#math-pedagogy

1 messages · Page 24 of 1

zealous kettle
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Tell that to your students who are also parents, or caring for their parents, or in abusive relationships, or etc etc

quasi musk
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I do tell them that. I tell them that college is a sacrifice. You must decide what is important to you. If these things are in the way, then you must make a choice

zealous kettle
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Only because of the system, and a typical grading scheme with few-no retakes is a great way at enforcing existing social inequalities.

quasi musk
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So if everyone was born with a bachelor's degree, would that be better?

vestal tangle
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colleges are worthless in theory

zealous kettle
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It's not quite that simple, but if jobs only required degrees when they required skills in those degrees, that would be better.

vestal tangle
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everything is free online

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but you need that piece of paper

quasi musk
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My father died in the middle of my masters. I had to take incompletes, the stipulation was I had an extra term to get it done. Should I have been handed my masters without passing my quals & classes because it was just so hard for me?

Should I have gotten special treatment from everyone else

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There is flexibility for extenuating circumstances

zealous kettle
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I mean, ideally you don't need to take a full extra term when you're only missing the back half of the material from the semester.

quasi musk
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A grade of Incomplete just meant I had to do the homework & exams at a later date to be arranged by the end of the next quarter

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There's a lot of flexibility in the system already for these extenuating circumstances. Outside of that, everyone must make a choice if they have the time, the willpower, and the energy to get the education

tardy ember
quasi musk
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Knowing the material as measured by the assignments in the class

tardy ember
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no. if they know the material.

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if you give people the qualification "iff they know the material and can also deal with attending a class about it etc", then you are saying that there is a class of people who, even if they know the stuff, should not be able to get qualifications in it

quasi musk
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Yes, that is what I'm saying. If a class requires attendance, and you don't have the time to attend, then you don't pass. Could you imagine medical doctors that didn't attend clinical rotations in med school because it was hard for them

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Even if they "know" the relevant medicine

zealous kettle
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That's not applicable to showing up on exam day, or having a rough week and catching up later

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It's not applicable to math or CS or physics

tardy ember
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if the idea is that they wouldn't otherwise have the knowledge, then uh, come up with a better test, my proposal is not that you should give people qualifications saying that they have a skill purely on the basis that they managed to pass a test that tests a different skill

quasi musk
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Yeah, the point is if a class requires attendance but you know everything and don't attend, then you don't pass. Because the class requires attendance. There are plenty of classes that don't require attendance.

tardy ember
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ok so your point is that that is how the world currently works?

zealous kettle
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And many of the classes that require attendence really don't need to

tardy ember
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i knew that that's how the world works, my point was that it shouldn't work that way

quasi musk
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Then on what basis should it work? Who is this qualifying whom for what purpose?

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What is your test of knowledge & mastery of the material?

tardy ember
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well you should get qualifications iff you know the thing the qualification says you know

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i don't have a particular proposal for how to test this, but the existing system also has the problem that it's unclear how to check if someone knows things, and i don't see how attending a class makes that any easier to check

quasi musk
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It's not unclear. It's very simple. You can have take home assignments, in class assignments, finals, projects, and even oral exams/presentations. It actually works quite well at educating people

tardy ember
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well alright, that's how you test people's knowledge then

quasi musk
tardy ember
quasi musk
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(on average, of course)

quasi musk
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High School by comparison is extremely rigid

zealous kettle
quasi musk
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I hope you teach a class on these principles, I'd love to hear how it goes

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I am genuinely curious on how these experiments go. I'm usually a negative nancy, because I've been subject to many of these experiments as a student, and as a professor's TA. They're very rarely done well

tardy ember
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i mean none of this is really ideas for how to teach a class

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that's, like, kind of my point

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how the student got the knowledge is irrelevant, the qualification should just be the fact that they do know stuff

zealous kettle
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Teaching and testing are kinda different skills, yeah

tardy ember
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of course you also need to have some system where people who want to know the stuff have somewhere to learn it from, but they don't need to be tied together the way they currently are

gray dagger
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The structure of assignments effects the learning of the students. A single exam allows the student to only “know” the information for a single week.

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Regularly scheduled assignments allow (or force in some cases) students to spread out learning, which is confirmed to increase long term learning

long pelican
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If this discussion is viewing mathematical knowledge as knowing information, keep in mind none of this is specific to math and so this falls in "general" pedagogy, and probably has more people studying and writing about this

gray dagger
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Students who study for 4 hours in one day, and 1 hour per day for 4 days score similarly right after, but the students who learned over time are much better a week or two later.

long pelican
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than just mathematicians and math educators

zealous kettle
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I'm reminded of Angela Collier's STEAM video

gray dagger
long pelican
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I've always viewed math similar to teaching a superpower, rather than just teaching information and a well-defined set of skills, but that's just me

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To those who view teaching math no more than teaching information, general pedagogy literature may be of interest to this discussion

zealous kettle
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I suppose there are 3 components to math:

  • Knowledge: Knowing times tables, powers of two, formulae and identities
  • Skills: Being able to add and multiply multi-digit numbers, add fractions, solve for x
  • Puzzle solving: Combining previously-learned skills and knowledge to solve novel problems, e.g. being able to derive the pythagorean trig identities from the pythagorean theorem and SOHCAHTOA.
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I have some students learning thru khan academy, and some thru beast academy. Both give practice for the skills and knowledge, but Beast also pushes problem solving.

long pelican
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Yeah, the puzzle solving aspect is the closest to what I'm thinking of by "superpower"

gray dagger
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I mean, any academic discipline is focused on puzzle/problem solving

zealous kettle
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I think my students learn the puzzle solving aspect more in coding class than in math class.

long pelican
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Well, some of them, at least 😉 😔

zealous kettle
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Engineers often do more problem-creating

gray dagger
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There are some disciplines where the PHD is the point, imo

zealous kettle
cold trench
gray dagger
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Exact opposite

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Business is designed to get you out into industry as fast as possible, with a broad (and shallow) knowledge of applications

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Take pure math specifically. If you’re not learning how to model, then a BS is essentially all just background material. What is the goal of studying pure math? It’s not directly applicable. The goal, in my mind, is to get to a phd and apply your wide base of knowledge towards new learning

cold trench
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(I was joking lol)

gray dagger
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Sorry I was locked in lmao

cold trench
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I am a business major but I used to be active in this server

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well, was - I just graduated lol

gray dagger
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Congrats

zealous kettle
gray dagger
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Nah, I mean that academic style research is the goal of learning. I get how you read it that way tho

next sinew
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This is absolutely bewildering to me. If you don't learn to read in first grade, should you be discouraged from ever learning to read? For that matter, I see no issue with going back and updating your high school grades. Where are these ideas coming from?

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I've never failed academically, and maybe you haven't either. But some people do. And I think kicking them while they're down is just unnecessarily cruel

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This general idea that academia should be a ruthless and competitive space is a big reason for why I'm drawn away from it, even as a "gifted" student

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Do we really immerse ourselves with the technicalities of our world in order to enrichen our experiences and to make the world a better place, or to rank people under some sort of hierarchy? Because currently it feels like school and academia attempts to do the latter

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I feel like only having to sit through the exam once should be the reward in and of itself. Having to go back to studying the same material when your peers passed the course is already frustrating as it is. And if the student actually puts in the time and effort to learn the material better than the other students the second time around, then maybe they've deserved the higher grade, because they ultimately probably put in more effort

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This sort of deadline way of thinking is the type of toxic productivity mindset that leads to people taking shortcuts and not understanding for instance technical proofs in detail, because generally technical proofs are not tested for in the exam (because then no one would pass)

It leads to all kinds of metagaming and grade optimization behavior which is poison to genuine learning. Not to mention some people are biologically more able to pick up surface level understanding quickly, while others take more time to understand more detail. It's always the surface level understanding that's tested in exams. Insisting on strict deadlines is putting tremendous strain on certain types of students through no fault of their own. I think it's ridiculous nonsense

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I've felt discouraged because I couldn't complete as many credits as some of my other highly gifted peers could. It almost made me think I was stupid or broken. Then I got around to becoming a TA, and suddenly people started praising me for my in-depth understanding, commending my ability to engage students and to give them digestible narrational stories about the course material. It restored my faith in myself, but it also completely destroyed my faith in the system as a whole.

Certain types of students get screwed over even if they study around the clock, and that's very, very disheartening to me.

The problem is that if there's an instructor who insists on a certain level of understanding by a certain deadline, then a student that needs more time to either digest the material in depth, or simply needs more time for whatever reason, will get unfairly punished. It's not only making academia inaccessible, but also promoting short sighted learning goals that probably aren't fruitful to advancing humanity's overall scientific understanding

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I feel like modern education is based on the philosophy that an instructor creates a holy Exam and a holy Deadline and anyone who doesn't pass the Exam by the Deadline is labeled as stupid, and everyone else is good and obedient

I think the whole system is quite frankly silly and based on very ableist premises, because the Exam tests for surface level understanding in a high-pressure environment with huge RNG elements attached, and the Deadlines are much harder for some people to work with due to neurological features that have nothing to do with the person being lazy or stupid

Of course these problems are very hard to fix, but I feel like change starts from acknowledging the issues and acknowledging that we need to uphold Learning, preferably generalizable and in-depth Understanding, as the holy end goal that we should all cherish. Right now we're cherising the Exam, which is a helplessly unfair and silly thing to hold to such a high standard

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The Exam is a necessary evil, it's not something to brag about, and it's certainly not a measure of anyone's academic ability or worth as a person

quasi musk
# next sinew The Exam is a necessary evil, it's not something to brag about, and it's certain...

I think exams are a measure of some degree of academic ability. Moreover, for documented medical cases in the US there are ADA exceptions; e.g. extra time for deadlines. This all currently exists within the system.

If someone is getting a professional degree for civil engineering, and part of the requirement is being able to build a computer model for a bridge, then make a physical model of that bridge to train them to build real life bridges; then, would you agree that this deadline has a purpose: to train and qualify civil engineers to design safe, effective bridges in a time efficient manner?

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But there's also a limit. If a student requires 10 years to pass an introduction to calculus class (assuming all pre-reqs were passed with an A) because of biological factors, then at that point when does someone come in and say "This is taking you too long to remain a student here". This isn't an abstract question, as many Universities give you three tries at a fundamental class, before you're kicked out of a major or suspended due to poor academic performance

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Here's another case: 3rd quarter senior year PDEs class. Student doesn't know the method of integrating factors, or how to integrate by parts. If these are fundamental pre-req skills for solving certain types of PDEs which the student doesn't know, how then can they even solve basic problems?

It's a sign that the student is unprepared for the class, and they can try again in the future

next sinew
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this deadline has a purpose: to train and qualify civil engineers to design safe, effective bridges in a time efficient manner?
I don't agree with this. You learn the skill once, for some people it takes longer than for others, and then you keep honing that skill regularly

quasi musk
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I'm asking you what happens when it doesn't get honed

next sinew
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I don't understand your question sorry

quasi musk
next sinew
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Then the instructor has probably failed

quasi musk
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Even if it's multiple different instructors over the course of 5 years?

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At what point is a student responsible for their education?

next sinew
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Well I'm not saying to pour endless resources into every student, instead I'm saying that we need to re-evaluate how we teach and evaluate our students. Some students are maybe lost causes, but the vast majority are not

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Right now I feel like students are not treated as individuals, instead they're all put through the same assignment + exam regime

quasi musk
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That is certainly the truth. A little bit of my background. I nearly failed out of High School, and I only stayed for sports. I went to my local CC initially for sports. An injury prevented me from going on, and I decided to take math classes to occupy my mind.

