex]

Yes, that was my suggestion too.

[quasi musk]

Definitely a super interesting read

[turbid zenith]

This seems like the perfect place to share it 🙂

[quasi musk]

I've always wanted to delve deeper into how my students are thinking about math

It's hard to forget what I know now and put myself in their shoes

[calm egret]

TIL! Turns out if you're really careful, you can recover the 0.999.... notation for a term whose difference from 1 gives an infinitesimal:

https://math.stackexchange.com/questions/281492/about-0-999-1

(but of course it's still not true that 0.999... < 1)

[austere delta]

I think it can be useful to argue that the way we represent numbers was in fact made by us.

Sarah has seen how to add and multiple numbers with several digits, and this procedure also makes sense for infinite digits. As this is a useful procedure we define our numbers in a way such that the procedure works. We could have defined it differently, but this is useful.

If she can also accept that 10x = 10 implies x=1, then it must be that 0.99.. = 1

And to her point that "the only reason we have 0.99... is for it to be different from 1" can now be answered. The reason we have 0.99.. is that we wanted numbers with infinitely many digits and we didn't make a specific expectation for 0.99...

[calm egret]

But Sarah already knows the standard approach isn't how she thinks about it.

The whole game with teaching is getting your thoughts into your students heads, motivation is great but it's only the start of the process.

The issue is why aren't the standard Real numbers sticking. I'm not sure there's enough information in that article to infer what's wrong.

[austere delta]

Sarah knows that how she thinks about it is different from what she's been told, but it doesn't seem to me that she actually understand what (or why) the standard approach is.

And I think "number system where the addition and multiplication algorithms you've been taught work" is a good description.

[cosmic ibex]

Right. She may be aware that the standard conception reaches results she considers absurd, but I don't think she has a good idea of where the standard conception departs from hers.

She's presented as:

a sophomore whose high school calculus course excused her from taking Calculus I at the university. At the time of the interviews, she was taking Calculus II, and had taken a course on Finite Mathematics the previous year
so most likely she has never seen either an axiomatic definition or a construction of the real numbers. Just pencil-and-paper arithmetic and some handwaving about how the decimals can continue forever.

[calm egret]

Ahh that's a good catch, then walking through actual proofs would have a good chance of sticking for sure

Time is always an issue, but it'd be fun to compare and contrast how different definitions changed proofs with her.

[turbid zenith]

Then again, how many students WOULD see an axiomatic definition or construction of the real numbers

All math high school and below is hand waving.

[midnight scarab]

In my experience (with classmates), emphasising the distinction between a sequence and a limit, and that the limit is not something "close" or that "tends to" but the actual number

And that when writing down a decimal expansion, we're writing out a sequence, but to designate its limit

0.999... means the number that the sequence of 0.999...9 approaches arbitrarily closely

[turbid zenith]

Yeah, it's difficult that the symbol "..." suggests a process, but the thing we mean to use it for is the object at the "end" of this process

[strong hearth]

Seems that I am late to the discussion, but I wanted to offers a perspective as someone who works in education research.

My sense is that many of the reactions here stem less from truly "controversial" ideas from the inequity in maths PDF, but more from a mismatch between disciplinary paradigms. To put it bluntly education as a field and mathematics sit within very diffferent traditions of knowledge. Mathematicians (and sciences included) work within a positive paradigm, characterised by:

• Truth is axiomatic, abstract and context-agnostic;
• Correctness is binary, readily observable and measureable;
• Elegance, parsimony and rigour are disciplinary norms; and thus
• The discipline itself is widely understood to be "culturally neutral"

From this worldview, concepts that are brought up in the PDF like "perfectionism", "objectivity", "only one right way" and so on being framed as features of "White supremacy culture" can understandably feel like an attack on the discipline of maths itself. It is very much the case from when we saw Tropo's leap: "if objectivity is white supremacy then you're saying mathematical truth is also white supremacy" yielding "so 2+2=4 is racist now?"

But this is a categorical mistake. The paper isn't interrogating claims about mathematical truth, it is instead interrogating pedagogical practices, assessment norms and classroom power structures.

This is evident if you look at the early sections, whether they make their conceptual framing quite explicit. When they discuss characteristics that reproduce inequity, these are applied to institutional practices i.e. how we design curricula, set expectatins, evaluate laerning and structure classrooms. So then when they say "White supremacy culture shows up when there's a greater focus on getting the 'right answer' than understanding concepts and reasoning" they're critiquing instructional emphasis, not the existence of correct answers. Similarly when they say "Studenst are required to 'show their work' in standardised prescribed ways .. [T]eachers should offer multiple ways to demonstrate thinking and knowledge" they are critiquing the standardisation of expression, not the epistemic validity of solutions. Another really strong illustration is about language, where they say "language acquisition is equated with mathematical proficiency". A student's understanding of maths is very often evaluated through their ability to process highly contextualised, linguistically dense problems (especially in school settings). Even I may struggle with a simple alg problem that reads like "Five raccoons go to a spring break and order fifty IPAs". The inequity is coming from linguistic gatekeeping rather than the validity of e.g., 2+2=4.

The synthesis here, then, is that inequity does not arise from the content of mathematics itself, but from the institutional norms that goven how maths is taught, communicated, assesed and valued. This is also well-established in maths education research field and aligns strongly in literature (see e.g., works of Gituerrez, Nasir and so on).

To me, the widespread misreading seems to arise from an ontological origin. One may very easily (and intuitively) assume the paper is about challenge mathematical truth but it is really entirely concerned with pedagogy, as in how knowledge is taught and assessed, not what mathematical knowledge is.

It is, however, understand that people feel defensive when terms like "white supremacy culture" appear, which can feel accusatory for those who 1) see their discpline as cultural neutral (see above); 2) have not had adequate training in sociology of teaching and leanrning; and 3) conflate "objectivity" as an institutional, epistemic norm with "objectivity" as a mathematical property. Evidently, many interpreted the paper as moral judgement about mathematics and mathematicians within it, when it is in fact an institutional analysis.

What saddens me the most is how quickly some here have dismissed the possibility of inequity in mathematical education, despite substantial evidence, and despite not having enough training in pedagogy, assessmeht theory or the sociology of teaching. Ironically that is precisely the dynamic this paper brings up --- Well-meaning educators who at an ideological level separate their teaching practice from the broader structures within which that teaching takes place.

Fuck me that is a lot of typos I am not willing to fix.

[calm egret]

You'd probably find the article I referenced above interesting:

#math-pedagogy message

[strong hearth]

holy fuck I'm sorry but already I can see so much F*ucauldian influence in this paper 🤣

From my quick reading
it's not the problem that 2+2 is 4, it is everything that has historically been attached to the authority to declare that 2+2=4 in a classroom, perversely examine it, grade it and weaponise it to distinguish who is "competent" from "incompetent"

[cosmic ibex]

For what it's worth, I suppose I'm the "some here" you're talking about. I'm not -- or at least I'm not consciously trying to -- dismiss the possibility of inequity in mathematical education. It is well documented that an inequity of outcomes exists. What I'm saying is merely that it doesn't feel plausible to me that this inequity is caused by the particular bad teaching practices singled out in the original document, because it seems to me that the harm they do would be more or less color blind. (I suppose an argument can be made that socially privileged students have a better chance of succeeding in the end despite such intellectually crippling teaching, though, having more resources at home to draw on). So by all means let's work to reduce those teaching practices (and stop encouraging teachers to stick to them anyway because in the short-term perspective that helps their numbers when the standardized tests come in), but it feels wrong to claim anti-racist valor in particular while doing so.

[strong hearth]

thanks for the clarification, i think it makes it easier to find our common ground and also locate where our disagreements sit. I don't think any will dispute that poor teaching practices harm all students. The point being made here is not that these practices only harm the students of colour (in the way US operationalises them), but that the harm is not evenly distributed because these practices intersect with existing structures at the linguistic, cultural and socioeconomic levels. I'll try to expand on three points.
First, many wordy maths questions which to my understanding are common in the US systematically disadvantage L2+ students (i.e. immigrants, those who use another lang at home, myself included as L4). Sure, we can claim two students may have identical maths competence, but one is disproportionately penalised because of the language load. So then the inequity is from the interaction between pedagogy and the demographics, rather than pedagogy alone.
Second, context embedded in problems, like "baseball cards", "apricot chicken" (idk anything more culturally US things ;-;) further advantage students who happen to share that cultural capital (in Bourdieu's sense), which as you suggest is not about race per se it's about which cultural background the curriculum assumes as the norm—which this paper builds upon the established dominance of the "white Americans", hence the white supremacy culture argument. (But I must caution, since I'm from a country that does not utilise "race" and "colour" to this extent, my view differs here).
Third, it is about access, but also somewhat relates to point 1, in that perfectionism, "show your work in this way only" etc. also disproportionately disadvantage students whose families may not so-easily decode the hidden rules of schooling, help with hw etc. Again the practice itself is neutral but the impact is not

So I agree with you on some grounds, but (ran out of words)

but the point ins't that these practices are racist in isolation, but that in these educational and social texts they compound existing inequities in ways that associate strongly with, as they argue, race

also i shall return later after i pick up my husband ❤️

[dire harbor]

It seems like theres a kind of snowball effect that anything that could be detrimental to anyone is even worse for people that are already disadvantaged, and 2+2=4 thinking is no exception

Tho i think i react negatively to that pdf and things like it because they usually feel like more of a cheap way to look deep and heroic

[austere delta]

But if any problem that exacerbates racial differences (e.g. any problem that can be mitigated by money, power or familiar resources) is "white supremacy", doesn't the word kinda lose all meaning?

