math-pedagogy

[tacit adder]

so assignment works very similar to homework; assignments are graded tasks that have a slightly longer duration (i.e. 2-3 weeks as opposed to weekly), so the difficulty of the problems tend to be greater since students are given more time. Homework tasks are more guided and give students the chance to practice problems that are slightly easier than assignments

[lethal leaf]

ah

yea we do like

weekly 2 written problems and 1 online problem

the online problems are easier

more like guided walkthroughs

and then the written problems are actual algorithm design questions

[tacit adder]

ah I see

[lethal leaf]

the adv. algo class has 3 written problems a week, no online problem

[tacit adder]

yeah we've found that students are just struggling to even get the basic problems so we're trying to do a huge re-evaluation on how we teach the course

[lethal leaf]

Scroll down for HWs https://courses.engr.illinois.edu/cs374/fa2021/A/

[tacit adder]

OHH UIUC

nice nice

[lethal leaf]

ye ye

took algos with the GOAT Erickson (for both intro and adv algos)

[tacit adder]

man

his book is so good

😭

[lethal leaf]

it is

he has a new one coming out soon 👀

his HW problems are similarly good

[tacit adder]

oh let me know when it gets released; I was planning on buying his latest edition

[lethal leaf]

https://github.com/jeffgerickson/1dct

he taught a topics course on this last sem and I couldnt' take it 😭

[tacit adder]

man

we don't have courses like this

this is so tragic

[lethal leaf]

I am very lucky here for at least my interested in TCS

taken some cool courses

Adv algo HWs https://courses.engr.illinois.edu/cs473/fa2022/

[tacit adder]

we are severely lacking in tcs courses

I've basically taken everything

verification, parameterised algorithms, theory of comp

[lethal leaf]

I've kind of run out of interesting TCS courses for me

well, interesting to me

oooo parameterized algos seems cool

next sem we have randomized algos and algorithmic game theory and basic theory of comp

[tacit adder]

it was the most fun I've had

[lethal leaf]

none of which are interesting to me next sem

there is a cool course on streaming algos tho next sem

[tacit adder]

algorithmic game theory sounds fun

[lethal leaf]

might take that time permitting, but time will probably not permit

[tacit adder]

I should ask for a course on algorithmic graph theory

[lethal leaf]

cause math classes

I'd kill for an algorithmic graph theory course

I took an algebraic complexity course last sem

now that was fire

[tacit adder]

I was decently surprised to see a lot of extremal combinatorics pop up in complexity theory

second time I saw the sunflower lemma come into fruition

[lethal leaf]

I'm taking an extremal combo class next sem

hopefully that'll patch some gaping combo knowledge holes that I have

[tacit adder]

I struggled with it but man, it was really fun

[lethal leaf]

too many cool classes

not enough time

I'm definitely gonna take a bunch of classes while in grad school

[tacit adder]

Oh yeah same haha

need to start reaching out to professors to see if they want to write me a letter of rec

(funnily enough, I'm planning to apply for UIUC for grad)

seems like a pretty cool uni for tcs stuff

[lethal leaf]

ooo nice

if you have questions DM me

[vagrant meadow]

at least the connotation to me is that assignments are graded and hw is not (but that's just the connotation that i associate with those labels, i think they really are the same thing at the end of the day).

i think maybe i'm a bit different from a lot of math people in that i generally value the destination over the journey (in a lot of areas, including beyond math). while struggling a lot to finally get an answer is great in a lot of ways, the goal is still to learn. and i think math should be a collaborative process. i know a lot of people who disagree with me, and who despise it when they're given hints or any help at all. but i don't really care about that, personally. i'm not saying just give everyone a homework solution. i'd hate to just have my assignments handed to me, myself. i like to work hard, but i also just get bored and frustrated when i don't understand and the text/prof are unavailable to help. but i think this stigma against providing anything above a vague hint in the right direction is overkill.

there are times when i've asked a question and someone gave far more than a hint: they gave me an answer that explained far more than the solution, with the purpose of providing a better and more whole understanding to the deeper connections of the topics involved. and i learned way more from that than i would have just finishing the problem and calling it a day. maybe this really only applies to students who truly have passion and care about learning as much as they can, and what i'm saying misses the point about the students who don't give a hoot and just want solutions to copy and turn in. but i think those people will hit a wall they won't be capable of skirting around eventually. i also think one should teach for the students who want to learn, rather than waste their focus on actively making things difficult for the students who don't (and in the process, also inconveniencing the hard-working students).

one other thing i've noticed: i think subconsciously, i'm more amenable to just giving solutions to students who display a clear drive to learn over those who seem to just want an answer and don't care. perhaps because i think the passionate student will be less likely to simply take advantage of a free answer, and will instead use it to further their understanding?

jeez great. another wall of text from me. sorry yall

[cosmic ibex]

The journey vs destination metaphor can be applied in several incompatible ways here:
A) get the concrete number the problem is asking for,
B) write down a computation that produces the right number in the authorized way,
C) memorize how the authorized computation goes for problems of this particular form,
D) understand enough of the underlying math to be able to come up with a procedure on the fly when you see the problem,
E) achieve that understanding by the particular strategy of sitting in solitude and staring at the problem.
I hope we all share the opinion that (D) is the real goal. But how to express this can differ. You can either say that you're valuing the destination (D) over the journey (E), or that you're valuing the journey (D) over the destination (A or B).

Conversations about homework help ethics often start with a desire to avoid students sabotaging themself by getting someone to tell them A or B and never caring about D.

Where things tend to break down is the fact that for a motivated and self-aware student, seeing someone else's B can be a perfectly legitimate path towards D, especially if honest attempts to find a solution themself have gotten nowhere -- yet it is hard to convince a bystander (or would-be helper) that their goal isn't just to hand in the B as a way to pretend they've achieved C or D.

[cosmic ibex]

Hmmm, how about this as an ideal of how homework should work, for motivated self-driven students:

1. The "homework" period is about trying to solve the problems from scratch, without outside help, or perhaps with vague hints. Each student will either succeed or not, it's the attempt that matters.
2. After the deadline, a classroom discussion of solutions to each problem. This lets students who didn't find a solution at first backfill their understanding from seeing a solution. And students who found a convoluted solution get to see a neater one.
3. If someone feel they didn't understand the solution discussed in class, then they can come to places such as this server and ask questions -- hopefully they'll now be well focused questions because they are informed both by solitary attempts to solve the problem and by the subsequent discussion.
   Unfortunately that's too idealized to have much chance of working in general. Grading the homework handed in (and thus rewarding/punishing students based on how well step 1 worked for them) will provide an irresistible incentive to try to shortcut the process by getting a B answer, by hook or by crook. And ill-motivated students who are in it just for passing rather than for learning will be unlikely to do steps 2 and 3 no matter how much they're told to do it for the sake of the exams ...

[cloud zealot]

you could make homework not count for a grade and have more frequent quizzes in addition to midterms and finals

also, the midterms could be broken into an in-class portion and a take-home portion

in-class should test fluency with concepts, while take-home should test depth of understanding

i know teachers intend homework to be for learning, but making them for a substantial portion for the grade AND checking for correctness makes students feel like it's a summative assessment, when it's really supposed to be formative

you could make certain homeworks open to submission for feedback for a little bit of extra credit

[vagrant meadow]

sure i suppose there are many ways to go about it.
definitely i mean D over E.
never just A. that's just infuriating. sometimes B, depending on the simplicity. as an example, often in calculus, seeing the product/chain rule in action for specific functions is more effective than drilling the formulas if you're still shaky on applying abstract formulas to specific problems.
i also tend to discourage attempting to accomplish C directly. but i think achieving C indicates a sense of mastery of the concepts, and can function as a good goal for how many practice problems to do.

[vagrant meadow]

this is interesting, but i really dislike part 1. i get and admire the idea, but i think it'd be discouraging and frustrating for many students.
and often, there isn't time for step 2 in the curriculum. especially in a higher level class, often you have to lecture every class meeting or risk failing to cover major portions of the class material.

i think working with others, or getting help from servers like this during step 1 is good.
if one desires to struggle in solitary attempts, that is totally up to them. but i personally do not find that to be very helpful to me. that said, i think one should at least attempt it on their own before asking for help. it's annoying when students come in for tutoring as if we're paid teachers there to teach them everything their profs already lectured and walk them through every single problem on their homework.

again, i personally feel it's best to teach for the passionate students who care. i know that isn't practical in real teaching situations, though :/

[cosmic ibex]

What I say is that because needing some help after thinking about the problem for a while is normal and expected, educational institutions should meaningfully let that happen within their own walls. It's an essential phase in the learning, and just punting it to volunteers in servers like this is not scalable or sustainable.
And yes, not grading the intermediate products of the first state is a prerequisite for this.

It's simply not right for a school to tell students, "take this problem and don't come back until you've found some unpaid volunteer online you can talk into doing your teacher's job".

[zenith slate]

I can only speak from anecdote and experience here but British high-schools have gradually been adopting step 2 for the last couple years, and it has broadly been very successful.

More than just fulfilling the stated goals for the student, taking time to listen to students discussing their answers and understanding can be very insightful for the teacher. Each mark on a homework script is a binary outcome, and it can be very difficult (especially as students get younger) to evaluate more important factors like confidence and fluency when simply grading a script.

(Interestingly, this has come about almost entirely for political reasons - teaching unions have successfully reduced teacher marking time down to nearly zero and this has forced schools to get creative with how students' homework gets graded)

[zenith slate]

I wholeheartedly disagree with your suggestion that teaching should be done to cater to the passionate students who care.

Take a sample of passionate students and you will find that they are overwhelmingly more likely to be students from better socioeconomic backgrounds than their non-passionate peers. I don't think pedagogy should be seeking to reinforce the already enormous attainment gaps in education.

[vagrant meadow]

I'm thinking more about higher levels where nearly all students have selected to specialize in math. in those classes, they've chosen math because they like it, or sometimes they're just there for the degree so they can get a job they won't be qualified for or something. I'm saying i dislike when classes are structured around the latter type of student.

but that's an interesting point.
while i still think lower level students (like in high school or lower div college) who like math should be encouraged, I'd assume they're an extremely small minority. i think that's why I'd probably be miserable teaching high school math. it'd be like being a dentist. like nobody wants to be in your class. ):

[twin shell]

teaching is a thankless job

[zenith slate]

I teach high-school and in my experience it is fairly easy to get students enjoying math class if you teach them well and treat them like human beings

A majority of the students I teach enjoy their lessons with me

It is all about building positive relationships with your students and appreciating that fact that they are teenagers

[zenith slate]

Public perception of teachers is very poor, for sure. But my wife and I have received a few dozen cards from our students who are finishing school this year. We've had potted plants from two students, one of them bought my wife some earrings, loads of flowers, mugs, all sorts of little gifts that have made us feel really great

Students are humans and they wanted to be treated fairly and compassionately - do this, and the job is (most of the time) a delight

The idea that highschoolers are lazy and disinterested and full of angst is extremely cynical and completely misses the point that we put students through a tremendous amount of stress at a time when they're still figuring out who they are. All that without even thinking about the shitty home lives and abuse and bullying that some students have to tolerate day-in-day out. Us teachers like to complain about bad public perception - but no group is as frequently and openly degraded as teenagers.

[vagrant meadow]

kind of a random question, but there have been professors that i and other students have wanted to give gifts, but we don't know when is appropriate. would you feel weird getting a gift from a student before final grades have been submitted?

[pastel horizon]

I don't think it matters

Grading is usually anonymous anyway

The safest bet is probably one gift that you all contribute towards

[pure coral]

another problem with catering to “passionate” students (who might be passionate for all the wrong reasons, like ego!) is that, if every class did that, then students will just end up with narrow interests

[zenith slate]

Not at all

Anything to show appreciation to a professor/whatever will be greatly appreciated

One thing I will say - often the best gift is just a card or a written note. Items like food or drink or whatever get used and then they're gone. But cards - you can keep those forever. My department head has been teaching for decades and his pride and joy is a big box full of all the kind notes and cards he's received over his career. He sits down and reads through all of them over summer break every single year.

[vagrant meadow]

can you explain what you mean by narrow interests?

[vagrant meadow]

makes sense. yeah maybe i should go ahead and write some cards, even though i've graduated.

[light garnet]

Just wanted to mention that NYT put out an article today regarding the use of tutorbots in the classroom, for when the teacher isn't immediately available. They highlight a specific one by Khan Academy (Khanmigo) which has mixed reviews: https://www.nytimes.com/2023/06/26/technology/newark-schools-khan-tutoring-bot.html.

On Wednesday, a reporter asked Khanmigo the same fraction question. In student mode, the tutoring bot explained the steps and then directly provided the answer: “the fraction of consonants in the word ‘MATHEMATICIAN’ is 7/13.”