It was interesting, and I got a job on campus as a math tutor/grader. Since then I graduated from UCLA, got my MS degree in mathematics, and have probably helped thousands of students in their math classes at this point. I used to run a math & science resource center at an HBCU, where I recruited, hired & trained student tutors. I've worked as a supplemental instructor at several different colleges, and am currently a PhD student

next sinew
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I'm glad you turned your life around

quasi musk
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I used to teach 5th grade - 12th grade at an after school program. So I understand people's frustration with the system, but more often than not I see student's not engaging with material in an honest way, not making effort, and having issues structuring their study habits

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I've worked with blind students, deaf students, veterans, illegal migrants, and everything in between on how to form proper study habits to pass classes. Even in alternative education places

next sinew
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It's easy to blame the students but as educators we never ask ourselves whether the material is engaging or whether the exercises are engaging or whether there's a visual computer game one could make out of a central learning objective

quasi musk
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More often than not it is not 'the system' but student's lack of understanding and accountability. About 5% of my students had genuinely awful teachers/professors with unreasonable expectations. But that's not most of the case

quasi musk
next sinew
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Kids these days spend all their time on TikTok and yet we expect them to have the patience to read through textbooks

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like at a certain point if you want to see solutions you can't just keep blaming the youth

quasi musk
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This is a fair point, parents provide them with infinite distraction but ask them to focus. I've even worked with parents on limiting phone use, and I collect phones when I teach k-12

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But by and large, in university/college they are adults or will be treated as such. They must be responsible for studying, doing homework, and passing exams given a reasonable class

next sinew
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I think a student will be able to read a textbook eventually after they've slowly learned to familiarize themselves with a subject. But a lot of things that we ask of students could be made a lot more accessible by chopping exercises down into smaller pieces, giving visual illustrations, making short lecture videos, etc.

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I think a lot of lecturers are simply not giving it their best effort

quasi musk
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Yes, reading is a skill that we are taught. In my MS program I was specifically assigned projects to read papers, summarize, and present. It was excellent training for my current PhD program

next sinew
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We keep asking students to read without giving them a reason to read

quasi musk
# next sinew I think a lot of lecturers are simply not giving it their best effort

This is 100% true, but again a student can only control their own actions. I always focus on optimizing student's routines, habits, and strong study/problem solving skills. If after all that, they cannot pass then there's no shame. The student gave it their best and the student was either not up to snuff, or it was an unreasonable class

next sinew
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There are so many ways to get a kid excited about math that are not just giving them a copy of Rudin's analysis

quasi musk
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I 100% agree here

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Maybe I'm coming off as more negative/disagreeable than I am lately, but that's because I hold my views passionately

next sinew
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There's a certain type of academic negativity that I don't mind at all

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I hate it when people try to sugarcoat things when our entire modern society is getting flushed down the toilet as we speak

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To speak about our modern problems honestly requires a little bit of negativity

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Clearly there are things we disagree on but if you wanna be blunt about the way you feel then that's just the price I pay for getting to be blunt about the way I feel

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I think getting our thoughts out there is worth it for both of us and I do appreciate what you have to say

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We've clearly both noticed the same pattern: certain students are horribly underperforming

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This sounds really bad but I'm honestly not too concerned about the most underperforming students because I think most jobs in the end are not actually that deep, like a lot of the stuff you learn at university is not really job relevant

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So I talk from a very privileged standpoint, I talk on behalf of students who show great academic potential but who are still struggling. These people exist and a lot of the time it has to do with the system not being fit for them

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I think a student that wants to learn stuff in detail is honestly, generally speaking, not gonna have a good time in college. And I think that has a lot to do with the rigid course structure

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But even when it comes to the weakest students, I just don't see the appeal in punishing them for no reason

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You say we shouldn't let someone keep retaking calculus for 10 years, and my European brain immediately thinks yeah that makes sense because we don't wanna waste resources

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But the US actually has a natural system against that kind of thing, you have university fees, so no one is gonna retake calculus for 10 years unless they're super rich, and if they're super rich then they can do anything they want anyway, that's a part of the game

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So with that realization, I don't see the need for ANY restrictions in the US, because it's not a burden on the state, the student pays for their own education

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If they wanna spend that money retaking calculus for 10 years, why not let them?

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I mean, in Finland, you can literally keep retaking calculus for 10 years, and here it actually costs tax money to make that happen

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so why is it that in Finland you're allowed to "waste taxpayer money" while in the US you're not even allowed to waste your own money when it comes to university studies? doesn't make any sense lol

quasi musk
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In the US there are many kinds of institutions. I know someone that took Calculus 2 ten times before he passed it. He attended different community colleges. The idea behind the 3 strike system is to say "Ok this isn't working out for you, go do something else". But in California CC is very low cost or even free for most in-state residents

next sinew
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Isn't that just a remarkable thing though? To me that's someone with perseverance

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Even if they "wasted resources", I think that's only through the narrow American lens of short term payoff

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If passing Calc 2 was what this guy wanted in life then presumably it gave him a boost of motivation to keep working on himself

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It probably didn't actually cost that much money to the state

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There are many things that cost way more...

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Even if the guy never ends up graduating, I think setting a goal for yourself and reaching it is a life lesson whose worth can't be easily measured in dollars

nimble estuary
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What do you think of the "math pathways" approach?

next sinew
# quasi musk I think exams are a measure of some degree of academic ability. Moreover, for do...

Also I think this hidden assumption that the medical system works perfectly is a naive one to make because for instance I was born in a rural area where my parents didn't really believe in diagnoses so I never got a diagnosis that could've potentially helped me in life. We can't assume that people's brain chemistries always fall under "documented medical cases" and that whatever deadline extensions one gets are actually the type of support that is sufficient and most beneficial

quasi musk
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My claim is that in university it is the student's responsibility to notify or seek appropriate diagnosis, then report to the relevant office. It is not my responsibility as an instructor nor TA to make such decisions or handle such cases

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My job pertains to teaching & grading

next sinew
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I know but I'm talking at a systemic level

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It doesn't do society any favors to ignore the ableist nature of academia

next sinew
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A typical feature of neurodivergence is to get caught up in details and to have one's focus be internally as opposed to externally motivated.

These are horrible qualities to pass exams, because you don't need to know how the binomial formula allows you to differentiate polynomials, if you just know the end result, which is the power rule. Additionally, if the lecturer says to solve these exercises, and you can't focus on them because you want to do something slightly different, well you're just gonna be miserable

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I believe that by making teaching more inclusive, we can also make it better for everyone

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Because "neurotypicals" face the same problems too, just to a much smaller extent

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There's no reason that we couldn't make our school more flexible and give students more opportunities to demonstrate their understanding, but for some reason we insist on these outdated practices because we had to suffer, so let's make the kids suffer too

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Surely there can't be any better system than making students cram surface level understanding a few nights prior to the big exam, right 💪

midnight scarab
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That just means the course/hw/exam is bad because it doesn't develop/encourage/test actual understanding

tawny slate
# next sinew Kids these days spend all their time on TikTok and yet we expect them to have th...

i totally get the whole "cant just keep blaming" no matter which side it is coming from, what matters is solutions and outcomes. i generally assume benefit of the doubt that most anyone who is even here agrees with that, given that its a pedagogy channel

that being said, probably more than anyone else, I am a person that believes that there are loads of simple solutions to complex problems we havent found or considered yet, and i dont think pedagogy is this silver bullet. what we are seeing today is a bunch of serious systemic problems far beyond anyone's control intersecting

you mentioned that kids are using tiktok, and there is pretty strong evidence that this reduces attention span pretty permanently during developmental periods and attention is a core requirement for learning to happen

whatever power teachers have, pragmatically their resources are limited and they cant fight certain physical truths, and i think we need to be realistic about that

if we are going to go down this route, we should then also put some responsibility on the administrators, on the parents, on the government, etc too, no?

vestal tangle
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i don't think kids in public school should be learning anything more than arithmetic and basic algebra

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so their tik tok-induced small attention span shouldnt be an issue

lunar mica
quasi musk
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California CC is closer to $50 or $60/unit

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Yeah, resident is $46/unit

turbid zenith
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Does anyone happen to have any good lessons to specifically teach about the value of abstraction in mathematics?

vestal tangle
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that and the fact that abstractions are simpler than the things from which they are derived

turbid zenith
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Right, but by lesson, I mean something that students can DO.

gray dagger
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What level of student? demonstrating how matrices can replace linear transformations and make linear function calculations easier is pretty impressive imo

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and that can lead to/from how linear systems can represent real world systems

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I dont have currently existing materials on that though

tawny slate
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i dont currently have specific examples but i do have a general approach to designing something like this

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if you want examples where abstraction helps, find examples where specificity does not help

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a basic example: many people find the monty hall problem very unintuitive, not convinced it is 1/3 chance instead of 1/2

however, by simply trying an example with 100 doors instead of 3, this makes the intuition immediately clear for many students

the problem was not easy to see when you are at n=3 doors, but generalizing gave us a clarifying lens

turbid zenith
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So college level, but all different majors

tall bolt
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I dont have a specific example either right now but there must be something nice with like the dihedral group, where working with the idea of rotation and reflection abstractly helps

turbid zenith
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(Many of whom haven't taken math in years and are card-carrying members of the "I Hate Math" club :P)

tall bolt
# turbid zenith (Many of whom haven't taken math in years and are card-carrying members of the "...

Unsure what you count as abstraction but another nice idea could be showing how “abstracting” problems into the key information can make seemingly different questions the same

The idea I have in mind is a question about wether or not two people in London have the same number of hairs on their head and some problem I vaguely remember doing about a turntable at a dim sum restaurant and telling if everyone can have their dish in front of them (both applications of the pigeonhole principal, despite seeming different)

tawny slate
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gridwalking might be a good one, where it is easy to solve some simple cases, but more extreme cases will require a general method

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formulas for arithmetic and geometric sequences are very different, but understanding the derivation generally using functions/bijections might help

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what are you allowed to do on "both sides of an equation"? you can memorize specific rules or understand why those rules work to know when youre dropping a solution or have extraneous ones

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but these probably wont motivate people who already dont like math hmmmm

midnight scarab
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Hmm I'm thinking maybe some examples of abstraction as in building simple mathematical models of things relevant to their majors

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Like idk mendelian genetics, or some economics or game model (I remember there were some nice examples of using invariant or monovariants to prove there's no winning strategy for the 2nd player or something)

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Probably hard to have something for every major though

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But hopefully the examples still feel somewhat concrete even for other majors

vestal tangle
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it's important to pin down what you mean by abstraction as well. the term is very overloaded. there are distinctions between abstraction, generalization, and idealization

turbid zenith
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So what I usually mean is the idea of removing parts of a problem that don’t matter to get to the core of it, such as ignoring parts of the Bridges of Königsberg problem

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That’s what I’ve usually used to teach it, but what I don’t like recently is “and then a really smart guy named Euler came along”

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I don’t like the message that sends

midnight scarab
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Well maybe if you can turn it around by showing how it can naturally be arrived at

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Like "by learning how to approach problems, you too can do it"

next sinew
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With k doors, the probability of winning after switching is (k-1)/k * 1/(k-2)

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To be honest, I'm not sure why it's obvious that that's bigger than 1/k

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there's probably some wisdom you have in mind that I can't really think of

next sinew
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The other one is the construction of integers, rational numbers, etc. but that can easily feel pointless and abstract

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When I think of abstraction, I think of going from small molecules to living organisms, or going from simple logic gates to modern computers

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when mathematicians abstract things, I think most of the time they do it because it feels cool and not because it actually helps anybody

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Something that may or may not count as abstraction are derivatives. How to differentiate a particular kind of function is a bit of a tedious problem at first, but once you get a few basic rules down, you start being able to differentiate all kinds of functions even if they look totally horrible and impossible to work with

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Obviously math grows in complexity as you get further but it's very incremental and sometimes you maybe work really hard to build up your theory of differential equations and at a certain point it feels very abstract and then after a while you move on to studying graph theory and you get back to the integers and concrete proofs and so on

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I think there's a huge difference between how math is generally done and how math probably should be generally done in my opinion

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so yes mathematicians are very eager to abstract everything into axioms and what-not but based on my honest experience I have not found it terribly useful for actually understanding math. Like it's not really abstraction per se that is helpful, it's a certain interplay between abstract and concrete thinking. Like obviously you need to be able to reason about an object in terms of its properties instead of what it actually represents, and that's a very abstract skill. But at the same time, in order to make sense of anything in our heads, we still need to build mental models of what we're actually talking about. And the process of taking something abstract and turning it into something digestible is what I view to be at the heart of mathematical abstraction

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Kinda like with computers you understand what an algorithm is doing even if you can't keep track of the individual logic gates, I'd say it's the same with math where you need to understand what's going on even if it would take a horribly long time to reduce everything back to its most fundamental constituent parts

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And I think a good mathematician is not necessarily always able to recover everything back to the most fundamental principles, just like how a computer scientist is not necessarily able to understand every part of how a computer operates on the different layers of abstraction, but I still think a great mathematician has at least some vague idea of how the whole house of cards holds together, even if there are gaps lower on the abstraction chain

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And for sure, of course there will be gaps because some people study different foundations for mathematics for a living and we already know that there's not just one way to build mathematics from the ground up, but in any case I think mathematical abstraction is not really about taking a problem and abstracting it for the sake of abstracting it, but rather it's an inevitability due to the complexity of our logical realm, and any time we have to abstract stuff away I would see it not as something to feel good about, but rather as a bit of a sad goodbye to the beautiful mathematics that we leave behind, only to be abstracted away from sight

next sinew
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An analogy I came up with that occasionally applies is that you get to meet a super interesting person, perhaps a three dimensional vector in R^3. You talk, you get to know each other, and it feels good that you can do so many great things together. Your co-ordination is impeccable and you make an amazing team. Then, one day, because you're a mathematician, you make a mockery of your new friend, a stereotype of how all objects like your vector ought to behave, and you teach everyone this new truth and wisdom that you've learned about a very general class of objects.

