math-pedagogy

[tight star]

For that, I think I’d say there’s a tradeoff between definitions which are easy to check, and those that are easy to use

You can add more conditions into your definitions so that they’re harder to satisfy, but then their use might be clearer

Or you can add fewer conditions into your definitions so that they’re easier to satisfy, but then it’s harder to see why those definitions would be useful

In general mathematicians seem to prefer the latter

But this isn’t necessarily made explicitly clear to students

Even if two things are logically equivalent, it doesn’t mean they’re literally equal in all aspects - one set of conditions might be genuinely easier to check/prove

Which I guess comes back to a broader point about equality in math

Often it’s not just the case that two things are or are not equal

Instead there’s a sense in which they’re the same, and a sense in which they’re different

And it can be useful to draw attention to both of these senses

[lyric lotus]

Fahrenheit > Celsius

I like the other metric units though

[midnight scarab]

But Fahrenheits are not graduated like Kelvins

[tepid smelt]

There is so much. I don't know how many here have actual experience in public schools but it's a shit show to say the least and only has been getting worse in my near decade teaching.

Some huge safety issues are the lack of adults on campus and huge classes(talking about 40 kids in a room and I have seen more then that). Almost every school has vacancies so you get subs for a year where kids learn nothing..(not uncommon in hard to fill positions like math/science)

The issue is also you get these near 40 kids in a class and on average some are years behind and some that might be a few years ahead so you're trying to teach kids who might be 10 years apart in the same room..

Having more qualifications is not going to fix these issues. In fact the more qualified teachers often have the worst classroom management which is 90% of the job.

I don't know exactly what it's going to take but it will require a lot of money especially at the earlier levels to ensure kids are staying at grade level. This seems to be the opposite of what politicians want. They are totally content with public schools collapsing because many don't want upward mobility for all. They are good with the public schools being trash as long as the private schools their kids go to remain strong.

It's really quite sad how bad public education has become. Then we are shocked somehow with rising homelessness or crime when all you have to do is look at the schools to see why it's happening.

[wary python]

question, i want to learn about chaos theory to apply it to physics, any advice on where to start or available resources like videos or books

[pure light]

this channel is for teaching math, not learning. you may be better off in #math-discussion or #book-recommendations

[tepid smelt]

That's what Finland did but I would argue that everything else they did during their education reform had a larger impact. Like all the extensive support systems in play from counseling to free meals and healthcare to small class sizes etc.

Singapore made some big reforms also and I believe they do ability grouping for students. I mean that is very common for say music or sports it's strange to me we don't do it for math.

America does have some fantastic private schools. I wish all kids could receive similar support. It only be a net positive also in that your supporting everyone to contribute to society. Yet again I do feel public education is slowly being chipped away at as teachers struggle to keep it alive while barley being able to support themselves with continued decrease in wages with rising costs.

[tawny slate]

curious why you think so

[pure light]

the answer for either preference is pretty much always "because i'm used to it"

[vocal phoenix]

Indeed, the best temperature scale is the one I was raised using.

(same for metric vs imperial measurements)

[midnight scarab]

Metric is more systematic than imperial tho. Like the conversion from m to cm is simpler than from yards to feet

Plus compatibility with our base 10 number system

[tawny slate]

I would say unless there is an explicitly advantageous reason to choose one over the other, C > F simply because it is the standard the rest of the world uses

less conversion is less prone to error, like the mars climate orbiter failure

[lyric lotus]

Celsius is just as arbitrary and Fahrenheit is more fine tuned control

Celsius is temperature for water Fahrenheit is temperature for humans

The difference between 30 and 40 Celsius is massive

[tawdry venture]

also stop calling it military time tbh

[daring gulch]

I suppose the biggest problem with primary math, from my observations in these discussions, is that it is simply not individualized enough. It seems as if even interventions for the worst performing students can only do so much for primary math classes as a whole. Perhaps if there were a program to provide individual 'tutoring' for all students, as well as a systemized way to correctly place students in the class with the optimal difficulty, many of the problems in primary math would be alleviated.

[pale basin]

As you say, "everything else they did during their education reform had a larger impact." Another one of the things Finland did was ban tuition for schooling. Then they gave public schools appropriate funding. Private schools could no longer compete. And that was a very good thing.

[pale basin]

Is it an individualization issue, or a pushing kids through when they aren't ready issue?

[daring gulch]

Well, at some point you have to have a minimum standard of things. Actually, the minimum standards currently in primary and secondary school are already pretty ass low, so 'pushing kids when they aren't ready' is not really an issue at this level. Maybe in college it is. The amount of effort you put into something basically determines how good you are at it. Discrepancies will inevitably arise between students such that some put less effort in than others, whether that may be because of time constraints, study methods, interest, etc etc etc. Generally, the harder a task is the less one will want to do it, making poorly performing students fall behind even more. [Insert elaboration on individualized learning]

[pale basin]

Totally agree. My thinking was along the lines of lack of enforcement of those minimum standards and pushing kids through anyway. Teaches them they don't actually have to learn things. Could hold them back instead.

[long pelican]

Just a thought, eliminating private schools gives a way to explain why performance in public schools is higher, without actually having any concrete improvement in their public school education. Because the high SES private school type kids (who would get higher grades) are now included in the public school stats, whereas they would be excluded previously

[pale basin]

The bump is too big for it to be that alone tho. They also unified funding instead of the property tax funding BS we have in the US. This makes it so parents can't have "pay to win" for their child's education, and instead all the schools have to be good enough for the rich kids. This makes education better for everyone.

[long pelican]

I can't help but think that "that alone" is a huge bump

But that addendum is nice

I like it

[daring gulch]

Holding back kids is a bit of an issue and schools nowadays like to avoid that as much as possible. In Florida, for example, they keep lowering and lowering the standards required for a diploma so kids don't graduate with a GED and have to go back for adult education. The worst performing students (assuming they are capable enough) won't care about being held back as putting in little effort is what got them to that point in the first place. Investment into better teachers, curriculum, and the like would be a better long-term solution, but of course that's easier said than done.

[pale basin]

Why do they like to avoid it? Summer school or repeat year if you aren't ready. That used to be the norm.

[tepid smelt]

One big issue with summer school is nobody wants to teach it. So districts adopt these online programs that anyone can pass.

Year round school is something that should be the norm also. Low income students fall further behind with each summer break.

I never understood how it is ok for kids to be home not getting all the services schools provide for months every year. College should be year round also. You could get people done quicker also.

[pale basin]

Based on US History, one might think the system was intentionally designed to fail kids with a lower SES... >.>

[bronze stirrup]

becoming a tutor

how do i not suck at it

[tawny slate]

uhh can you be more specific? what exactly are you struggling with? or did you just want general tips? what motivates the question?

[bronze stirrup]

general tips

[vagrant meadow]

practice

i remember the first day i did a calculus review session, i was devastated. i thought i had completely failed. it just takes practice and confidence.

i would recommend shadowing someone more experienced. i got my start working at my CC's tutoring center, and that was really good practice to have done before i started private tutoring

to develop that confidence, i recommend tutoring subjects you are genuinely confident in.

[daring gulch]

Like OldandSlow said, summer school can be a manpower issue, though it depends on the school. Some teachers, like my sister, like to teach summer school since it gives them money outside of the regular school year. Interventions with students and parents also cost resources which can be a headache at times for teachers and administration, though in an ideal world it shouldn't be. Conversely, it's much easier to let a very poorly performing student simply pass than hold them back. In Florida, a D grade is passing, and it is only until you reach an F grade that you fail. There are some pros and cons for this.
I suppose 'letting poorly performing students pass' is essentially 'pushing kids through when they aren't ready,' but I would argue holding back students generally causes more problems than it solves. Instead, interventions should be provided as the student progresses to the next class, to ensure they achieve a more acceptable grade and they are able to achieve that same grade on their own from there on out.

[bleak skiff]

giving tutoring another try, last time i tried was a year ago and went not so well, ehehe, but i think i've better mindset this time

[bleak skiff]

[cloud zealot]

in early proof-based courses, how do you teach students to be confident their solution is correct

[tawny slate]

(really funny because i teach younger students who have the opposite problem)

[vagrant meadow]

yeah that is kind of funny. most students get super humbled by the time they get to proof based courses

[vagrant meadow]

this is really hard imo, and one of the reasons proof based courses are so difficult. it's so easy for us to say "well, if their proof is correct, then they should be confident "
this is where the value of a teacher really comes in. trying to self study and only being able to consult Google (or god forbid, ChatGPT) to see if their understanding/proof is correct is kind of awful and potentially dangerous. asking on MSE or discord is better, but imo doesn't compare to a teacher's email/office hours (where the interaction is more personal and potentially less public).

what I often do is ask questions about their proof. like ask them to repeat definitions, question certain aspects of leaps of logic to make them more confident that leap is correct, or give a scenario which would nullify the proof and ask them to explain why it can't happen.

otherwise, I don't really have a fantastic answer to this, and I'm curious what other people will say. I feel like it's possible to come up with a convincing proof that is subtly and nuance-ly flawed at pretty much any level of math. and by nature of learning the material for the first time, it's kind of impossible to not have some doubt. even the one line proofs (or maybe especially the line line proofs) can make a student second guess.

[tawny slate]

i cant speak for the college level courses

for my students who are usually around middle school level, and thus they are getting their first taste of writing proofs in geometry, i try to clearly define as much as possible: the axioms they are allowed to use, the logical reasoning, and once they justify and prove a theorem, the theorems as well, and how to stitch them together

i go over the common errors like (p->q) =/= (q->p)

so if there is something wrong with their proof, generally speaking they can go validate it themselves for the most part

basically i have them treat it as a construction problem where the basic building blocks are given

not sure how useful that is as a response though

[vagrant meadow]

yeah logical rules are super important. that stuff is definitely not obvious to every student

[quasi musk]

I tell them to write out the definitions of what they're working with and to write down the goal. Then I ask if there's any theorems that can help you get to that goal

I also tell them that their proofs should look similar to mine, or what the professor/book does

I haven't had the chance to TA at my current place something proof based

[vagrant meadow]

yeah I always mention that it's one line, and usually say "don't overthink it". the thing I really want to avoid most, on this server especially, is to ask a leading question (which should have a one sentence answer), only for the student to spend minutes trying to compute something pointlessly

[torn valve]

[bleak skiff]

i just went on and on in a tutoring session about like rise over run and how to find the function for a line between two given points in the plane for like the better part of an hour, only to hear at the end that the student didn’t really understand what a function or its expression is 😭 now i feel really stupid for not checking if she were comfortable with that before talking about further things

[tight star]

Did you manage to explain to them what a function/expression is?

[bleak skiff]

this was like just as the session ended so i tried but i think she had completely given up by then

[tight star]

i see…

yeah it’d be helpful to know that confusion earlier

though metacognition is a skill and not just automatic

[bleak skiff]

mhm… i’ve a lot to figure out of like basic pedagogy, i’ve no idea what i’m doing and just try to explain the math in as simple terms as i can

[tawny slate]

yeah thats a mood

it happens

the more practice you get, the better youll be

youll know what to check for

youll remember to check at all

youll give the student heads up to speak up when they are confused by anything, knowing they probably still wont (because of the metacognition comment)

youll pay more attention to the student and check understanding more frequently

i have certainly been there before

[bleak skiff]

i sure hope so, thank you

[vagrant meadow]

I once spent all my time before the quiz in a discussion session explaining how to interpret the roots of the characteristic polynomial of a 2nd order linear homogeneous constant coefficient ode, only to find out that a substantial portion of the students didn't know how to use the quadratic formula when I went to grade the quizzes.

[mint lark]

This is literally what nightmares look like

[wind ginkgo]

In my experience as a student things only become obvious after a certain level of maturity in the subject is reached. If a student struggles with an obvious proof either they haven't internalized the definitions of objects/maths they are dealing with or lack fundamental insights.

At least that was the case for me

Every time I missed an obvious proof

Sometimes it happens that I have an unintended and much more complicated solution which works but still shows I lack the necessary knowledge.

Sometimes it takes really long to notice the obvious thing.

All of these are from lack of practice and internalization of certain things me thinks.

[vocal phoenix]

One year in a measure theory exam I set a problem that involved using the DCT to calculate a limit of integrals, and the hardest part turned out to be actually computing the resulting integral (it wasn't anything particularly horrendous, something like x^2exp(-3x) or such).

[tight star]

yeah this kind of stuff always confuses me

i’ve met people who can do super abstract stuff and find calculation the hard part

and it’s difficult for me to understand how exactly that’s possible

[vocal phoenix]

My guess would be simply lack of practice; in particular math curriculum often focuses on proofs at the expense of computational exercises (that's certainly the case at my university).

I think the general idea is that anyone in the position to take a proof-based real analysis course shouldn't have any difficulty calculating a Riemann integral, and that's probably true in principle (i.e. a math student will probably learn to calculate integrals faster than an architecture student), but learning to do these kinds of things quickly/without pausing for consideration, as part of larger problems, is still something that just takes practice.

To complicate matters further, it's practice they'd have to be doing on their own anyway, because there's only so many class hours in a semester, and you tend to discuss the harder (proof-based) problems in those.

And an astonishingly small precentage of students do much practice on their own unless forced to (but forcing them to would mean more homework/tests, which would mean more checking of homework/tests)

And in the case of homework, particularly in this day and age, also more cheating, so a lot of them wouldn't learn much from it anyway.

[bleak skiff]

i don’t think i’ve computed like a single integral over anything more complicated than a polynomial since i took calculus like three years ago so i’m definitely in the lack of practice camp

that said i wouldn’t have any trouble computing the particular example you gave there

[tawny slate]

my dad had stories about teaching phd elec eng students that didnt know how to add fractions

[midnight scarab]

Arnold has this story about a student doing the analysis of the stability of some fixed point (as taught) and concluding that the point is stable iff 4/7 < 1

[noble dirge]

How does anyone think of the mathematics in secondary school ( or before )

I’m from America and I’d like to see how other people think

[wet slate]

there's also a stigma around computations in general, math majors will work their ass off to show something without, god forbid, invoking an ugly numerical bound, or a calulator

thankfully this trait seems to get beaten out of people at around the phd level

when the proofs are no longer quite as nice and easy

of course anybody can appreciate a pretty non-computational proof, but if you need to bound with an integral and it'll take an hour or two to perfectly evaluate it, just use numerics and call it a day

[tight star]

Ah, I’m a physicist so I don’t really have anything against computations

[tawny slate]

i will say that although it seems really silly in retrospect, one of the most important things i learned in undergrad studying elec eng is that sometimes (maybe even most times) exact form answers are not only stupid hard but nearly impossible, and numerics do the job just fine

probably something a lot of people take for granted but it is such a good thing to learn if you dont know/accept it

[tight star]

Oh yeah for sure for sure

Exact form can be nice for symbolic computation

If you have some family of integrals, for example, then it can be useful to see how it depends on a particular parameter

And numerics alone makes that challenging

But if you can extract out the dependence on the parameter and reduce it to computing an actual integral

Then by all means do it numerically

[tepid smelt]

What do you mean exactly? Like the content or just the state of education now? I think it's pretty important everyone knows basic algebra. I don't necessarily see anything wrong with the focus being a track to calculus. I do wish we had more options for math electives like they do for the arts. Many kids could greatly benefit from acceleration. I have an 8th grader in my algebra 1 class this year who is by far the strongest student I have and they could honestly be several more years ahead but are already going as fast as the system allows. This happens every year for me in the public school system. Strong students need to be allowed to go further and weaker students really need embedded support classes.

[wispy slate]

Hello, I am in high school and I hope to become math professor. I am really interested in math (and they tell me I'm pretty good at it 🙄 ) and also teaching math. I am wondering if it would make sense to try tutoring, and how I would do that. I really love explaining and helping others with math, and I need some money for college.

I applied a little while ago to a math tutoring company, where I know one of my friends older brothers used to work but I was turned down.

I was planning on asking my Calculus teacher about this on Monday but I just wanted to hear what you all had to say.

[quasi musk]

Just gotta work on your interview skills. Do you know why you were turned down? Another option is to just advertise your services. If you don't have experience, then you get clients just by advertising for cheap

Word of mouth will spread pretty quickly, but just know that tutoring can be rough going

[wispy slate]

Thank you. I just sent in my resume and cover letter they emailed me what seemed like a pretty standard rejection letter, so I don't really know. I think I will just talk to my teacher and some friends next week and see what happens. Do you have any tips for how to advertise or approach people?

[late vortex]

Im not sure, but maybe because youre still in highschool? You could check their requirements.