In teacher mode, which is designed to walk educators through problems and answers, the bot provided a different — incorrect — response. Khanmigo said erroneously that there were eight consonants in the word “mathematician.” That led the bot to provide a wrong answer: “8 consonants/ 14 total letters = 8/14”

[lethal leaf]

incredible

[quasi musk]

I've noticed that many students in Calc 1, 2, 3, Linear Algebra, Diffy Qs at colleges/universities prefer online homework nowadays with the "Stuck? View an example" option nowadays

I think using technology in the class can be a good thing; however I worry when education becomes too attached to propietary software

[zenith slate]

Open source teaching resources are few and far between, unfortunately

Though it is worth bearing in mind that education is already inexorably linked with proprietary products

Textbooks, school administrative systems, yadda yadda

Propriety aside I think technology can be fantastic, though I don't think we're anywhere close to the point where teachers become replacable with software

[pure coral]

you go into a classroom not knowing anything about a subject, but you don’t have an opinion about the subject yet

and then suddenly the teacher assumes that you’re some prodigy that already knows the subject

that’s how you get people to hate subjects that they’re not only bad at, but subjects they’ve never even experienced before

it would be like if you get an entire class of grade 3s to read hamlet or macbeth because 1 or 2 prodigal (read: rich parents) students read shakespeare for fun

[vagrant meadow]

that's definitely not what i was referring to. an example of what i meant was my ODEs professor. he taught far beyond the textbook and went into niche applications of concepts that the curriculum didn't deem important enough to list. he was able to get through so much material, it was amazing. and while there were students who bitched and moaned because they were learning things they didn't have to learn (and the exam questions were very difficult, because they expected conceptual understanding), other students were being really well prepared for how engineers actually use ODEs to do their jobs. and it inspired me to love the subject. it also really well prepared me for my subsequent classes that used the material, which i probably wouldn't have taken if i hadn't learned from him.

he taught for the people who were going to use the course content in their careers. not the people there to get an easy A and a requirement checked off. that's what i mean.

[pure coral]

oh i see

that’s just a good teacher

even a less passionate student should appreciate the interesting presentation, i think the other students were just unwilling to be at college at all

it’s a reflection of how much learning has been devalued in college, culminating in one huge final exam where crammers usually have an advantage because it’s all about answering basic questions you could put on a flashcard, and not applying knowledge

[vagrant meadow]

well, no. it feels like a more systemic problem.
i had to take ODEs again when i transferred. and it was a joke compared to what i had taken before. we learned barely half the material he covered. in the place i finished my undegrad, over and over most of my professors were cutting course content because exam scores were bad (i.e. people cheated during covid and were completely unprepared for the level of course they were taking). so there i am, preparing to go into research and a phd program, wanting to learn as much as i can, and the professors just start cancelling lectures because everyone is doing too poorly. imagine taking an advanced linear algebra course where you don't even MENTION jordan canonical form. i just had to teach it to myself, and then what was the point of the thousands of dollars i spent on my education.

that's what i mean by teaching for the unpassionate students, and that's what i hate. i think the professors should have taught the material anyway, because there was at least one student who wanted to learn it.

anyway sorry rant over. i'm just still salty i had to self study a bunch of the material i was supposed to learn in my undergrad.

[pure coral]

it seems like a problem with the school not maintaining expectations, and the professors being lazy and just cutting content to create less work and raise their student ratings

the first course (with the prof who was actually motivating the content) should also be more friendly for the lower performing students, since they get a lot of examples of how to apply that knowledge as well

but the apathetic students are pretty much unteachable anyways

so yeah i do think they should just forget about the apathetic students, but a good teacher will be good for any student that cares about learning even if they hate the subject itself

but a good teacher would also set high expectations and not lower them, knowing they’ve done everything they could to help meet them

[pure coral]

nah i get the frustration, the price of college is obscene compared to the product you get

with so few math majors, surely the education can be a tad more personalized instead of a tired professor reading off the textbook in (somehow) packed lecture halls

i was mainly referring to k-12 where it’s infeasible to ignore a huge number of students, but on the other hand, college shouldn’t be catering to everyone

[pastel horizon]

I find that for maths the community tends to be very good at sharing resources

AI will soon be enough anyway

[quasi musk]

Buying a book is different than a subscription to an online learning platform that expires after x amount of time

I think a good example is graphing calculators ~ rather expensive, rather useless outside of that specific class

[shy crag]

Exposure comes down to it. Being around mathmaticians and people who can regularly compartmentalize this information helps a lot

That understand, that information is often lacking in the boonies.

[shy crag]

Hardly, I know a few uses. Pretty useful for bars.

[light garnet]

Big news in the K-12 world: the most recent draft of the California Mathematics Framework was put online very recently (https://edsource.org/updates/state-releases-revised-and-probably-final-math-framework) and is expected to get approved by the board of education in CA. The most notable change since the previous version was the removal of the proposed K-12 data science pathway.

For those of you who missed reading on this proposed pathway, here was some of the text outlining its rationale, with emphases mine:

The traditional sequence of high school courses—Algebra, Geometry, Algebra 2—was standardized in the United States following the “Committee of Ten” reports in the 1890s. The course sequence—which was primarily designed to give students a foundation for calculus—has seen little change since the Space Race in the 1960s. With the rapid expansion of information available to all in the form of data, far more students pursue statistics classes than calculus, and may be better served by a data science course as a culminating high school mathematical science experience. In addition to the importance of the data science content—to twenty-first century jobs and to a wide range of college majors—many students are more engaged by open-ended explorations of important data sets, drawing upon important mathematical principles and tools, than by many traditional courses organized around mathematical techniques.

[cold trench]

makes sense

I like this

with the prevalence of data and tech in the world, this seems like a smart move

as long as there is still the option for calculus for the stem kids, this seems good

I also believe there should be options for data analytics or AI related courses

[sharp nimbus]

yea

[winged urchin]

What kind of things would an AI or data analysis stream look like at high school?

First thing that comes to mind personally is linear programming

I don't recall students doing a whole lot of inequalities really at the highschool level. Especially not inequalities on the Cartesian plane

But the simpler kinds of 'machine learning' need that I suppose

[light garnet]

You may have misread, but this option was removed in the most recent revision

[light garnet]

Given that DS hasn't been figured out at the college level yet, I think it's fair to say it's not been figured out at the HS level

[cold trench]

I only read what you wrote sorry

[light garnet]

Yeah, I stated it was removed above

[cold trench]

oh what the fuck

sorry I'm dumb

but that's stupid

why would they remove it?

budget?

[light garnet]

There was a ton of backlash from college professors in the University of California system

[cold trench]

inequalities are taught in high school

[light garnet]

I can't find the website right now, but I'll share it once I find it

[cold trench]

about students knowing how to use data?

[light garnet]

No, it's more nuanced than that. Let me find the site.

[light garnet]

Found it: https://sites.google.com/view/publiccommentsonthecmf/

[cold trench]

maybe statistics -> programming (python) -> data analytics -> ML/AI

[cold trench]

thanks, I'll read soon

[sharp nimbus]

U take stats as a Freshman?

[light garnet]

Here's a more detailed one: https://sites.google.com/view/mathindatamatters/home

[cold trench]

good point, maybe after alg 1

and the last two I listed could be combined into one course

[sharp nimbus]

Yeah, some algebra is not replacable,

[cold trench]

I don't mean replace

I mean add on

[light garnet]

Keep in mind, the original proposal was to have a data science track in lieu of Algebra 1, Geometry, and Algebra 2

[sharp nimbus]

No I agree thats why I asked "stats as a freshman?"

[cold trench]

okay, I'll read the articles in a sec

[cold trench]

ohhhh ok I see now, didn't realize it was replacing alg 2 - I think it's fine to replace calculus with statistics but not alg 2, thanks for the clarification

[empty gull]

why replace anything?

why not have both

[vagrant meadow]

I didn't even really like math that much until calc I. idk it seems like a lot of potentially great math students may never even consider the subject because all they know is the intro math, which (relatively speaking imo) is unrepresentative and not nearly as interesting. :/

[cold trench]

most students will do one or he other

[empty gull]

Why not make math mandatory?

I'm coming at this from a pretty eurocentric POV tho where I really didn't have any choice in the classes I took

[cold trench]

one class per year is fine

[light garnet]

I am not saying I agree with this perspective, but I've worked with many people who do: people are noticing that failure rates in math courses are extremely high and can prevent people from graduating, so the proposed solution is just to get rid of the traditional curriculum entirely. There are only so many teachers and faculty available to teach, so it's very much an all-or-nothing situation

Though California did not pass this proposal, I've seen this play out in different ways in other institutions

One trend that seems to be taking off is, at the college level, getting rid of remedial classes entirely because people have observed that being placed in a remedial course is a good indicator that you won't graduate with a college degree

My former colleagues will be going through such a change in 2026

[pallid night]

Surely they'll just crash and burn in the regular class anyway then ?

[light garnet]

I don't make these decisions. But that's what the administrators decided.

[pallid night]

Not Maths admin taking decision for Mathematicians yet again 😔

[light garnet]

It's really a game of "pick your poison." Do you stick with the status quo, where people are graduating at a rate of 25-50% with their degrees, or do you try to make math courses easier to try to give your graduation rates a fighting chance of increasing?

[pallid night]

Also ... can I have an opinion on this Riemann Sum notation ?

Cos I've seen some good notation and some utterly diabolical stuff that is pedagogically unhelpful.

So I came up with this.

[pallid night]

Yeah Maths has an obscenely high drop out rate at my uni. Cohort goes from 350-ish taking end of Y1 exams to 300 in the 2nd year.

I don't know how many make it to 3rd

Fail rates on the exams regularly cross 20%

[stray grail]

"obscenely high drop out rate" of 20%? I almost never have over 50% pass rate in my courses, and they are obligatory for many programs with about 400 people every year.

(and consitently get great student evaluations)

[pallid night]

It's obscene for the UK

And especially obscene for an entire degree

[stray grail]

ok 🙂

Sweden here

[twin shell]

I love grade inflation

[vagrant meadow]

ah yes, of course. the purpose of school is giving degrees, not making sure students learn things. who needs math anyway? haven't you seen all those epic memes about how you never use the Pythagorean theorem as a McDonald's employee

[vagrant meadow]

so this is in sweden? maybe there's a prevalent cultural aversion to failure we have in the US. failing a math class and taking it again is often the best course of option for a student in the long run. but failing any class is taboo. so students would rather skip the class entirely, or cheat their way through it.

[stray grail]

I guess a big part of it is that university is free here, and failing a class does not have much impact (here grades are 100% given by a final exam, and if you fail an exam there is a reexam after a few months.) There is like zero stigma to fail or skip the exam of a course. The main course I have had the last few years is the last programming course most engineers take, but I have had courses in scientific computing, numerical analysis etc. The same applies to pretty much everybody here also in pure math. Something I have noticed the last year is that like 30% drop out without doing anything in the course, they just disappear.

[winged urchin]

That sounds so nice geez. ⭐

[stray grail]

Another thing is that we have no obligatory attendance, and I have all lectures prerecorded. Lessons are only for getting help. So many people just take the course by themselves. I guess a certain percentage of students can not handle that. Setting deadlines maybe added 10% more people finishing (I had none in the beginning of Corona).

[light garnet]

Having been faculty myself before, pre-recorded lessons - though I loved them a student - are heavily burdensome on faculty, which discourages them from pursuing them as a primary mode of instruction. Not only do you need to figure out how you're going to get someone to record your lecture, schools have decided that in order to comply with the Americans with Disabilities Act (ADA), faculty are usually the ones responsible for dealing with things like captioning. Auto-captioning, as I was repeatedly told in my faculty position, is not compliant with ADA and was a huge reason in making me leave academia entirely.

Frankly, I spent most of my time back when I was teaching online dealing with the editing and captioning of videos, which burned me out

[stray grail]

I am the lazy type. I just record in zoom with beamer slides. And if I make a mistake I just take a pause and cut it out after. Maybe spend 5 mins for each lecture editing. I never listen to the whole thing (just assume students complain if something bad) I reuse maybe 50% of videos unless I update anything. I use autocaptioning, I think I am the only person even doing that. Again, nobody has ever complained.

[light garnet]

Yeah, we were told to never do that because if the school had to deal with litigation from someone suing due to ADA, we'd get in trouble

[stray grail]

:/ that sounds bad, sorry for that situation.

[pallid night]

My lecturers only turn captions on if they get a specific, anonymous request 😬

[shy crag]

I'd want to understand why the language is so complex that many people are failing. At its core math is about data flow among numbers.

[light garnet]

I understand how you're viewing this, but I think, given my experience, your comment is a bit too simplified: especially post-pandemic, I've been hearing (and have seen myself, since leaving teaching, in online forums) of people taking college-level courses who don't have basic fundamentals down. Back in February, I actually had to ask a student the following question because they could not figure out how to calculate a percent for an intro stats class:

If there are 5 data points in this group out of 30 of them, what percentage is that
to which they responded with an online widget and asked what numbers should be inputted in there.

[shy crag]

Fair

Fundamentals are often overlooked at higher levels.

Ironically when you get to the top, a lot of new work and understanding comes out from just retreading basic math.

Its sad she isn't thinking about how the math will need through through those numbers.

Part of mathmatics is number engineering to a 'degree'

[twin shell]

there’s a review I read https://www.maa.org/press/maa-reviews/principles-of-mathematical-analysis that basically says that college math standards have plummeted significantly over the years

[light garnet]

From chatting with my former colleagues, imagine having to teach a college intro stats class and coming to the realization that your students don't understand what a percentage is

[twin shell]

while I can’t verify if that’s true or not I do believe that math in the US is far weaker than in other developed countries, namely in europe and southeast asia
in my opinion there’s a lot of factors that lead to this

• part of it is cyclical incompetence: bad teachers -> bad students -> bad teachers (in the K-12 education system, that is; professors can also be bad, but at least they understand what they’re trying to teach)
• a lot of it, naturally, is social issues; if your parents are shit you’re just not going to learn well
• another problem in K-12 is the overreliance on testing—teaching to the test, not to learn. this happens I think because school funding depends on it

the product of all of this is students who have no idea how to get gud when they enter college

[light garnet]

another problem in K-12 is the overreliance on testing—teaching to the test, not to learn. this happens I think because school funding depends on it
I actually used to work with this data for a living: not only does school funding depend on it, it is moreover federally required ever since the passage of No Child Left Behind in 2000; do some searching on "educational accountability" if you wish to learn more about the specifics.