What you're doing is not wrong, not mathematically. But anyone reading these new stories written in completely abstract language will not have any idea of the fun times you and the vector in R^3 shared together. They will only see the cold and lifeless end results that bitter experience and occasional failures taught you. And then we end up teaching something that, while technically correct, lost a part of its soul.

midnight scarab
next sinew
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right, of course (although to see that, you have to do an algebraic trick, so I still suspect that Cozmo had some additional insight that I'm missing, because personally I find the 3 door version significantly easier to analyze than the 100 door one. Like sometimes if you increase the amount of doors or whatever you're studying, whatever effect you're hoping to see will also become more noticeable, but in this case I'd say it actually gets less noticeable)

next sinew
tardy ember
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oh wait i see the issue

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the intended generalisation wasn't that one door that isn't the prize gets opened, it's that 98 doors (or in general, k-2 doors) get opened

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which does make it a lot more obvious in the general case that the chance of switching being correct approaches 1 for large k, specifically it's 1 - 1/k

tardy ember
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ok actually no it's neither, it's "you seem to have misunderstood what abstraction is to roughly the same extent that the education system misunderstands what mathematics is"

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like, they're not completely wrong, it would be somewhat bold to walk into a high school "maths" class and claim that no mathematics is taking place, because it is true that numbers are mathematical objects, and the things that are being done with them are (probably) mathematically meaningful and justifiable

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...but also like

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  1. mathematics really isn't "the study of numbers", there are mathematical objects that have nothing to do with real numbers
  2. the way that "rules" are presented, with no explanation or justification, and you're just expected to accept them, is not how mathematics is done in any other context
rapid tusk
#

“study of numbers” is an incredibly shallow and reductive way to describe math 😭 id provisionally call it “study of structure and patterns”

heavy trail
heavy trail
heavy trail
# next sinew Obviously math grows in complexity as you get further but it's very incremental ...

The thought that integers are "concrete" gets close to the heart of the issue, IMO, because it took humans thousands of years to think of integers like we do today. It's at testament to the power of the abstraction called "the integers" that we can teach them to young children.

And teach them to a high level of familiarity/automaticity.

Being able to conceive of 0 or negative numbers as "numbers" is a significant achievement of human intelligence spanning thousands of years and dozens of cultures.

For example, IIRC, when Cardano was writing about solutions to cubic polynomials, there are places where he says expressions like "3 - 5" are simply nonsense. (He doesn't use that language, but talks about subtracting larger from smaller and so on.)

This complicates any kind of linear, bottom-up story where we progress from concrete to abstract and "abstractness" is a property of the concept per se. There was a time when humanity's conception of cube roots was clearer than our conception of negative numbers. Are cubic polynomials therefore more concrete than negative numbers?

Would someone in the 16th century be right in saying we should focus on more concrete things like magnitudes and areas, and not on these empty algebraic abstractions?

cosmic ibex
heavy trail
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(This isn't to defend the way abstractions are often parachuted in out of the blue in math classes.)

vestal tangle
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abstraction is when you have a bunch of specific things and you isolate a collection of common features that they all share. and then you go on to treat that bundle of common features as an object in its own right. e.g. you look at real numbers, oriented line segments, functions, etc and create the "vector space" abstraction from them.

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and then generalization is when you come up with a new object that captures of a bunch of already known objects as special cases.
e.g. you have vectors, bivectors, 1-forms, linear transformations, etc and you invent the tensor to capture all of them as special cases.

midnight scarab
turbid zenith
#

We’ve got abstraction, generalization, and axiomatization. We can always subdivide them further, nudge definitions, and tease out nuances.

midnight scarab
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It's stripping off something that doesn't affect the answer at all

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Idk if you want to call that idealisation or abstraction

heavy trail
#

A thing I wrote years ago and share with my (computer science) students (long-ish):

Most people confuse the idea of abstraction with the idea of information hiding. Abstraction does not mean "putting all the details behind some interface." Abstraction is the process of isolating and identifying patterns, giving those now-recognized patterns names, and thinking and acting in terms of those patterns per se vs. instances of those patterns.

Numbers are an abstraction we deal with every day starting from a very young age. Consider the number 5. You can't point to the number 5. You can't touch, smell, or taste the number 5. You have never once encountered the number 5 walking around outside.

That said, the number 5 is an abstraction of many concrete every day experiences. You can't touch the number 5, but you can touch 5 apples. You can't smell the number 5, but you can smell 5 cows (and how!). You can draw me 5 smiley faces or 5 hearts.

What is 5 except something that all these groups-of-five have in common? It's an abstraction that isolates some aspects of these groups-of-five that we find relevant and only those aspects, discarding the particulars. It's powerful because it encapsulates 5-ness per se without needing to answer "5 of what?" It's not 5 of anything — it's just 5.

Even that symbol, "5", is still just a picture of the number 5. 5 is prime whether we write is as 5 (decimal) or 101 (binary) or V (roman numerals) or IIIII (tally marks) or 五 (Japanese). The number 5 is not an interface. It is not "hiding details." It is a pure expression of a pattern we've isolated and elevated to a thing-in-itself.

Did you know the first model of computation — the lambda calculus — was first written down in 1936, long before we had a computer that could ever possibly implement it? What "information" or "implementation details" could this model have possibly been hiding? The first fully programmable digital computer wouldn't even be invented for another decade!

Instead, the lambda calculus was an attempt to isolate and define what we meant by "computation." It allows us to think more clearly about computation per se. It allowed us to start making statements about what computers could and could not do (even absent a working computer).

vestal tangle
#

it's more like a kind of idealization. i don't think all idealizations necessarily involve approximations

halcyon glade
timid shard
#

I haven't read the entire conversation but to go back to the original question, I think bridges of Königsburg is a nice accessible example of abstraction

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although not necessarily that mind blowing

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bars and stars too

turbid zenith
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Welp I guess what I've been calling abstraction for the past 5 years isn't abstraction then lol

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Either that or I'm mixing idealization in with it

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Of course the annoying thing in this book is that nowhere does the author define abstraction ... she just gives lots of examples

heavy trail
turbid zenith
#

How To Bake Pi by Eugenia Cheng

heavy trail
#

Ah

turbid zenith
#

It's a popular press book rather than a mathematics-with-a-capital-M book

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Which was a deliberate choice to keep things accessible to non-majors

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Actually looking through the chapter she definitely talks about idealization as part of abstraction

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She does also give the example of numerals being a kind of abstraction, trying to get to the essence of underlying structure

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So when I'm thinking of the bridges of Königsberg as an example of abstraction, I'm thinking of it less as making the shape easier and more about only focusing on what's connected to what

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Especially with the benefit that now that you're studying what's-connected-to-what-ness, you can apply whatever you find along the way to a whole bunch of other problems

heavy trail
# turbid zenith So when I'm thinking of the bridges of Königsberg as an example of abstraction, ...

I don't really have a sharp distinction between "idealization" and "abstraction", myself. I think the sense of "idealization" varies a lot more between fields.

For example, I wouldn't say you're abstracting if you assume an idealized "spherical cow" in order to make some calculation easier, or if you assume an "ideal/perfect vacuum". I'm not forgetting shape or pressure when I make those assumptions, I'm just picking values that make my calculations tractable.

In the case of Bridge of Königsberg, I'm not assuming my islands are perfect circles, I'm erasing the very concept of shape from the entire situation. I want to say that shape isn't even germane. Banish it.

Could you call points "idealized shapes"? Yes, absolutely, I think that'd be fine.

In Eugenia's book, IIRC, she wants to draw a distinction between abstraction and generalization. The former is more like forgetting and the latter is more like extending.

midnight scarab
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Yeah Koenigsberg is a typical example of "making abstraction of"

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Like geometry of say an enclosure makes abstraction of its thickness (though maybe that's also idealisation in that it's only valid if it is indeed relatively thin)

#

Maybe making abstraction of colour or material is better

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A generalisation of the Koenigsberg example is how topology makes abstraction of distances

heavy trail
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I think of it more like...you have a situation with a bunch of parameters, dials, knobs, etc.

Are you setting those knobs to one specific value to make things easier, or are you removing the knobs entirely?

Sometimes both can be viewed as the result of a limiting process.

midnight scarab
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I like the analogy!

heavy trail
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For example, imagining a point as a nested sequence of closed, positive-width intervals. If I have no conception of a zero-width interval, a point becomes a kind of "idealized interval". Any such sequence "points at" something and I could say that even if my only conception of "interval" was of something with a positive width.

But then maybe I say, well, let's live in this universe of sequences of closed, positive-width intervals and then define an equivalence on those sequences if they "point at" the same thing.

Are points abstractions? Idealizations?

I dunno, I'd have no issue with either characterization.

midnight scarab
#

There's also the fact that in common usage abstract often means hard to grasp/unintuitive

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So people would call the notion of limit "abstract"

rapid tusk
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ahem taps sign

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wrong channel.

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<@&268886789983436800> sry if this doesn’t warrant a ping

serene apex
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!help

coral copperBOT
#

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

wary ether
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I see you did – thank you – but also worth pinging them too!

true granite
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Hi; I'm willing to prepare an activity for a class for next week about how Eratosthenes computed the circumference of the Earth. He used the stick and shenanigans to compute an angle, but back then they didn't have calculators (for obvious reasons); how did he compute the angle without using reciprocal trig functions?
Say you know lengths a and b, and that their ratio doesn't equate a well-know value for the tangent; how to proceed, using the tools available circa 200 BC?

image.png
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In fact, I'm thinking: did he "just" draw several copies of that triangle, try to assemble them until he got a full circle, and divided 360 by that number? (you get the idea)

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Again, I'm thinking for something that's doable without a calculator (and this would be for students that don't have lots of background)

midnight scarab
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The idea I have is to draw a circle centered around the angle, and then just take some lace to measure the length of the arc

true granite
#

Ah that's not a bad idea using ropes!

cosmic ibex
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Eratosthenes's own work is not preserved, and is only known through summaries written by others, who would not have been particularly interested in his precise methods of calculation.

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Archimedes lived before Eratosthenes and was famously able to give numerical estimates for the ratio between side length and radius for a regular n-gon for high n. We don't know exactly how he did it either; only his raw results plus later reconstructions of "here's how it might be done with tools that were probably available to him".