[cosmic ibex]

I agree that a difference in language or cultural assumptions between students and the teacher (or authors of central tests the teacher is required to teach towards) must be a problem. On the other hand "wordy problems" are not readily avoidable, because a mathematical skill that one can only apply after it has been reduced to (supposedly culture-neutral) symbols rather than a real-world situation is not really learned.

[dire harbor]

I dont think that makes it lose all meaning, it would just have a much less inflammatory meaning than it used to. Which makes it a prime candidate word to stick into your pdf so that its technically not wrong but riles up people for no reason

A journalism classic

Tbh thats probably WORSE than having no meaning lol its actively confusing

[tardy ember]

...i'm not convinced that the standard approach with word problems actually works to get students to the point where they can actually apply the mathematical skills to the real world instead of just to a different set of symbols, but i guess that's a separate issue

[cosmic ibex]

I see your point about "show your work in this way only". Though it sits ill with me to make it the individual teacher's moral duty to stomp it out if the broader systemic context of the classroom will demand such standards at the end-of-year exams and punish both students and teacher if they've failed to acquire them.

[dire harbor]

Thats another topic i wanted to ask about in here, am i the only one that sees standardized tests as a hurdle/distraction from meaningful learning?

[cosmic ibex]

No.

I feel that view is fairly broadly accepted among both mathematicians and teaching professionals -- though not necessarily among the administrators and politicians with the power to demand them.

[tardy ember]

leaving that aside, it feels like you could probably get somewhere with at least requiring less cultural/linguistic knowledge? the underlying skills these problems are supposed to test aren't any harder with cultural references that you get and someone else doesn't than an isomorphic problem that's accessible for everyone, so it seems like fixing the problem here would mostly just involve being more conscious of the possibility of it being an issue

...of course that's a big thing to shove into a "just" if we're also talking about changing the tests in that way

[dire harbor]

Thats the kinda vibe i got from some of the (good) teachers i had in high school, tho i never heard it said outright and i kinda got a side eye for bringing it up in my classes now

[calm egret]

Usually these arguments are framed as criticisms of curriculums not individual teachera perse.

And I've been consistently surprised at how popular standard curriculums are. They are public services.

[dire harbor]

Standard curriculums as like a resource for teachers seem great but micromanaging and requiring teachers to adhere to them strictly doesnt seem so great. Maybe id feel differently if i saw even a single good one

[cosmic ibex]

Yeah -- but no matter what the right solution to a real-world connection is (I don't have much of a suggestion, except agreeing that many word problems seem to be fairly inane even for students with all the expected cultural baggage), whatever it ends up being it's going to involve lot of words too, so fixing that is somewhat orthogonal to the language barrier problems.

[cosmic ibex]

The side eye could easily be because it inadvertently came across like challenging the teachers to defend a system they personally disagree with (but cannot say outright because that would be criticizing the hand that feeds them) yet are powerless to change ...

[tardy ember]

well you might be able to take an analogous approach of just... phrasing the problem with simpler grammatical structures? or something? i don't really have a good model of what it's like to speak english well enough that it's not utterly absurd to sit in a class that's being taught in english but still relevantly struggle with it (i don't speak any language that well, other than english (my native language) which i have no problem with, so i don't have personal experience here) but it feels like there should be something you can do to make it easier

[cosmic ibex]

Yes, I'm not dismissing the possibility of doing better there, just that it seems to be somewhat independent from other reasons word problems suck.

[tall bolt]

I still dont see it as a claim that this is actually white supremacy but that it is a symptom of a culture that has normalised it. Fundamentally, as I probably would in fairness, I see this as a class issue, but you cannot spererate class from race (or other identities, but race in the US is by far and a way the prodominant factor to consider). Like all of these ideas have been discussed at lenth in the UK but simply with the words class here because we have much bigger, and much more I guess culturally noticable class division (by that I mean, its not just race, its the way you speak, dress, conduct yourself etc. I of course recognise this is true everywhere, but its very much so here, and very ingrained). Like the issue of the lanague used in word problems is something that has been dicussed and researched at length here, because the languge of maths, and I guess more generally academia, is the language of the upper and middle classes, which is just yet another barrier for working class students. Of course, thats not to say working class students are less capable or whatever, but its not their common, natural langauge that they go home and speak.

The paper is certainly inflammatory in its wording, but I think theres a clear difference between the usage of the word as a cultural power vs the useage of the word as a label for people and an ideology. This is often the case, but we just have slightly better distincitions in the words, like the patriarchy vs sexists/misogynists/misogyny.

Not to like start this all again though, I feel like everything which could be said has been, at least without everyone starting to write actual essays, and I think leo puts it very well.

[tight star]

<@&268886789983436800> spam

[haughty latch]

What kind of set up do you guys have for teaching math online? I am looking for solutions that don't rely on an ipad/drawing tablet

[tall bolt]

I would think if you want to do so seriously, the small investment into a cheap drawing tablet would just be worth it. Not being able to write things down when youre teaching just kinda sucks

Either that or get very good at drawing with your mouse lol

[slim path]

Or, if you are okay with investing heavy amounts of time just preparing material, Manim and OBS

[tall bolt]

I took teaching to mean like tutoring, where you want something a bit more fluid

If you want to do like youtube videos that is a "good" option

[slim path]

Another option, slightly less time consuming is to use Beamer presentations with appropriate pause commands and use OBS with a pointer.

[slim path]

Aye, Manim is far too much work to use in live teaching. Tried it. Got exhausted in a week and came back to using a tab. That said, Beamers are doable for weekly lectures/tutorials.

[tall bolt]

For lectures and stuff certainly, I just took this to mean like 1 on 1 tutoring where I think being able to write stuff down live is 100% needed. I guess you could live tex in like obsidean or something but like, drawing tablets can be really cheap

[slim path]

Indeed. The best option for online one on one tutoring for sure is a drawing tablet. Was just suggesting alternatives depending on the need.

[haughty latch]

What software do you use to write stuff down (with a drawing tablet) while teaching?

I managed to get my hands on a crappy one and I'm having a bit of a boomer moment (or alternatively: an ipad kid moment) with it's pressure sensitivity

[tall bolt]

Zoom has built in whiteboard stuff though I’ve not used it myself. There’s Microsoft whiteboard and one note which I’ve used and are fine.

This isn’t something I’ve had to do more than once, and I had an IPad though so I don’t have the best recommendations unfortunately sorry!

[haughty latch]

ok, thanks anyway

[molten void]

Do you all think that ableism is prevalent in mathematics education like from secondary to undergraduate level?

even advanced pure mathematics

[strong hearth]

Could you contextualise?

[molten void]

Like, for example, my diff eq professor once said in an almost negative tone that the poeple in my proofs class "are hardworking but not talented"

... what is talent?

Like how do you get "good" at something is it just natural or there are other factors that make someone more "proficient" in a subject

Anyways that prof is a creep weirdo so i am able to brush what he said to me to the side to some degree but it makes me concerned that do at least these old mathematicians who are tenured professors believe that doing mathematics is an innate metaphysical ability and it should be privileged over others putting sufficient effort

Well it could be true that he just said that to be antagonistic towards me

On the other hand sometimes the amount of effort someone can put into these subjects can be variable within people bc different people deal with different issues

Imo I think it's about effort and passion if anything

Like when I say privileged like morally privileged

Like this professor was exhibiting favoritism or it seemed like he'd be more happy towards this one student who seemed to get all the answers right more quickly than others

[strong hearth]

I understand how this feels because what this prof is doing is genuinely harmful

but since this is pedagogy, what you describe is different from ableism in the disability or moral sense. It fits more closely with what we call a fixed mindset and deficit discourse, you know, believing that your ability to do maths is somehow innate, fixed, selectively distributed and so on. This prof's framing of some students being talented while others hardworking is exactly this imo, it's this toxic cultural (culture of practice) narrative that considers brilliance as the norm and effort as something like lower-tier - contrary to the evidence that proficiency develops through instruction and support - but then it also makes those people who don't fit the genius narrative to feel left out

[warm valley]

as a (visibly) physically disabled person, it is certainly extremely present in academic mathematics culture in my experience, and this has rubbed off into my experience in my education

I think it's present in pedagogy as well, but less blatantly so and more "these things were not designed with me in mind"

I can expand more on either if you'd like examples

[strong hearth]

thanks for that, and your perspective does add a lot of richness to the discussion, @warm valley.

I would say that the key difficulty in answering the original question is that "prevalent" can mean very different things, like whether we're talking about lived experience, cultural norms or at a more structural level. In disability studies for example there is a clear line between individual experiences of exclusion and the broader systemic patterns that (re)produce them.

So what you describe is indeed a form of structural ableism, but also the "talent vs effort" framing above is another (but related) issue. There are overlaps, yes, but not identical concepts

[quasi musk]

I'd say talent in math is being able to apply things you've learned from one setting or scenario, to a new one. It's not so well-defined, and people will disagree to some extent on this, but when you're working with a talented student, it becomes obvious that there is something different or special about them

[quasi musk]

That sucks, but some professors do have their favorites, and will treat them accordingly; I think this is a separate issue as to where talent comes from in Math

I once had a 4th grader that was better at math than most high school students, and knew algebra, trigonometry, pre-calculus, etc.