[twin shell]

yeah I heard of NCLB in my intro ed class
it was very funny and sad

[light garnet]

Well, then you probably know about ESSA too. I was responsible for helping draft ESSA-compliant accountability plans for one state

[twin shell]

I’ll admit, I did not, but I’m interested now

did it, uh, work

[twin shell]

I think it wasn’t included in my class just because it’s relatively recent

then again 2002 is also somewhat recent

[light garnet]

These laws have a cumulative nature about them; as you might guess, when you pass a law like this, you don't want to just scrap the whole thing because it's likely that sort of revision will never pass. I might've been involved in drafting such a law, but that was subject to approval by state and national officials

[vagrant meadow]

oh man that sounds amazing.... jeez.
this raises a good point... the prospect of paying thousands of dollars to retake a class does make the decision to cheat or avoid more understandable. not saying i think it's good, but it's certainly at least logical. not to mention the schools want pass rates to be high.

[light garnet]

and tons of people end up joining in, muddying those waters further

[twin shell]

rip

[light garnet]

It taught me a lot about education policy, undeniably

but I will never return to that work unless I really have to

[twin shell]

understandable

[stray grail]

Definitely a difference in behaviour from paying international masters students, both in complaining, grade grubbing and cheating. It has zero impact on me or my grading and I report all cheating I can prove (100% punished so far). Out of a class of 200 I maybe give 2-3 people some kind of extension on one assignment for a few days due to some extraordinary circumstance. Sounds exhausting the pandering and "customer attitude" that is going on in other parts of the world.
I think this works great for high performing students, and the "bad" students that still want to learn and put in the effort. The people that just want to do minimal effort typically fail.

[pallid night]

These are my uni's first semester grade statistics by course.

Green is 1st year level
Yellow is 2nd year level
Red is 3rd year level
Pink is 4th year level

UK grade boundaries :

70 -1st
60 - 2.1
50 - 2.2
40 - pass/3rd

Most grad jobs I've found and almost every Masters programme I've found ask for at least a 2.1 average, some of the top ones a 1st

All of these except the first and fourth Y1 course are courses for Maths students.

[noble hare]

Interesting i don't think prob and stat 1A had such a high fail rate in my first semester there

[pallid night]

It and Algebra 1A were
both around 17% in my first year

[noble hare]

Higher than I thought tbh

[pallid night]

I don't know if Kari was teaching it back then.

[pastel horizon]

You know what else, I think we need a serious introduction of some basic IT skills. It's amazing how many young adults don't know you can use excel as a calculator

Like they didn't even know the basic * / ^ operations

Why don't they teach this things? I swear I've seen 50k jobs advertised for people that can do a Vlookup lol

[light garnet]

The people who learn this content learn Excel over a semester in an IT course. I took such a course circa 2012. It starts off with cell formulas and ends with PivotTables, and is entirely auto-graded based on whether or not you followed the "book" (more like instruction manual) click-by-click. No mention of lookups is in there.

Anyone can get a 100% in that course if you simply do what the book tells you to do

[pastel horizon]

Yeah so essentially they get a pass for turning up. It's a lot better than nothing though

I mean I wouldn't say lookups are essential or even complicated. It can be useful if you need to retrieve different information from another sheet based on some given inputs however

Otherwise you may as well just link a direct reference

But you should expect people to at least know the absolute basics before they hit 16-18 right

[light garnet]

It's not at all taught in K-12

so no

I doubt the teachers have any idea what a lookup is, anyway

[vagrant meadow]

at my HS you could take an optional microsoft office class

but it should probably be standard

imo

[light garnet]

In the US, K-12 schools are generally not accountable for anything outside of English, Reading, Writing, and Math. I don't see it happening.

[pastel horizon]

They will be forced into being made accountable for IT literacy soon

[shy crag]

Sometimes you can get by even not knowing basics

Recently I’ve gone over addition again and gained new insight.

[tawdry venture]

my tutee and i meet 2 times a week. about how much homework should i be giving him?

[cold trench]

it depends - what's the course/field? it also depends on each specific topic, and how he does during each session. for things he finds easier, don't give him as much homework. I'll give the unhelpful answer of saying whatever is enough practice to help him understand the concepts, but not too much that it is draining. no more than a few hours a week, especially if he has a lot of other things going on in his life

[tawdry venture]

speedrunning preparations for 1st year of uni

[cold trench]

so like algebra/calculus?

[tawdry venture]

which in his case means trig, vectors in 2D and 3D, basics of calculus (limits, derivatives, integrals), and if time permits, basics of matrix algebra

[cold trench]

oh ok

[tawdry venture]

we are currently on trig

[cold trench]

wow that's a lot

[tawdry venture]

yeah

and i only have 2mo for it

[cold trench]

how are things going so far

oh fuck

[tawdry venture]

decentish

i mean ok it's more like

refresher rather than from scratch

cause he's dropped out but wants to go back in

[cold trench]

ohhh ok that's good

did u plan out the schedule for the topics youll talk about each session?

[tawdry venture]

approximately

[cold trench]

ok that's good

[tawdry venture]

i have a roadmap that ive made for myself

[cold trench]

ok, so I assume it's a lot every week

so like idk maybe a few hours of hw per week

it's hard to tell

ok so he has an idea which is good, has he told you what he's better at or wants to focus on more?

[tawdry venture]

at the beginning he confessed to me that he's forgotten a lot of stuff and can't tell what he is and isn't good at

[cold trench]

hmm ok well ig assume everything is new

[lethal leaf]

^^^

assume everything is new

[rain rain]

This isn't really a discussion but I'm not sure where else to put it that people might actually read/care lol

But I appreciate so much what 3B1B has done for math education, particularly when it comes to making math more accessible

Every time Grant comes up on my YouTube feed I'm just so glad lol, maybe cringe how much of a fanboy I sound like but idk I can't help but feel that appreciative

Like look at what SOME has done for Mathtube

3B1B is the main reason why I fell in love with math and I'm sure it's had a similar effect on so many others

Thank you to all math educators <3 y'all my goats

[vagrant meadow]

yeah same here. I don't know if I would have stuck with my math major without him

has anyone ever been in a tutoring/teaching situation where the student is completely wrong about something but completely convinced they are right? and it feels like no matter how you phrase the numerous reasons they're wrong, you just can't figure out what to say to convince them?
like they just keep repeating variations of the same thing, or moving the goal post/conditions of their statement, waiting for you to cave in and say they're right (at least under a billion different special conditions)

something like they're choosing to die on the hill that 0.9999...≠1 or claiming 0*infinity=0

[light garnet]

I've run into this in different respects. When I was faculty, my job was to give them an informed opinion of how to do things. Whether my students choose to follow my advice is not my problem.

[vagrant meadow]

so are you saying to just tell them they're wrong and move on? without wasting time arguing

[light garnet]

Not as directly as that, but yeah. I'm paid to give people my expertise, not change people's minds.

It is the student's decision to do the latter.

I dealt with plenty of that when I was teaching data science.

People asking "Clarinetist, why are you teaching us how to clean data when we could be running neural networks in Excel (yet we have no idea what a derivative is)?"

You make your point and move on.

This also happens too, in music - especially for piano playing. "Why can't I learn Chopin's Fantasie Impromptu or Liszt's La Campanella as my first piece?" The implication there, compared to math/DS, is that you can seriously injure yourself due to your ignorance.

[vagrant meadow]

i can see how that would be prudent in a teaching scenario. there are other students and it's not worth holding them back getting stuck mucking around with someone who isn't actually there to learn. but in a 1 on 1 tutoring scenario, i feel like this might not be as constructive.

[light garnet]

but in a 1 on 1 tutoring scenario, i feel like this might not be as constructive.
What's the point in arguing with someone if they don't want to listen to you?

See also my music example

If anything, large-group teaching allows you to easily avoid those conversations. I was a tutor for nearly a decade before I started teaching.

[vagrant meadow]

lol yeah true. but as an employee at the college tutoring center i don't really get to choose my clients. and if someone is paying me $50+/hr for private tutoring, then i guess that would be the reason to do so.
i'm not disagreeing with you at all btw. i'm just trying to push a bit for the sake of the conversation, in the pursuit of me being a better educator.

[light garnet]

Yes, I dealt with that too. If you want a different take on this, I don't think the student would appreciate paying for an extended argument

Some students are more receptive to that sort of thing than others, but it sounds like you're dealing with the latter, not the former

I dealt with statistics tutoring in my undergrad with people who were clearly just looking for answers to their HW, not much in the way of learning

You can do all you want to steer the conversation the way you want it, but they're paying for you to be there at the end of the day, and some of them have you there for a very narrow purpose (like giving HW answers)

[tawny slate]

This came up in a discussion with one of my students

They claimed that imaginary numbers don't exist

Obviously, this is a philosophical question and not a math question

But nonetheless I explained very simply that generally speaking in math, as long as it can be a concept that can be conceived, it exists

Then the student asked if there can be an idea that doesn't exist

And here I told them that if something's existence is inherently contradictory, then it doesn't exist

In the moment I think I may have went with my gut instinct and didn't think through it too deeply

But I was reminded of the subject while watching a video and I kinda wanna address it

What do you guys think? How do we define existence of mathematical concepts?

Is there a "correct" definition? Is this definition meaningful or useful? And these questions are obviously connected and nuanced so any input relating to this is appreciated

[twin shell]

recall goedel's incompleteness theorem

there will always be true statements you can't prove, like consistency

that doesn't render your system of axioms useless

actually idk where I was going with this

but the important part is that all these different number systems exist because people found them really useful

a lot of them took a long time for people to accept

like negative numbers

but people understood that the potential benefits and applications far outweighed any feeling of unrealism they carried

imaginary numbers are super useful in electronics, or so I've heard

maybe try researching some of the history behind topic X

in the specific case of imaginary numbers, more so than to solve x^2 = -1, they actually arose when people like cardano were trying to solve cubics using the cubic formula
there is a certain third order polynomial equation that evidently has a real root
but the cubic formula always included a square root of a negative number
if you accept for a moment that these square roots exist, then by carrying out the calculation you do indeed get a real solution

https://www.ms.uky.edu/~sohum/ma330/files/eqns_4.pdf this isn't the page I had in mind but it still looks relevant

so this hints that imaginary numbers might be consistent and useful

[vagrant meadow]

well this statement is just kind of vague in general. it depends on what you mean by "inherently contradictory".
1000 years ago, a mathematician may have called hilbert's hotel contradictory because "it doesn't make sense" to them how a hotel can have infinitely many rooms that are all full. after all, how can every room have an occupant if there are infinitely many rooms (this is an actual objection i heard someone make against it)? i know a number of people who refuse to believe its validity today (non-mathematicians).

arguably, hilbert's hotel is a less useful mathematical concept than complex numbers. but nonetheless.

[tawny slate]

I should be more clear

I understand that imaginary numbers and infinity are useful, I am far past that

What I want some opinions of are:

• what definitions of "existence" are there when referring to mathematical concepts in the abstract?
• are these definitions useful or meaningful? Is there a reason to use one definition over another?
• the reason why I ask these two above questions is to be able to give a more optimal answer to students, whether it is to adjust their philosophical notions or to give them practical advice

If a student asks me "do imaginary numbers exist?" I can comfortably say yes
If a student asks me "does 1/0 exist?" I generally say no, but it's pretty complicated

Is this the right attitude and approach?

Im also not referring to technical easily logical statements like "there does not exist an integer whose square is 2."

That's the easiest one to handle pedagogically

[vagrant meadow]

imo if you can define it, and it doesn't conflict with any mathematical axioms, then it's a valid definition and can "exist". this surely let's a lot of stupid concepts exist, but i think mathematics has generally moved past unanimously dismissing concepts because they're philosophically concerning (ex. banach-tarski, analytic continuation/interpreting divergent series, etc.).

mathematics is a human invention imo. i don't think it's a given that an alien species will have every math concept we do, and i think they would likely have fields of math we never even dreamed of. so the question of if one definition is better than another is subjective. but my take is that the definition which provides the most conceptual intuition is often the best one. but it's totally a case by case basis. as for if a concept "should" exist, that's also a case by case basis.

[tawny slate]

Haha as I read this the feeling I get is "well this is nothing new to me, but essentially I'm just overthinking too much"

I think I'm going too far down a rabbit hole, missing the forest for the trees

Thanks @vagrant meadow

[tardy ember]

i would say all mathematical objects exist

the set of complex numbers exists, because it can be defined, and the imaginary numbers are elements of that, therefore they also exist

"1/0" by itself does not uniquely identify a mathematical object - the most obvious interpretation, that it's a real or complex number that gives you 1 when multiplied by 0, doesn't describe any actual real or complex number - but if you define some extended number system where "1/0" means something, that exists

similarly if you interpret "sqrt(-1)" as meaning a real number whose square is -1, that doesn't exist either

the thing that makes the imaginary numbers interesting isn't that "they exist", because that's an extremely weak statement; it's that they have so many nice properties

[rugged hemlock]

Not sure if this is the best approach, but telling kids that imaginary numbers are used to describe physics shuts down a lot of skepticism

it might just be that they go from "there's no way this silly thing exists" to "how can this silly thing actually be used in a useful way"

[winged urchin]

I've heard black holes modelling through complex numbers (or would it have been quaternions because 3D I forget) and I thought that was pretty cool at the time

[pure coral]

that's the wonderful thing, you can create things you wish existed just because you feel like it 😄

coming up with a negative square root just because you want it to exist is the stuff of fairy tales

it's a selling point

[tawny slate]

my approach is that i explain how numbers are usually used to describe quantity, but they can also be used to describe order/position
once you understand that real numbers can be intuitively thought as all possible positions on the real number line, it can be understood that real numbers encode a 1-dimensional position

once i've established this, i explain that complex numbers encode 2D positions on the complex plane, and that arithmetic operations on complex numbers give us all of the linear transformations, so the most obvious straightforward application of complex numbers is 2D computer graphics

[marsh compass]

Why not just R^2 for that?