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Similar calculations are fairly commonly encountered in astronomy, and surely the ancients must have had some way of performing the calculations -- we know they had sexagesimal fractions available for notating intermediate results, for example. But to the extent anyone in antiquity wrote down a description of how those methods worked, apparently nobody bothered to keep copying those descriptions (just for the benefit of future historians?) after better methods became available.
Hipparchus (a bit later than Eratosthenes) constructed the first trigonometric tables we know of, and certain knowledge of how they practically did without those tables before Hipparchus seems to be mostly lost.

true granite
#

Oh, what a shame! I find this very intricate how they did all those computations at that time!
I suppose the scribes didn't bother writing this knowledge down because they assumed it was easy enough and nobody would ever forget?
Regardless; thanks for your output, that was interesting! I'll mention that to the students!

Again, what a shame Eratosthene's writings weren't preserved, that was quite something

muted nebula
#

Heya. I'm wondering how to best teach proofs to undergraduate students? Here are three 'paths' I've been exposed to:

  • I personally read Velleman's How to Prove It and he explains clearly proof strategies to try for proofs with certain structures (proving negated statements, iffs, disjunctions etc.). This was an excellent book for me.
  • A mandatory proofwriting class I had in the 3rd semester wasn't much help: they taught us what the proof strategies are and not really when to employ them. The course outcome was somewhat dubious. I hardly believe anyone benefited much from it. The exam wasn't exactly difficult either.
  • A professor of mine adopts a different belief system, that these proofwriting classes are actually futile and should not even be there in the first place. Instead, force them to prove statements by taking abstract algebra, real analysis etc. Though I believe this is a somewhat decent proposition, I don't believe many students will be able to withstand these classes in the first place without prior exposure to proofs.

I'd like to hear your thoughts! And I'm not really looking for 'tips' to learn proofs, more about teaching them to a class of undergraduates with only an exposure to calculus in maths (so quite new to higher maths)

cloud zealot
gray dagger
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My UG class really treated it like a writing course, and the prof had us turn in 4-6 proofs, she checked them for "math readability" and then returned them notated. We were able to then redo them and resubmit for improvement. (iirc if a proof was bad enough she asked you to come to office hours to really break it down) Thats a really high cost format for the professor, but I think it was really helpful in building an understanding of the common language. The proofs were pretty common set and number theory stuff, show that two evens add to an even and so on.
Personally, spending a lot of time on propositional logic was helpful, since I could break statements down to their logical format and then have insight into how to proceed. But, that was barely part of my actual course, and was more from a mathematical logic course that happened before.

austere inlet
# muted nebula Heya. I'm wondering how to best teach proofs to undergraduate students? Here are...

one option is following a book like Velleman's or Hammack's and that's fine, it has the added benefit of giving the course some structure which most students expect in uni (some complain when courses are too open-ended, lol). If I had to teach something like this on a whim I'd definitely work through the first two parts of Hammack + some selected topics from the latter half of the book, making sure to hand out HW and thus feedback every two weeks or so, since what's important here is to make sure the students get the basics down properly and then work on their prose

my old undergrad dept. has an interesting approach, that's a bit unlike your three options: in their second semester, math majors take an "intro to research" course where students can work around some REU-like problems in an open-ended manner (e.g. they're encouraged to solve particular cases or generalize, to find connections with other results, to find references by themselves, etc.), and along the way they learn to document their progress by handing periodic advances and a final report, all written in LaTeX

this complements the set theory, logic and basic proof-writing they see in their calculus (= one-dimensional real analysis) and algebra sequences, which imo is really the proper way to go about it since those topics shouldn't take a whole semester really -- the issue is that people in HS aren't exactly taught logical reasoning, there's this pervasive idea of math being about "the answers" rather than about understanding and the process of getting there

dark zinc
#

Hi there I'm a math teacher in IB school. Do you have experience with that sort of exercises?

twin lichen
# heavy trail A thing I wrote years ago and share with my (computer science) students (long-is...

I don't really find this persuasive. I appreciate your view of what an abstraction is, but I don't think you really succeed in establishing that an interface is not an abstraction.

I am wondering here if, in your view, an interface is meant for one specific object, because you talk about "hiding the details" of that object.
In the ML module system, a module can be sealed with an interface which hides its implementation details, but we can also write code that is generic with respect to an interface, i.e., the same code works for any module implementing the interface. This is similar to the idea of typeclasses in Haskell, where a type can be a member of a typeclass, and we can write code that is generic across any type implementing the typeclass. Importantly, we can write the code which is generic wrt the typeclass / interface before we implement a backend for the interface; we can build things on top of the interface first, and then instantiate it later (or never, leaving this to the client of our code.) So the interface is an abstraction, a "pure expression of a pattern we've isolated and elevated to a thing in itself", as you say. All the traits you ascribe to abstractions can also be said of interfaces.

In the lambda calculus, "abstraction" refers to the operation by which a term, t, can be transformed into a function of one of its variables, lambda x.t. We can instantiate t with any such x. Programming code generically against an interface is exactly the same operation, we abstract out the specific implementation we care about and just write our code to be parametric over any implementation of the interface, i.e., as a function of the module implementing the interface.

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Anyway, your distinction is in contradiction with the way some computer scientists seem to use the word.
Note the use of "client-side abstraction" to refer to generic programming against an interface and "implementor-side abstraction" to refer to information hiding, so they consider both uses of interfaces as a kind of abstraction
https://www.cs.cmu.edu/~rwh/papers/mtc/short.pdf

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In particular, I object to your claim

The number 5 is not an interface. It is not "hiding details."
It is indeed hiding details: the details of whether we are talking about 5 smiley faces or 5 hearts, i.e., the details of what the things are that we have five of. Any argument that deals with the number 5 (and other positive integers) can be instantiated to talk about 5 smiley faces.

heavy trail
# twin lichen I don't really find this persuasive. I appreciate your view of what an abstracti...

The process I'm describing is basically partitioning some set into equivalence classes, modulo concerns about what a set is.

It's not really "my" view, cf. https://iep.utm.edu/abstractionism/

There are other accounts, but the contemporary ones I know are responses to the (neo-)logicist account I gave. Maybe you have one in mind!

The point, I think, is to give students any kind of handle on abstraction. It's not just another name for putting things behind a function call or adding indirection.

The fact that ML uses abstraction or interface in a particular technical sense is a bit beside the point, I think. All jargon is like that. One language makes a sharp internal distinction between "function" and "method", another doesn't.

"Interface" as a term of art is a lot like that. Abstraction has more of a philosophical pedigree, IMO.

Internet Encyclopedia of Philosophy
IEP Author
Abstractionism
#

(I am not a logicist myself, but still think it's a good avenue into capturing something important about what we mean by "abstraction", morally.)

twin lichen
#

The point, I think, is to give students any kind of handle on abstraction. It's not just another name for putting things behind a function call or adding indirection.

This is a weaker and more defensible claim. What you actually wrote above is something much stronger that draws a sharp distinction between "abstraction" and "interface", and does not contain the phrase "put things behind a function call" or "indirection." It reads to me like you are trying to argue for a philosophical distinction between abstraction and interface that just does not hold water. The thing you are trying to say would be argued more effectively if you focused on the positive characterization of abstractions and stopped going in a different direction to argue that interfaces are not abstractions. Interfaces are abstractions.

The fact that ML uses abstraction or interface in a particular technical sense is a bit beside the point, I think. All jargon is like that. One language makes a sharp internal distinction between "function" and "method", another doesn't.

This is a bit unfair, no? You're citing your IEP reference to justify your use of the word abstraction, and when I cite a CS paper to justify my argument that this is how computer scientists talk about abstraction, it's just an idiosyncratic use of jargon used by these guys

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The ML module system is the mother of all module systems, that has influenced basically all research in modular program design, interfaces and abstraction in CS. It is absolutely fair for me to cite a paper on ML-style module systems to demonstrate that computer scientists interested in abstractions and interfaces use the words in a particular way

heavy trail
# twin lichen > The point, I think, is to give students any kind of handle on abstraction. It'...

I'm drawing from a historical tradition that goes back to at least Locke. This is material I wrote for first-time CS students at a liberal arts college, many of whom were familiar with thinkers like Aristotle or Plato or Locke from other classes.

In any case, it is also meant to tie CS into those other traditions.

They might have also some bad technical habits, e.g., thinking renaming or indirection are ipso facto abstraction.

I shared here in case folks found it helpful in situating the concept of abstraction in their own teaching practice. It's not the only thing I say on the subject.

It's not an argument. It's pedagogical, meant to throw something vague into sharp relief.

#

That's a risk of transplanting material from one context (a particular class) to another (this discord channel), I suppose! 🙂

rain hawk
#

why do elementary school teachers tell students that fractions with a square root in the denominator are bad style? in actual math fractions with square roots on bottom are super common

pure light
#

see here for previous discussion

austere delta
gray shard
tawny slate
#

small additional comment i want to make related to rationalizing denominators:

i think there is a fine line between between saying "students should always rationalize denominators" and saying "not rationalizing denominators is bad style" and "rationalizing denominators is good style"

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obviously, the first one is not what we should be teaching

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sometimes you want to explain the math concept before explaining the motivation, why we care about it, because it is difficult to explain that until you have a baseline understanding of what the thing does

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i dont think this is one of those cases, but i dont think this isn't one of those cases, depends on the student imo

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even purely just as style, there are cases where we sometimes might not want to rationalize the denominator, such as $\frac{1}{\sqrt[5]{x}-1}$

burnt vesselBOT
#

Cozmogrgdfschkipkhrshtensi

148282132781858816.png
tawny slate
#

so saying it is good style encourages it but doesnt give it a "you have to avoid this" feeling, whereas saying not doing it is bad style does

halcyon glade
rain hawk
#

I don't remember much from pre-university math, I didn't pay very good attention. It's possible it was late middle school.

gray dagger
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<@&268886789983436800>

vapid kestrel
#

Greetings people, how was your experience in high school like?

halcyon glade
#

Could you be more specific? I assume you're asking specifically about learning math, but what aspect?

vapid kestrel
#

What topics were covered in your country, in your school, whether it was difficult or not, did you get straight into university after that, those kinds of things.

#

My personal experience was quite unusual, cosidering the context of my country.

tepid smelt
#

I'm the states I have taught at a few public schools so content is pretty much the same everywhere.

It's generally algebra 1, geometry, algebra 2 then pre calculus followed by calculus. Many schools do integrated math 1-3 then pre calculus and calculus. The popular curriculum is Illustrated mathematics, Pearson, CPM then a traditional pre calc and calc textbook.

A few schools have accelerated where they just cover the same stuff faster.

Private and magnet schools can differ wildly though where you might get many doing college level math there.

austere delta
# vapid kestrel Greetings people, how was your experience in high school like?

As I recall, first year was trigonometric functions and the exponential function, lots of derivative formulas. Second year was inclusion exclusion, and other basic combinatorics stuff and more differentiation. Then last year was integration, proof by induction, vectors and cross products, and linear differential equations of degree 1 and 2 (it's possible I remember more of the last year than the first two).

I wouldn't say any of it was particularly difficult, and I start uni after high school.

vapid kestrel
austere delta
#

Well mostly just the solution to
y' = ay
is e^ax
you see this by plugging it in. Moving on

vapid kestrel
#

And I considered myself very fortunate to have a calculus course in high school, which is quite rare in Brazil.

tall bolt
#

You do basic enough LA and enough calculus to do so, there’s nothing particularly involved though

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IIRC nothing involving finding eigenvalues or anything but I don’t particularly remember at this point. I would also say I found maths rather easy though high school

vapid kestrel
#

I studied at a technical school, in my case it was the electronics course, where I studied C programming, 8051 assembly programming, analog and digital electronics, as well as the usual high school subjects and calculus at the end of the 4th year.

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The usual high school lasts for 3 years here.

tepid smelt
#

Calc bc has first order differentials and that is a common class at every public school I have taught at.

Most students work with matrices starting even in integrated math 1 with solving systems and transformations. I have not taught pre calculus but I believe they do s bit more with matrices also but nothing too intense.