Asking your teacher sounds like a good Idea. You could also advertise through friends and family, local bulletin boards and in e.g. facebook groups or craigslist-like websites where people in your area see your job posting.

Im only doing the latter (local craigslist-like website) and it works okay.

[tepid smelt]

Tutoring is great it's like all the fun parts of teaching. At my school we hire students to run the after school program. So asking your teacher can help. I also get emails every year from parents looking for tutors so your teacher might already know someone looking. You could also try nextdoor app to see if anyone in your community needs a tutor or Facebook.

Tutoring can also be great money in the right areas and if you have either good qualifications or experience (like 100+ an hour is common)

[bleak skiff]

me earning 20 an hour

i’ve a freelance contract with a rather shitty place, just to actually find students

[noble dirge]

That is honestly what is happening to me, I'm more advanced than all of the math classes and still I'm stuck in some algebra 2 class

[tepid smelt]

https://blog.evanchen.cc/2016/05/27/fill-in-the-blank/

Your story is quite common.

I encourage my strong students to either do
https://mathdash.com/
Or https://artofproblemsolving.com/alcumus

Some just read quietly. We try to focus so much on inclusion in public schools we end up just doing harm to both lower students and stronger students.

[wet slate]

sorting students by indicators of early ability does a significant amount of harm as well

would i like math if i wasn't put straight on the advanced track in kindergarten? i am not sure

would kids who weren't on the advanced track have liked mathematics if they were? i think some of them would have

i was very bored in several of my classes but those were the classes i could pass without effort, and since my teachers knew that, i was able to do whatever i wanted

which imo was much healthier for me than the alternative of being pushed into successively higher math classes, if not for my education, then for my well-being

[astral zinc]

I will be TA for a Logic Programming course.
I want to spend like 5 to 10 minutes in the first seminar to motivate the discipline.
What would you guys say are the biggest reason people started developing formal logic?
The first thing that comes to mind are the paradoxes that come with using a natural language.

[tepid smelt]

I often find that the motivation to learn something (like why it's important) doesn't connect with the vast majority of students. I often found for myself that my appreciation for a subject often came much later in my own studies. As your learning something for the first time it's just hard to really appreciate what the bigger picture is.

Also what you find important a student might not etc. As a TA are you going to be responsible for HW help or exactly what will you be required to do? I would be mindful of your students time and if your job is to assist with HW to maximize that time.

[tepid smelt]

I understand your perspective, but I believe ability grouping in math, like in sports or music, benefits students by providing the right level of challenge. Placing a student in a class they've already mastered can lead to boredom and disengagement. In my experience, mixed-ability classrooms can hold students back advanced students lose motivation, and struggling students feel overwhelmed. Grouping by ability helps ensure all students are learning at the right pace .

[lucid heath]

Motivation is an interesting subject. I think a lot of educators fall into this trap of thinking that applications and usefulness are the best motivators. But in reality it seems to be a weak motivation.

Actually, I think 3blue1brown did a Ted talk about it.

[vocal gate]

Echoing above sentiment but I think motivation is super complex, and varies across students hugely. Full respect for you for trying to motivate your class but also it's v hard to do in what experience I do have.

[dapper flume]

In my opinion, I think the biggest reason a mixed-ability class holds back advanced students is because the abilities of students are not wielded well. The point of mixed-ability grouping is as much a social-emotional one as it is academic; students who fall behind benefit greatly when they have supportive, highly-abled peers to collaborate with and learn from, while advanced students can be deeply enriched by reinforcing their abilities with peer teaching.

That's not to say I'm 100% against ability grouping, but I think doing it so early sets a precedent for for the student that their teachers have a lifelong expectation for them that may be unreasonable. Side effects of that include students who grow up resigned from math because they've decided they're not a "math person," as well as advanced students who become "know it alls" in their adult life because in school, they've rarely been around people who genuinely struggle with things they take to be easy.

TLDR: lots of things are relevant in the discussion of ability grouping, and I think math ability alone isn't enough to make a sweeping judgement call

[mint lark]

Fostering classroom communities where people are constantly working together and invested in each other’s success is really the key to teaching mixed ability classrooms :)

[tight star]

i got some advice here a while ago about whether to focus more on something being useful, or something being cool

and the answer was actually neither - the priority to focus on was clarity

#math-pedagogy message

making things feel less arbitrary etc

[feral spade]

I would say the decision to promote coolness or usefulness of the subject depends mostly on the audience. If you are teaching a class that is an elective, it might help for you to focus on coolness of the subject more than in a class thats compulsory for the audience.
People choose electives for the interest in the topic and as such are more open to look at the cool stuff as it might inspire them to further their interest.
On the other hand a compulsory subject or a core course might be a chore to some of the students, and in such a case, impressing the usefulness of the topic on students' minds might lead to more active participation.
Clarity remains essential in both as it is the essential part of teaching something. Conveying things with clarity is a necessity for any teacher

[stuck obsidian]

I have started working part-time as a teaching assistant this fall, and I sometimes have a problem with attendance during exercise-sessions. To some extent I can sympathize with this, since I rarely attend exercise-sessions myself.

Do people with experience have some general advice for how to become a better teaching assistant? Things you definitely should not do/definitely should do or just general things to think about?

[grim spindle]

I need outside opinions

do you think this phrasing is passable?

I would write it as "we will prove that [formula] for all n\in N"

This reads to me like we are proving it for that single (arbitrary n), while that's not true, induction proves the statement for all naturals simultaneously (And we have a "let n" in the induction step as well)

[viral pike]

It's not expressing the slicker, stronger statement, but I'd say it's still formally correct in a roundabout way.

"Let n in N. We will prove P(n) using induction:

• insert proof, using induction, deriving
  forall k. P(k) *
  Hence P(n) in particular."

Tho we could debate whether "Foo by induction" would always need to mean that Foo is a forall statement. The "by" then carries a lot of convention and would not e.g. be replaceable by "via".

The sentence should not be left like that either way, given it's off style at best.

[noble dirge]

Honestly, I am quite poor at test taking, and competitive math in general, I don't like being put on the spot with pressure on me, and to pursue competitive mathematics means taking valuable time away from my self-study in my other mathematics, time I won't get back. To be honest, I never liked the idea of competitive maths, it always seemed far astray from the totality of the rest of classical mathematics. Sure, it helps you think in new ways, which can open up new perspectives once arcane, but I find these can be solved by doing mathematics in the " right " way, in the language of understanding.

[daring gulch]

Disagree. There is an optimal class that has enough difficultly to challenge a student while also not dissuading them from doing math. By the latter half of elementary school (3rd-5th grade) it is easy to see which students excel in math (say, consistently scoring 90%+) and those that do not perform as well. If a student that has been placed in an advanced class is not doing well, they can just be pulled out. Done.
Being 'able to do whatever' you wanted in your classes may have been easy on your teachers, but in classes like math where you have to work hard at some point, it hurts you in the long run because it teaches you that you don't have to study.

[tawny slate]

im not going to claim that competitive math is somehow integral to math education, but i think there are a lot of people for whom it helps a lot, like myself. competitive math, i have found, allowed me to more appreciate the human aspect of "design" in math, and given me motivation to learn math. I consider competitive math to have a similar function as recreational math

[midnight scarab]

Yeah, exactly, it's fun

But of course it depends on the person

Some people don't have a lot of affinity with olympiad-style problems but enjoy learning higher math / doing research

[tepid smelt]

I get where you're coming from, and I agree that deep, self-guided study is essential. But I also think that contest problems offer something really valuable. They often teach techniques and approaches that aren't part of the standard curriculum. These problems can challenge students to think outside the box, exposing them to ideas they might not encounter otherwise. That's why I like using resources like Math Dash or Alcumus. They allow students to practice harder problems on specific topics, without me needing to create or grade them manually. Plus, it’s helpful for them to immediately check if they're on the right track. I see working on more challenging problems as a way to build problem-solving skills and reinforce concepts.

[lyric lotus]

speaking as someone who myself never did contest math, I kinda think a lot of people who didn't do contest math are weirdly insecure and defensive about not doing it and act like it has no benefits for becoming better at math when that's just obviously not true

I don't think it's that important and sure you can do fine without being a crazy cracked imo prodigy, but there is an obvious correlation between being good at contest math and being good at math later, especially if you compare to the general population

[hollow musk]

but couldn't it be the case that the ones who are good at math later on were already very into it at a young age, and got pushed into doing contest math due to their interest?

I feel doubtful that the correlation good at contest math -> good at math later is as strong as you seem to say it is

to me, it could easily be the other way around

[lyric lotus]

I never made any comment about how strong the correlation is. I would say it is about medium strength correlation going purely off of vibes.

All I’m saying is that I see so much evidence of many of the best mathematicians being ex imo prodigies and contest math kids that it’s silly to pretend there’s no relationship or that it’s a negative thing.

[midnight scarab]

The main reason imo is that being good at competitive math as a kid is a good indicator of both interest and talent in math

But of course no one claims that olympiad math doesn't help with research math

[torpid dew]

How would you approach tutoring someone who doesn't have a good idea of what functions are in Calculus?

[midnight scarab]

There's even some overlap in content with uni math

[midnight scarab]

I was under the impression that on the contrary people who haven't done contest math, which is generally synonymous with started studying math seriously only later, feel like they've missed out/like they have to catch up.

Well especially as students

[midnight scarab]

First get them more comfortable with functions?

[torpid dew]

Fair enough

[heady yew]

Hello, not to interrupt or anything, but could someone proofread this for me, please? I did proofread it myself, but then again, you never know. And I want to post this on my school's channel to help others, so it has to be accurate
https://docs.google.com/document/d/1ok8dzMk7EZiZaVRB61zGDlxRSDoelX3z6ixaCRlg0yM/edit?usp=sharing

Aside from the statement where it says "you'd need to multiply the matrix by its inverse" (the correct thing to say is "you'd need to multiply the point by the transformation matrix's inverse)

Other than that, please feel free to absolutely destroy those three pages! The more feedback, the better

[noble dirge]

I feel the same, I believe that those who do contest math already like math, and in doing so will be better at math later on because of that

because of how much dedication contest math takes as well, it can be easily transferred to the classical mathematics.

[noble dirge]

While I do believe that contest math does have merit, and does allow people to think more critically, I also think that a lot of it is memorization / pattern recon

[candid oar]

i agreed with this, but only up to middle school in my opinion, many of the high school math contest requires creative thinking and as well as vast knowledge in different fields of mathematics, and of course, including memorization and pattern recognition, and also, appreciation and enjoy the beauty of mathematics

Which will help knowing your math-self better and developed a more mature thinking on which Field you'll be investigating and interested in in their future

[noble dirge]

Yeah, of course. Though from what I've seen, it is a lot of small tricks instead of thinking critically.

[fickle current]

How would you approach teaching ideas of infinity and cardinality to high school students?

[vagrant meadow]

i like how vsauce approaches it in this video.
https://youtu.be/s86-Z-CbaHA?si=QmI_4ej6hvTDbm0J

but i would emphasize bijections and functions. make sure they understand what a bijection is.

there's this one guy in #math-discussion, who kept throwing a tantrum about cardinality, and claimed he could construct a bijection from {1,2} to {1,2,3}. so like... no wonder he didn't understand cardinality.

[vagrant meadow]

emphasize how naturally it makes precise the idea of "how many elements are in a set" via a sense of counting. this function perspective is what generalizes to infinite sets more than our human notions of "size" and "counting". you really want to make sure you don't lose them when you suggest that a set can have "as many" elements as a proper subset.

[tawny slate]

i would personally also try to spend some time on the motivation

like why do we care about infinity? is it even real?

most students think infinity is a number, and that's not really the best way to think about it?

[tight star]

I wonder whether, for subsets of N, it’d be useful to additionally talk about how you can make sense of things like “there are twice as many whole numbers as even numbers”

By talking about things like upper and lower densities

Again it’s like - there’s a sense in which there are the same number of naturals as there are primes, or evens, or squares, etc

But there’s also a sense in which there’s not, and talking about things like asymptotic densities is meaningful

So maybe… trying to point out both of these senses, rather than saying “you have to accept there are as many evens as naturals, there’s no other option” could smooth things over

[tawny slate]

is this just toxic masculinity/mansplaining but in math form lmao

[tawny slate]

one application where both ideas are used here is information theory

so like suppose youre making a programming language

you have literal strings that have quotes to denote their start and end

but what if you wanted a literal quote character in your string? you would need to escape it

\"

but what if that was literally what you wanted to type?

you would need to write \\\"

by escaping the escape character

but this goes on infinitely, how can you prove that by using an escape character, all possible literal strings can be represented somehow?

here, the countable infinite cardinality and forming a bijection here is a good way to explain it

but if you needed to know how efficient an encoding is, the asymptomatic density idea is good

[tight star]

I guess it’s just a more general thing about trying to offer both perspectives and the tradeoffs of each

If you’re just proclaiming one perspective as the “correct” one, especially if it’s unintuitive, it might be harder to reach people

[vagrant meadow]

I think "there are more" is kind of troublesome wording (just in general, not for you specifically). in the sense that one is a proper subset of the other, sure. but, as you obviously know, the cardinalities are the same.

I'm not sure what the solution is. cardinality is kind of a rigorous way to define "size" or say "how many" elements are in a set (that's how it's often taught at least). but that notion is contradicted by our intuition about proper subsets with equal cardinality.
is there a better way to explain cardinality? or should we just accept that our intuition on proper subset cardinality is a noble sacrifice to rigorously define a notion of "how many" elements are in a set?

I imagine if we say a proper subset must have a smaller cardinality we start to run into issues.

[vagrant meadow]

actually im pretty sure you can prove this can't be an axiom as it would violate the definitions of cardinality. you can easily map Z into N injectively, so |Z|<=|N|.
so i guess i would actually emphasize that injective map from A to B implies |A|<=|B|, so that you can show that proper subsets don't follow intuition when it comes to cardinality

[next sinew]

A while back people discussed ChatGPT and I said I used it to aid me with chemistry. Let me give you a small update.

Today I was confused about a table where we had standard entropies in units J/(K*mol). I thought, there's no way, why would entropy increase linearly with both temperature and amount of substance. If entropy is somehow related to the amount of microstates, and the amount of microstates obviously doesn't change linearly with temperature nor amount of substance, then what's going on

Then I asked ChatGPT, and I got a nice explanation saying that with some reasonable simplififying assumptions, the entropy is actually directly proportional to the amount of substance due to taking the natural logarithm in the Boltzmann equation. As for temperature, it turns out that the relationship isn't strictly linear, in fact it's logarithmic, but for small temperature ranges, especially over high temperatures, approximating the logarithm as linear is a fair approximation to make

I don't think this was explained in our textbook because an introductory chemistry textbook likes to brush over statistical mechanics but for me it gave a huge sense of security that otherwise would've been challenging to acquire. (Or maybe it would've been a trivial google search, I don't know)

Anyway, I'm warming up to the idea of ChatGPT sometimes genuinely being able to help students.

[next sinew]

I'm super late but isn't it the norm that "Let x in S." is equivalent to saying that whatever follows holds for all x in S? So therefore your phrasing would be completely equivalent?

I mean not only is that completely standard but I also can't really think of anything else it could mean

[midnight scarab]

Let n in N means that the variable n has already been introduced. Writing
"Let n in N.
We prove that for all n in N, P_n"
is incorrect

It's either "Let n in N. Prove P_n", or just "for all n ..." without the let

[next sinew]

The former is what they have in the image

[midnight scarab]

That entropy is essentially proportional to the amount of matter is precisely the fact that it is an extensive quantity, something that should be explained in any half-decent chapter on thermodynamics

And the J/K are just the units of entropy

And has nothing at all to do with entropy growing linearly with temperature or a linearisation of a log dependence

It's just that at constant volume dU = T dS

Or any equivalent thermo relation defining entropy like dQ = T dS for a reversible process

[midnight scarab]

I'm pretty sure Shin's point was that what the statement that is proved by induction is not that P_n is true for the chosen n, but rather "forall n, P_n"

[next sinew]

Actually I misread the exercise and I don't need to make a log linearisation approximation, I just need to use the fact that entropy is an extensive quality

[midnight scarab]

Also I think the logarithmic dep on temperature (which again has nothing to do with the units issue) is only for ideal gases

[next sinew]

Depending on what you take as the basic definition of entropy it is actually true that there are easier ways to see that entropy is extensive than the boltzmann equation, thank you for pointing that out, but doing it through the boltzmann equation is also satisfying in its own right

[midnight scarab]

All you need is indeed the dependence on qtity to be linear so that the notion of molar entropy would make sense

[next sinew]

Probably yeah

[next sinew]

But if you prove that P_n is true for an arbitrary n then that is the same as saying that P_n holds for all n

[midnight scarab]

Btw the dep on N is not exactly linear anyway, even for ideal gases it's N log (V/N Lambda^3) + N*5/2 with Lambda the thermal wavelength

(Sackur-Tetrode)

[next sinew]

in fact I think the notion of "an arbitrary n" is actually potentially harmful for students because I've had my fair share of students who've asked me what an "arbitrary" element is, and I just tell them it's just a way to not have to say "for all" in every sentence

[midnight scarab]

There's a subtle difference between "for all n, we prove something by induction" (one proof by induction for each n) and "we prove by induction that for all n"

[next sinew]

What is the difference?

wait

yeah no what is the difference

Is this some metalogical situation where like for each n there exists a proof vs. there exists a proof that for each n

and then some axiom of logic says that the existence of the first implies the existence of the second

If the course is a super hyper formal math logic course and not an introductory proof course then you would probably forbid the student from using any words at all in their proofs right...?