[tawny slate]

i mean sure, but complex numbers make the arithmetic operations on them more natural

R^2 alone doesn't imply that

[vagrant meadow]

“Schrödinger put the square root of minus one into the equation, and suddenly it made sense. Suddenly it became a wave equation instead of a heat conduction equation. And that square root of minus one means that nature works with complex numbers and not with real numbers.”
-Freeman Dyson

[wintry lichen]

Hey guys, I’m trying to teach my 16-year-old cousin basic naive set theory

[wintry lichen]

he’s almost got a working proof now

that |Z| = |N|

[wintry lichen]

have any advice?

[marsh compass]

not sure why it would be an advantage to work with complex numbers in this specific application

[shy crag]

Even if imaginary numbers aren't real, their 'math' still is. Almost like well written social interactions in a fantasy novel.

[tawdry venture]

is he comfortable with basic set theoretic operations like union and intersection, and perhaps the formalism leading up to functions

[pastel horizon]

Yeah there isn't really a practical application for having a "value" of 1/0 like with √-1

Maybe in some situations where you have inverse square laws I guess

But even then say if it's the intensity of a light source it doesn't make sense to have that at a distance of 0 anyway

[pastel horizon]

Also the sad thing is having to teach this to mechanical engineering students when there isn't really a use for complex numbers

It's just something that ends up on the syllabus because it's bread and butter for electrical

[cold trench]

I love that this channel is re-aliving itself now that school is starting soon :D

[pallid night]

Idk if this is entirely true on a wider scale but I feel from my pov like it's been fairly active for a while

[cold trench]

less so over the summer

[pallid night]

Definitely noticed a slight increase in longer convos recently though

[pallid night]

Surely Mech Eng people need to learn Resonance and Harmonic Motion under driving and damping regimes ?

And that definitely needs complex numbers

[tawny slate]

I still prefer complex numbers over R^2 for representing 2D points

And when abstracted in this way, I think everyone should learn complex numbers

To me, complex numbers are about two kinds of structures and how they are interconnected: rectangular form and polar form

Both of these are "natural" ways to represent points in the 2D plane and complex numbers allow us to do "natural" things with them pretty easily

I understand that in order to explain some practical applications, we need to pull real world examples, and often times this comes from what is popular

So almost everyone will mention electrical engineering and quantum mechanics

But these examples all have a bias, because then a student might think "well I don't plan on being a scientist or physicist, why do I care to learn this then?"

[wide ice]

There are two approaches: you learn what it takes to solve a problem, or you learn for the sake of learning. If you are of the first kind, and your problem never uses complex number, I think that's fair.

To be honest, the majority of stuff I learn in school is useless outside school.

Tbh Idk in what situation a doctor or a driver would ever need complex numbers. Sometimes I think current math curriculum is a bit overkill, and at the same time not sufficient for those who know they want to do engineering stuff.

It should be divided from high school: if you know you don't want anything to do with math, then you just don't learn math

[pastel horizon]

I think this is more important for civil I'm not so sure

My major was chemical anyway

I don't see it anywhere in the mechanical principles unit though

[pastel horizon]

This is fair

[tawny slate]

i 100% disagree

i wanted nothing to do with humanities when i was in college, but at the end of the day, looking back, im glad i took it

i wanted nothing to do with chinese when i was young, but my parents forced me to learn it and i am infinitely grateful

my major and area of work have nothing to do with either of the two directly, but both of those subjects inform how i do my work

and that's strictly referring to work

outside of work it greatly affects how i understand the world and how i understand philosophy

i think there is a realistic tradeoff you have to do

obviously, it would be better if everyone learned everything, but that's obviously not possible

but the attitude shouldn't necessarily be "if you don't want anything to do with math, then you should have nothing to do with math"

i completely ignored art as a subject because I didn't think i would be good at it and i didn't think i would ever need it

and now i need it and i wish i knew how to draw

[twin shell]

so in other words you are a proponent of teaching the liberal arts

[tawny slate]

absolutely? LOL

who isn't

for instance, knowing how calculus works in terms of foundational concepts only may not help us solve difficult integrals, but it will help us understand how we should drive in a way that is the most fuel-efficient

knowing how linear algebra works in terms of foundational concepts might not help us calculate determinants of large matrices, but it will help us understand that each additional unknown requires an additional relation to solve the system

knowing how complex numbers work in terms of foundational concepts, for example, that well-orderedness is a property that cannot hold under isomorphism, helps us understand that "the best" is not something you can metricize for anything in which there are at least two independent properties you want to maximize

to call someone "the smartest" or to call a food "the tastiest" is mathematically impossible by any metric

is it abstract? yeah that's kind of the point
is it niche? sure these examples are, but they are far from the only application of these ideas in the abstract

the concept of derivatives and integrals on polynomials helps me understand that when im playing a economic board game, some resources/builds/actions increase my income rather than increase my money, which in the short run is worse but in the long term is better, and i can calculate (or estimate) by how much better

partial derivatives tell me that if i want to maximize my caloric intake for a day, i shouldn't be necessarily be constantly stuffing my face 24/7, because the rate of digestion could be affected by other factors, such as how much food is in my stomach

[twin shell]

as with all things in life, there are good and bad ways to teach a subject
english and math are both super important (your average American is spectacularly bad at reading, writing, and thinking logically)
but if your students end up not appreciating the value of the subject then you have essentially failed as a subject teacher, because the only thing that will actually make you learn is a level of appreciation that motivates you to learn on your own
(of course, there is more to teaching than simply teaching the subject)
I want nothing to do with my english or history classes because I do not enjoy them and there’s really nothing you can do to make me like them
as a result I know like nothing about literature or history

[tawny slate]

i agree with the general gist of what you're saying, but that's a separate point. my point is that, for instance, math should still be taught to "non-math people"

how to teach it well is another topic entirely

there are more reasons to teach complex numbers beyond "it's used in engineering"

[twin shell]

I think they are inseparable
if you can’t teach a student to love a subject then I’m not sure if you should teach that subject at all, because they’re not going to learn that way

having said that my ideals will never be realized because of systematic limitations

[tawny slate]

In the ideal case, yes, but I think this lacks a ton of nuance

There are a ton of cases where a student doesn't need any appreciation of the subject whatsoever but they still gain something important through that learning

For example, music education should start as early as possible

Once you reach your teenage years, your brain is not plastic enough to learn many aural processing modes

By the time you reach high school it's too late

And it's not just about understanding music for music sake, your entire aural processing system is permanently shaped by your early years

Of course, this is one of the most explicit examples, but it's not directly related to math

[twin shell]

wrt music I can't really say I agree there, I hated music as a whole until I started band in HS and I can hear intonation and rhythm pretty well now

[tawny slate]

It's not everything, there are many things you can still learn throughout teenage and early adult years, for sure

That's why we still learn music beyond elementary school

But let me be even more specific here

Perfect pitch is an example of a skill that can be learned early on but not later

[twin shell]

that is true but in practice it really doesn't matter whether you have it or not

my view is that I'm not against teaching liberal arts per se--I'm against teaching it badly, which is like universally the case in my experience
of course there's only so much the teacher can do, because the situation is heavily influenced by curricular requirements, testing, students' personal situations, and other bad things
but mandatory education typically makes people hate subjects, which in my opinion vastly outweighs the little benefit you gain from forcing them on students--an unhappy learner is not a learner at all, and the students who come out this way usually forget everything they learned and have to relearn it anyway come the time they need it

by the time you get to college (assuming you don't hate every subject; otherwise, you really would be better served by just going to trade school), you usually have an idea of what subjects you enjoy
in my opinion we should lean on that, because that is what produces effective and happy learners

re:chinese
if someone likes chinese then let them learn it
if they don't then don't make them, especially if it's your family language--don't make them hate your cultural lineage
learning a language is a massive commitment if you actually want to get good--we're talking several hours of reading and listening a day

if they decide they need it then they will

if you really want to teach your kid something, what you should do is be encouraging and compassionate the entire way

I'm pretty sure the purpose of mandatory learning in the first place was to make sure kids go to school rather than the workforce

which they would have done if their family's financial situation demanded it

actually that might have been the no-underage-working law

they're connected I think, whatever

either way they create just as many problems as they solve

[tawny slate]

Then I don't completely agree

I do think there is a time and place for mandatory education

You can't make everyone appreciate every subject they need to learn

Also what I appreciate and what I like are independent of each other

Which is also independent of what is useful to me

[pastel horizon]

I think this is an idealistic point of view

Sometimes you are just teaching them because they need a qualification and you have to be ok with it.

That said I'll always try to teach it conceptually so they can appreciate why it works

If it's an undergraduate course in pure maths, sure you've probably done a bad job if you turned them off the subject

But you can't win every battle either

[tardy ember]

...well it's possible to teach so badly that some of the people being "taught" actually learn nothing, or even end up knowing less than they did before
(source: this has happened to me)
and at that point i'm not sure there's a point in it (why have people sit in a class where they're not learning anything?)

i think expecting people to deduce general lessons about life from "meaningless symbol manipulation" is ridiculously optimistic
if they can just write out the teacher's password from memory - the string of random symbols and words that gets them marks - i don't see why they'd be motivated to go beyond that unless they care about the subject itself and not just grades or whatever

[vagrant meadow]

i think there's just a deeper systemic issue in higher education right now (at least in the US) where, to many students, college is not a place to learn. college is a place where you play the game of working just hard enough to get the grades you need such that you get your magic piece of paper which is the key to getting a "worthwhile" job. and what else would one expect from making it standard and expected to get a higher education? if you say "i don't plan to go to college" there are people who look down on you for that. so no wonder students don't care.

maybe i'm just incredibly cynical, but i don't think it's possible to "make" students care or motivate them if they aren't taking your class for the right reasons. you can lead a horse to water, but you can't make him drink.

[rugged hemlock]

The issue is that US college tuition can leave someone broke for life

which is why actually being able to pay your loans back is #1 priority for a lot of people

[lethal sapphire]

If you were to teach someone algebra (high school level), what are the most fundamental concepts they need to know? I don't mean like factoring, quadratics, etc, I mean more broad conceptual/thinking skills. For me I found algebra 2 incredibly easy because it felt like I learned everything in algebra 1 already and now it was just extending those ideas a bit, but for others they didn't have that same grasp on it. So that makes me curious, are there certain fundamental concepts that the rest of algebra and (furthermore) calculus are built upon? How do you teach someone that intuition?

[winged urchin]

The most important thing for a student to learn is very difficult to teach in my opinion.

A lot of the times the student might say they remember the exponent laws or remember FOIL or whatever and while those are your tools you need to also be able to tell which tool to use. And also remember there isnt just one way to solve something

They need to be able to look at an expression or an equation and be able to manipulate it, see it in different ways, until something clicks and they see a path towards the answer. That eureka moment is the hardest thing to really teach because it's more about perservance, open-mindedness, and flexibility from the student

I had a question with a student the other day that was essentially this

"Cindy and her mom were both born on January. On March 15th, 2015 Cindy adds up her birthyear, her age, her mom's birthyear, and her mom's age. What is the result of this calculation?"

Granted this was a contest math level problem so the level of 'jump' in this problem is probably a bit higher than typically expected of students but how do you 'teach' them to see the solution?

I don't think you can honestly lead them to the solution without giving something away. They need to know how to study a problem. How to break down a problem into simpler parts to understand the full picture

[lethal sapphire]

It's more about the process, I guess? Noticing patterns starting from concrete examples

[winged urchin]

So maybe just.. general problem solving techniques are the most important thing to teach

And the ability to test their hypotheses!

I actually was elated today when a student told me they checked a step they were unsure about

I told them they could try different values of x if they weren't sure. And if they got one value where their initial expression and the final expression are different then their step was wrong

And they did that! They were able to test their own idea

It really is a combination though. If they don't know the exponent laws, the elementary derivatives, how to evaluate a limit, whatever... then they won't know how to proceed if they can even write down a statement from the problem

[rugged hemlock]

I mean despite the fire that contest math seems to come under it's really not that bad if you wanna teach problem solving

[winged urchin]

But if they don't have that problem solving basis they wont know how to navigate a problem to begin with

The rules are the toolbox and the uhh... problem solving... critical thinking part is the 'wisdom' to use that toolbox I suppose

[winged urchin]

Yeah for sure! The nice thing is usually contest problems don't require you to write out equations at least at elementary level

[rugged hemlock]

I mean you kinda need to solve problems to gain a problem solving basis

[lethal sapphire]

what do you think about purposefully trying to put them in situations where they can't know the answer? Like giving them the tools but not the knowledge of how to put it together, maybe when introducing a topic?

[winged urchin]

They're usually designed in such a way that if you just see the eureka then you can solve it without much trouble

[winged urchin]

Yeah you're right aha

[winged urchin]

I think it's good for students to feel out of their depth but it also can lead to disillusionment

[lethal sapphire]

haha honestly been thinking about trying it myself, ive never really liked it much

[rugged hemlock]

I mean the nature of contests can be questionable but I think the problems themselves are gold

[lethal sapphire]

ive always felt like they're too random, or dont really have a purpose. liked learning new concepts more than finding small tricks to solve simple ones

but perhaps you need a bit of one to do the other

[lethal sapphire]

why do you say that?