Most strong students in public schools just take college classes at a local community college from what I have seen. In rare cases I had one student taking like real analysis at the local college but he was an outlier

vapid kestrel
#

And I was also enrolled in a private english course, during the middle and high school years.

tepid smelt
#

When I taught in middle California I didn't see as much acceleration compared to North Cali though. Many kids up here are in programs outside of school like RSM is a really popular one and it's just more common that kids are taking community college math like linear algebra, multi variable calculus etc.

I think one disappointing idea is there isn't really 'honors' math offered and in the public schools it's just doing the standard content twice as fast so two years of math in 1 to speed run to calculus.

vapid kestrel
#

The only regret I have is that I wasn't involved in extracurricular activities like research projects or astronomy olympiads during high school

tepid smelt
vapid kestrel
tepid smelt
halcyon glade
#

I was lucky enough to be at a public high school that taught linear algebra and multivariable calculus as options. That's definitely not typical in the US. Some students also took ODEs or abstract algebra at a local college while enrolled in high school.

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I didn't have too much trouble getting into a university after high school, and I never really seriously considered the idea of either not going to uni or taking a gap.

terse isle
# vapid kestrel ODE's in high school? wow.

here in india (you probably heard this many times) we begin trigonometry in 9th grade, learn differential calculus in 11th and integral calculus in 12th. Alongside this stuff we learn ODEs and PDEs(not as much as ODE)

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The rest of the syllabus is algebra 2 and 3, linear algebra, 3D geometry etc

minor plover
twin lichen
#

rectangles and circles are not as important an area of research as they once were, modern algebraic topology mostly focuses on triangles

zinc dove
#

Which math resources do you use for teaching abstract algebra?

turbid zenith
#

Does anybody happen to know any educational research about how students keep track of what they know versus don't know at a particular time when building an axiomatic system in a class?

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Like issues with not remembering "well I can't use A to prove B because we didn't know A when B was introduced"?

turbid zenith
zinc dove
lavish briar
honest elm
#

Hello, I've got quite an odd question. I'm in my early years of math curriculum, and amongst the top of my class. I gladly help others understand when they have any questions, and I also like to help them solve their problems (PSET equivalent, maybe) but I don't want to give away the answer immediately (takes all the fun). It's worth adding that I'm considering becoming a math teacher.
All in all, my questions are:
How do you (tips?) explain the concept intuitively so they can get it and then add some formalism?
How do you help or guide someone intelligently, in a way that helps them think more than just finish? Mostly thinking here about oral exams, when the student is trying different things, how to approach the "giving pointers without giving the answer" problem?

lethal hornet
# honest elm Hello, I've got quite an odd question. I'm in my early years of math curriculum,...

usually when students are unsure about a problem, there are a few very precise spots where they are misunderstanding something.
i always try to pinpoint exactly where the student is getting stuck. even more crucially, you should try to point out why their reasoning/attempt is incorrect. it saves everybody time and frustration if you can do that.
typically this involves discussing previous attempts, writing some things out, getting a common vocab understood, etc.

once you identify where they are stuck, you need to understand the problem well enough to correct any misunderstandings and give them good hints on how to proceed that don’t give away the entire solution.
usually this is like the first one or two steps of your own reasoning, rephrasing or setting up a question or subquestion with the right vocab/suggestive tone, or pointing them to a relevent source.

its also up to the values that your math community has. if the students are generally self-motivated and prioitize personal understanding over completion, then the interaction will usually go without a hitch. eventually, you may need to explain the entire solution, but at least you both got thinking about it more deeply. on the other hand, it can be quite frustrating and uncomfortable if the person you are interacting with is trying to use you to get answers. gauging this can queue you in on when to continue and when to disengage.

sometimes it happens that you both get stuck, or that you explain something poorly/incorrectly. it happens. just work through it together and don’t get too upset with yourself

honest elm
# lethal hornet usually when students are unsure about a problem, there are a few very precise s...

How do you pinpoint those spots? I'd try when they seem to block on a concept, to make them write the definition, and to try to explain it, to understand what part they didn't get, and help them there. The part about pointing out why a reasoning is incorrect is very clear, but at the same time quite hard, you need a strong background to feel if something is failing because it's the wrong way or because the student made a mistake. Imo the hardest part in an oral when the student is trying something else than your solution.

The second part I understand a bit, it's like show them a thing in the question (an assumption for example) that's important, but that they could have missed or misused, to make them try to get more out of it, to lead them in the way of reasoning. Setting up a sub-question isn't always easy, I supposedly have the same level (as in we followed the same course, more or less) so adapting to them should come with time.

My math community is the prépa class, quite famous in France, if you know the system. Students are motivated, they want to understand to finish, not to finish for the sake of finishing. Usually they won't ask for direct answer, but more "how did you tackled this question", expecting a pointer more than a solution.

vapid kestrel
wary ether
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I've been doing some marking for a course where I TA'd for about 5 out of, idk, 500 students (those 5 were the only ones that showed up!) and I forgot how soul-crushing marking is.

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I just want to complain a little bit and maybe I'll put something worth thinking about at the end.

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I keep seeing students that just do not understand the content just barely scrape a passing grade by 1 or 2 marks

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I keep seeing the same mistakes in the same places by, idk, 19 out of 20 students

tight star
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500 students on a single course @.@

wary ether
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This is quite normal

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I have a proof that I'm marking where I think maybe 3 people have actually managed to get a correct answer? Might have overcounted by one

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Anyway, how do yall who do marking deal with this?

tight star
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why is this soul-crushing...?

wary ether
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Imagine spending several months working to try to get a handful of students to understand just a little bit what's going on

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And then you have to spend several hours every single day seeing vast swathes of people get it wrong again and again and again

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This isn't exactly cheery stuff!

tight star
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ohhh i misunderstood your job

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i didn't realise you were teaching too

wary ether
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TA = Teaching Assistant

tight star
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when i did the equivalent here i just marked their homeworks and led problem classes

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but i didn't teach per se

wary ether
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That is what a TA does

tight star
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must be called something different over here

tight star
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what were you teaching

midnight scarab
quasi maple
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https://en.m.wikipedia.org/wiki/Teaching_assistant

Teaching assistant

A teaching assistant (TA) or education assistant (EA) is an individual who assists a professor or teacher with instructional responsibilities. TAs include graduate teaching assistants (GTAs), who are graduate students; undergraduate teaching assistants (UTAs), who are undergraduate students; secondary school TAs, who are either high school stude...

tight star
quasi maple
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Oh I just meant nation-wise but fairs

spice matrix
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500 is a lot. That’s like CS course numbers or intro weedout calculus course numbers.

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and even those usually don’t go too much over 300

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though I did see an intro physics course get at least 700 and close to 1000 at some point

lethal hornet
# honest elm How do you pinpoint those spots? I'd try when they seem to block on a concept, t...

usually by asking guiding questions or drawing pictures. referring back to basic definitions or key theorems etc is usually a good idea too.

with regards to your difficulty about understanding another person’s solution to a problem: i try to remind myself that when someone asks me for help, the interaction is not about me, it is about the person asking me and their pursuit of personal understanding. im not saying that you are doing this, but it helps frame your approach/mindset and gives you some direction, specifically, for understanding that person’s perspective on the matter. understanding different solutions is a difficult skill to acquire, and even more difficult to do efficiently and effectively. i like to think that i have this skill, to a certain degree, in certain subject areas. i think that you should invite the opportunity to explore other solutions. it is one way for you to gain a deeper understanding of what you are learning as well as an opportunity to help someone complete their idea, or figure out where it fails. just keep doing it and you will slowly get better. but also don’t be afraid to be honest and say that, “we are both going to stumble around in the dark about this for a while until we get it, or until we don’t.” at that point you can move on to some other approach or asking somebody else.

sorry if this isn’t a super refined response. im not an expert on this, i just thought i would share my two cents and hopefully offer some insight from my experiences in undergrad

honest elm
honest elm
# wary ether Anyway, how do yall who do marking deal with this?

We're quite less in class, about 40 students at most. We all have weekly oral tests in maths, in groups of three, lasting an hour. Those orals always start with a basic question on the course, proofs or important exercises we need to know how to redo. I do think the fact that it's an oral exam helps us understand better, as you can be explained where it fails exactly on the spot.

tight star
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hm that makes a lot of sense actually

cosmic ibex
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It sounds very work-intensive for the teacher(s) to conduct 10-13 hour-long oral group exams a week, though.

midnight scarab
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Well there are many TAs lol

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And most importantly the grades of these don't matter

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(And the grading is extremely non-uniform and somewhat vibe-based, but again it doesn't really mean anything)

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But the context is very different from uni

honest elm
vocal phoenix
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That does sound like a great way to do things, so long as the school has the resources to make this possible.

honest elm
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Although I have to say, it's a system that's quite often re questioned, because as you can guess quite expensive, even tho it's only for the best students. It's sometimes also seen as an elitist system, and thus not very equal... But it has done its proof in the maths and physics community, the ENS seems to be quite famous outside of France, with students of the level of Harvard or Oxbridge, and counted as one of the institutions with the most Nobel prizes and Fields medals

midnight scarab
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It's elitist in the sense that's it for the "better" students, but it's also very much a way for people of modest backgrounds to get an "elite" education

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(Of course it's not truly background neutral, in particular a good background can compensate somewhat, but I do think it's true that a good background is not necessary [you just need a good enough background to do well at school, which should be the barest minimum anyway])

honest elm
# midnight scarab (Of course it's not truly background neutral, in particular a good background ca...

I totally agree on this, but it was more on the part of accessibility: best prépa (LLG, H4, iykyk) are in Paris, which is an expensive city, and students coming from a more modest background can't always afford to live in Paris for a year or two, on the other side, prépa also allows good students from poor backgrounds to get a really good education, and allow them to rise from skills and not money gifted at birth

midnight scarab
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Not to say that everyone gets a place in the dorms, but many do

night badge
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hi folks 🙂

in the context of explaining how proofs are written, im trying to tell students how, when looking at a proposition, we can understand what we can use and what we cannot use, by trying examples. if we have to prove forall x in X. P(x), we modify X a bit to obtain Y, and find a y in Y such that not(P(x)), then we must use something that is different between X and Y (since otherwise our proof would apply to Y as well, and there's the counterexample y). meanwhile, if we see that the proposition is true for everything in Y, then it must be true for something that encompasses both X and Y, so we can try use things from that "lowest common ancestor" object (but don't need to).

the example i have now is to prove that every nonzero q in Q has an inverse. i show how it's not true for Q[x], but it is true for C and R and Z/pZ, so while we don't need to use order or infiniteness of Q, we have to use something that's different between Q and Q[x]. i don't love this example.

can you think of others?

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(the context is computer science students, second year. can assume some basic graph theory, combinatorics, some algebra but nor really abstract, calculus, some linear algebra)

honest elm
quasi maple
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Surely the example has an easier proof?

night badge
quasi maple
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Let q in Q be non-zero; then there are a, b in Z such that gcd(a, b) = 1 and q = a/b; consider b/a

quasi maple
night badge
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yeah, i dont love that because im not super using an interesting difference between Q and Q[x], im using something very strong which is a bijection between Q and N^2

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and even though the proposition is true for R and C, im not using anything remotely applicable there

quasi maple
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Right, then I think it's your theory explanation that you need to tweak slightly

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You don't want Y a "modification" of X, you want Y a strict superset of X

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You can analyse Q as a subring (of sorts) of Q[x] but Q[x] doesn't have that inverse property

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Q as a subring of R (or indeed of C) gives you that inverse

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Because R (or C) have inverses for all its elements (0 excluded)

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Though I still think this is delving into some Ring Theory that I'm missing atm

night badge
night badge
quasi maple
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tbf Q itself is also a field

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oh you said that already lol

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Okay, but that still leaves you with trying to show that the whole of X satisfies P

night badge
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yes, and so that guides your proof strategy, you can use something that's common to X and Y

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(you dont need to, but you can)

quasi maple
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Showing, so to speak, that a "nearby" set Y also satisfies P leads you only to that X intersection Y satisfies P

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You still need to consider the set X\Y

night badge
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well in this scenario youre asked to prove that forall x in X. P(x), so emotionally you already know it's true, and so you emotionally know the union of X and Y satisfies P

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you dont know this formally of course, you're trying to prove it for X

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but that's how you get information about how to structure your proof

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there (probably) exists an argument that is good enough for the union of X and Y

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at that point you think about what the "lowest common ancestor" between X and Y is, and why P might follow just from that

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e.g. you change Q[x] to R[x] and P still holds, maybe it holds for all polynomial rings?