But clearly this cannot be the issue because the validity of an inductive proof has nothing to do with whether you use "an arbitrary element" or the \forall quantifier to express \forall

[tight star]

The first would be like trying to prove $\forall n \forall m P(n, m)$ by showing that, for each fixed n, you can construct an induction proof by inducting over m

[burnt vessel]

Pseudonium

[tight star]

The second is just standard proof by induction

[tight star]

E.g. this crops up when trying to prove commutativity of addition in Peano arithmetic, I think

[next sinew]

But doesn't an induction proof do this under the hood?

And then there's some additional metalogical axiom that allows you to deduce that there exists a proof of the original statement

[tight star]

It’s like afqt said - the former is that there’s a proof by induction for each n, whereas the latter is a single proof by induction

Those aren’t the same thing

[next sinew]

What does a single proof by induction look like for a statement of the form \forall n \forall m P(n, m) ?

[tight star]

Induction is essentially a proof strategy to show that a predicate $\mathbb{N} \to {0, 1}$ always outputs “true”

[burnt vessel]

Pseudonium

[tight star]

I wasn’t saying that? You’ve mixed up former and latter here

[tight star]

This is many proofs by induction

[tight star]

In the former, we have a predicate $\mathbb{N} \times \mathbb{N} \to {0, 1}$

[burnt vessel]

Pseudonium

[tight star]

Taking $(n, m) \mapsto P(n, m)$

[burnt vessel]

Pseudonium

[tight star]

The way afqt suggested to show this always outputs true is

For each $n_0 \in \mathbb{N}$ you can define a predicate $P(n_0, -) : \mathbb{N} \to {0, 1}$

[burnt vessel]

Pseudonium

[tight star]

Taking $m \mapsto P(n_0, m)$

[burnt vessel]

Pseudonium

[tight star]

The suggestion is to show this always outputs “true” using induction on $m$

[burnt vessel]

Pseudonium

[tight star]

But importantly, there was no suggestion to use induction on the value of $n_0$

[burnt vessel]

Pseudonium

[tight star]

Again, I would recommend looking at this for an example

[next sinew]

Didn't the original image contain only one variable, so why are we considering predicates with two free variables?

[tight star]

Ah I’m not sure what the original image was, I was just explaining what I thought afqt meant here

[next sinew]

This is the original image (which according to Galois is apparently fishy somehow)

[tight star]

Seems fine to me ig

[next sinew]

it is somewhat interesting that I don't understand Galois' image, afqt doesn't understand my misunderstanding (and thus presumably understands Galois' image) and you understand afqt's opposition to what I said even though afqt sees a problem with the original image while you don't

[tight star]

Bear in mind I am a physicist so I perhaps don’t care as much about the minutiae of phrasing proofs

At least, the way I interpret the first “let” statement is specifying the “type” of the variable n

Rather than saying it refers to some specific natural

[next sinew]

I interpret "let" as a shorthand for "forall" written on every line, what do you think of that?

[tight star]

Then sure it’s a little weird, because fixing $n_0$, $P(n_0)$ is a proposition, not a predicate with a free variable

[burnt vessel]

Pseudonium

[tight star]

In the sense that $P(3)$ is a proposition, and not really something you prove by induction

[burnt vessel]

Pseudonium

[tight star]

But you can prove $\forall n P(n)$ by induction so I let it slide

[burnt vessel]

Pseudonium

[tight star]

.

[next sinew]

The way I interpret this is that "for all n0, P(n0) is a proposition" which I agree with but I just fundamentally don't interpret the words "let" and "fix" the same way as you do, in fact I don't really understand how you interpret them in terms of symbols of some logical system

[tight star]

Well idk much about logical systems

In fact logic/foundations stuff is often just inscrutable to me

[next sinew]

What frustrates me is that Galois showed the image as if the issue with the image was completely obvious, and maybe to some people it is, but to me it absolutely isn't

If we are trying to construct a proof in first-order logic assuming some set of axioms like maybe the Peano axioms then that context would be helpful

If we are trying to construct a not-100%-hyper-rigorous proof in first-order logic and we don't actually need to make it formal at all and it suffices that it's clear enough that the steps make sense in the standard model of ZFC that we use in mathematics or whatever, then I firmly stand by the claim that "Let x in S" is the same as appending "for all x" and "\land x\in S" to all the proceeding statements and I would love to be told why I'm wrong if there's something I'm not understanding because this is certainly a very important thing to get right

I read this
https://math.stackexchange.com/questions/563348/when-is-first-order-induction-valid
and I think it's quite related to this discussion

For context, I don't know much about logic, nor foundations, nor anything really and I'm just trying to understand the issues that people see with the original image

If there's someone who's at all familiar with logic, then one thing I'd like to ask is what rules of inference we're allowed to use when constructing a mathematical proof and are those rules of inference inherently a part of first-order logic or is it something on top of that

Because from what I understand, the fact that a proof by induction actually proves what it sets out to prove is a result of somewhat nontrivial rules of inference
EDIT: posted in #foundations so maybe answer there

[tawny slate]

quick side comment about the infinity and cardinality discussion earlier, it often helps to first give students this idea that there are a lot of ways to define things, like for instance, a parabola

now you can explain that there are lots of clever ways to define things like the property of being infinite. one of my favorite definitions of an infinite set is a set who can form a bijection with a proper subset of itself

this definition feels intuitively impossible at first, but does two things in particular: it highlights how a seemingly unimpressive "finite-sounding" definition can describe something as vast and obtuse as infinity, pointing out some key trait that makes some structure "infinite". secondly, it motivates why we care about bijections even in the infinite case

so under this view, i like to treat cardinality as simply "does a bijection exist? if so, they are the same cardinality" and thats really the only intuition you need. anything relating some notion of size is merely an analogy

[next sinew]

I think bijetions are only "finite-sounding" because we are mortal finite beings who can only imagine finite mapping diagrams :P

[vocal phoenix]

On an entirely different note: this semester I'm going to be teaching linear alegebra to non-math UG (think people in the first year at a STEM university, so they only have high-school math background and most of them won't be hugely interested in math as such), and I'm wondering how best to approach determinants, to avoid them seeming too much like "magic"

I'm thinking of focusing on the geometric interpretation, but I'm not sure how to deal with the sign issue.

So far the best I've come up with is that we want the determinant of a sum of matrices that differ only by one row (or column) to be the sum of their individual determinants, because you can make a convincing 2D drawing that it shoould be the case in 2D

[next sinew]

When you say "magic", what exactly do you mean by that? Isn't it pretty much "magic" that the laplace expansion gives something related to volumes and areas?

[vocal phoenix]

And a consequence of that is that if you flip a vector, then you need to change the sign of the determinant.

[vocal phoenix]

It is, especially in dimernsions higher than 2 or 3, and some degree of "magic" is inevitable at this level.

But I'm hoping to demystify and motivate it as much as possible.

[next sinew]

oh yeah I forgot that you can actually do it pictorially in the 2D case

[vocal phoenix]

I'm not really expecting them to grasp the intuition behind the Laplace expansion, I just want them to see the determinant as something more than "magic number that turns up in some formulas/criteria"

And I quite like the volume interpretation, except for the issue that it's signed volume

[next sinew]

Do you mean like if you have one column where you break each element into a sum then the determinant of that matrix is det(A)+det(B) where A contains the first summand and B contains the second summand

[vocal phoenix]

Yeah, in proper terms the determinant is a multilinear form

But I can't tell them that

I mean, I can, but it's unlikely to help

[next sinew]

yeah I don't think it would help

[vocal phoenix]

But the property as such is helpful (and part of the curriculum), and has a geometric justification in 2D

[next sinew]

I've studied math for 3 full years at the undergraduate level and while I have some familiarity with that a multilinear form is the only example of a multilinear form that I know is the determinant

[tight star]

I think it would be good to motivate why we care about multilinearity

If you just care about areas, then the modulus of the determinant is really what you need, and it’s not obvious why focusing on signed areas is important

[vocal phoenix]

Yep, that's my issue

[tight star]

The main advantage of signed areas is that they’re multilinear, and so easier to deal with in the context of linear algebra

[vocal phoenix]

(side note, we're not covering linear maps at all, so I also can't really talk of matrix multiplication as composing actions, and of the determinant telling us how a linear map changes the volume and orientation of a cube)

[tight star]

You always have the option of tacking on a modulus sign afterwards, but it’s a good instinct to keep things “as linear as possible” when working with linalg

[midnight scarab]

Do the students know about the vector product?

[vocal phoenix]

Not likely

[tight star]

And indeed this gives you nice properties like the determinant being a polynomial, so in particular smooth (which allows you to differentiate it)

Whereas the modulus of the determinant is not, in general, differentiable

You also get a clean recursive formula for the determinant because of the multilinearity

Trying to directly work with areas, and not signed areas, would prevent having such a formula

[tight star]

This is generally the approach I take and it works decently well

[next sinew]

May I see the drawing?

[tight star]

It’s essentially the proverb “when in Rome, do as the romans do” but applied to linalg

The way I usually introduced signed areas is just by showing that multiplication of real numbers is bilinear, and you can visualise this with them being the width and height of a rectangle

That at least explains what bilinearity is supposed to mean, and why you might want to consider it

[vocal phoenix]

It's from a book I have, the notation and actual graphics can probably be better (for my lecture I'll definitely do a color version)

But my main point with this is that if k2' is the opposite of k2, then the LHS will be 0, so the two determinants on the RHS must have opposite signs (in particular, one of them has to be negative)

I suppose the whole thing is in a similar mental space to what happens with "definite integral is the area under the graph of the function" when the integral ends up negative (although they're unlikely to know much about definite integrals at this stage)

I wish I could do an Axler and avoid determinants altogether

I have no choice either way, I'm bound by the curriculum.

[next sinew]

wow that diagram might be my new favorite diagram

It's probably gonna be kind of confusing for students but I like it

[vocal phoenix]

Well, in the actual lecture I would walk them through it.

And maybe even do a simplified version where the two summands are parallel, the point is to justify the whole thing a bit

Just not to make it not seem entirely arbitrary

[next sinew]

Would you emphasize the fact that all three rectangles share the same base, and therefore the areas being equal just becomes a statement about expanding out the parenthesis?

[tight star]

The main difficulty is that you can’t just translate the two parallelograms to make the third one

You have to chop them up

[vocal phoenix]

Yeah, pretty much, you've got three paralellograms with the same base so you just look at the heights

They do know the ways to calculate the area of a paralellogram from high school

[next sinew]

A potential area of danger is that some students have seem proofs of the pythagorean theorem where you rearrange pieces

here the result actually requires not only an understanding of geometry but also an understanding of algebra

[tight star]

Maybe mention this explicitly

[next sinew]

as Pseudonium said

[vocal phoenix]

Well, I have about two weeks until determinants, since I'm starting with the basics of matrices and related operations

So still time to mull it over

[tight star]

Otherwise they may assume you’re trying to make an argument of this type, when you’re not actually

Honestly I think it’d be helpful to go over why “shearing” doesn’t change areas/volumes

I don’t think it’s immediately obvious, and it also shows an important way in which vector addition interacts with areas

Which is usually the harder part of the multilinearity of det

[vocal phoenix]

Good point

[tight star]

There’s neat arguments you can do here with e.g. cavalieri’s principle

[next sinew]

Do you have an end goal of things you want the students to understand about determinants?

Maybe something you're building up to?

[next sinew]

Because this property on its own actually isn't super useful to non-mathematicians who generally see determinants as a number that tells you whether a matrix is invertible

So is that just a stepping stone towards eventually proving some kind of matrix invertibility result?

Or does it bridge the gap between the laplace expansion of a determinant and the way determinants are related to areas somehow?

I suppose you can use it to prove that you can multiply one of the columns by a natural number and it scales the determinant by that number

[vocal phoenix]

They're only really going to feature in matrix inversion and the Cramer method, which is why I don't want to get too much into the weeds of motivating it and establishing all the properties.

I just don't like introducing things as completely arbitrary

I'm perfectly happy to just do some handwaving along the lines of "this seems arbitrary, but if you followed this thread to its conclusion, you'd find out it's inevitable"

[pure light]

the sign of the determinant reflects the orientation.
https://mathinsight.org/determinant_linear_transformation
gives some visual insight

[vocal phoenix]

Ohh, this is actually a very good idea

The curriculum doesn't include linear maps, but I do want to allude to it, and I also later on I want to explicitly talk of the interpretation of multiplication by real numbers as scaling with possible change of direction (to motivate the imaginary unit in terms of rotation by 90 degrees)

So matrices being "positive/negative" is something I can tie into this.

[cloud zealot]

are you allowed to choose the textbook? is this course mostly computational?

[cloud zealot]

oh i guess not

but i like meckes and meckes

they also defer determinants to the end

[cloud zealot]

@vagrant meadow

[vagrant meadow]

when I was TAing for linear last spring, the prof didn't even COVER determinants. so it was up to me to try to teach them what they needed to know between the topics. in 3.2 and 4.3 of this document I tried to give a very concrete algebraic intuition for why ad-bc is so useful, but I avoided focus on the geometric intuition beyond mentioning it, and I didn't really extend it beyond 2x2s.
https://smashmath.github.io/assets/pdf/linalgsolutions.pdf

last year I spent a lot of time trying to find a nice way to explain determinants without "magic" to like undergrads without getting too technical, but I wasn't particularly successful beyond 2x2s.

I think thinking in terms of row reduction might be the best way... but idk. I don't like cofactor expansion as much (in terms of intuition).

[cloud zealot]

@turbid zenith

[pliant yacht]

Just use the interpretation as signed volume?

show how if u want a notion of signed volume

It has to satisfy axioms 123

And this uniquely characterises the determinant

[brittle cedar]

hola vv

gracias por ayudarme

tqm

casate conmigo lindaa

no perdon

no puedo, tengo novia

yo se que querias pero no puedo

es un amor imposible 😔 😔

[pliant yacht]

Totally…

(I don’t know what you are saying)

[brittle cedar]

tqm

[halcyon valve]

Hermano que haces 💀

[bleak skiff]

"the number of mathematics courses taken"

what an incredibly bad metric

[tawny slate]

thought about this for one of my math videos, and i partially regret not doing a first principles proof for this

if you can go show any simple shear preserves area, then you can shown that any shear is just a composition of shears

so then just prove the simplest shear geometrically

[turbid zenith]

So ... longish post incoming. Curious what y'all think...

My partner just taught me how to cook shrimp. 🍤

I promise this is relevant.

Cooking is something I've had trouble with for the longest time because I never really learned it and it felt like something out of reach. I've tried to learn it by experimenting, reading, watching videos, and asking my partner questions as I'm trying to make something. But what invariably happens is I screw something up and get really frustrated and disheartened and think I'm never going to be able to learn.

But this time we tried something different, where he essentially just showed me step by step what I need to do from beginning to end, I took notes and asked questions (why are you doing this instead of that, what's the reason for doing that step, etc.). And I feel SO much better having seen it done right the first time. I haven't made it myself yet, but I'll be doing that in a couple of days to see how that goes.

The common wisdom from math education nowadays would be that I need to "learn by doing," to "investigate" and "explore", etc. "Nooooo you're not supposed to TELL them what to do!!" "I do, we do, you do is sooo passé..." But in this case, the frustration that I feel from trying to learn by doing paralyzes and demoralizes me.

I know that sounds like an oversimplification — "Surely nobody's saying you can't EVER do direct instruction" — but it really does seem like in many circles there's a kneejerk reaction to dismissively label any mention of direct instruction as "teacher centered" and "sage on the stage." And it's made me like I'm doing something wrong if I'm ever lecturing. This just adds to that conflict.