[twin shell]

there's a thing called guided discovery learning
pure discovery learning is bad according to modern research iirc but guided discovery is very good

[winged urchin]

Because the students who go into academia or really competitive markets need to not just give up when they fail once or twice or thrice

I think at least, they need to learn that some problems will take perservance

[rugged hemlock]

this is true

[winged urchin]

I remember one of my friends in undergrad complained that she spent all day working on an assignment problem and didn't get anywhere

And thought it was a waste

[twin shell]

i think my math really started to improve once i started spending several days on the same problem

[lethal sapphire]

what's the difference?

[rugged hemlock]

most of the people around me were very used to failure even before college

[winged urchin]

But it wasn't really, they spent that time checking possibilities, ruling things out, and learning all along

[rugged hemlock]

Although there's the other argument of "suffering builds character or not"

[twin shell]

just nudge them toward the solution with hints every so often

probably

[winged urchin]

Yeah exactly that

I do that a lot, actually

[rugged hemlock]

Isn't math kinda just finding large tricks to solving large problems

[lethal sapphire]

yeah thats what I generally try to do as well

[winged urchin]

I'll be sitting with a student and watch them write and if they get stuck or seem to be wandering

I'll ask something about what they're doing or try to lead them a bit

It's very socratic tbh in my application

"You think 9/4 becomes 3/2 huh? Why is that?"

As a simple example aha

[lethal sapphire]

pretty much haha. But I found competition math to be a tad boring. Some of it's cool but it's a lot of banging your head against a wall for the result to be just a bit of manipulation. Struggling with a proof for a few days and finally figuring that out is a lot more satisfying to me

[winged urchin]

Yeah sometimes it can feel like the answer comes out of nowhere. Like if a question requires a student to see a difference of cubes type formula

My students rarely ever even remember that that is a thing

So when it does ever show up it's like... when did I learn that?

[lethal sapphire]

yeah, I feel like it's just you have to memorize a bunch of little tricks or formulas that can be cool but its a lot of work for something that doesnt really produce much of value

but maybe im just looking at the wrong problems

[winged urchin]

Although proofs I think can be more intimidating for students. Sure algebra might require some mystery manipulation or formula you forgot or maybe never saw. But with proofs it's very very difficult to actually feel 100% with your answer

[lethal sapphire]

my only exposure to it is through AMC

[winged urchin]

As a young student at least without much experience under your belt

[lethal sapphire]

oh absolutely

[rugged hemlock]

I mean when that happened to me my attitude was like "oh that was very cool why didn't I think of that?"

Although I probably shouldn't talk because I only made it to AIME

[winged urchin]

The worst imo is like upper year undergrad problems where they're the solution is just like...

"Notice that blah*blah^blah + blah = the thing in your problem plus some other stuff"

[lethal sapphire]

oh totally, I still get that aha, but I just can't imagine commiting hours and hours to learning that

[lethal sapphire]

yeah ive seen a few of those, definitely feels manufactured

anyways back to that discovery learning thing

what's a good way to actively use that as a teaching tool?

that's mostly how I learned math tbh, I would look online for a problem I didnt know how to solve and research it and explain it to someone else

[twin shell]

ye this is what learning math should feel like imo

[lethal sapphire]

like if you're trying to teach something, should you first give them the problem and have them try to work through it, and then when they get stuck just have them ask questions and I'll answer/explain what they need to know and work their way up to solving it

I like that idea but im not sure how well itd hold up

[winged urchin]

I think that's good 'if' you have a 'good' student

[rugged hemlock]

this is fine, at the beginning I would maybe just do one contest math problem per day

[twin shell]

my physics teacher did something like that

[rugged hemlock]

like a "challenge problem of the day" or something

[lethal sapphire]

well im saying you could do that with the material youre trying to learn anyways

if you're trying to introduce logs, start with the definition and jump into a problem and try to work through how to solve it

try to get them to come up with a change of base formula on their own from experimenting + a bit of guidance type thing

or the formula for the reimann sum based on the idea of it

[rugged hemlock]

I had a bit of a random memory but while I was helping students review for calculus tests I would give them problems way harder than the exam material and give them small resin ducks after they solved the review problems

[lethal sapphire]

haha

[winged urchin]

ducks?! ahah

how many ducks did you have?

[lethal sapphire]

I had a TA do that for a class I took this year and it was amazing

[rugged hemlock]

https://www.amazon.com/Landscape-Aquarium-Dollhouse-Ornament-Decorations/dp/B0B8Z2DJG1/ref=asc_df_B0B8Z2DJG1/?tag=hyprod-20&linkCode=df0&hvadid=632148078901&hvpos=&hvnetw=g&hvrand=16549251151965266329&hvpone=&hvptwo=&hvqmt=&hvdev=c&hvdvcmdl=&hvlocint=&hvlocphy=9031352&hvtargid=pla-1900672695591&psc=1

ducks like these

[lethal sapphire]

exams felt so easy but the class was the hardest ive taken

[rugged hemlock]

I think once I made it into a contest and the grand prize was a giant stuffed duck plush with a few baby duck plushies stuffed inside

[lethal sapphire]

lmaoo

thats great

[winged urchin]

the potential worry here is maybe covering material that the student won't need for their final

Especially if they're the type of student who just wants to survive first year calculus and then they're done with math

[rugged hemlock]

I mean it wasn't any new content, it was just problems of harder nature

it's not like I penalized them for not participating, some just tried one or two and I explained everything in the end

[rugged hemlock]

This was an AP calc class in high school

so probably a different demographic of students

[lethal sapphire]

actually that brings up another question, how do you make hard problems?

or make problems at all for that matter

[winged urchin]

Honestly it kinda just comes to me at this point

I can usually make up problems and also change them on the fly if I notice it'll highlight something important

[lethal sapphire]

would a 'way harder' calc problem just be something like iterated integration by parts?

[rugged hemlock]

physics competitions were my friend

plus there's plenty of hard problems I pirated from the textbook

and changed it a bit so they couldn't tell

[lethal sapphire]

i guess im just thinking about it and like i cant imagine what a hard problem is

like there are problems that take more time/steps, but how do you make the same concept harder

[rugged hemlock]

For one problem I gave them the "pythagorean identity" for inverse hyperbolic functions and had them find the derivative of one

[twin shell]

word problems

make them figure out what the problem is

[lethal sapphire]

thats actually a good one lol might steal that

[rugged hemlock]

some of the hard problems were troll problems

like these

shoot

this was to make sure they knew conceptually what a limit was

[twin shell]

one issue that a lot of math books have (by design) is that they don’t interleave section material
so if you know you’re on the u-sub section, you know you’re looking for a u-sub somewhere
and the problem with this is it doesn’t really teach you when to look for a u-sub

[rugged hemlock]

I know another problem where I had them solve the F=-kx equation with seperation of variables which took more time, it was mostly a lesson on why you don't forget the +C when doing a diffeq

[lethal sapphire]

yeah that helps a lot actually

[rugged hemlock]

Also physics-y problems like finding the moment of inertia

[lethal sapphire]

its more trying to find something they may not understand and creating a problem around that?

[rugged hemlock]

Yeah

I think what I cared more was making sure they understood the derivation of everything

my hyperbolic function problem was there so they knew where the derivatives of sin^-1(x) came from

for the rotations unit the problem I gave was a thing rotated along a diagonal

which in retrospect turned out to be a bit too brutal

[lethal sapphire]

oh man that brings me back

when I learned about rotations I spent like a full week writing out a model for rotating functions about one another

[lethal sapphire]

yea that makes sense

i think questions designed to make you derive something can be super helpful as well

my alg1 teacher basically had us all derive the quadratic formula without even knowing what it was lol

[rugged hemlock]

but most of my review problems were either 1) filler ap calc frq problems, 2) problems with more steps, 3) troll problems, 4) derivation problems with twists

[rugged hemlock]

Ah you had a good teacher

[lethal sapphire]

i did

he was prob the best teacher ive had tbh

[rugged hemlock]

the filler ap calc frq problems I only put there for my own sanity

[lethal sapphire]

haha

alright ive gotta sleep but thanks for all the help

if anyone thinks of anything revolutionary ping me lol

[toxic talon]

i like this problem a lot

and I would absolutely have gotten it wrong in my calc 1 class hehe

[shy grove]

class participation is good in math teaching

[pallid night]

This is evil lmao
Based exam question

[winged urchin]

I like those kind of questions tbh.

They don't require much calculation. Just understand what limits mean and use the graphs

Graph problems actually usually are pretty dope imo

[rugged hemlock]

Oh I have more evil questions

[pallid night]

I tried to write a 1st year (UK) undergrad Real Analysis exam

I think some of it was fairly tricky

But I thought it was fair

[rugged hemlock]

Oh these questions were just from a AP calc ab course

[pallid night]

No

[rugged hemlock]

Most people in the us don’t do real anal 1st year

[pallid night]

None of them were

[rugged hemlock]

I meant the ones I have

[pallid night]

Let me send the syllabus I based it on

[rugged hemlock]

Most of my troll problems were written for AP calc ab

[pallid night]

This is the syllabus of my uni in y1 analysis

There's some overlap

[rugged hemlock]

Dang that’s like calc bc + proofs

The us isn’t that rigorous when it comes to hs math

[pallid night]

It should be harder cos it's 1st year Maths undergrad

[rugged hemlock]

I mean there’s no proofs in bc

Calc bc is just calc I + ii

[pallid night]

Some of the proofs in this Analysis course would literally destroy someone at that level.

Proofs of this kind require a sort of ingenuity that most late school age students haven't yet developed

[rugged hemlock]

I still have yet to take a real anal class

[rugged hemlock]

I didn’t teach any AP calc kids proofs but I taught induction to some middle schoolers

I have the weirdest priorities when teaching

[wispy slate]

ANAL CLASS?

[vagrant meadow]

analysis

[wispy slate]

ohye

[tawny slate]

please share

i want to compile these

that last one was great

[rugged hemlock]

I don’t have access to my computer now but I’ll fish them out when I do

^^I posted that other problem in another discord server once so I’ll share that for now

[vagrant meadow]

i really like Hilbert's hotel and related infinity shenanigans related to cardinality as a mini intro to more advanced math and mathematical thinking. but i sometimes get pushback about not just if the results have actual applications, but if there is any application to even considering the idea of infinity at all (this was a particularly smart neuroscience major who studies philosophy).
does anyone know of applications? or what do you say to someone (or a student) like that (especially if the applications are flimsy or nonexistent)?

[twin shell]

infinity is in the end an idealization, and math is about ideals
you typically can't attain an ideal but it gives you a goal to work toward

see approximations

with that said, maybe this neurophiloscience major would enjoy https://www.youtube.com/@njwildberger

[vagrant meadow]

right. one argument i bring up is that the world is technically discrete. so in populations, for example, it would seem like you would have to use difference equations to tell you anything worthwhile. why would a differential equation, which is for continuous functions, tell you anything useful about a population? but it turns out that differential equations are much easier to work with than difference equations, and are basically just as useful for modeling. supposing that the world is continuous, makes things easier without greatly sacrificing utility.

[twin shell]

perfect

[echo lodge]

The idea that the cardinality of the reals is greater than that of the rationals has important implications to compatability theory, specifically that almost no real numbers are computable (or even describable in any first-order language)

Caveat: any first-order language with a countable set of symbols

[twin shell]

griffiths says something similar

Water, too, “really” consists of discrete lumps (molecules); yet, if we are dealing with reasonably large quantities of it we can treat it as a continuous fluid. This is in fact much closer to Maxwell’s own view; he knew nothing of electrons and protons—he must have pictured charge as a kind of “jelly” that could be divided up into portions of any size and smeared out at will.

[echo lodge]

Even in (most models of) QM, space and time are continuous, as are the values taken on by wave functions

Whether space and time are ontologically continuous is not yet a settled question

[pallid night]

This reminds me of how much easier I found it to work with integrals than summations

And thus how much easier I found working with CRV's than DRV's

In my first year when I was learning proper Probability Theory for the first time

And you've made a good argument there as is

[vagrant meadow]

right, this is another perfect example. imagine if integration was outlawed because it used the idea of infinity. you just had to do Riemann sums with the specific delta x value mandated by law. 😆

[pallid night]

That sounds utterly rancid

[tardy ember]

as a completely different perspective: some mathematical objects just are infinite if we want them to behave nicely

take the natural numbers for instance
there are natural numbers so large that they cannot be counting a number of objects within the universe, because the universe just doesn't have that many objects in it

so in one sense these huge numbers are completely useless for modelling the real world

but if you try to say that they don't exist, that means there must be some point where the natural numbers just end, some number that you can't add 1 to

and aside from that also not matching our experience of the world... it's just annoying. it would be this extra detail you constantly have to keep track of... and for what purpose? what would be the point of doing that?

a set of natural numbers that goes on forever is simpler than a set of natural numbers that ends somewhere

and since they both give the exact same correct answers to problems about the kind of numbers that exist in reality, why not use the one that's simpler?