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so you explore that

burnt vesselBOT
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e
Compile Error! Click the errors reaction for more information.
(You may edit your message to recompile.)

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stuck oxide
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How I would approach it is to define $$L(x):=\lim_{h\to 0}\frac{x^{h}-1}{h}$$

burnt vesselBOT
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oz1442

936146127424737302.png
stuck oxide
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Then we show that $L(xy)=L(x)+L(y)$ and $L(x)$ is continuous by sequences so we can write $$L\Big(\lim_{n\to\infty}\Big(1+\frac{1}{n}\Big)^{n}\Big)=\lim_{n\to\infty}nL\Big(1+\frac{1}{n}$$

burnt vesselBOT
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oz1442

936146127424737302.png
stuck oxide
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$$\begin{aligned}L\Big(1+\frac{1}{n}\Big)&=\lim_{h\to 0}\frac{\Big(1+\frac{1}{n}\Big)^{h}-1}{h}\&=\lim_{h\to 0}\frac{hc_{n}^{h-1}\frac{1}{n}}{h}\&=\lim_{h\to 0}\frac{c_{n}^{h}}{c_{n}n}\&=\frac{1}{c_{n}n}\end{aligned}$$ where $1<c_{n}<1+(1/n)$ by MVT. In the end, we have $L(e)=\lim_{n\to\infty}n\frac{1}{c_{n}n}=1$

burnt vesselBOT
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oz1442

936146127424737302.png
quasi maple
night badge
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so if it's true for Q and for I, then it must be true for a union of both, and indeed it's true for the reals. we conclude that it's true for the reals, we don't use it

quasi maple
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So, "If it's true for Q, then because it's also true for I, then it would be true for R; if we can then prove that it is true for R, then we have that it is true for Q"?

night badge
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no

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"then whatever proof we use does not need to use things that are exclusive to Q or exclusive to I"

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and

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"there might be a proof that tackles all of Q U I at once"

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(moreso if the union of Q U I has some structure that seems relevant to the proposition we're proving)

quasi maple
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Again, then this boils down to "prove for a superset" (that you're checking Q u I does not make this not-a-superset)

night badge
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of course it's a superset, i said so from the get-go, but what you're saying is that Y is the superset

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i am saying that one creates the superset by finding a different set, Y (in this case the irrationals) for which it's also true

quasi maple
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You've gone "kind of" every time I've said "superset" though, so...

night badge
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yes, precisely because what the proof-writer plays with and looks at is Y, not Y U X

quasi maple
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Oh, you're trying to build a superset (by finding some Y)

night badge
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yes

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the superset is Y U X

quasi maple
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That's one step too detailed, I think, especially if you're trying to teach this

night badge
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we are tasked with proving P(x) forall x in X. we see that it's true for all y in Y. thus we intuit that whatever proof we use for X, probably should work for everything in the superset Y U X.

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and can sometimes make our proofs cleaner

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what students always do is try with small examples, and then abstract from them

quasi maple
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Ideally you should abstract from the set you're working in

night badge
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and this is trying small examples that are outside of X, to see if it breaks, or keeps being true

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if it breaks, your proof must use something that is different between Y and X

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if it doesnt break, your proof may (but doesnt need to) use things that are common to X and Y (i.e. that are true for Y U X)

quasi maple
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"
Suppose A is a proof (e.g. in strict first-order logic) that states that for all x in X, P(x).
If A also holds for all y in Y, so that P(y) holds, then A is a proof that works for the union X u Y.
If there is some y in Y such that A is insufficient to prove P(y), then A relies on some property in X that is not in Y.
"

is the gist I've got

night badge
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yes, and proofs that hold for more abstract things tend to be cleaner than proofs that work only for smaller cases

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thus the heuristic "try to abstract"

quasi maple
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I believe "abstract" is less the word you want and "general" more

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This is a form of generalisation

night badge
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(there are cases where this fails, and the proof for the abstract case is just "proof by cases" on two totally disjoint sets of things)

quasi maple
night badge
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im less interested in the philosophy part, more the proof-writing-strategy part

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if the heuristic saves them time on average, it's good

quasi maple
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Ah

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Then you want to rephrase how you define "Y"

night badge
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how do you suggest?

quasi maple
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e.g. the set of irrational numbers is not a "modification" of rational numbers

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It's literally distinct from them

night badge
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modifications are usually different haha

quasi maple
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"Modification" implies there is still some shared property (and of a set, that usually also includes at least one element)

night badge
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mm idk, i'd think Q and Q + pi are modifications of eachother, and they share nothing

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(no element, i mean)

quasi maple
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Okay so

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What you want to explain is

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If you want to generalise X to a set Z, you want to check if a property P holds for some z in Z but not in X

night badge
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yep

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and i suppose also the motivation that generalizing to a set Z is a good heuristic, since it sometimes makes your proof shorter

quasi maple
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Don't introduce "Y", not as you have done, you're just gonna conflate these ideas

night badge
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sounds good

quasi maple
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Z, a larger set, is what you need - and to test whether it's viable, you need to consider whether elements in the difference still apply to P;
beyond this point, you then can consider proving P for Z, which makes it a generalised proof;
this proof would still apply for X, which is what we're aiming for

night badge
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sounds good, your point is that i should say "the difference" without naming Y

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and to put the focus not on Y, but on Z

quasi maple
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Yeah, you want to explain the motivation behind your generalisation upfront

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(if you like, your "Y" is actually Z-X)

night badge
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yep

honest elm
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<@&268886789983436800>

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most likely in quite a few other chan

wary ether
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We managed to nip it in the bud, thanks

serene apex
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Thanks

honest elm
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yw

tepid cypress
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Hello guys! I'm preparing a talk for new students at my university, I was assigned the topic of combinatorics. What are some classic and entertaining/engaging problems I could present and solve?

warm grail
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Six partygoers/Ramsey theory could be a good one

tepid cypress
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this, right?

image.png
warm grail
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Yeah!

tepid cypress
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I'm going to look into it! thank you

cosmic ibex
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Ramsey numbers is more a shock-and-dazzle thing (something with such a simple statement can still be obscenely hard), but that can also go counter to a desire to present the topic as something where we can actually prove things, solve things, and know what we're doing ...

tight star
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wait you can know what you’re doing in combi??

tepid cypress
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I thought about calculating every possible pokemon team lol

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but idk if freshmen have played pokemon like i have

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😭

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i wanna have it be engaging

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there's also the futurama theorem, that one i really like as well

lapis cosmos
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Jk

austere inlet
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please don't post video call links in the server

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and stay on-topic -- this channel is to discuss tips on how to teach mathematics, not a place to e.g. find tutors

tawny slate
tepid cypress
rain hawk
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A parabola with vertex at the origin has equation y = x²/4p where the focus is at (0,p). Is there an intuitive explanation for the factor of 4 here, the algebra isn't too hard but a high level explanation would be nice to have in the back pocket

tawny slate
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Choose n points around the circumference of a circle, and join every point to every other with a line segment. Assuming that no three of the line segments concur, how many regions does this divide the circle into?
The sequence starts 1, 2, 4, 8, 16. What comes next? (Hint: it's 31)

There are more possible games are chess than atoms in the observable universe, and it's not even close. This is a consequence of combinatorial explosion (which arguably is a function growth thing and not a combi thing, but what is combi really). Other applications of this include the idea that if you shuffle a 52-card deck uniformly randomly, you will always shuffle it into a unique order that no one on the planet will ever see, + rice/chessboard story

https://oeis.org/A250001 This sequence is not so hard to work out manually for small n values: (https://oeis.org/A250001/a250001_3.pdf) However, we have no clue where to even begin to write a program that can find the value of the seqeunce in general, let alone have a formula of some kind.

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kruskal's tree theorem could be a good example in theory, but might require too much work to figure out how to make it accessible to be a "quick example". TREE(n) doesn't take too long to explain probably, but getting students to understand even at a cursory level just how huge TREE(3) is might be futile. numbers that big are far too large to comprehend for even many professional mathematicians, so they'll probably just think "oh ok big number shrug"

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i guess i should explain that these examples are "demonstrated" more than they are "solved"

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stars and bars, as well as this problem:

image.png
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i think might be good candidates for demonstrating that it serves a purpose to abstract objects in real life into something in math

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like stars and bars turns boxes into the regions between dividers

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the screenshotted problem takes empty parking spaces as objects we can just insert in between other filled parking spaces, which is clearly absurd, but helps us count them

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if i think of any more examples i'll share them

gentle slate
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hey all

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i'm starting a math/science club next year at my hs

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and the plan for it is that several people (including me) will be free to teach their own interests to other people on designated days

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such that on one day, the organic chem expert teaches organic chem, on another the linear algebra expert teaches linear algebra, etc

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and there's one rule, you can't teach something that's already a course in the school

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calculus is a course already but it's mostly derivatives and curve sketching

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how would i go about teaching integration?

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keep in mind i'm grade 12 next year but i've got a solid base in multivar calc to teach lower levels of calc

tight star
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May I suggest not using the typical “area under a graph” approach? 😅

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I think viewing integration as an ODE solver can often be more fruitful

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Where “area under the graph” is a special case of this

tight star
burnt vesselBOT
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Pseudonium

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tight star
vestal tangle
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4p is the "focal length" of the parabola, which is the blue line in that picture

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y=x^2 has a focal length of 1, so if you know your focus is p then you need to stretch y=x^2 horizontally by a factor of 4p

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which you do by dividing

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hopefully that's intuitive enough

rain hawk
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Why would it have a factor of 4

vestal tangle
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line is level with focus point so the height of the end points above directrix is 2p

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so each half must have length 2p so total is 4p

rain hawk
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Oh that's good

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Thanks

placid timber
gentle slate
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work with basic ones where the answer is obvious, etc

tight star
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but you can introduce the idea of an “accumulation problem”, as that video suggests

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for example, trying to figure out how long a mars rover travelled given its velocity measurements

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it helps introduce integration as this “local to global” translation

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you don’t know the precise trajectory the rover took as a function of time, which would be the “global” description

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but you do know how fast it was travelling as a function of time, which is a more “local” description

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integration is then the attempt to translate this local information into global information

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“finding the area under a graph” is then just a special case of this kind of accumulation problem

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of course, any other accumulation problem can be transformed into an “area under a graph” problem

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but unless the origin of it is that specific geometric question, it’s a little confusing to think of all integration this way

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as that video explains, students often get confused at “how can distance be an area”, and I don’t necessarily blame them

gentle slate
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so in general explaining integration as a classical mechanics fundamental?

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that’s actually a pretty nice intro to it

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it goes against my idea of anything low undergrad and below having a geometric intuition but this actually sounds better

tight star
gentle slate
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aspiring physicist here 😭

tight star
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oh, nice!

gentle slate
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struggling with multipole expansion atm but that’s not a convo for here

tawny slate
placid timber
willow crater
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idk where this belongs. but i found a way to plot prime numbers using just one axis and square rules translating you into the complex plane and rendered the paths/points taken for composite numbers as it navigates the space and ive discovered what seems to be a predictable distribution of how primes and composits are arranged in a volume. If taken to infinite points, this would form the remainder of a cubic volume with y=1/x being the function at the center of every plane.

8 of these can be arranged to create a cubic area with a sperical void in the middle.