There's also students that score consistently high on Math competitions; surely, they are talented. Being talented at math research seems to be a different ball-game. Also, being a talented student in high school at math would be different than it is at university; and being a talented undergraduate is distinct from being a talented graduate student. This is because each institution, diploma, and setting demand different things from their students

[molten void]

Meh I think his viewpoints of this contributed to this weird favoritism treatment haha

[molten void]

Okay does good at math mean talented but why does one get good at math and also why would being hard working be viewed as a negative oppose to talent are they dichotomous or is talent really just passion and hard work

"good"

[quasi musk]

Talent is probably a combination of talent, hard work, and pre-disposition

[molten void]

What is talent tho

Like innate ability

Why does innate ability exist

Besides being good at something

Explain about pre-disposition

[quasi musk]

I'd say pre-disposition is initial success in an area that gets reinforced over-time through a social setting (e.g. parents telling you that your art is great, and then you spend a lot of time working on art, and getting good at that)

[molten void]

Should those people be given more praise and encouragement compared to someone who is struggling but still is passionate and still enjoys the challenge

[molten void]

Idk sureeeee that opens a whole load of questions tho

But idk i guess yea

Maybe my problem isnt exactly with definitions but really this mindset about innate ability being the driver or idk

[quasi musk]

I don't have an answer to this. Digging down deep into this, it becomes very vague, and ambiguous. We each have 24 hours in a day, to eat, to sleep, to exist and co-exist. Work on something you find interesting

[molten void]

I rlly believe in tabula Rasa

Okay like for example, I am struggling in Real Analysis

I find the course material very interesting and engaging

I am taking Real Analysis next semester

[quasi musk]

Yeah, I do have an issue when people say "This person just has an innate ability". I have less issue with someone saying that "This person is a talented student"

[molten void]

But don't people also use it synonymously

Also

Like the weirdo prof who used it as a contrast to hard work

[quasi musk]

Innate ability to me seems the most bogus of all. I'm not sure what "most" people do, as the people I know, and how I use the phrase would not constitute most people

[molten void]

Maybe he is kinda dumb

the professor

and mean

[molten void]

I am not the "best" according to my grades and exam scores but I totally feel like it is not reflecting all that I've learned and put effort into

Like u can have a shitty day and mess up your exam

and what not

or be tired

or have a demanding life outside of school

[quasi musk]

A talented student in a classroom is one that doesn't need as much guidance/supervision to get the work done, and can come up with creative ideas, some of which work out. That's what I'd say more or less a talented student is. Innate ability, I take to mean a person who is requires no training in a subject to be proficient. They are self-directed in a way without any training. I find it impossible to believe that "innate ability" is anything tangible beyond what I call "initial disposition"

[molten void]

But like

are they better than the other students

What about the students who struggle

I think people who still put their best in even if its struggling still deserve some praise

or its just the case be relatively neutral to all

[quasi musk]

If you have two students, and you are teaching them Calculus: One requires ten minutes of instruction to understand how to compute limits, and another requires 1.5 hours of instruction to be able to compute limits, which one is better at picking up the concept of limits?

[molten void]

But like morally

[quasi musk]

I'm not trying to tell anyone what their morals are

[molten void]

Does the time matter as long as they get it

I'm not saying who is faster

[quasi musk]

The time absolutely matters. If you cannot solve the problems in a reasonable time frame, then you have not sufficiently learned the subject

[molten void]

Like idk personally so what? As long as they can get to it eventually

Ofc our daily lives work in a time frame

[quasi musk]

If you're talking about learning it before being tested, in some sense it doesn't matter. But teachers are always keeping track of 1. Who is excelling 2. Who is keeping up and 3. Who is falling behind

[molten void]

Like do you just let them fall behind or you encourage them to work on it even if it takes them more time

Like what if they are asking for help and what not

[quasi musk]

Are you asking what I personally do? Or are you asking what a moral person would do? Or are you asking what the ideal should be?

[molten void]

I'm thinking about morally and what would benefit all students

In an ideal world I guess...

[quasi musk]

Oh, then I have no idea. I'm doing my PhD to conduct research, not really to teach

When I'm in front of students, I do my best to be helpful. But outside of that, I try not to spend too much time on answering questions, or thinking about how I could better teach

[molten void]

Idk I think its unethical that a professor picks favorites bc your job is to teach and mentor and that means giving all your students the same opportunity and not making others feel left out in anyways like its just weird and can be detrimental to the learning environment of others

[quasi musk]

Yeah, professors do many unethical things

Kind of a fact of life

[molten void]

I lowkey have to report that prof cuz of other things

It kinda sucks he's a popular prof

I shouldnt share that much tho

[quasi musk]

Yeah, it seems that you have a personal issue with the professor. Hope this helped decompress. It sucks when you're in the thick of it

[molten void]

as a 19 yr old undergrad who is/was very insecure it was so disorienting

it kinda sucked cuz I was interested in that profs research and he just was SO WEIRD

I like my real analysis professor he doesn't do any of that even if he is kinda serious eastern european mathematician but is a good lecturer

Like idk he isnt weird and just does his job

Kinda bare minimum

[dire harbor]

Teachers and professors suck thats why you gotta take matters into your own hands and go into a career in education 😎

[molten void]

my real analysis examis gonna cook me

[abstract grove]

I think most working mathematicians understand that hard work is paramount, and few would be caught dead taunting their students for not being talented enough. Sorry your prof seems to suck

[molten void]

Real

I was going through a tough time in my life so i wasnt doing the best in school etc. but barely floating and he hit me with that comment

[abstract grove]

I have had a small number of teachers who had an edgy comedic sarcastic personality, and I could imagine them making a remark like that but not meaning it

I sometimes wonder how they've kept their jobs

[molten void]

He is kinda like that but i dont think he meant it as a joke I think it was a impulisive stupid slip of the tongue moment wrong time wrong momen tsituation and causing people harm

Kinda impulsive for a 60 something year old man

I was in his office discussing what classes I will take the following semester and he decided to make that remark

and I didnt do the best on the previous exam cuz I was dealing with some health issues and he says that

Anyways idk im gonna stop sharing

[abstract grove]

all good, I'd be bothered too

[dire harbor]

I feel like recently ive been learning that giving somebody any kind of unchangeable label is usually gonna do more harm than good

Like the existence of "talent" is an interesting topic but if you label a student as either having it or not you risk either making them feel hopeless or making them feel like they never have to put in effort

[molten void]

Yea it was dissappointing to hear the professor say that

Idk especially when youre a young vulnerable insecure and fresh undergrad

New to everything

[dire harbor]

Yeah in that situation its probably best not to even mention something like that cause then the student starts filling in the blanks themselves

[molten void]

I didnt believe it ofc but it was disappointing to hear a professor that i had say that to me

and nevertheless hurtful and reflection of them not me

Yea I asked him what it means and he couldnt even come up with a real answer

And then I left

[dire harbor]

The question of nature vs nurture plagues every one of my interests lol and ive never been able to come to a satisfying answer

[molten void]

I have issues so it made me very sad for a whole weekend tbh and it affected my motivation to study more for the final

I was going through a pretty tough time

[dire harbor]

Yeah honestly i feel like if somebody mentions something negative about somebody else its pretty natural to immediately think "well do i fit into that category??"

[molten void]

Yea

Like now that I am older id be like theyre problematic and its on them

I like many looked up to the prof in some way he is popular

Idk some teachers/profs put on acts for some degree cuz its their job I've learned how they are at work is different probably otherwise which make sense but this professor made his classes not feel as impersonal for some reason

[dire harbor]

That kinda thing is a gamble

[molten void]

yes

[dire harbor]

I feel like being a little personal is good but too much is offputting

[molten void]

They're just faciliators of my education

not really supposed to be anything more

[dire harbor]

I dont think thats entirely true, technically yes but i think being a little fun can be good for engagement which is part of facilitating education

[molten void]

Ture

True

But idk there can be a fine line sometimes

[dire harbor]

Yeah it can become distracting or weird

[molten void]

Yeaaaaaa

ANYWAYS thats enough of me venting hahaha

imma sleep

[dire harbor]

Lol dont worry personally i like seeing the examples of what not to do

[warm valley]

Would you recommend any sources on this topic in education (math or otherwise)? I find these kinds of things very interesting (I have taken a few electives on philosophy of disability and critical theory, which I have very much appreciated)

[strong hearth]

which concept are you interested in? deficit theory? community of practice? bourdieu's cultural and social capital?

[molten void]

can u give resources for all

i need for project

im taking a class on race genes and society so it talks about disability discrimination and assumptions about intelligence based on race and

[dim arrow]

ramanujan 👀

[dim arrow]

placebo is a powerful drug

[molten void]

what's the difference between deficit and anti-deficit approaches in education? is deficit thinking typically victim blaming and kind of a suck it up get better approach where anti-deficit tries to think of things more holistically and pose better solutions

im trying to go off of what this is saying

https://quod.lib.umich.edu/c/currents/17387731.0001.110/--what-is-deficit-thinking-an-analysis-of-conceptualizations?rgn=main;view=fulltext

[molten void]

yo

im reading this article about how white supremacist color blind narratives affect mathematics educatoin. im reading a paper which refers to students of color experiencing more emotional labor due to opppresions it says this: "Additionally, students must manage the ways in which they express emotions to
avoid deficit discourses about being perceived as argumentative, angry, aggressive,
and a multitude of other negative associations. When students of color are expected
to handle experiences that they consider unfair in a calm, dispassionate, and disconnected way, whiteness is restricting acceptable ways of grappling with the emotions of discrimination and racism (Moore, 2008)."

[strong hearth]

thanks for the link, let me have a read and get back to you

but as a preamble, I'm not a race theorist nor am I a proponent of it - I think it has its merits but it is very much only popular in the US and not much else where, which yields this weird archipelago

also Gorski 2011 paper (which this paper cites) is also a good paper too!