[twin shell]

well there's the programmer's perspective that the largest number is 2^some power - 1 and it wraps over after that

[devout fable]

Not really?

you can represent large numbers on computers

[twin shell]

i just recalled this example from tao analysis

yeah i was being a bit facetious there

[devout fable]

hate to be that guy but this is true and not true

i dont know how its actually done but you can always do arbitrary precision arithmetic with, say linked lists

i think big int should be default on python whereas not on say java

[tardy ember]

i think you'd probably use a normal list instead of a linked list but yeah arbitrary-precision natural number arithmetic is a thing

[devout fable]

what is a normal list

[tardy ember]

but the "normal", and faster, way to deal with integers in a computer, is that they're some fixed power of 2 bits long (8, 16, 32, 64, occasionally 128), and arithmetic wraps around

[tardy ember]

generally it's implemented as a contiguous region of memory, where the address of the nth object is the address of the start of the list, plus n multiplied by the size (including padding) of each element

[devout fable]

okay so basically array list in java

[tardy ember]

most programming languages provide this as a language primitive or as part of the standard library

[devout fable]

and the implementation of this still makes it faster than linked list

as long as you resize the memory allocation smartly like say double each time or something

[tardy ember]

yeah linked lists are kind of slow

[devout fable]

haha okay vaguely remember this from freshman year comp sci

[tawny slate]

one of the definitions of something being infinite is that you can find a function onto itself that is injective but not necessarily surjective

for instance, the map from x -> x+1 on the positive integers shows that the set of positive integers is infinite

here's another useful example I like to use relating to computer science: you are designing a programming language where one of the primitives that you have are strings. this is obviously a common thing, but suppose you want your string literals to include special symbols like line breaks/carriage returns. you might be familiar with the idea that the classic way to do this is to type "\n" to obtain the line break character. however, this means that if you want to literally type the characters "\n", you are now unable to. you might also understand that in general, special characters will start with a backslash, such as \r, \t, or \s. one one hand, we have sacrificed the backslash character, but we have now expanded our character list significantly. if we continue this process indefinitely, we can add as many characters to our list as we like, but we need to sacrifice a character in the process. if we wanted to type "a\b\c\d\e", this is now not possible. is there a way to achieve this without any sacrifice of possible strings we want to write?

the solution is to say that the literal backslash character can be written by writing "\". this way, if we want the string "a\b\c\d\e" we just type "a\b\c\d\e". okay, but what if we want to literally write "\" then? simple, type "\\" by escaping each backslash. in this way, we have fixed the issue of character sacrifice. every possible string you could have written before can now be written, plus more using our newly added characters. this is similar to the x -> x+1 map. we proved that our strings can be injected onto a set of strings that is slightly more complicated due to the backslash definitions. without the concept of infinity, we wouldn't be able to do this

i do like the computable/real number example, but i think this one is a little bit more directly relatable

someone could make the argument that the distinction between computable numbers and real numbers isn't important because all of the numbers we care to ever use are computable, and if we ever needed a real number, we could always just approximate it with a computable number

not only do i disagree with the first statement, the second might be wrong in some cases, but i can totally understand why many students would think that way

[tawny slate]

if i recall correctly, this guy, while obviously well-qualified on a multitude of mathematical fields, has a lot of weird math takes (that aren't necessarily wrong)

i think he said that the real numbers don't actually exist and i THINK he's an ultrafinitist

[twin shell]

ye

it's funny

[tawny slate]

yeah that part makes me feel super weird

because on one hand he isn't wrong, math is rooted in our axioms

he can just choose to reject the axiom of infinity

and that actually makes it more likely that his axiomatic system is consistent

but from a pedagogical perspective, idk if this is a healthy thing to teach without context

i remember one teenager i was talking to who almost follows this guy's teaching religiously

and flat out rejected anything and everything that had any reference to anything infinite

without being critical

[fickle current]

I think it depends on how it impacts the way you do math. You can think that there actually aren't any infinite sets, but study them as constructs that are useful or just plainly interesting. If we got rid of infinite sets, we would get rid of calculus and that would have impeded great advances in science and engineering. So I think you could think about them as nonexistent but at the same time recognize merit to the idea, which is a perfectly sound and pedagogically useful, in my opinion.

Ofc you could also just disregard all infinite math, but even if infinite sets don't ontologically exist it's just false that there's no reason to study them

[vagrant meadow]

yeah my neuroscience friend says the same thing. since the universe is discrete, the universe works in rational numbers up to some degree of accuracy. that's his belief anyway.

[hollow kernel]

Beutelspacher had a nice line on this somewhere, don't know if I can piece it together

Something similar to "The engineer believes their equations to approximate the universe. The physicist believes the universe to approximate their equations. The mathematician does not bother with this distinction in the first place."

[wide ice]

kids are too young for that

we math students are mature enough not to favour one axiom system over another. One should be aware and keep the possibilities in mind.

I think it's dangerous to teach kid these things without a proper approach. Belief on the superiority of an axiom system can lead to belief that someone knows the absolute truth and thus may refuse to accept otherwise.

[twin shell]

what do you think grade school is for

[wide ice]

Most math teachers I know are not that careful about precision or rigour. Unconsciously or not, they can inject their belief

One of many reasons why I favour extra rigour. You can never be too cautious.

[twin shell]

my AP calc teacher always wrote = for calculations that should be approximate at best and I've never really formed a solid opinion about it

[wide ice]

My 8th grade math teacher made a dubious proof in a geometry problem, and I argued for 2 hours straight.

Really pulled a Sheldon moment right there. But these kinds of things taught me to have faith in my own math skills.

It's unlikely that these things make a difference. But you never know. And if teachers aren't careful about it, who will?

[wide ice]

One guy actually wrote a book to criticise Newton and his "dubious" fluxions. You just reminded me of him 😄

[wide ice]

I remind you of Lorenz, he said about chaos theory:

Chaos: When the present determines the future, but the approximate present does not approximately determine the future.

There are things that are continuous, things that are discrete, and things that we just simply don't know and have to try both ways. But it's wrong to say the world is technically discrete. DEs sometime are not handy tools, but they are simply the way things are.

Or in sum, continuity is sometimes not only convenient, but essential.

[vagrant meadow]

sure, i agree with that. if the world is not "technically discrete", then what is it? is the world actually continuous, or is it a mix?

[wide ice]

well, what do you mean by "the world"?

Someone studying Navier-Stokes says air is simply a continous fluid, a flow. Zoom closer, and you say "oh, but it's a bunch of molecules". Zoom even closer, and now quantum stuff starts to get in the way.

Hmm... this is a bad example. A better one should be wave-particle duality. It states something will behave the way you want it to be. An electron can be a wave, or a particle. It can be discrete, or continuous.

Measure it, and sure enough it's particle. Turn away, and it becomes something else, not measurable anymore.

[vagrant meadow]

let's say reality as we can perceive it.

[vagrant meadow]

yeah it's interesting that we keep finding that things aren't as clear-cut as they once seemed. quantum mechanics seem to turn a lot of what we know on its head. things are waves and particles at the same time. you cannot know both position and velocity of a particle. there seems to be an inherent uncertainty in the nature of reality itself. but i don't know much physics, so maybe i will say differently when i finally go back and take more physics classes. then i won't be talking out my ass so much.

[wide ice]

there seems to be an inherent uncertainty in the nature of reality itself
Yes, Heisenberg's uncertainty principle

I think I can refer to Bell's theorem too, but i don't recall it refers to uncertainty explicitly, only that the uncertainty is inherent and cannot be explained by hidden variables

Anyhow, I'm not Physics student 😄 gotta leave it to someone else

[vagrant meadow]

oh shit i've never heard of that. looked into it, and that's awesome. i remember watching this video a few months ago, i don't know if it's related.
https://youtu.be/lu4mH3Hmw2o

[wide ice]

But this is beyond pedagogy at this point. We should move it to somewhere else

[vagrant meadow]

lol was just gonna say

"i hope this still counts as #math-pedagogy"

[tawny slate]

we teach limits and calculus in school, which assumes the axiom of infinity

I think we do have an implicit bias of an axiomatic system in our education, because we establish an accepted convention for our axiomatic basis, and we also don't use each axiom equally

we don't employ axiom of choice often, but that's taught to implicitly be true as well

we also almost never teach large cardinal axioms in school, because they are both not necessary for almost all applications, and we don't even know if they are even consistent, but that doesn't mean we teach this in a non-biased way

i think understanding that the selection of math axioms can be arbitrary is a good thing, but we can't just pretend that we don't have an actual bias when we teach math

[wide ice]

and this shouldn't be the case. While it's too much for students to understand the proof that there is no non-measurable set assuming AC is false, or to find where do we need AC explicitly in Hahn-Banach and how it affects anything, I still think it's profs' job to point it out.

We can have a bias when we do math, and we often do: we pick the one that's most convenient for us. But at the same time we should be aware of the choices we make. It's important for students to understand this

[tawny slate]

right, but let's not pretend that we don't teach with a bias

we bias towards ZFC in most cases imo

we certainly bias towards accepting axiom of infinity

[wide ice]

I don't say we don't teach with bias, nor do I complain about studying with bias. But it's another thing to teach it as "the truth" and not mentioning that it's but our choices, and there are other possibilites

Many teacher entirely omit this part

[tawny slate]

so this is where the pedagogy part comes in

imo i think many times it's okay for the teacher to omit things here and there

perhaps the teacher at some point mentions that axiom of infinity is arbitrary selected to be true or false before getting into an entire unit on limits

but this may instill some doubt in the students of "okay, well if this could be false or can't be shown to be true, then why are we learning it and why is this still useful?"

[wide ice]

I think if you reach a point where you can mention an axiom, you should also be able to warn that it is still an axiom

[tawny slate]

this could trigger a meta-discussion about math philosophy

and truth

which could implicate ideas far deeper than the students are ready for

[wide ice]

I don't think so. If students are ready to hear about an axiom, they should also be able to, with much guidance, think about it deeply. Sure, not to the depth of a mathematician, but to the point being conscious about it.

A good example of this is Euclid's postulate and the relevance of non-Euclidean geometry. It's simple enough to visulise and demonstrate what can go differently if you don't assume an axiom

[tawny slate]

Maybe in the ideal case sure

[wide ice]

For undergrad, proving the existence of non-measurable set using Zorn's lemma is also a good homework (and I had this one actually)

[tawny slate]

Obv I'm in favor of the idea

I'm simply wondering in large classes if teachers have the resources to handle this kind of thing

[wide ice]

And i'm also wondering if teachers have the capacity at all. The ways they ignore rigour have made me furious

[vagrant meadow]

i think even just mentioning that it is an axiom is helpful. it leads the student to think about why it is not provable from the other axioms, and to speculate on what undesirable things might happen if we choose to omit it. then the interested students can potentially ask about those things if they want to (maybe after class) which wouldn't take time away from necessary material.

[wide ice]

We are still in perfect world here. In reality... imagine a large class with students mostly uninterested, or even having beef or smth, and the workload as a teacher, thinking about these kinds of things would probably the last thing on my mind

[vagrant meadow]

I think that's increasingly actually the case, at least in the US. but I still believe one should teach under the assumption that at least one student cares about being well educated in the subject

[tawny slate]

i want to be clear that i think rigor is important for obvious reasons, but i think this kind of statement is a bit dangerous. as a teacher, sometimes i do ignore rigor, but not because i don't think it's important, but because it's too much of a detour from the topic of discussion and/or it's not pedagogically optimal to bring it up currently

we begin by teaching that real numbers just exist de facto. we don't send kids through an entire real analysis course before doing any kind of arithmetic with them. sometimes it's better to first teach the conception of the ideas, and then refine them later on. i think pedagogically, this is sometimes more efficient/optimal

another example: take for instance the fact that

if x > y > 0, a > 0, then x^a > y^a for all real numbers x, y, a.

this is actually pretty difficult to prove from first principles, but it's intuitive enough that we just throw it at students. we treat it functionally as if it were an axiom, but we gloss over the details that show it's provable

another example is the fundamental theorem of algebra. pretty difficult to prove from first principles, but it is so powerful if students just accept it first and use it to guide their intuitions it helps them immensely

[shadow flower]

i’ve heard that no proof of that exists using only algebra

[tawny slate]

the entire point of a pedagogy channel and pedagogy discussion i think is to discern the difference between cases where rigor is important vs rigor can be pushed to the side to an extent for the sake of pedagogy

it's not that we are saying that rigor isn't important, it's that in terms of the process of learning, it can be argued that helping students understand WHAT rigor is, how to refine intuition, how to check for rigor, these are more important than the rigor of the subject itself

[tawny slate]

ive never looked into this in depth before, but now im pretty fascinated and i really want to find the most primitive proof

[shadow flower]

I feel like I have read at some point that there exists over 100 different proofs, none of which only use algebra. I’m not sure how based this claim is but it’s worth looking into I suppose

[vagrant meadow]

oftentimes, rigor can actually get in the way of learning. but sometimes that can go too far, and things get too loose.

[devout fable]

Most don't care, neither do I

[devout fable]

Most people don't care about this either. If you want it to be discussed or taught, then there would need to be compelling arguments as to why it matters. Fortunately or unfortunately, techincal things don't receive much attention

[devout fable]

Im not saying you are wrong for caring, and admittedly, i don't know much about the issue. But it really does feel as if this side is a different field than math, whether or not thats correct. To give an extreme example, it would be like saying that people in math to care about economics

[wide ice]

I think the problem is how you define exponentiation for real a. The classical approache in Analysis I is to define x^a as e^(x*ln(a)). I think I saw somewhere ppl defining x^a as limit of x^b where b tends to a over rationals, but I'm not sure how well-justified this way can be.

Either way, a primitive proof of this will be challenging because we don't use a definition of exponentiation with only algebra.

[maiden relic]

Check P12 of this sheet for the limit approach

Its pretty well justified and approaches it from first principles. Its a problem in Rudin iirc

[wide ice]

Of course we're walking on fine lines here, and I certainly don't like anyone walking through the whole theory of logic to construct natural numbers. But introducing Peano's axioms shouldn't be too big of a problem. Sure, one can drop the discussion about why we need each of the axioms.