Idk where to share this. I just was playing around in python and this was interesting to me. i can share the code if anyone wants

willow crater
rain folio
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I'm starting as a peer tutor for 9th and 10th grade math as a rising junior going into MVC. I haven't had to think about HS type geometry in ages

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Any tips for teaching it? Most of my friends found it really complicated and it kinda turned them off math

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I think the emphasis on just magically finding a solution or just somehow seeing a hidden relationship can be really hard on people who aren't experienced in math or aren't confident

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Any advice appreicated

rose dove
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you could look through a textbook, it would probably have some type of motivations/overarching ideas that would make math less about "miracle results" and give more direction

nova stirrup
tight star
nova stirrup
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sorry i'm meming xd

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u havent seen that meme?

noble hare
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0

tawny slate
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the results are basically exactly what I would have expected, but it leads to another question:

to what extent should educators allow AI-assistance in students' work?

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on one hand, the argument is that if AI is going to inevitably be a part of society, we should teach students how to use it. but if there is a cognitive cost to students using this tool, does it not defeat the point of education?

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I am now of the position that AI should be a topic to be explored, because of how deceptive and dangerous it is, but the students' work must not use AI in any form. there should be a class or lesson that outlines the dangers of AI, but outside of it, it should be de-emphasized if not avoided outright

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anyone else have any thoughts on this?

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search engines are obviously an entirely different matter, since there is no real alternative to finding external information and sources, and they have to be used to evaluate source quality regardless

rose dove
# tawny slate anyone else have any thoughts on this?

Here is an article arguing the opposite: https://doi.org/10.1057/s41599-025-04787-y
But I agree with you that GAI should be discouraged in all courses. Wang and Fan's analysis (I only read the abstract) seems short sighted in that it prioritizes academic outcomes over personal and social outcomes. The most important effects in my opinion are that GAI makes human-like interactions that lack direct social benefit and that it allows people to become worse thinkers as the article you linked indicates.

Nature
The effect of ChatGPT on students’ learning performance, learning...

Humanities and Social Sciences Communications - The effect of ChatGPT on students’ learning performance, learning perception, and higher-order thinking: insights from a meta-analysis

tawny slate
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thanks, skimmed through the article a bit, haven't read deeply into it yet, but the literature review section seems to outline a detailed description of the pros and cons of AI-assisted learning, and yet the abstract definitively says the effect is positive and recommends integrating it into education

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this feels like it's burying the lede, why are they doing this? or am i reading this wrong?

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on the other hand, while the study i linked isn't a meta-analysis, it does approach it from a neuroscience perspective, which to me, is one of the most direct ways we can measure how it affects brain activity, so it still feels like it's stronger evidence to me

rose dove
# tawny slate this feels like it's burying the lede, why are they doing this? or am i reading ...

well, GAI has real positive effects, especially related to academic equity. i think the problem is that between me and Wang, one of us is a bit out of touch. Wang seems to believe that regulated use of GAI will solve many of the problems it presents. i think that regulating GAI use is nearly impossible (also, regulating GAI). Wang is an empirical researcher and seems to get caught up in the positive results of their study and other empirical studies. the negative effects they mention in the lit review are mostly (all?) theoretical, more from a humanities perspective. i think humanities results are very important.

rose dove
# tawny slate on the other hand, while the study i linked isn't a meta-analysis, it does appro...

i think it probably is stronger evidence, and more targeted toward the long term. one big problem i see that makes this divide is that society makes things (including peers) super inaccessible, which makes the ability of GAI to make things accessible (albeit poorly and sometimes falsely) really enticing from an education-on-the-ground perspective, sometimes enough to outweigh the fear factor

tawny slate
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thanks for your input

my personal feeling is that society doesn't really have much of a fear factor to this kind of thing to begin with, but that's beside the point here lol

dapper flume
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The way AI needs to be handled should at least be based on the learning objective. If the objective is to find the volume of a cylinder, then sure. Teach the tool for rote assistance and searching. But if the objective is to analyze a scatterplot with some kind of sociopolitical significance, then it doesn't matter how well the AI does it. Even if the AI was amazing at this task, the verb of the learning objective is to analyze, not produce. If we care about anything other than the final product of a task, then we should be teaching AI independence with at least as much as we teach AI use.

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Especially since AI is at this moment not very good at that kind of insightful analysis, as much as it fools a layman into thinking it is

rose dove
tawny slate
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i still dont think AI should be used in things like volume of a cylinder either

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  1. its easy to understand conceptually so this needs to be taught
  2. even if you did need to look it up, you have to be able to verify the result, which involves using a search engine anyways
  3. maybe the AI could prove it independently so you dont need to go to a search engine, but why go through such a roundabout method when the top 300 results all give the same answer anyways?
#

i think "rote assistance" is a bit poorly defined here so i wont comment on it, but searching is something i think should only be assisted by AI if the user explicitly does not know what to even search for and plans on using search engines to check and verify

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otherwise, thats what search engines are used for

#

half the internet is AI output too so if students cannot learn to distinguish AI writing style in the wild and verify source quality, then we are really in big trouble

#

maybe you already agree and that was just a bad example, but i just had to clarify because i really think this is a common opinion and thats concerning to me, like as a society we collectively forgot what search engines are for and how to validate sources

dapper flume
# tawny slate maybe you already agree and that was just a bad example, but i just had to clari...

Yeah i was like struggling to think of a good example lmao. Maybe there isn't one. I was being as charitable as possible to AI positive attitudes but honestly I can't see myself ever appealing to using AI as a tool in my kind of classroom. We are a thinking classroom.

It's easier to justify for other subjects maybe. I know in my philosophy of education course, our final was to design a school based on personal philosophy. We had to write up the paper ourselves, but if we wanted to generate AI imagery to visualize what the school might look like, that was welcomed. I still felt dirty doing it but it saved a lot of time completing a part of my assessment that was not crucial to the actual knowledge I was supposed to demonstrate.

Maybe there's something like that for math. Some kind of performance task which calls for the creation of artifacts where the non-mathematical act of building the artifact itself could pose an unequitable barrier to completion for some students. Maybe then. But I dunno.

tawny slate
#

i think for generating "rote" problems, the kind where you apply rote steps to exercises as practice to develop muscle memory, i think thats something i may accept as an educator

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but at the end of the day, you have to verify and validate all of the output, so sometimes its still a million times easier to just reverse engineer the problems from scratch (usually a quick <20 min python script does this for me anyways, and i can be sure it works)

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anything slightly more complicated than that youre probably just shooting in the dark and hoping it hits, and maybe you get nice problems once in a while, but i still dont think it beats the efficiency of just pulling from actual sources

dapper flume
analog bloom
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When learning or self teaching math, is it better to...

  1. start with the intuitive/concrete approach, and then develop rigor later, or
  2. start rigorous and then restudy to develop intuition and visuals
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And how does that change between undergrad and graduate-level math?

tight star
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Of course I’m a physicist so I’m biased towards concreteness

analog bloom
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Lol fair

tight star
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I will say that I’ve gotten more comfortable with approach 2 for grad-level math

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This is mostly a result of becoming familiar with category theory

analog bloom
#

How much intuition does one need before proceeding to (2)? For example, would one want to be comfortable calculating limits, derivatives, and integrals or, would 3b1b be enough to attempt to start basic analysis?

tight star
#

I don’t think there’ll be a simple answer to this Q

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Probably what you should do is alternate between approaches 1 and 2

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Perhaps starting with approach 1

analog bloom
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Do I lose anything by defaulting to 2 and switching to 1 whenever I notice I'm out of my depth?

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Basically can I risk skipping the intuitive step without knowing?

tight star
#

Probably not much

#

Sure you can

#

The math police won’t stop you or anything

analog bloom
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😂

#

What are some precautions to make sure I have a solid understanding?

tight star
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Idk

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Having a solid understanding is incredibly subjective

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There are things I thought I had a solid understanding for, and now realise there was even more I could’ve understood about them

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Maybe you can think about how well you could teach it to someone else

analog bloom
#

Yeah that's a good metric I think

tight star
#

3b1b has a talk on “math’s pedagogical curse”

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In it he provides a prototypical checklist for pedagogy

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Perhaps you could check your understanding against that

analog bloom
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I'll check it out, thanks

tight star
#

https://youtu.be/UOuxo6SA8Uc?si=bX1XbqropF8CbyGM

YouTube
Grant Sanderson
Math's pedagogical curse | Grant Sanderson JPBM Award Lecture, JMM ...

What steps can the math community make to improve its communication?
Delivered Thursday, January 5 at the 2023 Joint Mathematics Meetings in Boston, MA.
Reposted here with permission from the AMS

Correction at 50:24 – The measure of transferability score for Class B (3.1) and that for Class A (5.4) should be swapped.

Podcast mentioned in the...

▶ Play video
tight star
#

How would you explain or motivate the exponential function to someone seeing it for the first time?

vocal phoenix
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I haven't had to do it in quite a while but I think I'd talk in terms of proportional growth, like population or compound interest.

tall bolt
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I like the population approach because it also leads quite naturally into the standard “there was 5 million bacteria at 12pm etc” problem. Compound interest is another good way to go if the students care more about “real world” stuff

vocal phoenix
#

In the other direction you have proportional decay; radioactive half-life being the most obvious example, but also pharmacokinetics

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Most pharmacological substances have their half-life stated.

tight star
vocal phoenix
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I have very little experience teaching pre-university mathematics so I don't really have worthwhile thoughts regarding the order of things

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In a calculus course it would probably be quite early on, the order I'm used to starts with general notions related to sets and functions, and then a refresher on some specific functions such as exponential, logarithmic, trigonometric and inverse trigonometric.

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In my country all of those except inverse trigonometric are in the high school curriculum, so at university we treat it more as a refresher/getting everyone on the same page.

tight star
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I see

dapper flume
# tight star Around when do you think you’d introduce it?

The concept of exponential growth (without graphing) is something my curriculum taught at the end of 8th grade, with the actual function f(x)=a^x being introduced in Algebra I or II.

We just motivated exponential growth at the 8th grade level by considering what happens when a genie doubles the number of cookies you have every day. Money or radioactivity references are accurate but a bit inaccessible to young students, many of which probably have not yet spent a dollar of their own.

brittle grove
#

Hi
Any one of you working on some sort of research paper or review paper?
Actually i need internships (unpaid will do, just some name or reference on the paper is what i need) .
I just finished highschool. I don't really know how research works.
I would love to work and learn.
I have access to laptop and internet. ( Which means i can read , analyse etc) .

If anyone of you is interested. Kindly dm .

rapid tusk
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<@&268886789983436800> wrong channel

serene apex
halcyon glade
serene apex
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oh yeah, that's perfect.

brittle grove
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Ok i will

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Thanks

tight ridge
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Im not sure if this is an appropriate question to ask, but if you were given 15 minutes to demo teach a statistical concept to be a high school teacher, how would you use the allotted time? I'm a bit stumped right now because im trying to apply for a teaching position haha

abstract magnet
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That or I’d go over independent events

tight ridge
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That's true. The time constraint's kinda messing with me tbh lol

tight ridge
dapper flume
tight ridge
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Just a bit nervous

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Im not based in the us

tight star
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I’m sure I must’ve asked this at some point beforehand, but I feel like my understanding has changed so much that it might be worth asking again

#

What do people usually find challenging about learning, or teaching, category theory?

abstract magnet
vapid trench
# tight star What do people usually find challenging about learning, or teaching, category th...

I started learning category theory relatively recently, so I can give my own experience. I while back I had heard category theory allowed you to generalize ideas from different subjects of math, but I wasn’t entirely clear how.

I started learning from some notes, called “notes on category theory” which were available online, and I could mostly follow, but I had a hard time understanding why we cared about “monomorphisms” and “epimorphisms” for example. This was made clear with examples (e.g. Z —>Q in Ring is monic and Epic, since we can localize Z at the nonzero integers to get Q).

I did a directed reading project on abelian categories last quarter, and ended up doing a poster on Adjoint Functors. I kinda overestimated how much I would be able to explain, which is probably normal. But the biggest issue I had with explaining this topic was that I first had to introduce the concept of categories/functors for what I was saying to make sense. If I had more time/room on my poster, I probably would have tried to include a lot more examples from various fields.