Any thoughts?

[cloud zealot]

https://www.rxddit.com/r/math/comments/1fw247h/if_an_average_msc_mathematician_from_today_would/lqbujln/

semi-related thread

[tight star]

One thing I did tend to dislike about my math classes was that there was a lot of expectation that you should be able to just reinvent math for yourself

This always confused me because, if you could already reinvent all the math yourself, why do you need to be taught…?

It ended up feeling more like they were merely sifting through the cohort for those who were capable of reinventing math already

I don’t think “ability to reinvent math” is unteachable per se, but they didn’t seem to put much effort into teaching it

And there’s obviously a sliding scale here, but I found that my math classes were very far in the direction of “reinvent this yourself”

On the other hand, my physics classes were much less in that direction - it felt much more like a collaborative effort, in that we were studying the hard-won fruits of physicists before us

In a collective goal to understand the universe better

We weren’t expected to reinvent physics, though of course we had to apply the ideas in class to new situations - that was on the problem sheets

[tight star]

And at least for me, it never felt like such “sifting” was taking place in the physics classes

This was a big reason I decided to go over to physics

[tawny slate]

I really agree, and Im happy for you! let us know how the shrimp is haha

i think it all just depends on what the student's needs are and how they are met. sometimes you need direct instruction, sometimes you need to give them a confidence boost. its not even always about the math

I used to think that poetry was one of my worst abilities, and even after being taught the basics by my wife, she still had to give me critique, which was necessary for me to improve but also very demoralizing

once she realized how little confidence i had despite learning so much and improving so much, she pivoted into confidence-boosting mode, by telling me what i did well and more importantly, instilling in me this feeling that, more than any other medium of art, "mediocre" poetry has more value than "mediocre" anything else. this built a passion for writing poetry in me despite me not being very good at it

it all depends on what you need, some kids just need to be told not to worry about being good or useful

[mint lark]

This is extremely relatable and just true

[cloud zealot]

@vocal phoenix @vagrant meadow you both may be interested in checking out nathaniel johnston's textbooks in linear algebra, one a first course in euclidean space and the other a second course with abstract vector spaces (although still putting heavy emphasis on matrices and matrix decompositions)

https://www.youtube.com/@NathanielMath/playlists

there are lecture playlists accompanying both books

[tepid smelt]

Direct instruction is necessary. I feel just like how we learned we need to teach phonics I hope we will likely see the shift more towards direct instruction.

[high skiff]

5th time taking real analysis and a researcher concludes “oh wow they must be a great teacher”

[pure light]

i think the intention is not to say "these are better teachers than other countries" but to dispel the idea that "these are less highly trained teachers than other countries"

[plain pebble]

This likely suffers from the fact that what counts as “high school education” in the US is below what a lot (all?) of the other countries there count for it

[empty gull]

Not going to lie I've always felt that it's actually the opposite?
But that could be a cultural thing
For years I've been hearing in math circles how we should be doing LESS direct instruction
That we should let students figure stuff out on their own because in a way that's how math research is done (I think this kind of idea mainly comes from math researchers ala Lockhart's lament type of stuff)
And I've always been taught by direct instruction but unfortunately I've found it lacking in many aspects
Mainly that it causes math to seem overly stringent to many students and unnatural if it's not done correctly
At the end of the day there should be a balance of styles
I'm sort of torn on whether or not we should tell students how to solve stuff algorithmically i.e. give them step by step instructions on how to do something because a lot of the time what ends up happening is that students just memorize the algorithm without understanding what's actually going on
I think there should be an aspect of discovery to stuff, like giving a few examples and then saying try it out for yourself on some more alien an less familiar examples yourself

[tawny slate]

i always considered it a balance

like you don't just give them steps, but you also don't have students completely rederive everything from scratch (top down results would be near impossible)

and thats where i think the default is to teach concepts, then tune it in either direction based on context

[shy token]

In my experience it was mostly about how good the direct instruction was, rather than if there was or wasn't one.
Back in elementary school I had a math teacher who had huge issues with proper verbal communication. That man couldn't articulate his thoughts to save his life, although he knew his math. I think it was a byproduct of his education, partly. When he was studying and practicing math, he never had to express himself verbally, so his primary occupation didn't "force" him to develop this skill...
I've met quite a few "STEM" people who don't like literature (as a subject), or reading in general. They are used to working on technical problems in silence, and reading only technical books, without having a chance to appreciate the value of good word usage. Their vocabularies are very limited, and I have a strong feeling that they would have the same issues while teaching as the teacher I mentioned.

[vocal gate]

I agree with you to the fullest extent possible. I think avoiding direct instruction is hugely damaging and the over-focus on class discussions during the introduction phase actively hampers student comprehension. Certainly one should not teach everything algorithmically, but I think the skew towards 'lets see if these students are clever enough' is not healthy.

[tardy ember]

yeah i can see what the concept of "don't do direct instruction" is trying to correct for but also you don't want to go too far with... really any one thing, to some extent

although there is also the point of like, if you just optimise for "it working" then you might end up going the wrong way

the whole "look at the problem and try to guess off of keywords which of the memorised algorithms (that you don't actually understand the purpose of) you're 'supposed' to apply here" behaviour is a local maximum, it's where you end up if (under "normal" maths education at least) you repeatedly make the smallest adjustment to what you've already learned that succeeds at the most recent thing you've been asked to do

and until you start to get to the point where ~creativity/actual logical reasoning is actually a hard requirement, somewhere roughly around the point where we don't know how to tell a computer how to solve the problem, going further into this attractor is in fact the lowest-effort strategy to learn the next thing

so you have a choice between

1. take as your goal just getting them to the point of passing the exam in the time you have, and prioritise the short term while adding more of a problem to be dealt with whenever they have to learn what maths actually is
2. take a massive step "backwards", making zero or even negative measurable progress as you show them the logic behind what everything they already "know" is actually doing, hoping that this will help them in the long run

[vagrant meadow]

(in a tutoring context or on this server)
personally, i find myself leaning towards direct instruction when the student seems motivated. for some students, i feel like it's sort of obvious that direct instruction is genuinely helpful, and it's not just like "giving them the answer" or "making it easy". they take it and immediately learn from it, and can apply it to other problems right away. and i don't regret leading them through the problem directly, step by step.

when a student is being passive, not putting in any effort, just wanting someone to solve it for them, etc. ex. just posting a picture of a problem in this server. i lean more towards just giving hints/pointing them in the right direction. even if i choose, against my better judgement, to lead them through the problem step by step, it feels like they often take nothing from it except the answer. they move onto the next problem no better than the previous. even if it's a very similar problem, they often still act helpless and can't even be bothered to go back to their work from the previous problem and try to replicate the steps. it's just a waste of time. and i usually regret wasting my energy and patience on them if i tried a more active approach.

[cloud zealot]

https://www.reddit.com/r/math/comments/1fxq66a/purpose_of_intro_de_classes/

@quasi musk how do you feel about more proof-based ODE books that are accessible to students who have only had some real analysis, e.g. perko?

[quasi musk]

Some Real Analysis? I feel that a proof based ODE class can actually motivate a real analysis class

It's like why someone should care about the details in analysis

And if after that they still don't care, then they can just go do applied math

[vagrant meadow]

would be cool if more analysis classes taught the existence and uniqueness theorem for first order odes. but it's such a dense subject it's usually just not worth covering.

[small glade]

Why isn't basic linear/abstract algebra taught in high school?

It seems like if you look at it very broadly(not formally) it should be accessable to algebra 2 students(as long as you don't do it too rigorusly or go too in depth)

[pure light]

issue is that the overriding priority is to get students into calculus as quickly as possible, so something that doesn't directly apply to preparing students for intro calculus is low on the priorities list. for example the AP precalculus curriculum includes basic linear algebra, but only as an optional unit at the end (liable to be cut for time)

[austere delta]

There are lots of things that would be accessable to students early though, so that isn't really an argument to teach it.

That being said, I think learning some basic linear algebra is appropriate for high school, and in my country they do teach that. Mostly just adding vectors and dot and cross product for physics stuff, but still.

[quasi musk]

My guy students can barely handle algebra 2/trig

[wet slate]

ugh i forgot the original reference but one of the big ode textbook guys wrote an essay about how poorly he things de classes are taught

[cloud zealot]

you mean gian-carlo rota

[wet slate]

oh yeah him

he's way more than a big ode textbook guy fr

[knotty arch]

is graduate school commonly done in the same school as undegrad or different

[cloud zealot]

not really a question about teaching (pedagogy) in math

professors often recommend going to a different uni for grad school, and the most common reason is to "get a different perspective"

for a lot of logistical reasons, however, that may not be the right choice for you

[knotty arch]

oops i meant to put this in the channel below

apologiesss

[knotty arch]

thanks!

[spark hull]

yo

on a scale from 1-10 how true is this

[spark hull]

I mean, sometimes students just want to get their homework done and they're doing an online class, so the teacher doesn't know they're (cheating or) getting someone to do it for them...which should be happening quite often..cause a lot of people don't really want to learn, but get by..........life

[vagrant meadow]

I have sympathy for the bio majors who have to take calc 2 but will probably never use it. and there are many others who just have the math they need to take because the school tells them they have to, but not because they'll use it.

there are many students like this, who end up being the latter type of student I described. the simple truth is that I don't want to tutor them unless I'm getting paid to do so. if they don't care, then why should I? if I have time, I'll help in this server, but I don't usually invest too heavily unless I can sense their motivation and initiative, and that they've genuinely tried hard. ||that, or I'm making bad decisions which leave me annoyed.||

again, this is a tutoring context. as a teacher, you don't get to choose your students. but tutoring is generally much more hands on, and can greatly wear on one's patience.

and patience is a limited resource one must spend wisely.

[spark hull]

earlier I had to teach fractions and it was crazy hard

cause like

there was no logic

there

for the student

I was tutoring/teaching

like I gave him
x+3=3+3, what's x

and he couldn't tell me

he might say something like, 2

teaching is hard

[vagrant meadow]

this is why I could never teach/tutor anything below calculus. I just can't deal with that lol I would be terrible

[spark hull]

😮‍💨

calculus is fun

linear algebra is more fun

I was teaching fellow tutors linear algebra today at work and we all had a great time

it's an outstanding (and better yet different) feeling going from someone where you spend 2 hours on one concept to 2 people who you go through everything very fast and have a lot of fun

[tawny slate]

imo anything beyond basic arithmetic is not terrible to teach

as soon as algebraic equations become a thing, you have to implicitly consider the meaning of the symbology, involve types, build and use nontrivial domains, create lots of edge cases and tricks and pitfalls

and thats just algebra

i found that the more i did competitive math and the more i learned univ-level subjects like set theory, abstract alg, etc, the more interesting it was to teach ~middle school range

its as if competitive math gave me the motivation and the univ courses gave me the philosophy foundation

[quasi musk]

I had this thought as well for a really long time, and then my work just pushed me to teach/tutor all the way down to 5th grade. I learned a lot of new techniques of how people learn and now appreciate algebra problems more

Without calculus you can really focus in on algebraic properties, finding roots, and connecting it to geometric pictures. E.g. completing the square is literally geometric diagram to fill in a square

[tawny slate]

also we teach that in algebra, we should be allowed to do the "same thing to both sides", but this is not a universal rule. for instance, you can't always square root both sides, and if you square both sides you can potentially get extraneous solutions. how would you explain to a middle schooler when this rule applies and when it doesn't and why?

this is just one of like thousands of really interesting problems that really force you to delve deeply into what is really happening to make your explanations crystal clear

[spark hull]

PLZ TEACH ME THE TECHNIQUES PLZZZZZ

[pure light]

i would talk about it in the context of functions, where the fact they have only one output for any input means a = b implies f(a) = f(b). so then the problem comes in when a function has multiple inputs that give the same output

[quasi musk]

Well nothing I know is unique to me! It's all out there somewhere in the ether

[lyric lotus]

Square rooting doesn’t introduce extraneous solutions, but it may cause you to miss solutions

Squaring is the thing that introduces extraneous solutions that may not have been solutions to your original equation

Unless I’ve gone crazy

Which is entirely possible

[tawny slate]

that's right

but it's questions like that, trying to figure out how to "eli5" them, that i like

[vagrant meadow]

should be teaching the ideas behind functions in gradeschool
preparing their minds for higher math

[spark hull]

plz teach me what you know

[fleet schooner]

Do you have a methodology to follow in mathematics? I would like to learn mathematics, but I don't know where to start

[candid grove]

Probably show them the graphs of the sqrt x and x^2 functions

[vagrant meadow]

not really what this channel is for. it's about teaching math.

[turbid zenith]

Does anyone in here have experience with guiding undergrad research? Looking to pick some people's brains in DM.

[stray grail]

what about it?

[turbid zenith]

I'm looking at applying for a tenure track position that may open at the university I've been teaching at for the past 4.5 years, and I know they're interested in the possibility of undergrad research. But I've never overseen undergrad research — the closest experience I have is overseeing high school students at a gifted summer program when they do their research projects.

(I also wasn't a math major in undergrad myself — had a bad experience with it and switched majors.)

So what I'm wondering is a few things:

1. How do you choose good problems for undergrads to research?
2. How do students get the background information necessary? In particular, if there isn't a course taught at the university that covers it, is it the sort of thing where you meet with the student and teach them the background, or you expect them to read up on their own, or somewhere in between?
3. What kind of time scale does undergrad research usually happen on?
4. What's the faculty member's role when doing undergrad research?

[stray grail]

I guess it is different for everybody, I treat everybody the same by default, if it is bsc, msc, phd or postdoc. after a while it is obvious how well it will go

1. I just pick any of the random ideas I have that seem interesting and could be done in a few months.
2. I have a few "lectures" for them + use discord during the work
3. 3-6 months?
4. give the initial ideas, then they have to be creative themselves

this has mostly worked out for me, people usually exceed expectations and we get real useful results.

My field is very easy to generate small or large projects that are doable.

I try to be very selective nowadays who I accept to work with (turned down a MSc thesis today for example)

[turbid zenith]

This would be as part of a BS program. Most of this seems totally reasonable, although (1) scares me 😛

(Trying to find something that's both actually novel and actually doable)

[stray grail]

what is your field? for BSc I does not have to be truly novel, but could be

[turbid zenith]

Combinatorial Game Theory!

So I know at least one thing that students can do is to find a new ruleset, investigate it, and characterize the game values. But often I'll think "oh this would be a cool game to analyze" and then look online and someone already has done it.

[stray grail]

i find it good to have a project being part of a larger goal, so it can be extended or shrunk without issues

to me it is the detours and unplanned things that are the best. and what people enjoy

[turbid zenith]

Hmmm I see

Might that larger goal be something that you're trying to publish yourself?

[stray grail]

yes, always

or rather things i want to know before i die

things we will not be able to do in at least 5-10 years

[turbid zenith]

That's certainly a way to put it 😂

I like it!

[stray grail]

but i can choose who i work with, maybe that would not apply to your position?

[turbid zenith]

What do you mean?

[stray grail]

like if it does not work out i can just fail people and dont work with them 😛

people i work with are very motivated usually

[turbid zenith]

Fair enough... wait, fail them? At your institution there's a grade for undergrad research?

[stray grail]

pass or fail yes

like bsc thesis

[turbid zenith]

Interesting.

Well as it is, my institution hasn't done undergrad research in math for a while, so that's something we're trying to start up in the first place

Thank you for your thoughts on this btw!

[stray grail]

i only do bsc theis, msc theiss, a project course for msc students, and phd projects etc

good luck with the position, if i can give some more input just ask (i am assistant prof atm, but will apply for associate any day now, when i get the time to write the application)

[hoary plover]

The following incident bothers me for pedagogical reasons. Maybe I am missing something.
In #help-16 message, someone asked for pre-university (analysis, I assume):

x = 3, 5, 7
f(x) = -1, 4, 6
f'(x) = 8, 3, -2

Based off the values provided, give a reasonable estimate for f'(4)
Notably without any more restrictions on f(x), like it being monotonous or differentiable anywhere else than x ∈ {3,4,5,7}.
I hope you agree that this is a shitty exercise.
Issues:

1. Without restrictions, every value is equally plausible since a function (over real numbers) is just a mapping of each number to any number.
2. There exists only a single polynomial function over real numbers of degree 5 or less with those values (and it entails f'(4) = 143/128; [eqs1], [eqs2], [eval]), so considering that in school you usually deal with such functions of low degrees, all "rough guesses" like f'(4) = 2.5 (as by the student) are not only implausible, but impossible within those boundaries.