[maiden relic]

The proof is too convoluted for high schoolers though

[wide ice]

No one likes technical things, but a better approach can be not to ignore the technicalities entirely, but to make them less technical.

[wide ice]

It'll be stupid to present the proof of each and every statement we introduce, lol. But it's another thing to give an outline and to point out where exactly the students are not ready to understand and must admit for the time being.

That's also a better attitude to rigour. Not to take the pain entirely, but to be mindful that it's there.

I agree however that it's not beneficial to put that much of emphasis on rigour when the ultimate goal for high school students is... to pass MCQ exams.

Or problems of particular types. I didn't care about definitions when I was in high school, only methods to solve particular problems. I doubt if things have changed since.

I feel like rigour should be introduced earlier than freshman undergrad, but somehow high school doesn't seem like a right place.

Rigour is way to think, not only just math, it should be beneficial for (almost) everyone to be criticial and rigourous, not only for math students

[twin shell]

here’s the problem
before you introduce rigor you need to get students to start thinking for themselves, else they’ll just memorize the procedure and forget it like they’ve always done
I suspect most American college students don’t achieve this even after graduating

[wide ice]

What a failure. What's the point of studying if you can't think for yourself?

[hollow kernel]

Could you elaborate on your last statement please? In what sense do you suspect that (American) college students didn't learn to think for themselves?

[pastel horizon]

Also sometimes being precise with vocabulary can help more with making people understand than simply "dumbing things down".

There's definitely good ways to refine intuition besides rigour I agree

[twin shell]

you can read https://press.uchicago.edu/ucp/books/book/chicago/A/bo10327226.html if you want
basically, your average college student here (grossly underprepared if not ruined by public education, and more concerned about making money than learning) ends up cheesing college by picking out easy classes and graduating without remembering a thing
(this is what I do with my gen eds tbh)

now let me talk about my own experience
I know I liked math in high school but my studying went as far as "I read this on wikipedia" and "I watched a bunch of 3blue1brown and blackpenredpen videos"; none of this requires particularly deep thinking, just a lot of exposure
in my algebra and calculus classes, I'm 99% sure everyone just watched videos, copied down notes, and emulated the teacher's solutions, because that's literally what the teacher asked us to do

probably the most rigorous class I had was my geometry class and even that was a failure
you know those point-and-click puzzle games where you're looking for a specific object?
just click everywhere on the screen and you'll get it eventually
that's what I did with my geometry classes, just try everything
this might sound good but the problem with this is "just try everything" doesn't really help you discriminate which strategies are useful for this problem and which are not
so in the end you still haven't learned anything, you're just memorizing solutions and passing tests

(also two-column proofs are shit)

/end{rant}

ok I lied, “just try everything” is only part of the strat

the real strat is rereading your notes—not to understand, just to check the hypotheses and conclusions

so people do this in elementary, middle, high school

guess what? they still do it in college

[devout fable]

Which is OK

[twin shell]

I’m afraid I don’t follow
my core philosophy is if you’re not thinking, you’re not learning
and to me I don’t see thinking in that

[devout fable]

You don't need to know why everything is true, and learning things for the first time is hard compounded with the pressure of due dates and grades

[twin shell]

yes, that is true

[devout fable]

besides, you'll eventually see it again , and if you don't, then who cares if you understand it?

[vagrant meadow]

they will, and we can't stop them, nor should we try to imo. but I also think we shouldn't tailor our courses to these specific students. you can lead a horse to water.
so why not teach as if every student is expecting to have to use the knowledge from the course in their career? instead of teaching as if every single student just wants to pass, get their degree, and never use anything from the course.

[devout fable]

Unless there is some kind of underlying reason the result is true that is a huge underlying theme in the field, who really cares why the result is true

I dont think I've ever read the proof of a techical result lol

[devout fable]

Not to mention that the only people who care about this math stuff probably looking at grad school, and that's a very insular group. Most people are just going to get a job and never see math again

If anything, it's more a debate on why on earth you need a college degree for so many jobs rather than a commentary on state of math pedagogy

[shadow flower]

that was the mentality the math education majors at my school had

“why do i care about this if i’m not going to teach it”

“the only reason you care about this is because you want to teach it”

[devout fable]

Pragmatically speaking, most people are going to never teach anything they learn in college

Whether or not they know the stuff they learn in college is not going to help them with the only metric that matters, getting kids to do well on some state standardized test

[twin shell]

most people are really better served by going to trade school IMO

unfortunately that’s really looked down upon in a lot of circles I feel

[devout fable]

Maybe but trade school doesnt get you a desk jon

job

those require bs/ba

[wide ice]

even if you don't become a teacher, thinking rigourously is still a good habit to have

Every year, a bunch of students BS on their application files about critical thinking, and yet they fail to think critically about simplest objects

[devout fable]

Math does not quite help you with thinking rigorously

In math, things are black and white

Nothing which is black and white in the real world would necessitate critical thinking

[wide ice]

almost nothing is black and white, but math trains you how to think about a complex issue

dissecting facts and connecting pieces if you're a lawyer, and that can be useful

Math is the simplest training you can have on thinking cleanly and clearly, because things are black and white. If you can't handle it, I doubt you can handle anything else that cleanly

[visual trout]

a question on the pedagogy of modular arithmetic, every time I've been taught it we begin with the definition that a is congruent to b modulo m iff m | (a-b). From what I've read this notation invented by Gauss was used to in practice to denote if two integers had the same remainder or not, which is really the core of modular arithmetic, remainders, however this fact is proved only later as Theorem. The typical definition is very confusing at first and I see no reason why we cannot instead use the much more insighful definition that a is congruent to b modulo m if they share the same remainders and then prove the fact that a is congruent to b modulo m then a-b is a multiple of m, which is a useful property in proving facts about modulo m, why isn't this the case?

[devout fable]

Not convinced by this

the essay i had to write in my high school english class is probably way more useful than the three years so far of my bachelors

[wide ice]

then that says more about your bachelor than math itself

[twin shell]

personally I’ve always understood it as “a == b mod m if a = b + mk for some integer k,” but this is not how it’s usually taught

[devout fable]

I don't see how these two approaches are fundamentally different

[rugged hemlock]

Isn’t math literally rigorous logic?

[twin shell]

have you ever had a student who asked, “why do we multiply instead of adding here?” and wonder how anyone could possibly confuse these two operations?
because I have; do you want this kind of person in charge of your finances?

it matters to properly understand why a lot of stuff is true, because if you only learn how to calculate things or apply theorems, then your understanding will fail to transfer to other areas
it doesn’t have to be a super deep, rigorous understanding, but you should be able to see where a statement comes from and get an idea of why it’s true

this is true beyond just math

[devout fable]

Read next sentence

[inland ferry]

One can hope…

[rugged hemlock]

I don’t get how objectivity means less rigor

[vagrant meadow]

I don't see how objectivity means less critical thinking

[empty gull]

I think this is kind of trivializing it a bit

[rugged hemlock]

I think Murray Gell Mann accidentally reinvented a portion of Lie algebra while studying particles because he wasn’t familiar with that field of pure math

Just because a math idea has no practicality today doesn’t mean that it still won’t tomorrow

[vagrant meadow]

not always. continuum hypothesis for example. whether or not something is provable is also not always known.

this simplistic view of math is an outdated perspective. we know now that we can't take certain things for granted. things that many people would like to be true, specifically (completeness etc.)

[devout fable]

dont you think this is, a bit contrived?

who cares about the continuum hypothesis? how many people actually do work relative to this

if your point is that there could be certain things in math which is not necessarily one way or another, then that is clearly outside the context of what I said

no one doing undergraduate math will ever encounter such a thing

[tardy ember]

regardless of "black and white"ness, a lot of the stuff that happens in maths is just a reflection of more general strategies and truths about how to think correctly

[rugged hemlock]

again, just because something seems useless today doesn't mean it'll stay that way

cough number theory

If math doesn't grow along with engineering and science that makes innovations in all fields limited

If you want to go down the practicality argument

[rugged hemlock]

Also I'm pretty sure a lot of undergraduates learn the continuum hypothesis (even if they don't necessarily learn the proof?)

that's what I learned my first math class in college

[tardy ember]

at the very least, logical arguments are something that happen a lot, and those are very similar to mathematical proofs

a lot of the ways an argument can be invalid are also ways a proof can be invalid - claiming a statement is definitely true just because it seems obvious, or because an authority said it's true; defining the same word two different ways without proving the equivalence of the definitions; claiming to have proven a universal statement without actually considering every possible object; using a word that hasn't been defined at all and acting like it means something; proving a statement weaker than what you originally claimed; getting morality involved when talking about a factual statement (admittedly i haven't seen this one in a claimed mathematical proof before); etc.

[devout fable]

this is not what I am referring to

at all

[rugged hemlock]

So what were you referring to?

are you saying undergrads won't encounter it after they graduate?

[vagrant meadow]

why not? as someone who wants to go into pure maths research, i want to learn about these things, and there's nothing wrong with a professor mentioning it because it's cool.
i still don't really understand what your position is. or how you can say that math doesn't require critical thinking

[vagrant meadow]

this just seems... lazy. like why are you even getting a degree in math? ...assuming you are. and, if not, that raises several questions. but, either way, i just seriously don't understand this aversion to knowledge, or how it could possibly benefit the pedagogy of math. what, are math teachers just supposed to make things easy because either the student doesn't actually need it (they just need the requirement checked for the degree), or if they do need it, they can just google it later, because math doesn't require critical thinking?

i really can't imagine that's what you're arguing

[vagrant meadow]

i maintain that the existence of students who don't (and won't ever) give a shit should not impact the teaching style of a lecturer. students who are struggling but trying, yes, absolutely.
despite the fact that many treat college as a game with the goal of getting a degree and a job, where they just need enough points to clear each level, i still believe gaining knowledge is the primary purpose of education. and, as shocking as it may sound, some people are actually in college for that reason.

[pure coral]

you do have a point here

real world reasoning tends to be pretty fast and loose and most importantly, thinking heruistically works well enough for most things

the only time where mathematical rigor is needed is generally, well, in math lol

also we make most of our decisions with our emotions anyways, would rather spend time managing those better

[hollow kernel]

Strongly disagree, claiming mathematical rigor does not have a crucial spot in everyday life is blatantly inaccurate, I believe.

[empty gull]

Tbh id say math helped with thinking more critically
Specifically because in math the idea is to question the rigor of everything
So that kind of mindset sorta gets translated to other aspects of life

[hollow kernel]

Obviously heuristics and emotions are crucial as well, none of those ways of thinking can be ignored.

[empty gull]

Like when I read a math proof I don't take it as fact
You question it and think through why everything written gives you what you want

[twin shell]

imo rigor is a means, not an end

we write rigorous proofs to guide our understanding

I think tao wrote something about this once

[tardy ember]

heuristics are useful if they're good heuristics, and you know that they're heuristics

heuristics are an approximation of rigor - faster to compute with, without sacrificing too much correctness

if you don't know that, because you don't know what properly careful reasoning looks like, then your intuitions seem like just the way the world is
meaning you will be overly confident in them, and also it's not obvious how (if it would happen at all) you would fix incorrect intuitions

[normal aurora]

https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/

[pure coral]

absolutely, and it reflects in a lot of modern math research as well

it’s a lot more common to separate the theory from the model nowadays as always thinking about the model just makes reasoning harder, and to put thought into representing things in a way so that the proofs are easy

[pastel horizon]

What if you teach rigour in the form of "spot the mistake" exercises?

[pallid night]

Yeah the number of times i've been able to call out logical fallacies in people's arguments cos of my awareness of logic from doing a Maths degree is pretty outrageous.

In particular I often see people get directions of implications mixed up during arguments and debates

They often confuse what direction of implication a necessary condition requires

They do all of this subconsciously, because they don't know this kind of logic anyway

[vagrant meadow]

i like those. there's always the possibility that one of the students will think "oh, boy, i would have totally done this without a second thought", and then they have to look closely and figure out what they would have missed, and where their conceptual deficits lie.

plus, if the mistake is subtle, it can still give the students an image of "this is what a proof should look like", even if there is a flaw in the reasoning. this is a minor point, but often a lot of students new to proofs have absolutely no idea what they should look like, how much justification to give, etc. seeing an example, albeit a flawed on, can still be helpful.

[pastel horizon]

You'll be surprised how many times people answer with "They are wrong because the actual answer is: ..."

They miss the point of the exercise initially but then you can train them to think critically and it will improve their rigour

Or maybe phrase it as "find 2/3 mistakes" so even if they say that answer they still have to find 1/2 more mistakes

[vagrant meadow]

well... that just sounds like the instructions are unclear (the teacher accidently omitting what they really expect). should probably put in the directions to point at the exact line with the error, and what the correct statement should be, etc. stuff like that

[pastel horizon]

You will usually see those types of responses on exams where it's phrased as "explain why they are wrong"

[vagrant meadow]

this can be good. but if there are three mistakes then it sounds like the proof is way, way wrong. i was imagining looking at something more subtle, but both can be good, i imagine

[pastel horizon]

So saying the alternative correct answer isn't necessarily answering the original question

[pastel horizon]

Ah I meant more for some basic problem solving type questions rather than formal proofs

But that's where you need to start I think because the pre university education system focuses on that style of delivery

Here are the examples now try the problems independently

[vagrant meadow]

now that i look, there was nothing you originally said to suggest proofs. idk why i assumed that lol 😆

[pastel horizon]

But it can be a good self differentiating task because some mistakes might be basic arithmetic error, basic algebra errors or more advanced errors

[vagrant meadow]

yeah. maybe try to get algebra students to explain where the 1=2 proof goes wrong. stuff like that

[pastel horizon]

Naturally students will look for errors based on their own perceived skill

[pastel horizon]

This can be interesting too once they get the idea of the drill

I just think it's something you have to train first like with any other mathematical concept

[vagrant meadow]

totally

in a spot the mistake exercise, could it be more or less helpful to have the final answer actually be correct? then they can't just say "the correct answer should be this".