[molten void]

why is american math academia so white and male

[strong hearth]

might take a while

[molten void]

dam ur typing in tex

thanks for calling me astute

[eternal adder]

So, I heard from Wrath of Math that the long subtraction method used to be criticized as "overkill". Was it true, why was it "overkill", and what was the original subtraction method?

This is in reference to the triangle method of subtraction, that many parents hate.

[abstract grove]

I guess long subtraction refers to the standard method, writings numbers vertically with borrowing?

That method is overkill for problems like 1000 - 997, where a little bit of number sense goes a long way.

What's the triangle method?

[eternal adder]

By long subtraction, I mean something like this:

Triangle method is this:

[abstract grove]

ah, I see

[eternal adder]

I still don't understand why would this method be useful, but Wrath of Math argues that this method is actually useful?

Honestly, if the problem is teaching subtraction of number between 10-19 and another number 1-9, I think you'd be better off with rote memorization.

[abstract grove]

for 17 - 9, the traditional long subtraction way doesn't really help. Because when you lower the 1 to a 0 and add 10 to 7, you again end up having to do 17 - 9. So, the triangle method is actually better for that problem

but I agree, it's ideal to eventually get all small subtractions like that solidly into long term memory

but for developing that skill I can see the triangle method being good

pure rote memorization is no fun for most young kids

[cosmic ibex]

Hopefully it's not an either-or. Possibly small subtractions should eventually end up memorized, but that doesn't mean a more first-principles way to see that the memorized outcomes are right shouldn't be offered too.

(Though on the other hand I don't think I recall all of those cases by rote myself, and I don't really feel that's holding me back. E.g. for something like 13-6 I'll mentally count 3 down to 10, and then the remaining 3 down from 10 to 7. That happens quickly enough that I've never felt a need to drill instant recall of it).

[eternal adder]

I asked my mom, who used to be a schoolteacher, and apparently, she heard that technique. Somehow it was taught for mental arithmetics? It's weird.

[turbid zenith]

https://youtu.be/UIKGV2cTgqA

I wonder if they’re saying the old way was what’s described at the beginning of this video.

Where they described “carries” applying to subtraction.

[cosmic ibex]

I think that's somewhat orthogonal to whether teens-minus-single-digits should be learned by pure rote or broken into smaller pieces. Lehrer's subtraction problem just does them. Huh, I'd forgotten how the first part of the song goes.

[zealous kettle]

I've recently been tasked with running a math puzzle enrichment program thing for elementary school-aged kids, and both of the sessions I've done have followed this pattern:

• I introduce the concept with a toy example that's as simple as possible
• The students are engaged
• One says "This is easy!"
• I take that as a cue step up the complexity of the examples
• The students swiftly lose interest and disengage entirely

[turbid zenith]

Hooo boy. Gave my students a group final project at the beginning of November, it's due next week ... they're complaining they didn't have enough time to work on it in class, AND a few groups are telling me that some people are slacking, with one person essentially having to do all the work in their group

Anyone have any recommendations on how to deal with unequal contributions? (I know this is less math and more pedagogy, but ugh I'm annoyed.)

They had to read one chapter of The Secret Lives of Numbers, prepare a 15 minute presentation about the people and the mathematics in it, and design a game related to the mathematics that can be played in 15 minutes by the other groups.

[zealous kettle]

How old are the students, and what exactly does "deal with unequal contributions" mean to you? Is the concern that grades need to reflect work, or that the students ought to feel proud of the projects, or that contributions should be equal in future projects, or what?

[turbid zenith]

They're undergrads, 3rd and 4th year.

This is, sadly, the final project of the course, so there's no chance of repeat interactions.

What I want to do is, if there's a group where some students did the bulk of the work and the others were slackers, I want to penalize the slackers and boost the students who worked.

So that grades reflect work.

But I can't figure out a fair way to do it.

It also amazes me how some students can ruin what was supposed to be a fun final project in lieu of sitting down for another exam. They're supposed to create a game for crying out loud.

[zealous kettle]

Figure out what each student's responsibility was for their respective project (perhaps by surveying each member of each group), and then apply a rubric to each student individually.

[turbid zenith]

Doesn’t that incentivize everybody saying they contributed more than they may have?

Prisoner’s Dilemma style?

[zealous kettle]

Yeah but the rest of the group can call people on their bullshit.

[turbid zenith]

Well imagine a group where one person has had to do the bulk of the work because the others are all slacking

Say A does all the work while B, C, D slack.

[zealous kettle]

Ask what, specifically, each person did for the project

If A says "I wrote flavor text on all the cards and designed the board", and B C and D say "I designed the effect for these 10 cards" then you can evaluate the flavor text, the board, and the card effects.

And then if multiple students claim sole responsibility for the same aspect, send them to the honor council or equivalent

[turbid zenith]

Hmm. Will play around with that…

Ugh.

I already hate that I switched to in class exams because so many students were using ChatGPT for problem sets and writing assignments.

[plain pebble]

It feels like there are multiple steps missing here, including but not limited to “get the students together and ask them about the discrepancy”

[turbid zenith]

Which would make sense except it's finals week :/

[zealous kettle]

fair

[cosmic ibex]

Uneven contributions seem to be a typical problem with group projects unless students have fairly uniform levels of motivation (and to a slightly lesser extent, skill).

Figuring out how to navigate the social pitfalls of a self-organizing team is definitely a relevant work-life skill, but if it's not a fixed element of the university's style, it can be a bit much to expect students to do well at it on their own, on their first attempt, and even where the stakes have been raised by being in lieu of an exam.

[midnight scarab]

As a student, my experience with group projects has been

1. ugh
2. do the important things myself - it's less effort than first getting them to do something, then convince them that it can be improved to ultimately still make the changes myself

There have been some exceptions where my partner earnestly did their work, and that was really nice

But if they don't want, you can't make them

That's why I don't think it's fair to make "team work" part of the skills you evaluate

[turbid zenith]

Yeah. I might just have to not do it again next time. Which I hate.

[tawny slate]

i personally would have loved to do that project solo

[tall bolt]

Yeah I have genuinley never had a positive experience with group work

I see that it can work well. Im currently taking a course on Lean and its supposed to be a group project (...) and I see that a couple of the other groups are working really well together and that seems lovely, but I think for every sucess story theres 100 car crashes.

It sucks because they should be fun, and it should develop important skills, but in my experience you tend to, in the best case, just have to do the work of 3-4 people yourself, but more often than not have to do that work, and try to fix other peoples nonsense, whithout getting anyone upset.

Im aware theres a reading of that second part that makes me sound like some sort of control freak megalomaniac, but I genuinely dont think it is that. I think most other "motivated" students have had similar experiences.

[turbid zenith]

Solo would have made the presentation part a lot more difficult

I could just split it, but the logistics would work really well IF people took it seriously

5 groups of 4 people each. 15 minute presentation of a chapter, then 15 minutes of having the other groups play the presenting group's game, with the presenting group dispersing to the other four tables to help them through playing the game. Do five of these, which takes 2.5 hours plus any transition time. Our final block is 3 hours.

Also who knows ... the final is next week. Maybe it'll go great.

[turbid zenith]

So I just found this guy

https://youtu.be/cFbvoLpEa9M?si=WZmKLMlKJXTdoS1T

And … wow, I hope I can be the opposite of this when it comes to the utter sneering disdain for applications

[wise onyx]

Yeah, that guy is ridiculous

[turbid zenith]

And in so many other ways

The only place I agree with him is that FOIL and PEMDAS are kinda dumb

But like… talking about how Newton would weep about the current calculus curriculum because supposedly it has applications instead of proofs, when the whole reason he worked on calculus was physics and his proofs would be considered ad hoc today, that’s just plain anachronism

[wise onyx]

[quasi musk]

I'm half-way convinced this guy is a dedicated troll

It's pretty funny, I love mathematicians so much. We're all so ridiculous

[dire harbor]

The random dips into flaunting his ignorance about linguistics and music are pretty good

[tight star]

same

but yeah, this is probably the most extreme example of the phenomenon of "pure mathematicians looking down on applied mathematicians"

[slim path]

Mathematics without a taste for its utility beyond a mere artform, is utter depravity among those who seek beauty in truth. -- Me

[native iron]

At the very least, the persona seems totally fabricated.

[tight star]

-# well in a sense all personas are fabricated

[turbid zenith]

Update... on that group project, four out of my ten groups have now had issues with people not doing their work :/

I am so utterly disappointed

[tall bolt]

Disappointing but not surprising :/

[vocal phoenix]

Four out of ten is fewer than I'd expect

[tall bolt]

I had to submit like a personal progress report for my lean """group""" project last week and fully just admitted that Ive never heard from one of the guys and he hasnt even joined the github, and that ive heard from the other guy once and hes not done much. Im over letting people just coast by in group work lol shit sucks

[slim path]

Lean is so cool tho. Why would ppl not wanna do it lol

[tall bolt]

Shit is painful, im having a bad time

[tight star]

Apparently the programming is quite annoying

Hopefully they improve it before it becomes mandatory

[tall bolt]

I dont think Lean will become mandatory, certainly not any time remotely soon

[tight star]

I thought that was the plan tho

[tall bolt]

Whose plan?