Have you ever seen something like this and what would you guess the teacher expected from the students?

I additionally assumed f(x) should be a polynomial and [solved it accordingly](#help-16 message) towards
f(x)=15/128*x^5-403/128*x^4+2105/64*x^3+-10693/64*x^2+53119/128*x-51683/128, thus f'(4) = 143/128.
Do you think this was what the teacher really intended? Unfortunately, the help session timed out and I could not clarify whether the given exercise was incomplete.

[midnight scarab]

It's very highly likely that polynomial interpolation was not what was expected

[hoary plover]

But? (And why do you think so?)

[midnight scarab]

Becauss that's still rather arbitrary (unless the lecture before was on Lagrange interpolation or smthg) and way too involved for a hs exercise

[hoary plover]

So the correct answer was essentially "Every value is equally plausible since a function (over real numbers) is just a mapping of each number to any number"?

[midnight scarab]

I do agree the question is nonsensical though

[tardy ember]

the values for the derivative look like a linear function (f'(x) = -2.5x + 15.5) so f'(4) = 5.5 is plausibly the kind of "reasonable" answer they were looking for

[midnight scarab]

Maybe what the helpee did was what was expected

[tardy ember]

the issue is that this is completely inconsistent with the provided values of the function

[midnight scarab]

They'd know best what their prof had in mind

[hoary plover]

So the teacher was just incompetent. 🤔

[midnight scarab]

Correct probably, expected most likely not, psychologically speaking

[midnight scarab]

Yeah that was also my first guess

[tardy ember]

i mean i don't know if "incompetent" is the right category of judgement when they might have just not even been trying to write a question that means anything

[hoary plover]

Then at least my remark

Well, in which way plausible then? This exercise is total crap.
was on point.

[turbid zenith]

I think this is a case of "I would need to see the original wording of the question."

These kinds of exercises come up often in AP Calculus, but they're asking things that really could be answered from the table.

But how often do students just look at numbers and formulas and entirely ignore the wording of any conditions?

(Answer: A lot.)

[vagrant meadow]

MVT. slope of the secant line (4-(-1))/(5-3)=5/2 so clearly, that's a reasonable value for the derivative

[spark hull]

is it better to take notes and never look at them again or to take notes and heavily rely on them.
for yes
for no

[hoary plover]

Are you joking or did you miss the issue?

[vagrant meadow]

two different options: yes or no

[teal warren]

What will you study from?

If you will study from a book bring the book to lectures and just add notes to it. If you like to study from your own notes make your own notes.

[spark hull]

lectures

so I listetn and write everythign down

and practice during hw

so I remember it

[teal warren]

And this works for you?

[cloud owl]

I don’t get why people take notes when they have a good textbook available

The textbook is my notes

During lecture I focus on the lecture

(This is a lie; half the time I’m on my phone or doing homework, but that’s neither here nor there)

[cloud zealot]

perhaps there is no assigned textbook and the professor is working purely from their notes

[cloud zealot]

i play games and half-listen to lectures in case there's an interesting insight

[tight star]

In my case taking notes often involves making flashcards, which textbooks don’t come with

I do also find that it’s useful to have a layer between the textbook and my brain, which notes serve the purpose of

[near turtle]

Can integral calculus be taught using Liouville's theorem and differential fields?

That way, not only can you find elementary integrals, you can also prove when they don't exist.

Liouville's theorem is also used in Risch's algorithm. Has anyone tried teaching integral calculus using a simplified version of Risch's algorithm>

[tight star]

I would think it’d be more important to focus on what integration is doing conceptually than to narrow down on how it behaves algebraically/symbolically?

[tawny slate]

hard disagree

[tawny slate]

i agree with what pseudo said here

the purpose of notes is not to log the information somewhere

the purpose of notes is to log the information for yourself specifically

sometimes it isnt even a matter of having a reference for yourself to look back on

often times the very act of taking notes is how your brain processes and understands the material

i believe this notion is fairly well-supported by neuroscience literature, iirc

also, being able to reframe and reorganize ideas when the scale of the material is larger is insanely important

i dare someone to try making it through A=B without taking notes

i had to take notes about the notes that i took, and then i took notes on those notes before i redid all of my notes

note-taking is a critical skill that is underrated and all students, with few exceptions, need to learn

[midnight scarab]

There are notes that I never look back on. Precisely because having written them helped me mentally organise and memorise the material

[near turtle]

Symbolic integration might be easy if you limit it to the $\mathbb C[x]$-module generated by ${1, e^{ax}, \log(a + x), \frac{1}{a+x}, \sqrt{a + x}, \sqrt{a + bx + x^2}, \frac{1}{\sqrt{a + x}}, \frac{1}{\sqrt{a + bx + x^2}}}$.
You simply use integration by parts a lot. You use u-substitution to handle the ones with square root.

[burnt vessel]

wlad 🌻

[near turtle]

I have a feeling that this handles more than 80% of the anti-derivatives a student would meet at the end of high school. And it would do so in a straightforwardly algorithmic way.

Maybe treat it as a differential module. That might make it even better by removing half of the above generators.

The point of this is to address the fact that integration does not in general behave well with respect to:

• Multiplication of functions f(x) g(x)
• Composition of functions f(g(x))

But if you limit which functions you can multiply and compose, you remove obstacles to using integration-by-parts and u-substitution respectively. The function space ends up being a module, as opposed to an algebra.

The payoff should also be a fairly short piece of Python code that does symbolic integration totally from scratch, and to nearly the level of a competent student.

And this could be pedagogically useful.

I'm wondering also about the fact it's a $\Bbb C[x]$-module. We know that $\Bbb C[x]$ is a PID. This might have some lovely consequences. 🤔

[burnt vessel]

wlad 🌻

[near turtle]

I'm pretty excited about this. It's almost certainly not new though.

[lyric lotus]

You know I thought this too but when I looked into it there actually isn’t that much evidence for notes as opposed to active listening

https://youtu.be/cRQqH18wJgw

That perspective is kinda explained here

There are many cited studies in the description

[tawny slate]

ooh yeah i think that video is great

i do agree that the most important thing is active listening/attention

the free recall method is generally what i am referring to when i talk about note-taking during lectures, and i believe is the proper way to take notes

i guess i should make that distinction clear in the future

thanks for sharing

[lyric lotus]

I wouldn’t call that note taking during lectures tho tbf

I expect most people who say they don’t take notes are probably doing something like free recall

[twin mirage]

Yeah. I rarely take notes on paper. I take notes in my mind

If you want to retain information. I think the most important way is to think about it like a lot

[tawny slate]

yeah i wasnt saying that that is what i was calling it, but that i emphasize that free recall is especially important for lectures because the lecturer doesnt stop, so you dont want to miss anything

[twin mirage]

Talking notes while the lecturer is talking just distracts me more, actually 😔

[tardy ember]

yeah i don't have much/any experience with lectures in particular but from my experience learning maths off the internet, just thinking about it seems to work, i don't generally take notes about it and i still end up remembering it a lot of the time

specifically i think it helps me a lot that proofs make sense, in a way where they're not just random data

so there's this spectrum where i don't quite remember it but i still have, essentially a lot of hints

enough that i can just spend some time rederiving it, partially from scratch (because the thing about a proof is that you can do that) but with all of the parts of it that i still remember pushing in the right direction

and then the next time around i remember it better because i did that, and so it's easier that time

[twin mirage]

Well. If you understand it, you never have to memorize

[tardy ember]

exactly

well

you do often want it to stick in your memory, but yeah it is easier to do that if you understand it

and also the way memory works is that anything you need often enough, you do just end up remembering, even without having made a deliberate effort in that direction

[twin mirage]

The interesting thing about human memory is that humans possess associated memory. Meaning one piece of information will be connected to other pieces of information. The more you connect the more you remember. So taking notes in your mind means connecting that piece of information to another piece of information which also connects to other pieces of information. Another piece of information can be sound, image, words, numbers, another memory...

So when I remember one piece of information the whole chain comes out which I find quite interesting. It is like a chain reaction

[shadow flower]

i agree with this. i find it more beneficial to sit there and not write anything down while trying to be as engaged as possible. i can’t think critically about the material if im trying to write it down simultaneously

[twin mirage]

Nowadays, everything is in the books or on the internet. So that will be my notes. If you know the keyword, you can always check it out later

That is why I just fully focus on what the lecturer is talking

[tight star]

it’s interesting the different experiences that others have here

I think this might’ve been somewhat true for me in school, but not so much from university maths and beyond

[tight star]

this is true, though I think it can be difficult to tell what you’ll need in the future

[tight star]

and I think in my case, having notes really reduces the friction with regards to looking something up, in a way that only relying on primary sources just isn’t capable of

[vagrant meadow]

it really depends on the kind of student I think. many of my cohort mates don't take notes, but if I was tutoring or had a student in office hours who said "I don't take notes" i would be pretty annoyed

if you can say "I don't take notes but I still learn the material" then that's great. but if one is struggling with the material, then taking notes would be my first suggestion. one of the first things I ask a student in tutoring is "what do your notes say"

especially in lower levels where approach can vary wildly. I don't want to be explaining how I personally learned it if it's going to be completely different from what their teacher is expecting.

[tepid smelt]

I always thought notes were overrated. I do think it's important if you're going to give a talk or present information though. It's incredibly important how you present material in a condensed format and I do get a lot out of a good set of notes or textbook. Yet I feel most of the learning comes through doing lots of problems. The notes that are already made by the book/professor are already good enough to reference.

[feral spade]

Notes are helpful, simply for the fact that writing your notes makes you go through the entire study material thoroughly. Otherwise, if you study from a prof's notes, you might tend to skip or casually skim over some parts of the notes that you feel like you really know/are easy. Taking comprehensive notes means you go over the entire material uniformly to not have holes in the notes, and that makes you spend equal time on sections you would have otherwise been overconfident about. And that imo is what some people notice when they feel the benefit of taking notes over reading prepared notes from a prof

[twin mirage]

As a programmer, I am more comfortable searching the Internet if I forget something. Just skimming through some articles is enough to regain your knowledge.

So if the Internet is down then I am doomed

[tight star]

I think it depends on how much prerequisites you need to understand the thing

In my experience with programming, it’s more breadth than depth

So I don’t keep notes on things like syntax details for example, that’s something I can look up relatively quickly

But for something like physics, you can definitely need a lot of prerequisites to understand something

So there’s a lot more friction associated with looking something up

[lyric lotus]

There isn’t actually much evidence to support this belief. The video I linked addresses this exact point

Free recall is significantly better for long term memory

[cloud owl]

I definitely agree with this - I would never give a talk without slides or notes

[feral spade]

I actually did watch the video you had linked. My statement corresponds to exactly the point he made around the 5 minute mark about paying attention to the right stuff. In my opinion, when you take notes, you kind of dont want to leave holes in it, and so you go through the entire script to make sure of that, ensuring you dont overlook anything. I do not anywhere mean to have notes have any comparable effect as free recall or anything such. All I mean is taking notes later on ensures you cover the entire syllabus properly and without overlooking anypoint that one might be overconfident about. This is ofc based on my personal experience, comparing times when I took notes as against the courses where I did not take notes.

[lyric lotus]

Taking notes later on I agree with (free recall). Taking notes during the lecture I think can be counterproductive. I think it’s a little extreme for me to say to never do it but I would definitely not be trying to get down everything the lecturer goes over imho

[feral spade]

My preferred way of studying that I have come to realise is to attend the lecture (in whatever format possible, be it in person, online or pre-recorded) without reading the notes in advance (shocking I know) and picking up as much as I can. And then once the lecture is done. I put myself in the shoes of the prof, and try to give the same lecture to myself. That way I know exactly which topics I would feel confident about answering a question. Only then I make notes about the entire topic of that lecture, such that it highlights the topics that I need to work on, and then go on about it by solving problems and so on. I would not recommend this for totally new topics, but as I studied more and more, at courses in higher undergrads and above level, there started to be a significant overlap between the topics, and that was when this method started being more productive for me.

[midnight scarab]

The thing is that personally I'm unable to think about things like maths without a pen and and paper

So like what I write down during say math lectures are the calculations but annotated with all the explanations I had to come up with to understand why a certain inequality holds or what the intuition for this or that result is (of course also using what the prof is saying)

Same with reading a textbook: it's useless if I don't fill in the missing details

Also I suck at remembering stuff I hear or see, especially when it's technical

Writing it down works much better for me

Also in maths/theoretical physics free recall can only be done if you've already understood at least the broad strokes of a particular argument

[vagrant meadow]

curious to get thoughts on this: how would you motivate the definition or idea of a general field to students who only know up to calculus? for the purpose of introducing ideas of linear algebra.

trying to consider a potential alternate approach to linear algebra where we use the reals to motivate the idea of a field and then use the idea of a field to motivate the idea of a vector space. if we consider vectors as functions to a field, a lot of the vector space axioms come pretty naturally.

[austere delta]

I think the best approach is to say a field is a system where you can add, subtract, multiply and divide.

Then rationals, reals and complex numbers are familiar examples.

The integers are not a field because you can't divide.

Then if you're feeling gutsy you can even show that Z/2 and Z/3 are fields.

[vagrant meadow]

haha, it's funny in my draft, i kind of do that exactly! i put Z2 as an exercise though

[austere delta]

Smart minds think alike, as the saying goes

[tawny slate]

a small idea because its what ive been doing recently

so im working on trying to understand all of the algorithms to fully solve hypergeometric identities

and one of the things you have to do is take these recurrences and solve them

each term can have a coefficient, but these coefficients are not real numbers but polynomials

so perhaps this could be one example of how to motivate not just the definition but also application

i wouldnt be able to do the steps if polynomials didnt form a field (technically i guess its the rational functions not polynomials but it is trivial to convert them in this case)

maybe this particular application is a bit hyperspecific but maybe it might inspire other ideas

[tight star]

I’m curious whether introducing the idea of “syntax vs semantics” and emphasising that fields are focusing on the “syntax” of arithmetic would be useful

I’ve not tried it to be clear, but I’ve found this perspective to often be useful

[heavy trail]

Have you read the introductory chapters in Spivak's Calculus? He attempts that, there.

[vagrant meadow]

i haven't. checking it out, it's quite interesting

[heavy trail]

It was my first-year college undergrad calculus textbook.

[quasi musk]

High school algebra holds: you can add, subtract, multiply, and divide (except by zero)

[vagrant meadow]

here i've defined F^n as being the set of functions from n:={1,2,3,...,n} to F (a field). i like this perspective because it makes the way we add and scale those vectors very obvious (based on a definition that (cf+g)(x)=cf(x)+g(x), which is a natural definition from the nature of a field), and it motivates and gets them thinking about coordinate vectors before even learning about basis (i do try to make it clear that this is merely the simplest choice and not the only way to "represent" f as a list of elements of the codomain).
it requires a heavy foundation and buildup, but i'm hoping the payoff is worth it and it does indeed make linear algebra more intuitive and feel more connected...

[turbid zenith]

So here's something I'm giving my students in my liberal arts math class

They're going to make mathematician trading cards, where they research two mathematicians and at least one needs to come from a traditionally underrepresented group

Any suggestions for what else would be good to put on said cards?

[quasi musk]

I'd say an interesting quote from them

(As long as its class appropriate)

[vagrant meadow]

"hocus pocus"
-Wittgenstein

[turbid zenith]

Also, wow, I am having a hard time coming up with probability problems related to casino games that ChatGPT can’t do

[vocal phoenix]

Mathgic: the Grothendiecking

[tawny slate]

curious, how would you guys explain the concept of infinity to someone? feel free to provide your own context, like is this towards middle school students or college-level students, etc

(i have my own opinion which maybe ill share after some others chime in)

[vocal phoenix]

I like to stress that "infinity" is a very broad and vague concept, and mathematicians don't really have a definition of "infinity" as such, instead talking of infinite sets or infinite limits.

[turbid zenith]

I often emphasize that there are many ways of thinking about infinity, and there's not one "correct" way of thinking about it. I often end up talking about the difference between potential infinity and actual infinity in some way.

(And I see both as valid in their own way.)