[pastel horizon]

Oh you should also be running provey questions concurrently too

Even if it's something simple like

Given 2×240=480

Show that 480/240=2

To start with

[pastel horizon]

You can possibly rephrase it to say "Even though this is the correct answer, what mistake did they make in their working"

You probably don't want them to get confused and think the final answer is wrong also

[vagrant meadow]

good point

[pastel horizon]

Yeah especially if they try the correct method see the same result and get discouraged

[pallid night]

This reminds me of this exam report

This is horrifying

[pallid night]

16 year olds took this exam

[scarlet pond]

Can someone explain to me what is so hard about substituting into functions? I do not understand how students cannot see that substituting 2x into f(x)=x^2 should give f(2x)=(2x)^2, and not 2x^2. Are the methods used to teach functions to algebra students that bad?

[twin shell]

the rule is, everywhere you see an x, you replace it with 2x, right?

so if I see x^2 I should replace x with 2x to get 2x^2

that’s one possible misconception, at least

a good way to try and address diagnose this is using numbers
if f(x) = x^2, then what is f(1 + 1)?
if you simplify first you get f(1 + 1) = f(2) = 2^2 = 4
if you substitute first you get f(1 + 1) = 1 + 1^2 = 1 + 1 = 2
oops, what happened?

[vagrant meadow]

yeah when i tutor students i make sure to use a lot of parentheses, because then things like this happen less frequently
so writing (x)^2 originally, and then the student will (usually) put the 2x into the () and get (2x)^2

i think a lot of the confusion/algebra mistakes with squares are just due to the lack of parentheses. even just writing things like sin(x) can prevent mistakes like that.

[twin shell]

that’s a a good part of it yeah

the other part is treating sin like multiplication
every function is linear, it’s a theorem of math education

[scarlet pond]

This is why I do not tutor anymore. I was one stupid algebra mistake away from banging my head against the wall so hard I concuss myself. I simply do not understand how teachers of algebra or arithmetic have the patience to stop themselves from telling students how mind-bendingly stupid their mistakes are. How in the world do you all do it.

[twin shell]

it’s a different kind of patience to what math students need
instead of interacting with the problem you have to understand and learn how to interact with shudder people

[rugged hemlock]

I'm not exactly a teacher of arithmetic and algebra (although I have done it a few times)

How I do it

I remind myself that not long ago, that was me

making the same mistakes

[tardy ember]

i think the issue is not understanding that lists of symbols aren't fundamentally what expressions are

like what expressions actually are is they recursively contain other expressions, so you end up with something that looks like a tree (this diagram represents "1 + 2 * 3")

and from this perspective, if you take x^2 and replace x with 2x there's only one thing that could reasonably mean (i'm using ↑ for exponentiation here)

and from this perspective brackets are just notation, they're not a built-in component of the universe

but if you're thinking of an expression as just a list of symbols with no fundamental structure beyond that

then the situations where brackets suddenly appear are a real part of the universe and a thing you have to "learn"

and if you don't happen to know that this is a situation where brackets might appear out of nowhere, then the most obvious thing to do is to just "replace x with 2x" as a purely text-based operation, which results in "2x^2"

[vagrant meadow]

that is a really interesting point. i never thought about how if you were literally doing this as a "find and replace" operation, then that is exactly what you would get: 2x^2. as mathematicians, it's like we automatically think of x^2 abstractly from the perspective that the brackets are there but invisible. and from that perspective just more obvious that we would interpret it as (2x)^2.

i suppose that is a part of teaching algebra. to promote that kind of abstract thinking. if one is simply told "replace x with 2x in x^2", then it's really not unreasonable to assume some people would do that literally.

especially if there's a disconnect between the notation and its language. like when i hear a calc student say "d y over d times x", that sets off alarm bells. same with x^2 as just "x two", to an extent.

i heard one calc student say that today actually 😆

[twin shell]

I remember I attended a regional HS math competition and one of the guys at my table thought AB was a product when A and B were points in a triangle

string operations are confusing

[vagrant meadow]

and while it's fun to laugh and say, as people with (or getting) math degrees, "wow i can't believe someone could possibly think that", it really isn't that wild. we denote products x times y as xy, so AB looks like the product of A and B (and that is literally what it is in linear algebra, if they are matrices, for example).
i don't really know what one can do when a student lacks such a foundational connection between math symbols and meanings besides just teach it to them. what else can you do? you can't just say "AB is not a product. guess what it is" because they aren't going to guess that it means a side of the triangle. how do you lead someone to "discover" that x^2 is notation for x*x?

[twin shell]

ye
unfortunately it can be contextual so you have to decide if “car” is one object or c * a * r

and like we train people to think of it the second way

in the beginning, when it’s necessary to employ concrete examples, we might say “apples + 2 * apples = 12”
here “apples” is a single string representing how many apples I have, and we think of it as one object
and that’s almost at the level of a word problem
then we tell people “but we usually write it as x + 2x = 12 because it’s more abstract”
and then people start forgetting what x means
then we introduce multiplication of variables
and soon we forget the original motivation behind using variable names, which is to make our work more convenient by removing irrelevant details
so now when we write a string someone may interpret it as multiplication because that’s what you’re taught to do with concatenated variables, and we’ve wasted the value of the notation

that was not a cohesive message but it’s 01:30 here

[zenith slate]

To teach successfully you need to avoid a mindset wherein you label mistakes that seem elementary to you as "stupid"

This does neither you nor your student any good

[scarlet pond]

I do not know how to do that when students come in and do not know how many months are in a year.

That student was an adult in college.

[vagrant meadow]

once I was tutoring someone in logic. so I used the example of mammals and dogs (ex. if it's a dog, then it's a mammal), and they didn't know what a mammal was.

[twin shell]

at my school we get people from really fucked up backgrounds

so our remedial classes go as far down as arithmetic

[halcyon glade]

I agree with rat here.

Labeling your students as stupid is grossly unhelpful, regardless of what basic knowledge you think they lack.

[rugged hemlock]

not everyone is fortunate to have a good childhood education

even you at one point didn't know how many months were in a year, you might've climbed out of that really quickly and much earlier but you still had someone to teach you

[scarlet pond]

That is why I do not tutor anymore. I refuse to live in a fantasy world where some people are not smarter than others. Not everyone has the same potential, we are not all created equal. There are people out there genuinely incapable of the most menial work. If one has gone through over eighteen of the twelve month cycles and not picked up how many months comprise it, then I do not want to have to be the one who has to explain such basic facts. If I have to teach mathematics, I want to teach mathematics, without having to act as a remedial primary school teacher for adults.

Is that unhelpful? Absolutely. I would be a horrid teacher. My choice to stop tutoring was also made to protect any poor souls from my extreme lack of patience.

[vagrant meadow]

honestly, that's kinda necessary. more and more colleges (even community colleges!) are cutting these remedial classes and I'm really worried what the consequences of that will be. because, yeah, some people really didn't get a good early education, and they need to start from the beginning. forcing them into algebra is often disastrous.

[halcyon glade]

If you don't have any interest in teaching, I'm not sure why you're talking in this channel

Also yeah I think having classes that can serve everyone is great

[scarlet pond]

Teaching is a necessary part of the Ph.D. program I am starting soon, so I have no choice. I need to have my mindset challenged.

[halcyon glade]

Well it is one of the core values of many teachers that anyone can learn if they put their minds to it

You're also bound to enjoy the experience more the more you empathize with your students

[fallow siren]

ESL, maybe? (hopefully?)

[vagrant meadow]

no 😦

i basically just said "uh... animals like dogs and cats and monkeys" or something. in retrospect, i'm not sure if i should have said "animals with nipples" 😆

[fallow siren]

I'd be worred the next question would be "what are nipples?", and now you're not in a good place

I just had a really weird experience in the help forums, and I want to discuss it a little (or maybe just get another set of eyes or maybe just vent). I hope this is an okay place. It does raise an interesting pedagogical discussions, I think.

Also, if someone came to the same conclusion that I did (and still hold), then I'm curious what exactly tipped them off.

[vagrant meadow]

so i'm a bit more gullible/easy to fool than most, especially by AI. usually i don't notice it until someone points it out, and then it's more obvious. but one thing i've heard people say is that putting extra space between lines of text is a sign of chatGPT use.
especially if there is confidence, but the answers are sketchy.
the "I apologize for that. You're right, [what you said rephrased]" is absolutely chatGPT

i'm 99% sure they were using chatGPT

[fallow siren]

The apology is what tipped me off. That explained (a) and (e) having different answers so confidently and without comment. And lastly, agreeing that S(P)>1, but making no attempt to figure out what it actually is.

I have a friend who puts lines between his paragraphs. It's a bit annoying because it makes it harder to reply in a targeted way.

[tawny slate]

You never know what you may or may not encounter until you actually encounter it, thus is the perpetual catch-22 of maths

Abstractly, the idea that some things are foundationally unknowable is an important philosophical idea that can be made rigorous using math

This is important to everyone because for any logical argument, we have premises (axioms) and we have logical syllogisms. Just because an argument is consistent/valid doesn't necessarily mean it's sound, and so understanding that this applies to math as well is an important reason why this distinction is ever the more important

Rigor is also critical because while math for the sake of math is whatever, often times we apply these mathematical concepts in applications such as engineering. When the question of whether or not a submarine can handle underwater pressure is a matter of life and death, suddenly the importance of rigor becomes all too real

[pallid night]

I type like this.

Cos I find it clearer to follow

[vagrant meadow]

same. also since i prefer to do one long message than sending multiple, personally. i'm an overthinker

[tribal tiger]

I'm on mobile so it's a bit hard to read but the first answer seems a bit strange like there's a lot of extra information I don't know people who type or answer questions like that. Then in the second message the recap is a bit strange. The biggest offender is the apology. Then to top it off when they do actually type it's nothing like their answers were

[tawny slate]

Ive overcome this attitude by simply being humble and realizing that even the most basic of questions are not fully understood by most people, and that in order to fully understand these questions, we need to try and really understand the people asking them. In addition, this is also a good challenge for yourself: how clearly can you explain a concept you take for granted? What is in fact the simplest (or one of the simplest) ways to describe or demonstrate something? How can we shape intuition in a place you haven't explored thoroughly enough?

If you still think this is not something worth your time or that you are skeptical about how much depth you can actually pull from the ideas you take for granted, here are some examples I have compiled from my experiences teaching. See if you can explain these answers in a way that is satisfying to students:

• Why is the order of operations the way it is?
• Why do we have names for both commutative and associative properties if every operation we learn is either both or neither?
• Why can't the square root of 4 be -2? ("Because the square root is always positive.") But (-2)^2 = 4, so why don't we define it that way?
• We can reciprocal both sides of an equation, why can't we reciprocal both sides of an inequality and flip the sign?
• Why is any number raised to the 0 power 1 and not 0?
• Why are 0/0 and 0^0 undefined? Why is infinity*0 not 0?
• Why do we care about learning point-slope form?
• How do we know that all equations of the form y=ax+b are linear?
• I know I can check my answer to an algebra problem by plugging it into the equation, but how do I check my answer to a geometry or counting problem?
• Why do the angles of a triangle add up to 180 degrees?
• What's the point of learning radians?

I have more but you get the idea. Some of these are obviously easier to answer than others, but if you simulate a curious person on the receiving end, these are now (at least in my opinion) very exciting and interesting questions, and suddenly teaching or tutoring becomes more exciting rather than frustrating. Take any and all misunderstandings or lack of understanding and frame it in this mindset

You'd be surprised how much you can learn through tutoring someone

[twin shell]

it’s not a good idea to think of learning as something that only some people can do

it’s more like most

because most of us are basically the same, we have a brain

some people are hopeless but this is usually not due to some kind of inherent mental incapacity

it’s usually a mental barrier that gets in the way

because when the entire world has been telling you for eighteen years that you’re a failure who will never be able to learn or accomplish anything, what are you supposed to do?

the reason most people don’t get good is because people are generally shit at raising children

one consequence is that many people end up taking on a fixed mindset, the idea that your skills are basically decided from birth

which is extremely unhealthy and inaccurate compared to the growth mindset that successful learners have

nurture vs nature, some shit like that

feel free to hold your own views on the matter, but in the end it’s best for your students if you encourage a growth mindset

somehow I’ve managed to leave this out of all of my messages, but the core point I want to communicate is that humans are predominantly shaped by their environments
most of us are similar in terms of intellectual capacity
the reason some people succeed and not others (apart from socioeconomic differences) is because the people around them are different and end up spreading different morals

and as a result people take different lessons from what they experience

[rugged hemlock]

I personally think that a grade-school teacher should care about their students more than the mathematics

It might be a bit different for professors since teaching isn't their main job

but any sort of teaching, you should do it wholeheartedly

[fallow siren]

I'm quite gullible too. Looking back, I just noticed another very obvious use of ai that I originally missed.

[lethal leaf]

it's exactly shit like that

What I will say you can't teach to some level is personal preference to some degree

no matter how much enthusiasm I show

or how well I think I teach something

or how much effort I put in

some students of mine are just gonna prefer other subjects than math

which is fine

and obviously to some level people put more effort into the things they prefer, we are only human

but modulo that

there is a big nurture aspect

but alot of it comes down to some students just not going to good schools growing up