I think there are certain people going for that, but Im not sure thats the mainstream opinion. Even within lean circles I think most people would agree thats like far away if its something people are going for

[tight star]

Hm maybe I’m misremembering from buzzard’s talks

Did he not want to ensure math is formalised so that he no longer has to have protracted battles with authors about whether their results are correct or not

And also in part to have a searchable library of math results

[tall bolt]

This is Buzzards goal yeah

But like this is far away and certainly not the concensus of most mathematicans as far as I can tell. Hell even just making mathlib searchable is far away lol

[tight star]

What do you think the main challenges are

As someone who’s done lean for a whole semester now

[midnight scarab]

As nice as that is, that doesn't seem very realistic. As long as formalising something in Lean is significantly more time and effort than convincing most reasonable colleagues

[tight star]

I guess I don’t have a good idea of what the lean community’s goal is

[tall bolt]

Theres like 4 different LLM powered search engines just to help you find stuff. A lot of the problem there is that most of whats in mathlib is like of no mathematical significance what so ever, but you need to prove just tiny tiny lemmas to actually make progress in lean. The other reason for that is like automation, you want a million basically identical statments of the same trivial thing so that the automation in lean can find something to actually apply to your specific case

[tight star]

Oh that’s fascinating

So mathlib is too “atomised” in your pov? At least from a user experience side

[tall bolt]

I think its largely that just working out how to actually state things in lean is incredibly hard, and also that just because something is mathematically easy does not make it easy to state formally. Ive spent most of the semester proving like a throw away lemma from a workshop lol. Its hard to even pin down the challanges because depending on what youre working on they can be wildly different.

[tight star]

I see I see

Do you think that’s something intrinsic to formalising math

Or could one have designed lean/mathlib better for this purpose

Maybe - do you have an example of something that’s easy mathematically but hard formally

[tall bolt]

Yeah for human search purposes I think so. They have quite strict naming conventions to try to help you with this but its still tough to know exactly the statement youre looking for (also because what you think you want, might not actually be the best way to do things formalisation wise)

[tight star]

I suppose the ideal would be that formalisation helps with proofs right? Hence the term “proof assistant”

But from what you’ve said, it seems like there’s enough formalisation-specific difficulty to prevent this

[tall bolt]

I think probably some of it is intrinsic, you could maybe have a curated "user side" mathlib but eh that would be a lot of work. I do think these like LLM based search engines do a pretty good job at fixing this issue though, theyve generally gotten me to if not the right result, at least related enough ones

[tall bolt]

Yeah Im not sure were anywhere near a point where we could prove anything genuinley new with the help of something like lean, like we still havent even formalised the "standard" array of UG results. I think its possible at some point in the future but I dont think its there yet.

I am also no where near an expert on this stuff though lol, a lot of my criticism of my course was actually that I dont feel like ive really learned lean or formalisation super well

[tight star]

The coolest stuff I’ve seen from lean so far is new math being produced as a byproduct of trying to figure out formalisation

But otherwise I don’t quite see how helpful it could be right now

Especially with what you’ve said regarding translating ordinary mathematical statements to formal ones

Even if a lean program compiles, it sounds like there’s still a worry that what you formalised isn’t actually what you wanted to prove?

[tall bolt]

Yeah thats a big concern lol

[tight star]

And it seems like a pretty fundamental one too

[tall bolt]

I am not entirely sure I have shown what I actually intended to show in what I have for my project right now

[tight star]

Without the connection between “lean program compiles” and “result is true”, it seems like you’re potentially no closer to deducing the truth of your result

Or to phrase it another way, you’ve traded “I don’t think your paper proves this result” for “I don’t think your lean code matches up with the paper”

So buzzard may not escape those protracted arguments even in his ideal world

I’d still be quite interested in seeing how the mathlib community tackles that interpretability aspect

[tall bolt]

I dont think this is impossible to get over though it just takes careful reading. Like the thing about type theory and I guess lean generally is that it complies iff what youve written down is all well typed. So like if it complies, what youve coded is true, its just a question of, have you written down the correct thing, and I guess this is verified easily enough by careful reading and knowing what youre doing. But yeah its definetly easy to prove something different to what you think you are (especially if youre not very good at it, like me!)

[tight star]

I admit I’m a little surprised because, as far as I understood, type theory was meant to be closer to how mathematics is done in practice?

Sure sure, I just think “careful reading” could also be applied to the existing mechanism for verifying papers

But I could believe it’s easier if you have a lean program

[tall bolt]

This is the type theorist claim yeah. This debated though, like im sure if you post this in foundations exo will quickly disagree lol. I think in a sense it is, but it doesnt really mean that its easier to work with

Its more just that maths in the day to day is very different to maths formally at the foundations, regardless of those foundations

[tall bolt]

FWIW I guess im inclined to agree with this, but I dont know enough about foundations to really take a position. They both lead to weirdness though, like set theory gives you things like $3 \in \pi$ being a valid question, but then type theory gives you stuff like theorem $\mathbb{N}$. Proof: 2. Which is also weird.

I think the whole ETCS thing is supposed to fix this, this is like Leinsters prefered approach that he writes a bit about in his category theory book, and he taught a whole course on ETCS at edi last year (kinda wish I had taken it), but ive seen foundations people say this is the worst of both worlds but I again dont know enough to comment. I go back to my position that I dont really care because I do maths not logic so everything works well enough for me

[burnt vessel]

Nope

[swift hatch]

"Theorem N. Proof 2" how

[plain pebble]

A theorem is a type, and a proof is an object of that type, iirc

[swift hatch]

As in, I don't understand how calling N a theorem when it isn't of a type itself of type Prop make

-s sense

[tall bolt]

IIRC, the type of the theorem is Prop, and the truth value of 2 in N is true, so that works

I may have the example slightly wrong but my point is that regardless you can construct weird statements in any foundations

[swift hatch]

I think it is wrong because Nat and Prop are in different Sorts

At least the way lean does it

If I understand it correctly

[tawny slate]

this might be pushing the issue to a much more unrealistic extreme, but just to point out a theoretical concern: technically it gets much worse than that, because even if you have that the paper proves the result and your code matches the paper, you still need to trade it for "i dont think your code runs the way you expect it to", because there is always a possibility that a computer is infected with a totally undetectable trojan, and who knows what that could do your computer or code

[spiral elbow]

I hate when trojans infect my computer and make my incorrect Lean code compile without errors

[swift hatch]

The true horror is just how likely that is

[torn apex]

@tight star @tawny slate what you are asking for is theoretically impossible. If you could automatically prove the link between one's semantic claim to the certificate of work in lean, you'd be able to solve the Halting problem or Godel's incompleteness

Automated theorem provers escape hatch out of these problems b/c we rely on the out of band human social process to determine "validity".
As much as we don't like it, math is ultimately a social construction. Math can't prove it's own consistency nor it's completeness

[tardy ember]

...how?

[torn apex]

not sure what the question is? Are you asking about Godel's Incompleteness or Turing's Halting problem? They're isomorphic but one is easier to understand than the other imho (It's easy to understand the Turing construction of how a machine can't prove it halts i.e. the undecidability problem)

[tardy ember]

the halting problem

suppose i have an arbitrary turing machine M, and i have an oracle that can, in your words, "automatically prove the link between one's semantic claim to the certificate of work in lean". how do i use this to determine whether or not M halts?

[torn apex]

"automatically prove the link between one's semantic claim to the certificate of work in lean"
The logical closure of this implies that you could automaticallly prove any statement P.
Ok, then the statement I give you is the halting problem or any homomorphism of it
There's the highlevel contradiction proof

[plain pebble]

How does the logical closure prove that?

You can’t produce a certificate in lean of any statement you want (assuming lean is consistent)

[tardy ember]

that you could automatically prove any statement P
well i would hope it doesn't imply that, because for instance you should not be able to automatically prove 0 = 1, even with hypercomputation

[torn apex]

assuming lean is consistent
😛 you literally said it yourself. That's the problem; you ultimately have to fiat that or apriori assume that

or let me ask it the opposite way, what do you think the Halting problem or Godel's Incompleteness theorems imply?

[tardy ember]

that no turing machine decides whether arbitrary turing machines halt, and that no system that interprets Q is both complete and consistent, respectively

[tardy ember]

anyway, could you elaborate on this argument

in particular

The logical closure of this implies that you could automaticallly prove any statement P.

what exactly do you mean by "you could automatically prove any statement P"?

[plain pebble]

When you’re talking about lean and automatic theorem proving, you’re assuming either your system is consistent, or that your theorem can be proven in a consistent fragment of your system
Like, this is literally just a “haha! You didn’t state an implicit assumption”

[torn apex]

Oh, I wasn't being arbitrarily hand wavey. This comes up practically. Fwiw, I wrote my own toy compiler for my own math domain specific language as well as well as working on the zig + LLVM compiler

dependent types which is what HoTT is based off is fascinating and powerful but at the heart of each meta-compilation compiler, there is literally a hardcoded timeout for how long the compiler is allowed to work on different parts of the compilation

it also shows up in parenthesis counting in programming languages for example. that has to be some hardcoded constant limit

[plain pebble]

Like, this is still literally just a “haha! You didn’t state an implicit assumption”

[torn apex]

yes, that is exactly what it is. I don't know why you think that makes the claim less important?

[plain pebble]

In the sense that like
You’re nitpicking about something that makes this conversation vacuous
Make the assumption that we’re not having a vacuous conversation

[torn apex]

I gave you two concrete examples that show up in programming languages in compiler theory. I may not be doing a great job explaining it but you can argue all day about it when the reality is what it is

[plain pebble]

In practice if this came up for a theorem people cared about, the result would be “modify the kernel to raise the limit”

[tardy ember]

well "you failed to state an implicit assumption" is not a proof of the claim you made earlier, that given an oracle that can check whether a given proof in lean is a proof of the statement you wanted it to be a proof of, it is possible to solve the halting problem

[torn apex]

ok, i'm wrong 🙂

[cosmic ibex]

Uh, is this disagreement (whatever it is) about pedagogy?

[tardy ember]

...no, it isn't

good point

[turbid zenith]

... so my first batch of students are doing their games, and actually the game part turned out really good in the end!