For potential infinity I give the example of repeatedly adding 1 to a number and pointing out that (at least within the relevant number systems) you can do so as many times as you want

For actual infinity I sometimes describe it as the "end" of the number line, whatever that might mean, and give the example of perspective drawing, vanishing points, and projective geometry

[tawny slate]

purely from a personal feeling, not sure i especially like the "potential" vs "actual" dichotomy here but i suppose it gets the idea across and as you said, no real wrong way to think about it

the projective geometry example does give it a strong motivation

[tawny slate]

i think i would give the same general gist, that there isnt a single way to define it

i would caution that infinity isnt usually something being bigger than everything that exists

its more about "structure" like being able to be paired with a proper subset of itself (id eli5 this part)

or that it is never ending

[feral spade]

I might or might not have hinted to people that infinity is analogous to etc. that we use in language. You might want to stop your list with etc, but you and everyone else knows that the list goes beyond what you listed and adding one more item to the list doesnt change the usage of etc.
On the other hand etc doesnt mark the end and similar to infinity it makes no sense to think of it as a concrete example of an item of the list

[vagrant meadow]

rather than explaining what it is, it might be easier to explain what it isn't. by properly understanding what finite means, we can understand what infinity is not, and hence what it is.

[vagrant meadow]

i've been looking through a couple linear textbooks, and i'm not seeing the result that every vector space has a basis prominently displayed. is it common to just... not mention that? it feels like a pretty central and important result.

[heavy trail]

Depends on the sort of linear algebra textbook it is. Does the textbook deal with non-finite-dimensional vector spaces, for example?

[vagrant meadow]

i was looking through a couple of each. FIS and Axler don't really stray away from that, but i couldn't find the result. i didn't see it in Anton or Strang either, which i suppose is more understandable.

[midnight scarab]

Idk it feels like such a basic result. My course definitely had a proof (for general dim with AC swept under the rug iirc)

But like picking a basis is a very natural reflex, especially for beginners
And then when you get to modules the first thing you're told is that they're not all free

[vagrant meadow]

i would say a basisc result, but the proof is kind of hard to present for most students in an intro linear algebra course. would you mind sharing the proof or how the prof did it

[midnight scarab]

Disclaimer: it was a second pass at lin alg (in my 2nd yr)
And also I lied we did the proof for finite dim (and then had a separate non-examinable part on Zorn)

Also the proof is meant to also prove you can extract a basis from a set of generators indexed by I and complete a (finite) set of independent vectors indexed by J [edit: I ignored J here but it's easy to include by imposing A maximal among J incl A incl I, and taking I' = union I_k union J]

If I is finite, pick A of maximal cardinal such that the family of (ei, i in A) is free

Then for i not in A, adding it to A would lead to a linearly dependent family by maximality of A

And the culprit for linear dependence is necessarily this new ei since the others were independent

Thus ei is in the span of {e_j, j in A}

So the ej, j in A are not only free but also generating

That's enough to prove a basis exists for finite dim vect spaces since this means a finite I exists

(Then there's a second part which tells you what to do if you want to extract a basis from an infinite set of generators)

[thorn sundial]

Hello guys

[midnight scarab]

Where essentially you take a finite generating family (which exists by finite dim) {s_k} which you decompose on the {e_i}

[thorn sundial]

I have a lil question

[midnight scarab]

Each s_k has only a finite set I_k of non-zero components

[thorn sundial]

may I ask it ?

[midnight scarab]

Which you put together to get an finite I' included in I with {e_i, i in I'} being enough to generate the sk and hence all the space

[midnight scarab]

Is it about teaching math?

[thorn sundial]

yes

please

[midnight scarab]

Well if it's appropriate to the channel then I don't see why not

[thorn sundial]

So I'm tutoring a new student in uni, who started in september

and He asked me a good question, I didn't have the answer

he said me "It can seem dumb, but how do you properly work on a chapter in maths ? Just started uni in september and I have a hard time being efficient"

And I think it's not as dumb as it seems

what would you say him ?

[thorn sundial]

is my question appropriate ?

[vagrant meadow]

so I guess I'm curious to get opinions. should this be mentioned to intro linear students?
I suppose one reason it wouldn't be is that the proof requires set theory, which the students have surely not taken or really learned yet.
still, it feels pretty important to be able to guarantee that a basis always exists. even if 99% of the vector spaces they study already have an obvious basis.

[heavy trail]

still, it feels pretty important to be able to guarantee that a basis always exists. even if 99% of the vector spaces they study already have an obvious basis

Why? Important to what end?

[vagrant meadow]

imo basis is the core of linear algebra. the idea that we can represent every vector in any space as a finite sum of a fixed set of vectors is one of the main things that make things so nice. we need not study all uncountable vectors in R^3, we just need to pick three nice vectors, and we understand the whole space. the fact this extends to any vector space, including those of uncountable dimension, is quite nice (and allows us to extend the logic of most of our definitions quite directly.

do you disagree?

plus, my main focus is trying to simplify the idea of what a vector space is. the fact that every vector space is isomorphic to a set of functions of finite support to a field makes a lot of the properties more intuitive (and follow pretty directly from the axioms of a field). however, that rests entirely on the fact that there exists a basis.

[heavy trail]

I think thousands of students learn linear algebra every year without getting caught up on whether every vector space has a basis.

What difference does it make whether they learn that on day 1 or day 100?

Like, what material difference would it make to their learning trajectory?

At the very least, the question should be held off until they're confronted with a vector space where they go "I wonder if this has a basis or not?"

[vagrant meadow]

I think the fact that vector spaces have a basis is a useful way to think about them as a concept. that's what I'm saying.

the fact they're over a field guarantees they have a basis. which is a pretty incredible property. and simplifies the way we can think about them.

[heavy trail]

I'm asking what difference it would make for a student learning linear algebra for the first time. How would it affect the course of the learning? What difference would putting that up front make relative to introducing it on Day 10 or Day 100 or Day 300?

It seems like if there's not a clear answer to that then it can't be so essentially, pedagogically.

When are they going to get their hands on a vector space where a basis isn't obviously at hand? When do they encounter their first infinite-dimensional vector space, for example?

[vagrant meadow]

the current pedagogy is usually

• Rn and unmotivated matrix reduction
• guess what? there are vector spaces that aren't Rn. guess we have to leave our column vectors and matrices behind...
• just kidding! coordinate vectors and basis allow us to treat any finite dim vector space as Rn. get pranked bro
  I feel like this misleads and confuses them.

[heavy trail]

Ok, sure.

I don't disagree. That's not how I was introduced to vector spaces.

[vagrant meadow]

I'm working on my own approach to the subject that tried to make a more holistic approach. deriving the intuition of vector spaces from fields and the existence of a basis.

[heavy trail]

If I have skepticism it's this: to really prove every vector space has a basis you're going to need to pull in something like Zorn's lemma.

What students live at the intersection of "I want to learn linear algebra" and "I'm interested in engaging w/ arguments involving Zorn's Lemma on Day 3 and see how this is connected to be interest in learning Linear Algebra"?

[vagrant meadow]

if B is a basis of V, then any vector v can be thought of as a map from B to F with finite support. where if
v=c1v1+...+cnvn
then v(vi)=ci (or 0 if not part of the sum)

[heavy trail]

I'm not arguing against a more rigorous or abstract approach. My first-year calculus course in college used Spivak and it convinced me to major in math.

Before that, I came from a rural school that hadn't offered calculus ever before, until my senior year.

[vagrant meadow]

right, this is one of the things I'm trying to figure out now. atm I'm starting with functions to a field and using that to motivate vectors, with the goal that eventually we can show every vector space is isomorphic to a set of functions from a basis to a field

[heavy trail]

I...

Well, ok.

What's your background? Have you taught a class of college students before? A linear algebra class?

(I've taught at the college level, albeit not linear algebra.)

[austere delta]

Is that the usual approach? My experience is more like:

• equations in R^n motivates matrix reductions.
• define linear transformations and observe that matrices are just that
• realize that there are other vector spaces than R^n that appear in the real world (like solutions to differential equations for example)
• show that we can use basis to study these spaces with coordinate vectors

[heavy trail]

My question is: who is this for?

[vagrant meadow]

not as a primary yet, but I tutored it for years and also been the TA a couple times. last time I basically was the lecturer for most of the students bc the professor was just reading the textbook to them.

I'm not so sure this approach should be the standard, or necessarily used as a primary text in a classroom (maybe supplemental material idk). the intention atm is for curious self study-ers and those wanting a different approach. I'm also focusing a lot on the linear ODE applications. way more than normal.

[heavy trail]

My introduction to linear algebra was very similar to this and I think you're over-thinking it. Focusing on this "every vector space has a basis" issue is a red herring.

I have the original handouts from the first time I was introduced to vector spaces:

I think Theorem 2.127 is about all that needs to be said on the matter for students w/ enough mathematical maturity to engage w/ the material this way.

[vagrant meadow]

this is definitely better than what I usually see

[vagrant meadow]

yeah this was something I was seriously considering. I don't think the actual proof would be particularly helpful.
I am explaining the idea though as an algorithm that uses choice

• take a nonzero vector
• if it generates we're done. else add a vector not in the span.
• repeat

(without mentioning choice)

[heavy trail]

Who is the intended audience?

[vagrant meadow]

my hope is that motivated students who only know calculus but are curious about LA could get something out of it. but otherwise more people who might be frustrated with the general linear algebra/ODE pedagogy

[heavy trail]

I think you need a clear picture of where they're coming from.

For most students, the abstract definition of group/ring/field is already huge leap in conceptual sophistication. It's like they're stepping on Mars for the first time.

Most students who "only know calculus" have not had to prove anything while doing calculus.

If you expect this to be students' first exposure to abstraction at this level then you're simultaneously teaching the content while shepherding them through that way of engaging for the first time.

That's hard and, tbh, one reason I mentioned Spivak before is because it's the only textbook I've seen that does it self-consciously and does it well.

[vagrant meadow]

yeah I agree. my approach for introducing fields is to just generalize the real numbers. like the idea is that the definition of a vector space doesn't change if you use complex scalars or rational numbers. the main thing we need is just to be able to divide scalars. so all they need to really understand is that fields are just the general concepts of nice numbers.

I have to go so I'm not explaining it as fully as I'd like but I'll come back here and respond more fully

but I try to emphasize that the real numbers should be the primary intuition. so thinking F=R or C is fine, and as much as they really need.

[heavy trail]

Sure. I know I directed you at the first chapter of Spivak before, but this is precisely what that chapter does.

There's a real artistry to it, too. Dedekind wrote a famous paper in 1888 called What are numbers and what should they be? where he first outlined ideas like Dedekind infinite, Dedekind cuts, etc. "What should numbers be?" is the question Spivak opens with.

And the book closes w/ the Dedekind construction of the reals, with all of single-variable calculus developed in between.

[vagrant meadow]

yeah I started reading Spivak I need to finish that chapter

[heavy trail]

There is a kind of student where the usual answers to "Why does -1 * -1 = 1?" ring hollow. There are metaphors and whatnot about reversing a reversed thing and it always seemed arbitrary to me.

But if you really grok the point of the "structuralist" attitude then you see it doesn't take much at all for -1 * -1 = 1 to be true.

[vagrant meadow]

but one of my main grievances is the pedagogy of linear independence and span. every quarter my students say they don't understand span. just saying "the set of all linear combinations" means nothing to them. similarly, linear independence is just a computation and they don't understand what it means.

for me, the idea of basis ties these concepts together. introducing span as "generation", and linear independence as "uniqueness of linear combination representation" (especially useful for those trying to understand linear independence in ODEs) makes a basis intuitively the simple concept that "every vector can be uniquely represented with a finite linear combination of these vectors". the focus being on how this completely simplifies how we approach vector spaces, and motivates how we deal with linear transformations: because they're defined to preserve the structure of a v space (linear combinations), we simply need to look at the images of a basis (because every vector is uniquely represented by them).
this is why we define matrix multiplication the way we do, taking linear combinations of the columns: the columns correspond directly to the images of the basis vectors. this also makes change of basis very intuitive, because it's the same concept just under a different basis.

[midnight scarab]

Isn't everything you've just said the way it's normally taught (except perhaps for the ode part), or was I just lucky?

[inner fog]

anyone seen MathAcademy? seems really cool on first glance — curious what people's thoughts are!

[vagrant meadow]

i think you were just lucky. most of the time, my experience is that

• subspace is the three rules (has 0, closed under addition, closed under scaling). no idea why the heck we care about that or what that means. showing something is a subspace is just rote verifying, and the intuition of the simple idea it's closed under linear combinations, or just a vector space contained in a vector space, is ignored.
• span is just "the set of linear combinations" (usually no direct emphasis that it's a subspace). maybe if they're lucky they'll see a geometric idea that it's a line or a plane. still, there isn't really a motivation for why we would care about span at all. it's just a weird, hard to understand, seemingly pointless set to them that keeps getting mentioned and used, and they don't understand why.
• linear independence is just "the only solution is the trivial solution" (why the hell do we care about that??).
• basis is just a set that spans and is independent. no emphasis on the consequences or uses of that. it's just another term. by the time it's explained, they feel overloaded with vocabulary, and when they see "basis" they either forget what that even means, or have the concepts of linear independence and span as separate compartments in their mind (and don't understand how those concepts interact or relate)
• matrix multiplication is just row column dot products because i guess that's how my professor says we do it; i don't know why. why are pivot columns a basis for the column space??? why do we care about the columns at all???

these are the issues i've seen tutoring and TAing the course over and over and over again

a lot of the time, span is talked about around week 3 or 4 (out of 10), but still even by week 7 i still get students asking "can you explain what span is again?"
and it's frustrating because we've been using span constantly since it was introduced. and either i spend some of the precious discussion time trying to explain it again (even though i've literally made my own handouts and explained it multiple times) or i'm forced to just leave it as "set of linear combinations. sorry, that's how the professor is teaching it"

can't really emphasize the direct connect it to subspaces without them getting confused because that's not how the professor is teaching it, and subspace and span are in separate compartments in their head.

same issue with the redundant vocab. null space and kernel are separate concepts to them. column space and range are separate concepts. why do we insist on bloating a subject full of new vocab with completely redundant terms?

[heavy trail]

Empty formalisms are bad, but given these students, how do you see your approach addressing that emptiness?

[vagrant meadow]

wdym?

for that course, i only had two 50 min sessions a week. not enough time to really go through and reteach the concepts.

i did try my best through my worksheets to connect them

[heavy trail]

Maybe I misunderstood. Do you see your approach as addressing/avoiding the issues you just laid out?

[cloud zealot]

should linear algebra and ODEs even be taught together? i've taken such a course in community college and the course felt very rushed and crammed. i didn't really understand linear algebra or ODEs at all. it's only when i've had to look up facts from linear algebra in my abstract algebra class and sorta watching videos that i started to get it

[vagrant meadow]

yeah, my goal is to connect these concepts directly. imo linear algebra is actually pretty straightforward, it's just taught super confusingly in a way that makes it seem misleadingly hard.

i guess i'd say the normal way is a more top down approach, but i want something more bottom up. the foundations and motivations behind the definitions of linear algebra are quite simple.

[vagrant meadow]

honestly, no. but my strong opinion is that to properly understand linear ODEs you need linear algebra.

so i focus on the foundations, and then go ham on linear odes with the perspective that it's essentially the greatest example of applied linear algebra. we just go as far as we can doing linear algebra on the differential operator

[heavy trail]

What are "these concepts", specifically?

[vagrant meadow]

the seemingly arbitrary rules and results (why do we guess an exponential to get the char poly? why do we multiply by t when the root is repeated?) are easily derived through linear algebra

[vagrant meadow]

linear combinations, vector spaces and their axioms (how they follow directly from fields being nice), generation and span=subspaces, linear (in)dependence, basis, linear transformations (and how matrices and their operations are defined to easily and compactly encapsulate/define them), image, and kernel.

[midnight scarab]

I think that what Jagr outlined here is really natural

Like both for a newcomer and from a "higher" perspective

[vagrant meadow]

if it's done well, i agree it can be good. but most of the time the intuition is ignored or not emphasized.
plus, i would say the matrix thing is backwards. you can define matrix multiplication to act in place of linear transformations. i don't like teaching them the arithmetic just like "that's how we do it" and then saying "hey look it happens to act as a linear transformation!" rather than consider the definition as being such that it acts as a linear transformation.

and also, i still dislike teaching "vector spaces actually aren't just coordinate vectors. jk they are"

but one of the other motivations of my approach is to prepare students for using the concepts in linear algebra to other subjects (like ODEs and abstract algebra).

the redundant terms and seemingly arbitrary choices and "coincidences" just really really bother me

[cloud zealot]

why specifically emphasize the existence of bases rather than discuss isomorphisms at perhaps a more abstract level?

[austere delta]

I think emphasizing that vector spaces are more general than just R^n, but that we can still study them through coordinate vectors is a good thing. And that sometimes the choice of coordinates is somewhat arbitrary.

You don't have to frame it as a gotcha moment of course, but I'm not sure it usually is

[vagrant meadow]

the current approach (not sure it's how i'm definitely doing to do it) is to

• define fields as generalizing the "nice"ness of R. using F to essentially denote a system of "numbers" which work in the way we've come to expect the real numbers to (but acknowledging that C and Q also have all those same properties, so there's absolutely no need to restrict ourselves to R)
• consider functions to a field (perhaps with finite support) and define the operations (cf+g)(x)=cf(x)+g(x) (generalizing how we do function arithmetic in R). the functions specifically from {1,2,...,n} to F (i.e. F^n) naturally lend themselves to being written as a column or list of entries in F, and the function arithmetic entirely determines how we define the arithmetic on these lists (rather than saying, "this is the most obvious way to define operations on column vectors: entry-wise addition and scaling all entries")
• define a vector space, motivated by the field axioms on these functions, and essentially consider them as an algebraic object where the "operation" is linear combinations. then F^X is clearly a vector space.
• emphasize the idea of generation to simplify how we deal with vector spaces, and specifically using a generating set with unique representations makes everything much nicer (and intuitively allows us to treat a general finitely generated vector spaces as columns in F^n, with the exact same logic we used before)
• to really cement that vector spaces are actually really predictable (they are isomorphic to some F^(X) (functions from X with finite support to a field)), then you need a basis. then X can simply be the basis.
  definitely not final, but i like this perspective because it makes the definitions and concepts feel natural to consider. further, there is no misleading or deception towards the students, nor do we have to retread old topics because we made the choice to define only R vector spaces. like why say halfway through the class (or never) "guess what, scalars can be complex and nothing about our vector space definition or basic operations changes".
  however, i do want to emphasize R^n as our primary example for intuition.

[vagrant meadow]

no i totally agree. but i think being upfront that though this is not a coordinate space, we can still treat it as such and use that intuition is helpful.

[heavy trail]

The devil's in the (pedagogical) details, but it's hard for me to see how a student who felt spans, subspaces, basis vectors, linear transformations, etc. were arbitrary abstractions and empty symbolic manipulations would see this and think, "Well, now I see how everything is connected."

I agree that a student who managed to successfully tackle the material along the lines you've suggested wouldn't have the problems you described before.

So in that sense this approach is sufficient to address the problems, when it works.

The question is whether it is more likely to work with the students who have "some calculus", not much experience with abstractions, and little-to-no prior exposure to linear algebra.

[tawny slate]

little late to this, but i don't think this actually works that well
for instance, let's say you have a student that doesn't believe that anything infinite really even exists. so they might then define finite as "anything that exists", and suddenly you have a weird definition of finite. they need to have a good idea of what is finite and what is infinite to come up with a prescriptive definition like this
another example is a student who disagrees with me that pi is finite. like, there's something about it that is infinite, sure, but it's a finite value, and now they say "decimals that go on forever are infinite", upon which now they are confronted with an idea that 1/3 is infinite but 1/5 is not, and now it just gets more confusing

[vagrant meadow]

right. this is the hope. that the time spent building up the abstract foundation will pay off. i'm not sure it will, and i expect to go through a lot of revisions based on feedback.

[vagrant meadow]

yeah, i'm skeptical as well, and i'm not claiming my approach is guaranteed to truly be better than the standard pedagogy.
but i'm working very hard to try to keep it very cohesive and natural.

admittedly, it would likely be difficult for a student already in a linear algebra class to open up the doc, skip to the section on generation/span/subspaces and be able to understand what's going on. what i have so far is kind of written like a narrative that builds on itself. i suppose that's a "benefit" to keeping the topics disconnected: it's easier to use the book as a reference.

right, this is a good point. in this perspective that every v space V is isomorphic to F^(X) (a finite support function from X to F), X might be impossible to write directly (though you could use an index set of size |X|). but, yeah, if you can't write the basis then it's hard to use it.

however, my hope is that by acknowledging that every v space is indeed isomorphic to F^(X), it provides a better intuition for what a vector space is, and the intuition behind them. i.e. we know for sure that V is "nice" because it inherits all the nice properties of a field. and the "structure" is still predictable (everything can be written uniquely as a finite linear comb of elements of some basis).

[cloud zealot]

a comment on this video really resonates with me:

Everything here perfectly dovetails with my observations in 20 years as a teacher. However, there is this huge issue with attention: our whole K-12 system (and maybe higher ed--I don't know) is now so focused on the idea of the product, and this whole transactional relationship where the student pays for grades by turning in assignments. For so many students, their attention span is so short that they really struggle to think past "what's due today?" and if you tell them to just listen and think, they don't take that seriously. There's no product. And even if they know a recall opportunity will happen later, that's a next-class-me problem, or even a latter-this-class me problem. This mindset can be fixed, but it's a process, not a one-time lecture at the start of class. It frankly takes most of the year for most students. So if I tell them not to take notes, but to pay attention, they just stare off into space (or look at their phones, or whatever).

Basically, the current model is deeply invested in the idea that teachers can use instructional design to "force" attention and processing: Cornell notes, graphic organizers, demonstrations of learning, etc., etc., all take it as given that higher order thinking has to be forced out of kids through some sort of product that we can assess. Students then, of course, play the game we set up: they find cheats and hacks to create the product without the thinking--from their point of view, this is being efficient. They think the product is the point, because no one has ever said otherwise. So we try to use more and more clever instructional design so they can't "beat us", and we become obsessed with stopping "cheating" instead of noticing that no one is really learning. And if there is no "product" due, they think that means that what we are doing isn't important.

[midnight scarab]

(Also for people who're actually interested in learning, having to focus on the "product" in order to get good grades (to get good learning opportunities in the future) takes away from time that could be spent more efficiently on learning

[flint wharf]

well yes, but you don't need proofs to understand this; 3blue1brown does this beautifully with geometrical explanations that i understood before ever taking calculus. to me, your approach is going too far the other direction: in wishing to share the beautiful connections you see having taken far more math than they have, you're bogging them down with even more vocab, notation, and structure. the connection is absolutely fundamental and graspable at their level, but the image i have in my head is fundamentally geometric, not based on the definition of a field and proofs that every vector space has a basis. if you just say 'look, we can get to every vector in our vector space by scaling and summing some of the vectors within it - look at this example with the cartesian plane' that's incredibly believable and comprehensible

[flint wharf]

...this is for a first course in linear algebra?? it's enlightening to you because you know the formalisms already and have the experience to get through the definitions, but to someone who has never seen linear algebra before this would, imo, leave a lot of students in the dust. start with what's going on in a familiar space like R2, generalize outward while preserving the geometric sense of the situation

like: 3blue1brown's videos taught me linear algebra as an 8th grader, and gave me a conceptual model that carried me through all of my linear algebra courses in college. when i explain linear algebra concepts to students using similar explanations, it clicks. to me, they are the gold standard of how introductory linear algebra should be taught. fields and general vector spaces aren't discussed until the very end! i'm not saying it needs to be in precisely the same order and a full intro lin alg course should go farther than they do, but goodness. if you have students lost in the formalism and definitions (and i agree, rightfully so - i think lin alg is not often taught well), the answer is not 'great, let's put in more formalism and start from what a field is!'

[vagrant meadow]

idk about a course. more of an introductory resource. i think the geometric approach is good, and i always encourage anyone i tutor or who is in my TA section to watch 3b1b's essence of linear algebra.

but my issue is that the fundamentals of linear algebra are actually more basic. the geometric approach describes just one aspect or a subset of this more general concept of a vector space, which is simpler than 10 seemingly arbitrary axioms of a real vector space. i'm not entirely convinced that it's impossible for students to be able to follow a more bottom up approach.

i am considering starting with the more concrete version of functions to R instead of a general field. but i do still think that learning linear algebra simply requires the idea of scaling by a "number" which can be added/subtracted/multiplied/divided. using R as a primary example could be just as good, i'm not decided.

[vagrant meadow]

im also still not decided about including the formal proof every vector space has a basis. nor am i writing expecting more than calculus. but i think any student who has taken calculus would be able to understand the basic idea of a field at an intuitive level. even if their understanding is just "F means R or C and maybe some other stuff".
and that explanation of basis you give at the end here isn't too far off from what i have.

again, this is an alternative approach which is untested and unfinished. i'm still not sure what my target audience is. but i disagree with the sentiment that abstraction is impossible for students to understand, and only a geometric approach will be digestible. still, i need to see where it goes as i work on it.

[quasi musk]

Interesting, I almost never recommend the 3blue1brown videos to someone in a beginning course in Linear Algebra. My main fear is that watching a video is too passive - providing a false feeling of concrete understanding that actually won't help them get through their assignments

Ideas are fun to play with, but I prefer something with concrete calculations done

[vagrant meadow]

mostly just as intuitive prep. to plant the seeds of intuition in the concepts they will learn.

but yeah you can't pass a LA class from 3b1b alone

[quasi musk]

My argument isn't that "this alone will do it", but rather the videos will lull people asleep to "Ohhh I get it now" and they don't really know how to compute anything. So it's not a neutral thing to watch, I think it's actively harmful for most students to watch in their first linear algebra class

[heavy trail]

If a listener nods his head when you're explaining your [proof], wake him up.

— Alan Perlis (https://www.cs.yale.edu/homes/perlis-alan/quotes.html)

[raw sedge]

I wouldn’t go so far as to say it’s actively harmful, but it’s entertainment not educational content. It should not be construed as something where “you can’t pass a la class from 3b1b alone” is even related. It’s like saying I watched dudeperfect throw a basketball off a dam now I know how aerodynamics works.

Unrelated, I tend to make up random numbers on the spot when giving demonstrations and that leads to some crazy fractions when I try to invert curves or find intersections. Does anyone have some simple tricks for keeping examples in integers besides ‘solve it then adjust the inputs to make the numbers prettier’ or work the problem backwards from numbers that look good and present those?

[vagrant meadow]

i disagree. especially when in week 7 of 10 students are asking "what is span again" (a week 3 or 4 topic, used continuously throughout the rest of the course). watching the 3b1b video and getting a good intuitive grasp of span is quite helpful imo.
it's obvious the normal/algebraic way the book teaches span/lecturer teaches span just doesn't cut it. many students greatly benefit from that geometric perspective.

[long pelican]

I’ve been liking Zundamon’s theorem lately (a Japanese YouTube channel). One nice thing about it is that it is dialogue while 3b1b is monologue. Anecdotally I process and remember dialogues when reading much better than exposition, for what it’s worth

[tawny slate]

i think this criticism is weird because i feel like this could be said about any lecture or textbook or really any media that explains a concept well

i think its more important to instill in students the idea that understanding how an idea works is only the first step, that practice helps iterate and solidify that understanding

[tawny slate]

i think youd need to be more specific about what kinds of problems, because i think abstractly these are the only ways to attack a problem

[turbid zenith]

I wish more channels did this!

[rare gulch]

I feel exactly the same as you. I dislike 3b1b videos. The relation between the concrete definition, and his explanation and illustrations, is very ambiguous.

I also remember seeing examples where the idea is too specific to one case, and it simply can't generalize to most cases. Even in a more abstract class like topology or analysis, you can imagine almost every concept in R^2, and it generalizes well.

So it's just a 3b1b thing, is not that intuition cannot be done visually or in simpler words.

[night grove]

I made up these questions. What do you think of them?

[turbid zenith]

Are any of y'all on bluesky? I've been meaning to move there since things got shitty on Twitter, and a bunch of the math education people I used to talk to on Twitter are all over there.

(It was honestly some of the best professional networking I've had.)

[tawny slate]

ive only heard good things about it

i try to stay off social media though

[deep kindle]

I'm still stuck in the ways of Reddit

[tawdry venture]

im doing 1 on 1 tutoring at my job (i work at a private school) as part of an arrangement w my direct supervisor, as the kid is a strong student in math in particular. we've arranged with each other to have a mock A levels paper (Paper 1 specifically) next Wednesday (13.XI) and we are going to meet the Tuesday before it. is it a good idea to dedicate Tuesday to revision of any topics my tutee has doubts with, as opposed to covering new material?

[quasi musk]

It doesn't have to be one or the other. You can dedicate some time to going over things the student wants to, and you can dedicate some time to going over newer stuff

[deep kindle]

I've never seen roman numerals used in a date format and I like it

13.XI

I'd like to assume that a typical student would benefit from review the day before a test as it keeps concepts more fresh in their mind, although this is also consistent with the traditional expectation of students to simply memorize things---Practicing problem solving strategies over rote memorization is always better

That trend is consistent with what I've seen in the US system, where class periods before a test are often spent in review as opposed to covering new material, unless the students are more autonomous (like in a lot of university courses)

[tawdry venture]

yeah ive taken it up recently. it used to be a thing in the soviets and i personally think its a good one cause it leaves no room for doubt which number is the month (it's the roman numeral)

[tawdry venture]

oh no we're not gonna drill memorization lmao

strategies and approaches is what i was thinking of too

[tawdry venture]

yeah ig we have a whole 5 periods together so we might do some of both -- i just don't want to hit him with a lot of new and heavy stuff right before a mock

though what i can do is message him on Teams telling him to prepare a list of things he's not 100% clear how to do, no matter how small

[wary thunder]

I'm not sure where else to ask this question. I am writing an expository paper that will be used as teaching material for a graduate analysis course. I've never written an expository paper before (though I will obviously be guided). Could anyone recommend some particularly good expository papers that they enjoyed? It will be rather short, maybe 20 pages.

Writing textbooks is an eventual goal of mine (I'm very passionate about writing). Do I have more freedom with an expository paper to put some style in? For example, Lang writes--while explaining the proof that \sqrt(2) is irrational--that "However, to find a rational number whose square is 2, the procedure is a bummer because of the following theorem." This sort of tone I find very useful for teaching. It's much more personable, playful, and sounds like you're talking to someone.

Basically, just general "what's allowed" for an expository paper.

[lethal leaf]

http://www.maths.lse.ac.uk/Personal/stengel/Sturmfels_PolynomialEquations.pdf

I cannot answer the latter parts of this as I have never written anything expository before, but you asked for some good expository papers so here's a good one.

[tawny slate]

not sure if A=B counts, and its also pretty long, but you probably only need to read the first chapter or so:

https://www2.math.upenn.edu/~wilf/AeqB.html

[lethal leaf]

I really need to check out that book

[deep kindle]

If $P(A|B) = \frac{P(A \cap B)}{P(B)}$ and $P(B|A) = \frac{P(A|B)P(B)}{P(A)}$, but $P(A|B)P(B) = P(A \cap B)$, why are we taught two separate formulas for conditional probability and bayes' theorem when they can be combined into $P(X|Y) = \frac{P(X \cap Y)}{P(Y)}$?

[burnt vessel]

m. frost

[deep kindle]

In both formulas, once substituted, the probability of some event X given some event Y occured is the events' intersection divided by the given event

I feel like that statement is more valuable to remember than both formulas, as students tend to mix up the placement of A's and B's

I guess given context $P(A|B)P(B)$ may be easier to identify than $P(A \cap B)$ and vice versa so they're sorta interchangable

[burnt vessel]

m. frost

[heavy trail]

What you've written is basically a proof of Bayes' theorem, given the usual definition of definition of conditional probability.

I'm not sure how to answer "why" both are taught. We teach lots of things are are "easily" derivable from other facts.

[deep kindle]

There's a perspective of developmental learning theory of course

I guess students who are more comfortable with concrete concepts would benefit from just being given the requisite formulas

[heavy trail]

I just don't think these things are actively considered in most teaching contexts.

[deep kindle]

While students more suited to abstract concepts would benefit from seeing the proof of bayes and extrapolating how both formulas do the same thing with slightly different contexts

They're things that I would like to consider thus the question here

[heavy trail]

I imagine most books that introduce Bayes also include the short derivation, but don't emphasize anything "metacognitive" about how students could or should make use of that derivation. How students make use of it is between them and their God, so to speak.

Maybe the professor makes the metacognitive point that they needn't "memorize" Bayes, but can re-derive it from the definition of conditional probability.

I suspect most professors don't say anything about that and, like the book, leave it to the students understand it in whatever way suits their temperament and prior habits.

Same with anything else like that, e.g., the quotient rule for derivatives.

Did the student commit it to memory as a fact unrelated to any other rules for derivatives? Or did they re-derive it using simpler rules, as a way to sanity check themselves?

Do we care? Does anything in the way we assess distinguish between those two students?