[tight star]

Oh sweet

[turbid zenith]

We'll see how the teamwork survey goes 😂

[untold spruce]

Hello

What kind of pedagogical methods are apt for mature students who have picked a lot of math through osmosis but lack a reasonable familiarity (which involves things like doing varied amounts of problems and engaging with standard coursework)? This is in the context of Undergrad Mathematics, not High School math.

[strong hearth]

https://doi.org/10.53761/w3x5y804

The emergence of widely available artificial intelligence (AI) tools has made assessment in higher education increasingly uncertain. Familiar (if problematic) assumptions about what assessment does or should measure, who or what is being assessed, and how judgements are made are all being re-examined. Educators and researchers are experimenting with new assessment designs, but the emerging research landscape is fragmented and difficult to navigate. There is little shared sense of what kinds of studies are most needed or how their findings might connect. To address this, a group of leading assessment scholars met in Melbourne, Australia in September of 2025 to develop a collective research agenda to help guide and connect future inquiry. This paper presents the outcomes of that collaboration, a set of guiding principles and framing questions –why, who, what, how, and where we assess –that together offer a structure for guiding and supporting the research we should be doing on assessment after AI.

PDF: https://open-publishing.org/journals/index.php/jutlp/article/view/2533/1175

inchresting perspectives

[warm vortex]

What’s your opinion on chromebooks in classrooms

[rapid tusk]

unless you heavily restrict them (a losing battle - any sufficiently determined student can break through the sorts of web filters that schools use)

absolutely not worth tanking your students’ focus even more than it already has been

[thick remnant]

A textbook/math book + pen and paper has always shown the best results for me

[torn apex]

this problem is much wider than just AI because fundamentally it's a socialogical behavioral problem with humans at scale. Even if you don't assume malicious actors and just incompetent actors, you have a decentralized network coordination problem (https://en.wikipedia.org/wiki/Byzantine_fault) so trying to solve O(N^2) network problems with linear steps is bound to fail.

This was the seminal paper on trusting trust

Even worse (I hope I'm wrong and just being cynical), I think the last 5 years have shifted where now malicious actors are active:

• there have been several high profile supply chain trojan attacks in very popular coding libraries: 2x Sha1-Hulud attacks 25k+ via npm Preinstall Credential Theft
• AI slop journal articles submitted to conferences AND AI slop journal reviews for submitted papers. I mean, the first I expected bc humans at scale are terrible but for the conference journal referees to AI slop their referee work.... smh
• the last 5 years has seen a wave of retractions due to intentional fraud in psychology by seminal researchers!

mrdosija

Shai-Hulud Returns: Over 300 NPM Packages Infected

https://www.aikido.dev/blog/shai-hulud-strikes-again-hitting-zapier-ensdomains

Hacker News · Nov 24, 2025, 10:40 AM

To bring it back to math pedagogy, it's the same epistemic paradox. How do you know the material you're ingesting is "true", especially in topics outside one's circle of competence? This applies to your professor, the textbook your reading, or the AI model that you're using.

And even then, one's own epistemics goes out of date as old theories are falsified or new discoveries end up reworking older theories. I think the AI dimension "merely" highlighted the problem because it made the cost of knowledge production orders of magnitude easier.

My guess is that some sort of sociological solution will emerge (i.e. i predict gatekeeping will make a comeback + new reputational norms so if one intentionally produces AI slop, there should be some reputational harm done to you. but also, good luck adjudicating! )

I can't imagine what it's like being a professor, journal editor, or even an undergrad right now.
I'm curious how other people have been managing over the last few years?
I'm finding myself resorting to dismissing anything that doesn't come from a "name-brand" or unless I know them

[turbid zenith]

It sucks because I’ve found AI to be so helpful as an educator, especially in creating materials

But a staggering number of people at all levels don’t seem to have figured out than you can’t just ask AI to do the thing for you and uncritically accept the output as the final thing

[torn apex]

So much this! I've been able to do the same for my nephews and I think of the possibilities for reaching underprivileged kids

[slim path]

People are fundamentally lazy, especially the so-called educated ones. Which is why the rate at which AI has been adopted across several places is dramatically high but the countermeasures to adapt to the change is laughable across the board.

[tawny slate]

dont forget that its not just ignorant people, malicious action using generative AI are exceptionally disastrous in their impact too

[turbid zenith]

Of course

There's always the capability of malicious actors

[torn apex]

@turbid zenith I'm keen to hear more about your experience with using it to create educational material! I've been doing this for 8 months now and can swap some insight. Idk if this is the right channel but happy to move the thread somewhere else to not annoy others. As much as I'm pro AI use, I have a hard revulsion to it being forced down our throats all the time everywhere

Maybe a good example would help? Here's a study aid I made for myself last night as I was reading John Baez' Counting With Categories series

My study aid on the 2-rig & Catalan numbers

The material was made just for me and conveys my literal understanding of the material so it's going to be as wrong as I am. So I look at it the same as my study notes on any topic but would not distribute them publicly unless I went back and really looked at it

and very auspiciously, terence tao just posted about using AI to solve an Erdos problem:
https://terrytao.wordpress.com/2025/12/08/the-story-of-erdos-problem-126/

I'd recommend it since he makes a much better use case than me, some rando on an internet discord 😉

[tawny slate]

people like terence tao are utilizing specialized AI tools, not chatgpt like everyone else is using

in particular, aristotle and alphaevolve, that terence mentions using, are able to do some form of verification by running the actual code and evaluating it, or by converting the proof to lean for verification, but the interpretability problem is still relevant and has to be hand checked

where he does use traditional LLMs is in deep search of literature, but again, it can just hallucinate results and the interpretability problem has to be resolved by humans

so im not sure how terence tao's use case here is relevant to this discussion

[cosmic ibex]

In particular even if Tao were using vanilla ChatGPT, that still doesn't really seem to be about pedagogy.

[tawny slate]

all it does is demonstrate that competent people can make good use of it, but competency is still a necessary prerequisite

AI, especially generative AI, is and always will be a slop machine in the hands of those who are not proficiently conscientious and competent

[cosmic ibex]

I suppose at some level it does speak to whether one should try to explicitly teach strategies for productive use of LLM, or on the other hand teach that that they are fundamentally useless and harmful for all purposes.

[tawny slate]

imo it is impossible to teach any kind of strategy that works, because fundamentally you must be proficient enough to manually check and verify the contents of whatever you ask it for

[cosmic ibex]

But that's part of a strategy too, isn't it?

[tawny slate]

i think it is completely misguided and dangerous to do this, it is imperative to first teach competency above all else, and only then is LLM a valid tool

well sure, i call this strategy "teach LLMs are bad until you're smart enough"

[cosmic ibex]

Hopefully by the time a student passes a proof-based course in anything, there will be an area of mathematics where they are proficient enough to manually check and verify a purported proof. (Otherwise they shouldn't have passed that course).

[tawny slate]

the operative word here is hopefully

we have before mentioned the criteria by which we assign grades like A, B, pass, fail, etc

i am not hopeful lmao

[turbid zenith]

Sometimes I'll use it to help design a lesson, where I have an idea of what I want to teach but I'm trying to think of more student-centered ways to teach it, or asking it to time out a reasonable amount of time for various things in a lesson I'm working out

Right now I'm using it to generate bulleted learning targets for calculus based on outlines I've written

[strong hearth]

sigh… where do i start

[turbid zenith]

With respect to what?

[strong hearth]

topics and utterances to reply to

which i may get to after my cup of tea

[torn apex]

Same! I've found it really helpful in making content specific to the student, especially in remedial math for 7th-12th grade. The LLM isn't a tool to help the student as much as it's a tool for me in as a "teacher" to generate content for said student.

I think there's two separate spheres of pedagogy that don't really overlap here:

• math pedagogy for the general public
• math pedagogy for actual STEM folks

[zealous kettle]

If we're distinguishing pedagogy between "general public" and "STEM folks" we kinda have to decide when/how to split the students.

I think the actual split is "Students who want to be there" and "Students who are forced to be there"

[torn apex]

I thought about this more and I think the meta is actually deeper. There's a split between math pedagogy for actual learning vs. pedagogy to know just enough math to use it/get by

the ideal example I think of is doctors/medical field. you want to make things as simple and easy as possible for them. so in effect, you want as many "cheats" or "aids" to help think for them because the objective function for that type of math education is not about learning math but using math.

There was that famous funny paper where some doctors re-invented the trapezoid rule from basic calculus and it made the rounds. Every field does a bad job reinventing/discovering things from all the other fields but atleast with math + computers these days being such an integral part of research, there seems to be a big ROI in at least spreading math around more

[lethal hornet]

-# that paper was hilarious

[cosmic ibex]

I remember that; in their defense neither basic calculus nor real analysis tend to use a lot of energy on practical numerical integration, and trapezoid/left/right/upper/lower sums are often left behind as soon as they've been used to motivate an integral concept that one can then approach with antiderivatives.
I feel the main source of backlash for the trapezoidal-sum case was that the rediscoverer wanted to name the technique after themself.

[solar sorrel]

Greetings!

I thought this channel would be interested in perusing https://www.justinmath.com/files/the-math-academy-way.pdf (in their own time, of course).

So far, the implementation of this focuses on multiple choice examinations, so I wonder if it would easily scale towards open-ended problem solving, and (most interestingly for me), proving things.

[The emergent popularity of Lean does suggest that logistics of more advanced mathematics education can be significantly streamlined.]

[solar sorrel]

Is this the work of Mary M. Tai?

In fairness, anything involving biology is on a completely another level regarding epistemic complexity and how easy it is to gather data from.

I am genuinely astounded by how the early mathematical biologists / applied statisticians managed to extract signal from noise.

[cosmic ibex]

I didn't remember the name, but that does seem to ring a bell.

[pure light]

what also irked mathematicians about it was the validation of the model on experimental data by comparing it to the 'true value' obtained by plotting the curve on graph paper and counting grid squares

[torn apex]

I agree but it's funnier to poke fun 😛 But yeah, numerical analysis is where mathematicians go to die.

For actual computers, integer division is a transcendental operation, floating point multiplication is non-commutative, and things like a * 1/x != a * rcp(x) so have fun memorizing all the arcane witchcraft out of agner's CPU manuals

Even worse than the doctors, there was a famous book called Numerical Recipes in C and I think a decade ago, some student implemented everything and....turned out none of the author's code worked! 😂

Imagine you write a seminal textbook in your field of study..... only to become infamous for the mistakes in said life's work. RIP

[molten void]

is that if jover if i get C in my field real analysis class

im taking analysis on R^n next semester with friends

[tawny slate]

is this still pedagogy or just chitchat

[burnt junco]

i’ve been working on a version of differential calculus that starts from sequences and defines the derivative as a linear approximation, with only the minimal linear algebra needed to make that idea precise.

i’ve written it up as a full text, which is here: https://math-website.pages.dev/

i’m trying to figure out whether this would actually be appropriate to teach from the next time i teach calculus 1, or if it should stay as a supplemental resource for students who want a more structural perspective. feedback from people currently taking, or who recently took, calc 1 or linear algebra would be especially helpful.

[torn apex]

what did you use to make it? It looks really clean

[turbid zenith]

Could work for advanced students

Are there any practice problems?

I didn’t see any but I’m on my phone

Also what demographic takes Calculus I where you teach?

[burnt junco]

built with quarto. math via katex, styling with bootstrap plus a custom css. plots and animations are generated in python (matplotlib). content is written in quarto qmd files.

[burnt junco]

currently working on adding problem sets (takes time to come up with genuinely good, interesting ones that aren’t just computation). will be full pdf version in latex with solutions for each section. oh, and it’s definitely better viewed on desktop.

[turbid zenith]

Will you have computations as well?

[burnt junco]

probably should

[turbid zenith]

Okay. I would be worried if there were none and it was all proof without any chance to build any kind of automaticity.

But again this depends on your demographic.

[burnt junco]

realistically, this probably wouldn’t work for the larger general calculus classes taken by all stem majors, but it might work well for math, physics, and possibly some engineering students.

[turbid zenith]

Okay. Then that seems like a reasonable approach

One question I still think is worth asking is, if I’m a student, how will this approach help me better learn calculus?

[compact isle]

the linear approximation perspective makes a lot of things easier to learn in multivariable calculus which I think is the main benefit

[turbid zenith]

What are the pedagogical advantages?

[turbid zenith]

Hmm, how so?

I do agree that approximation is a useful theme of calculus

[burnt junco]

preparing them for multivariable calculus, and it potentially lays a foundational perspective that aligns well with modern differential geometry

[compact isle]

well, in multivariable calculus the derivative really fundamentally is about linear approximation, and its hard to make sense of it in any other way

it also proves simple heuristics for why we should expect things like the inverse function theorem to hold

it explains the multivariable chain rule basically instantly

[turbid zenith]

Hmm okay

[compact isle]

it also explains why partial derivatives are a good description of the total derivative (because the approximation is linear, you only need to understand how it acts on a basis to obtain complete information)

and this can help when explaining things like the second derivative test

[turbid zenith]

I think at this point any objections I have are likely from a difference in philosophy

[compact isle]

I don't think its necessarily the best approach for a first class to be honest, but I do think its useful to emphasize in a multivariable class

[turbid zenith]

But it gives me an idea of where it’s coming from

[compact isle]

and perhaps just review single variable quickly using the linear approximation pov before starting multivariable

[burnt junco]

mhm, thanks for the feedback!

[turbid zenith]

I just ask that you make sure you’re considering not just the mathematical narrative but also the pedagogical one 🙂

If you’ve considered that then you’re solid!

[burnt junco]

well yeah, i’ve brought it up with colleagues and they seem to have the same view as you. i posted it on reddit, where people seem to have a similar view, maybe even a bit more skeptical, if you all want to participate there. but again, i appreciate your input and it’s definitely helpful.

https://www.reddit.com/r/learnmath/comments/1pjl731/differential_calculus_through_linear_maps/?utm_source=share&utm_medium=web3x&utm_name=web3xcss&utm_term=1&utm_content=share_button

[turbid zenith]

Hmm yeah that’s pretty much what I expected to see…

It’s an interesting idea but I would have a hard time seeing it working for the general population… essentially unless you’re already pretty advanced and have seen some of this before

I’m generally skeptical of rigorous approaches to introductory calculus.

[tight star]

Thank you for this, I’ve also been curious as to whether such an approach might work

There are a few things that have held me back - for one, I think the linear approximation pov really comes into its own when we get to multivariable calculus, especially functions of several variables

But calculus is quite a bit harder for such functions - limits are a lot more subtle, there are lots more counterexamples, PDEs are much harder than ODEs

When you’re working with functions of a single variable, linear maps $\mathbb{R} \to V$ can be identified with elements of $V$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

Hence why the derivative is just a number, or a tangent vector

So my worry is the extent to which students find this a helpful perspective at the single-variable level

Certainly these days, I find myself returning more to the linear approximation pov as a primary one, not even focusing that much on the derivative rules per se

[burnt junco]

thanks, i completely see that. i agree the linear approximation perspective really shines in multivariable calculus, but my thought was to introduce it gently even in single-variable calculus so students start seeing derivatives as linear maps early on. you’re right that in one dimension, a linear map from $R$ to V is just an element of V, so it’s not adding anything formally, but i think it can still help students understand why the rules work the way they do. the challenge is definitely to keep it intuitive without overwhelming them.

[burnt vessel]

CantorClosure

[torn apex]

@burnt junco Your approach actually reminds me of non-standard analysis. It's much more intuitive for most people and generally how most people intuit their way around calculus. Having that said, I don't know how I'd feel about a standard intro to calculus course using non-standard analysis (mainly because it's so foreign from a rigor pov)

[turbid zenith]

What do you mean, foreign from a rigor point of view?

Also the linear map approach looks very different from NSA to me, at least it’s not the usual way NSA seems to define the derivative

[torn apex]

by "foreign", I mean that it's very not standard! I don't know of anyone who has actually learned nonstandard analysis (including researchers)

[burnt junco]

mhmm, that’s interesting. my approach (as you can tell) isn’t actually using non-standard analysis, it’s really borrowing ideas from differential geometry

[turbid zenith]

Ahh I gotcha

[torn apex]

For me, the part that ties it to non-standard analysis is the usage of the hyperreals as the basis. This maps pretty well with linear functionals and differential forms from a intuition lens.

Ofcourse the rigor is very different

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

From my experience, even physist think in terms of "dx" or "dy" as being "small littler epsilons that "you can multiply".
Mathematicians understand it's a horrible abuse of notation but it is really effective as a mental shortcut for everyone else.

[turbid zenith]

I still think of them that way myself.

😛

[torn apex]

haha, me too (I'm not a mathematician)

Either way, @burnt junco i speed ran the site and it's really well done. I think you could even give it as an intro to calculus for senior highschool math

[burnt junco]

thanks, yall have been very helpful

[dire harbor]

I havent read enough to give an opinion yet but i was thinking the same thing. Calculus sits in an awkward spot of having a lot of potential for being super rigorous but usually taken by students that arent quite warmed up to that kind of thinking yet

[turbid zenith]

If you want to be rigorous about it you REALLY need to motivate that.

And I'm not convinced that the vast majority of students taking an introductory calculus class would buy that motivation right off the bat.

[tight star]

One thing I remember when I was learning calculus is being confused by the product, chain and quotient rules

If I viewed the derivative as “slope of tangent line”, it was not at all obvious to me how this would behave under products, quotients or compositions of functions

That’s actually a place where the linear approximation pov can shine if you figure out how

[tight star]

Is non-standard analysis really more intuitive for most people?

Personally I don’t find the concept of infinitesimals intuitive at all

[tight star]

Speaking as a physicist, I do indeed think of dx and dy as small things I can multiply, but I still treat it as a small finite quantity

[turbid zenith]

I think a lot of people are intuitively drawn to and can conceptualize the idea of infinitesimals even if their initial conceptions aren’t precise.

And the fact that so much of calculus was first built around that, even pre-Newton/Leibniz, before it was formalized and the infinitesimals were rubbed away, shows how evocative a concept it is

[tight star]

That may be true, but I’m not one of those people

[turbid zenith]

Sure but I was responding to this 😛

[tawny slate]

feel like we've had this convo before lol

[slim path]

Speaking as a physicist I doubt it. I much rather prefer the standard approach. Especially when modelling such things numerically.

[torn apex]

@tight star I think your conflating two things I was trying to say.

1. Most people I've ever met, including myself, intuit around differential forms e.g. "dx" or "dy" as tiny little infitisemals.
2. Non-standard analysis is an alternative approach to analysis that makes the notion above rigorous. The actual way it goes about with nilpotents etc may or may not be intuitive to most people, i wouldn't know b/c I never pursued it beyond mild curiosity.