[deep kindle]

¯_(ツ)_/¯

Just askin something I was wondering

[wispy slate]

https://youtu.be/3l1RMiGeTfU?feature=shared

[quasi musk]

https://www.math.toronto.edu/mccann/199/thurston.pdf

"How do mathematicians advance human understanding of math"

[agile mantle]

For those of you who are tutors, what do you use to write your notes?
I’m torn between purchasing an ipad or an e-ink tablet for tutoring purposes. I’d like some input on which you guys use and or prefer

[tawdry venture]

like online?

i use a graphics tablet plus whatever image editing software (but not really any specific note taking app)

[agile mantle]

Right
I’d like to share my screen as I write and then be able to save and share the file(s) with everyone once im done

[tawdry venture]

yeah i mean even paint can let you do that

share your screen through whatever group calling platform you're on and then save the entire canvas as an image file and share that

[austere inlet]

since I do online stuff either on my desktop or laptop, I use https://xournalpp.github.io

basically FOSS equivalent of MS's OneNote or Apple Notes

minus the cloud sync ig, I use other apps to sync my files

[marsh compass]

I love xournalpp except for the fact that the windows version is buggy as hell

[agile mantle]

Omg

Does this available for tablets as well? This looks incredible

@marsh compass @austere inlet

[austere inlet]

there's a flutter port in the works, I haven't tried it though https://github.com/xournalpp/xournalpp_mobile/

on mobile I'd prob just use other more estabilished alternatives

[twin shell]

i like xournal

[lethal leaf]

I have an iPad

worth it's weight in gold IMO

it only runs two apps for all intents and purposes: my notetaking app and Zoom for tutoring

and just for that I would buy it again and again

I haven't tried Xournal tho

[agile mantle]

Goodnotes?

[lethal leaf]

Noteability

I got it before they went the subscription model

So I don't have to pay a subscription to use it

[agile mantle]

Ayeeee nice

[halcyon glade]

I'm late, but maybe these documents will help

The second one is specific to a class I took, but it should be pretty easy to generalize

[wary thunder]

These are very helpful. Thank you!

[wispy slate]

Hi there,

I guess this is the right place to rant (apologies if it is not the place for that).

I am a statistician, mostly doing statistical learning theory work (complexity, covering numbers on neural networks, etc.) I have had my fair share of exposure to mathematics by now. I have studied Tao's Analysis I and II, Measure theory from Royden, Axler's LADR, some of Axler's MIRA, I am currently doing a course on Functional Analysis, went through some chapters of Boy'd Optimization book. I am also currently working through a book on probability theory.

Let's focus on Analysis, as I said: Tao, Axler, Royden, which made me think that my foundations of analysis are strong. I opened Rudin's (Baby, Papa, Grandma) books today. There were so many results, so many chapters there that just seemed like I don't know any analysis at all. I mean, Banach Algebras, H^p spaces, Tauberian theory, the list goes on.

Is this a general feeling? Does this insecurity ever go away?

[quasi musk]

Yeah, if you open up a book you haven't read then you will find topics you didn't learn in general. Rudin is known for having interesting topics included. It's not that you don't know analysis, it's that you haven't applied the analysis tools you do have to look at such things

[wispy slate]

Would this mean that many of these topics are context specific?

[quasi musk]

I'm not sure what you mean by context specific. There are lots of ways of doing mathematics, there are lots of research areas to go into. Usually people write a book because they feel they have a unique perspective, and they of course write with an eye towards what they think is important

You certainly don't need to know everything in Grandpa Rudin to do research mathematics in analysis

[wispy slate]

Thanks, this is a nice perspective. 🙂

[midnight scarab]

Yeah, you'd learn H^p spaces in an advanced PDEs course, Banach algebras in a course on spectral theory etc

[stiff oriole]

hey yall, so sort of by chance, I've found myself tutoring a person studying for their GED - algebra type stuff. As a calc 3 student, I felt confident enough to take them on, and things are going well.

Anyway, I've realized after this experience that math tutoring at a similar level (maybe up to calc 1) could be a realistic and lucrative opportunity - moreso than my attempts at comp sci tutoring.

So, my question: How can I network locally and put myself out their to find more tutoring clients at a similar level? Should I use reddit posts? Make my own website? Posters around town? thanks!

[austere inlet]

I think it's most effective to reach the students directly e.g. in online groups for specific colleges or uni

or post ads in campus if it's allowed

that's gonna get you a few people and then word of mouth often does the rest

a simple static website is a nice-to-have, and social media presence e.g. uploading solved exercises or notes may help but takes time

[stiff oriole]

right, that makes sense

how can i know if i'm ready to tutor algebra 2, trig, precalc, early calculus, that sort of stuff?

as i mentioned before i'm pretty srong with algebra 1, but i'm scared i'll falter and fail at my job as a tutor in more advanced math, even though i've done it for years

[errant notch]

I've been volunteering at a primary school recently, teaching 9 year old kids (1 to 1, or sometimes in pairs) about multiplication for an hour every week. For some reason (this might be related to COVID, or the fact this school doesn't set any homework, or maybe general levels of deprivation in the area), they seem to be well behind in terms of their maths education- I am paired with a different pupil each week, and in my experience, roughly half of them seem to not know their ten times table (or even if they have it memorised up to 12x10, they don't seem to be able to extrapolate the pattern of "adding a zero" to anything higher than 12, even with a fair amount of prompting, which I found odd); an even larger amount struggle to multiply things by 2; and some of them seem to be completely unaware that conceptually, multiplication by a natural is repeated addition. So I'm having a dilemma as to what to prioritise in these sessions- I would rather not prioritise brute memorisation, because that seems like a waste of a contact hour for such a short term improvement. But then how much time should I allocate to focussing on the pattern recognition (e.g. multiplying by 5 or 10, or, for the more advanced ones, by 9 or 11, has quite a few patterns), vs the methods (eg "double it and double it again" to multiply by 4, or "times ten and subtract the number" to multiply by 9), vs the conceptual understanding (what is multiplication?)? All of these seem so important, and I have so little time with each pupil. Bear in mind quite a few of them struggle even with things like adding single-digit numbers.

[lethal leaf]

The secret sauce is Facebook IMO if you can tutor high school stuff

Like if you can tutor AP math courses (assuming you're in the US)

Or just high school level math in general

Then go on local FB group pages

Because what usually happens is that parents of children get and pay for the tutors for their children

Oh also standardized test prep is huge (ACT / SAT)

Also don't be afraid to ask your current clients to refer you to their friends / family

[stiff oriole]

ah okay, cool! i'll definitely try that

[errant notch]

even if the removal of homework itself is justified, the school's reasoning for it was awful anyway
the teacher explaining it to me said "the only kids who end up doing the homework are the ones whos parents make them work all the time anyway and pay for private tutoring etc"
which a) is bullshit and b) removing homework is literally not a solution? the only thing it can do is make the disparity in education between high- and low- income households bigger because kids who cant afford private tutoring now have zero educational stimulation outside-of-school

but like
in primary school i never did the homework because my parents made me do it
i did it for fun

[twin shell]

I think the concept of homework is kind of strange because most working adults don't have homework as far as I'm aware
so if by "useless" they mean "it doesn't reflect the real world" then "err, ok?"

it's apparent to me this is not good for training children to become independent thinkers tho

[midnight scarab]

Most working adults also don't go to school

And if anything, hw reflects the "real world" more than lessons at school

[twin shell]

yea
does that contradict what I said?

[midnight scarab]

I agree with the "err, ok?"

[austere delta]

Homework is a point of stress for many students. For students with limited time or resources to do the homework the stress might make it a net negative.

And there doesn't seem to be much of a correlation between the amount of homework and performance on standerdized tests anyway.

You can encourage students to be curious and work on things outside of school without it being a set of exercises they have to turn in or present. Like you say, kids want "fun" not "work". Sometimes it feels like schools main job is turning everything fun into boring work.

[turbid zenith]

Has anyone here ever taught a "Business Calculus" class?

I'm interested in doing a final project (potentially instead of a final exam) and wondering if anybody has any ideas. Business/economics is NOT my strong suit at all but I'm slated to teach this course so I want to make it interesting.

All the students are business, economics, or accounting majors.

[quasi musk]

I had to TA for it last year. The primary instructor did no projects, instead had a lot of word problems that were business-y related

[turbid zenith]

That's what I figure is most common

And like ... I'm certainly going to try to use as many examples as I can from econ/business

But it would be so much cooler if they could use it in like an example scenario

[storm hawk]

Anyone has experience running student learning seminars? How much time commitment is it as well as how hard is it to administer?

Also, was it a worth-it experience for you?

[austere inlet]

I don't think I've ever ran a successful one but that's because it's been like, getting some grad students together and then they didn't commit enough

so it's probably good to agree to a bare minimum like keeping notes or rotating speakers

as well as having a clear goal, e.g. specific book chapters or papers

oh come to think of it I have been part of an online study group for over a year and that one's been successful precisely because we've committed to it and every meeting has a clear goal

like, each meeting ends with "okay next session we read section [x] of [y]"

it's just 3 people so that makes it a lot easier

[storm hawk]

ahhh I really like this. Giving people tasks even if they are not presenting to keep them actively participating. -Note taking, -Typesetting the notes, -Presenting, there's at least three tasks that can be rotated.

[storm hawk]

I was thinking of making it bigger, but maybe I was thinking too grandiose. Keeping it smaller might be better. Heck I might not even have to choose if no one shows up lol

Thanks for the input!

[turbid zenith]

Is anyone else seeing a big uptick in cheating with ChatGPT getting better and better? :/

[shadow flower]

yes. the algebra students i’m grading been leaning on it very heavily i think.

[rapid tusk]

apparently there's some GPT build that can do very well on AIME questions 😭

[long pelican]

Making instructors' jobs in writing novel problems harder and harder!

In 2022 (and before) most of my colleagues got away with reusing old problems or just putting textbook problems on problem sets

[stray grail]

1. don't have assignments.
2. have one test (no internet) in the end that gives 100% the grade.
3. no issues.

or have assignments, but they are just to give feedback what they have understood/misunderstood. it does not affect the grade

[turbid zenith]

I'm having to go back to in person tests because of it

[royal patrol]

maybe a stupid/common question but what is the purpose of teaching math in school. Like aside from the obvious goals in the early stages of school (every day usage of mathematics in life) and the problem solving skills, what is the point we try to achieve by going deeper? Is the point to give access to a "tool" that will be used by students in other fields or to teach mathematics for the sake of mathematics?

I had a conversation with a profesor about geometry, and he was telling me how a specific book used in highschool was explaining some terms by using a mix of different axioms and that was very wrong. But i wonder is there a point to be rigorous in things like that? Kids cant understand what axioms are or even if they can, they arent able to see how a theory is built from them. Is it that harmful to use some things "incorectly", for a mathematician, in order to teach a geometry class to highschool for example ?

[tardy ember]

to be honest i suspect the real reason is that the original reason got lost in the noise of everyone doing things that kind of sound reasonable without checking if they actually make sense in the bigger picture and/or just doing what the rest of the system incentivises or requires them to do

like if anyone actually believed that the real point was for the students to benefit in some way then the existence of cheating would be really weird and a sign that something is very wrong with the system

[austere delta]

I think the purpose of most things taught in school is to give the population a common lense of the ideas that explain the world / shaped society and culture.

Everything taught preuni (geometry, calculus, probability) says something quite concrete about how the world works.

I don't know the exact situation with the geometry textbook, but I guess there is harm in teaching people things that are completely wrong. If they later find out what they learned is wrong, then they'll wonder what else they learned was just wrong.

[tardy ember]

Everything taught preuni (geometry, calculus, probability) says something quite concrete about how the world works.
...does it?

[austere delta]

Idk, kids don't want to eat vegetables either. That doesn't mean no one believes the point of eating vegetables is to benefit kids.

[austere delta]

I'd absolutely say so.

What distances between things are, how rates change, what the odds of something happening is are all questions about real things in the real world

So many systems that society are built on uses such ideas

[tardy ember]

i can't think of anything in those that's in the intersection of "things the students did not already know" (you don't need to teach someone the basic intuitions about space and continuity that even animals already have) and "things that reliably actually get transmitted in a useful form, instead of focusing on symbol manipulation" (because humans' basic intuitions for coherent probabilistic reasoning are not very good, but knowing the "rules" for the symbol P(A|B) is not enough to fix that)

[tardy ember]

yeah i don't think the point of teaching geometry is that the students were not previously aware of the concept of distance

[twin shell]

which classes/subjects do you consider to be of questionable value

[austere delta]

Not just the concept of distance, but how to find those distances.

Just say you want to build something out of planks, you might actually have to use the Pythagorean theorem for that.

I don't think you can argue that kids just intuitively know all geometry that could ever be relevant to them.

[tardy ember]

obviously not all geometry that will ever be relevant to any of them, but like, you don't need to have been taught about geometry to figure out how to move around inside a building, was my point

"what if they're building something out of planks" feels like it's more in the direction of the "what if you specialise into a topic that requires this knowledge" argument

[royal patrol]

i can see that but i am not saying to teach something completely wrong. But like i dont think we have to be completely rigorous. Or like we dont have to teach everything. Like we can teach the pythagorean theorem but that doesnt mean we need to derive it from the axioms or doesnt even need to talk about axioms

[royal patrol]

i dont consider any classes of questionable value but i dont know how rigorous a class should be to not loose its end point

[austere delta]

I can agree with that. But is anyone doing that anyway?

[royal patrol]

yeah they dont do it. Its more of a thought about what "the people who made the curriculum" were thinking and about what was their goal for mathematics in school

[austere delta]

I don't see it that way.

We're not saying everyone is gone become a carpenter, but having just the absolute most basic idea of there being a way to figure out the length of a plank that would fit as support in a right angled corner would to me fall into "basic information about how the world works"

[royal patrol]

of course all of them are important but in which direction they should be teached ?

[twin shell]

too much rigor is bad, yes
on the other hand one of the main goals of teaching math, in my opinion, is to teach people how to reason logically
which is typically completely forgotten about, probably because not even the teachers understand it (and oh boy do I know some of those teachers)

I think a big issue here is society has different expectations from mathematicians about how patient you have to be when learning math

[austere delta]

I can even take an example from my own life.

Once I had to replace the rope for a new flag to a flagpole. Didn't know how tall the flagpole was, but could measure it with similar triangles.

I think most people should be prepared to find themselves in such a situation, even if they don't specialize in flag pole measuring.

[twin shell]

the value of a rigorous curriculum diminishes when you try to rush through it
which is why no one ever comes to appreciate rigor

[tawny slate]

my take is that a lot of it is not necessarily directly impacting a student's understanding, but it has deeper ways in which it shapes the way their brain works that is very important and meaningful

for instance, when students are taught not to use a calculator in solving the math problems in school, it's not because they want to discourage students using calculators at all, but because number sense is very important

if a student has an over-reliance on calculators, they will never build the basic skills they need to even sanity check statements that are very clearly wrong

that's a big problem when as an adult someone says something like "there are 300 million illegal immigrants crossing the border every day" or "there's a significant probability that i will be in a terrorist attack or major accident" and they have zero idea how bs that is

algebra problems can be checked by plugging in numbers (not perfectly, but ignoring these details), but you can't do the same for most geometry problems

so geometry is usually when rigorous proofs are first introduced

the proofs themselves aren't as important as getting the students comfortable to think in a logical manner, step by step, as if their thinking has to follow the standards of proof learned in school

this promotes critical thinking and the ability to independently verify claims and logic

in general, being better at math makes you less likely to fall victim to scams, such as MLMs or pseudoscience

a deeper understanding of math can help with other kinds of philosophy as well

basic understanding of chaos theory: universe is functionally not deterministic, future impossible to perfectly predict
basic understanding of game theory: understand how humans are motivated in decision-making, such as how cultural values manifest, improves empathy
basic understanding of prepositional logic: moral systems always have subjectively chosen axioms
basic understanding of proof theory: uncomputable problems means we can't rely completely on computers, always need humans

these are just a handful of high level ways in which education shapes our brains, it's not about particular specific skills or preparing for careers

[tardy ember]

uncomputable problems means we can't rely completely on computers, always need humans
...humans can't solve uncomputable problems either though

[tawny slate]

these things affect people in ways they can't comprehend because it is so subconscious and taken for granted

[tawny slate]

that's a nitpick, the point is humans can do metamath that we can't rely on computers to do

[tardy ember]

basic understanding of game theory
afaik describing the decision theory that humans actually use is still an unsolved problem, and to the extent that we have anything even close it's not particularly "basic"

[tardy ember]

...computers can do metamathematics though? unless by "metamath" you mean the Metamath language, but computers are also pretty good at that