Also, I doubt most people have ever read anything about non-standard analysis. The hyperreal construction is fascinating outside of analysis. I've yet to meet a physicist who knows what the hyperreals are 🙃

[midnight scarab]

I think implicit little o's is a more natural formalisation of "infinitesimal" dx than NSA

[tardy ember]

...NSA doesn't use nilpotent infinitesimals

[tawny slate]

hey does anyone have any good examples of sanity checking by plugging an "infinitely large" value

[tight star]

Happens in physics a lot

If you have equations for velocities of particles after a collision, it’s a good check to see what happens if you send the mass of one particle to infinity

[tawny slate]

yeah i was searching for physics examples

[tight star]

In general we like thinking about “asymptotic behaviour” as a sanity check

[tawny slate]

but ideally i want one that is fairly straightforward, requires minimal physics/math understanding, makes the sanity check crystal clear in demonstrating an initially-not-so-obvious answer is clearly wrong

yeah something like that

doesn't have to be asymptotic specifically, as long as it involves plugging in an "arbitrarily large number"

i have some examples, but they feel either too complicated or too contrived

[tight star]

You can use this

Helps for algebra mistakes

[tawny slate]

mmmmmmmmmm

im trying to find a convincing "false" version of this equation, something that it might be mistaken for

because otherwise i don't see much i can immediately use

[tight star]

Just make a sign error

[tawny slate]

v'2 being asymptotically double v1 is like

why is the asymptote 2 times v1? there's an algebra explanation but it takes a few steps

[tight star]

Or flip one of the fractions

[tight star]

Why don’t you send m_2 to infinity

Seems simpler

You’re basically reflecting off a wall

[tawny slate]

ahhhhhhhhhhh that's really good actually

i actually like that one

so now the question is

what would be a common mistake when say recalling this equation from memory for which this check could demonstrate that it's wrong?

$$v'_1 = \frac{2m_1}{m_1+m_2} v_1$$

[burnt vessel]

Cozmogrgdfschkipkhrshtensi

[tawny slate]

this one? it seems like people could make the mistake of thinking these equations are "symmetrical"

er except change m1 to m2

this might work, might use this

thanks

[plain valve]

Feels like the sort of thing one should not remember and should instead work out lol

[tight star]

Yes that’s what we do in practice

But it’s good to have sanity checks

Asymptotic analysis is one, dimensional analysis is another

[midnight scarab]

The way I'd memorise this is

1. v1' is prop to m1-m2 because for m1=m2, velocities are exchanged
2. v2' is propto m1 because if m1=0, then there's no momentum to transfer to m2
3. indeed m2->infty: bounce off a wall so v2'=0 and v1'=-v1

[tawny slate]

yeah im doing a thing on sanity checks, so that's why im specifically looking for a sanity check example

i think it's an underrated underused skill, an arguably one of the main reasons a general public should have math competency

[torn apex]

hmm..what do you think NSA constructs infinitesimels out of?
fwiw, here are two references:

• https://terrytao.wordpress.com/2007/06/25/ultrafilters-nonstandard-analysis-and-epsilon-management/
• https://en.wikipedia.org/wiki/Infinitesimal#Dual_numbers

I think maybe people have different definitions of non-standard analysis. I group constructions using the hyperreals, superreals, or dual numbers to fall broadly under the umbrella of "non-standard" analysis

[quasi maple]

"Hold on, lemme pull out my pocket black hole real quick to check my working out"

[tardy ember]

i was assuming you meant specifically the hyperreals, because that's how i've seen the term "nonstandard analysis" used before

if you meant "literally any approach that has been used to make infinitesimals rigorous" then what you said makes a lot less sense, because that is not one thing at all, it's several different approaches that have very little in common with each other

[compact isle]

I wouldn't count dual numbers as nonstandard analysis even though people frequently confuse the two from what I've seen

[tardy ember]

and so in particular "non-standard analysis is/isn't intuitive" makes very little sense, because how intuitive it is will depend on which approach you use

[torn apex]

am I confusing the hyperreals with duals? iirc, hyperreals < surreal numbers and they are formulated from notions of epsilons where e > e^2 > e ^3. I'm pretty sure I'm not too far off but i'm working from memory recall.

Also, I took "non-standard analysis" to mean any of the attempts to make the "Non-rigorous calculations with infinitesimals" rigorous. The seminal text being "Elementary Calculus: An Infinitesimal Approach" by Abraham Robinson

So i still stand by the earlier claim that non-standard analysis is meant to take the intuitionistic view of calculus and make it rigorous. I mean, is that not literally the Robinson's motivation for developing nonstandard analysis?

[torn apex]

sounds like you're an expert in NSA so how about enlightening the rest of us then?🙃

[tardy ember]

there are technically some provable relations between the hyperreals and the surreal numbers but afaik they don't really get you much and are difficult to prove so there's not really much reason to use them

the surreal numbers are a field that contains a copy of R and also contains infinitesimals but they're not really a particularly useful place to formalise calculus with infinitesimals, compared to just taking an ultrapower of R

generally people study the surreal numbers for other reasons

the basic idea of the hyperreals is the transfer principle - any first-order statement that is true about standard objects is also true about nonstandard objects

the main issue for intuition here is learning to identify which statements are "first-order" and therefore can have the transfer principle applied to them

the main other approach to formalising infinitesimal calculus that i'm aware of is SDG, which... i might be getting some technical details wrong here, but i think the basic idea is: you just use the real numbers, and the infinitesimals are real numbers that square to zero. in order to make this work you use intuitionistic logic, and so you can prove that there are no nonzero infinitesimals, but not that all infinitesimals are zero

which is counterintuitive in a completely different way, because you have to learn how to do mathematics without LEM

[torn apex]

Thanks for the informative clarification!

[stark heron]

i wanted to ask for anyones opinion on like how you help students that have no number sense or basic arithmetic skills and are far beyond the grade level they should be? i.e., in algebra 2, cant add fractions or divide numbers. the answer can be on an individual teacher level or systemic level. ik its like the age old question but still want to hear your individual thoughts

[tight star]

Can they add numbers?

[stark heron]

one student i am thinking of can, but i guess it was a more general question because i've encountered it in many different ways yk

[tight star]

My gut instinct would be to show them how to think of fractions as distances

I think the issue with fractions as “parts of a whole” is it’s not immediately clear what it means to add, say, 2/7 to 1/6

[zealous kettle]

Is it possible to get a 6 y/o student to understand why I want them not to call out the answer (in a 2- or 3-student class, situations where students are working out a problem at the same time)? Something like "keeping the answer to yourself is sharing the joy of solving the problem"

[zealous kettle]

I think systems need to be able to accommodate students learning at different paces, rather than assuming "This student has been given a lecture alongside dozens of other students and then assigned a homework on the topic, therefore they should know it."

In the situation you're describing, maybe all there is to do is try to remedially teach the student the prerequisite skills (That might be easiest with Khan Academy or similar), and to arrange topics that require those skills 'til later in the school year (so students have had ample time to get the prereq skills).

[zealous kettle]

Other possible framings or structures:

• "Always wait for the other students to finish, so we can say the answer together"
• "It's [other student]'s turn to give the answer and feel smart."

[abstract grove]

that's an unfortunate and all too common situation... in a more ideal world they would be assigned work at a level appropriate for them, but that typically isn't an option for them in existing schools. So to try to catch up they should be practicing more foundational things that they're lacking in, every day, alongside still trying to keep up with their assignments, hopefully with assistance from a tutor

[stark heron]

i definitely agree. as a tutor sometimes i feel a little guilty because i am the "make grades better" machine, and sometimes with such limited time and a lot of foundation lacking with the student i go the procedural route because i dont know how else to actually get them results im being paid for you know

[abstract grove]

I totally understand and have been there too many times

I used to work at a Mathnasium and we got loads of students who would be 5+ grade levels behind

[quasi musk]

I think helping students with where they are at is better than trying to shoe horn them into high level thinking. There's nothing wrong with teaching procedural math -- the issue is that the procedure is taken as the ultimate goal. If they're going to be longer term clients, then you can slowly move them up the learning totem pole, so to speak

[outer burrow]

a problem is often called an exercise (like lifting weights). when you do your exercises, you become stronger. if someone lifts my weight for me, do i become stronger? if i lift someone else's weight for them, do they become stronger?

i haven't explained that to a 6-year-old in reality, but i assume they can understand this

[slim path]

While the rationale of this is well and good, children, particularly those who feel like things are coming easy to them, tend to get a big head about it if not treated with care. Moreover, empathy with others at this level can be easy to understand but hard to realise.

When I was in uni, there was a guy in our class who was already really well versed with a lot of advanced topics and could write really neat proofs. Yet he kept hogging the class and the prof's attention when we were asked to solve something during the lectures. Imagine that, an adult doing this among other adults, many of who may suffer from some form of impostor syndrome as a result of this (which did happen to a few who switched majors). Thankfully I was not a math major to begin with lol but I felt pretty bummed out myself.

It's not as though this student was unaware of this basic fact but know-it-all's like to show off because they internalise a sense of superiority. It's also not that we as his fellow students were not aware of how big the world is and how many ppl are ahead of the curve. Which is exactly why you don't wanna accidentally pass a message to a kid reinforcing an idea of superiority he might have in his head already or an idea of inferiority in the others if this is a repeat occurrence.

[outer burrow]

i feel like this 6-year-old already has a pretty big head and seems a bit like the adult you mention 💀

i don't have the complete data here (Will does), but i'd feel comfortable saying this to, e.g., my 7-year-old niece. i don't see how superiority follows from my message

[slim path]

The analogy with strength can be misconstrued by children as ability.

[outer burrow]

again, not knowing the 6 year old, i think the 6 year old is already aware that doing the problems displays ability

i don't really see how my message would give him/her a bigger ego

[slim path]

True but would a six year old who is possibly competitive like the idea of an even playing field or prefer getting ahead by any means is hard to say without more information.