Something I have been thinking about more generally is how to tie math history into teaching. I have really enjoyed the way Richard Borcherds explains the origins of certain ideas or the people who were working on these problems. It gives the math more of a grounding feeling in my opinion, and if I want to explore more about the intuition for the topic it gives me a reasonable place to search for it.

I can elaborate more if you would like, just lmk! I am also very interested in how people learn math.

tight star
# vapid trench I started learning category theory relatively recently, so I can give my own exp...

ooh thanks, it'd be great to hear from someone who's been learning it more recently!

i can definitely understand what you mean about being able to follow but not understanding why we cared about things. i like to call such things "googits" - definitions i can 'parse', but not really understand. (incidentally, a googit is a collection of exactly 3 humans, 2 planets, 1 proton and 1 beaver).

and indeed, a lot of the difficulty with category theory is how nontrivial the language is - often understanding the statement of a categorical result can be significantly harder than understanding the proof! i think if one frees themself from needing to speak formally or precisely, there are ways to give intuitions for these concepts that don't require the full language of category theory, though. i've had success with this in this server (and other places!) with universal properties, for example

#

i'd definitely be interested in hearing more from your experience :)

lethal hornet
# tight star What do people usually find challenging about learning, or teaching, category th...

i think fitting it into the usual framework in, say, an undergraduate degree, is a bit awkward? like, a typical entry point is set theory/logic, calculus/real-analysis, and maybe intro to group theory or topology.
each of these has its own learning curve, framework, and language to get used to, so throwing category theory into the mix seems overwhelming, especially when you don't really need it for the introductory level material.
this also may make accepting it as a valid framework kind of hard, since there are some foundational issues that you often have to ignore unless you have the time to dive into them a bit.

in my experience, i thought it had a fairly steep learning curve in terms of getting the right language becoming more familiar with common ideas that are present within category theory, like (co)limits and the yoneda lemma.
slowly trudging through some more introductory topics in category theory myself, but i find it a bit dry on its own, at least the material i am studying rn. dry is the wrong word... hmm...

zinc dove
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Is it "bad" to make the analogy of a function as a machine that transform a thing into another?

dapper flume
tight star
#

what i would say generally is that you can probably split the viewpoints into two main ones

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the programmer viewpoint is more operational

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and the mathematician viewpoint is more like a lookup table

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specifically, in programming, it's very natural to think of a function as a sequence of operations one does on the inputs

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but this isn't really how functions are thought of in math

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indeed, you have function extensionality - this says that if $g, h$ are functions such that $g(x) = h(x)$ for all $x$ in their domains, then $g = h$

burnt vesselBOT
#

Pseudo (Cat theory #1 Fan)

547515245258735616.png
tight star
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you can think of this as studying properties of functions that are agnostic about their implementation

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so you don't actually care how x gets transformed into g(x) - you don't "look inside the machine" to see the operations being performed

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in this case, the data of a function is completely encoded in a lookup table - given x, you can look up the value of g(x)

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(of course i think i'd prefer function extensionality be expressed by a 2-cell in some sort of $(2, 1)$ category, but that's another story)

burnt vesselBOT
#

Pseudo (Cat theory #1 Fan)

547515245258735616.png
vocal phoenix
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<@&268886789983436800> unsolicited advertising

quasi maple
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The server equivalent of [redacted] DMs

turbid zenith
#

Randomish thought.

Are there any topics in the standard calculus curriculum (wherever you are) that you think shouldn’t be? Or topics not in the curriculum that you think should be?

vapid trench
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I feel like integral calculus should mention upper and lower Riemann sums. At least in my Calculus AB class, we just did “Left Riemann Approximation Method” and “Right Riemann…”. They do make calculation a bit easier I guess, but all you need is the notion of supremum/infimum (even just max/min since most functions in calc are continuous) and then you can explain how the upper and lower Riemann sums give us information on integrability.

This is just my opinion ofc. I just found it strange that this was never mentioned in my Calc class, and yet it’s how we go on to define Riemann integrals in analysis.

heavy trail
# vapid trench I feel like integral calculus should mention upper and lower Riemann sums. At le...

Because Darboux sums—as they're often called when trying to distinguish from left/right Riemann sums—involve sup and inf. Those concepts are more subtle and aren't part of the AP Calculus curriculum (AFAIK).

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

Lots of college-level calculus textbooks use left/right Riemann sums to define integrals. I think Stewart's Calculus does, for example. The two definitions are equivalent, so the only question is what practical or conceptual advantage each affords. Painting with an extremely broad brush:

  • Left/right Riemann sums make it more straightforward to find explicit approximations, especially of "nice" functions
  • Upper/lower Darboux sums are conceptually "cleaner" and make it more straightforward to prove things about integrals and work with (more) pathological/exotic functions.
Darboux integral

In real analysis, the Darboux integral is constructed using Darboux sums and is one possible definition of the integral of a function. Darboux integrals are equivalent to Riemann integrals, meaning that a function is Darboux-integrable if and only if it is Riemann-integrable, and the values of the two integrals, if they exist, are equal. The def...

#

I suppose the thing I'd want to emphasize, conceptually, is that we:

  1. Construct a partition
  2. Want to assign a rectangle to each interval in the partition
  3. Pick a point in each partition to determine the height of the rectangle for that partition

And we have total flexibility when it comes to the widths and heights of the partitiona intervals/rectangles. Generally we make each interval have the same width because it's easier, but there are lots of points in each interval we could pick for the height of our rectangles.

I might have students just start naming the sorts of points they could pick for (3). With a large enough class, some students would probably say max/min on that interval.

I might say that most of the sensible rules for picking a point have a name. In AP Calc we use left and right for ease of calculation, but you might see max and min in future textbooks. Those are called "upper and lower sums", so Google blahblahblah if you're interested.

vocal phoenix
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I would say that the upper/lower sums can probably be stated without the notion of sup/inf in a standard calculus course, since the kind of functions that would appear there will attain their extrema

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so you can talk of max and min

tall bolt
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Maybe they’re just introduced because they make for nice problems, because I do see how they can test geometric thinking and just giving you a nice vaguely motivated thing to integrate

serene dew
#

Hi everyone, I have a question.

Calculating the coefficients of a Fourier series can be difficult and time-consuming. How might a student be motivated/convinced to go through these (potentially tedious) details? Are there recognizable benefits that teachers and/or mathematicians can point to in the hopes of motivating students?

For example, consider a simple periodic function like a square wave. This function can be easily written (e.g. defined piecewise), but the Fourier series is an infinite series of sines and cosines, and we all know infinite series pose notorious conceptual challenges for students at all levels.

A student might ask: "Why bother with this format when we can quite easily deal with the given periodic function?" In answering such a query we might want to address similar questions: "Does this format tell us something useful? Does it reveal deeper behavior of this function and similar ones?", etc.

tight star
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That’s how they arose historically, after all

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The usefulness of a (complex) Fourier series is that it’s a sum of terms that behave very well under translation

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In particular under “infinitesimal translation”, I.e. differentiation

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You’re really just expanding into an eigenbasis of the translation operator

tall bolt
#

Yeah differential equations, signal analysis/processing, just kinda all over the place in physics. Fourier series are one of the most useful tools we’ve ever come up with I think

tight star
#

Which is useful if you wanna do translations

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:P

serene dew
#

I'd say Fourier coefficients contain useful information that is not at all apparent just from the shape of the periodic function itself. For example, when Fourier series are used for musical vibrations we may remove the terms in the Fourier series with coefficients having magnitude below a certain cutoff (kind of like taking out fundamental tones in a musical note at a frequency beyond the range of human hearing) and the resulting finite Fourier series may be indistinguishable from the original but require a lot less information to store it. That's the whole idea behind data compression. If students ever listened to an MP3 file then they know why this is important without even realizing it.

cosmic junco
#

Hello ! since i have no other place to go, I am trying my luk here. I am building a narrative universe to teach progressively middle school mathematics. I had a lot of conversations that made me think it's the right thing to do. Would some of you be interested ?

dapper flume
cosmic junco
quasi maple
serene dew
#

Hi everyone.

Normally, mathematicians teach calculus by presenting concepts in successive layers. What I mean is that, for example, we study limits in general, and then we tackle limits for various types of functions — polynomials, rational functions, trigonometric functions, transcendental ones, and so on. The same applies to derivatives, and so forth.

I was wondering whether, when dealing with students with a weak background, it might be more useful to proceed not by layers (horizontally), but by increasing the complexity of the functions studied (vertically). For example, one might start with real polynomials in one real variable. One studies their continuity (which is straightforward), their limits, their derivatives (without quotient or chain rules, etc.), and analyzes them as functions. Finally, one studies their integrals.

At that point, once the students have digested all the relevant concepts, one moves on to algebraic fractions—again with limits, derivatives (here introducing the derivative of f/g), and so on—then proceeds to exponential functions, trigonometric functions, etc.

Of course, all the (relevant) theorems are introduced gradually throughout the developing of the course.

What do you think? Has this approach been used before? Are there textbooks that adopt it?

dapper flume
# serene dew Hi everyone. Normally, mathematicians teach calculus by presenting concepts in ...

Besides pure speculation, what reasons might you have for shifting the perspective this way for struggling students?

There could be some benefits to this structure that come to mind. For example, since the Fundamental Theorem of Calculus will have already been understood by the time students reach the study of, say, the exponential function, they will be primed to think about antiderivatives right away. I think that opportunity for inquiry (such as the eventual need to discover u-substitution) is not to be dismissed. There's also a lot of opportunities for pattern recognition within classes of functions, like how trig functions tend to differentiate and integrate into other trig functions.

I also sense a number of potential drawbacks.

  1. The concepts of calculus are not quickly intuitive to most students. Many college calc 1 students need the entire week to really visually understand the difference between the limit of a function at x=a and the value f(a). Hopping so quickly between limits, derivatives, and integrals may leave a lot of holes in their knowledge.
  2. The course relies very heavily on a well-developed understanding of algebra, which from tutoring experience is the biggest barrier to success in calculus. Trying to apply all this algebra while learning all three major calculus concepts at once at the beginning of the semester sounds overwhelming for struggling students.
  3. Working with classes of functions might make it tricky to figure out how to include compound functions like e^x /(x² +1). Which part of the course would this function show up in?
cosmic junco
# quasi maple I'm interested to know what this "narrative universe" entails

The idea is to create a universe that provides an interesting, intrinsic, motivation to be interested in mathematics.

For example, imagine Full Metal Alchemist with chemistry instead of alchemy, or Harry Potter with math instead of magic, as the underlying narrative element. I'm sharing the first draft: It's 2 chapters in 1.
https://baatales.vercel.app/story/ch-1/index.html?lang=en

I plan to write shorter chapters of a few pages, regularly. And around that, introduce more theoretical stuff via the lore of the universe, through more interactive elements, closer to video games.

serene dew
# dapper flume Besides pure speculation, what reasons might you have for shifting the perspecti...

Thank you.

As a potential drawback of the method, don't you find that there is a risk of repeating similar proofs for each new class of functions, losing the efficiency of a general theorem formulated once and for all? The systematic treatment of limits would be postponed, although it remains indispensable for more advanced topics (series, uniformity, differential equations). As a compromise, I would keep the vertical progression up to the most common elementary functions and introduce the theory of limits as soon as cases that cannot be treated with algebra alone emerge (e.g. $e^{x}$ defined as the limit of $(1+1/n)^n$).

What do you think?

burnt vesselBOT
#

Mēdèn ágān

1236220929495924750.png
dapper flume
# serene dew Thank you. As a potential drawback of the method, don't you find that there is ...

I definitely sense that drawback, but would argue that an opportunistic teacher could see that repetition as a benefit. To me, the similarity between limits and derivatives of different functions highlights the power of calculus as a general, unified approach to the analysis of functions that otherwise are not alike from the high school algebra perspective.

As a student of the American university system, it seems to me that there was a sort of compromise between the vertical and horizontal approaches in the early days if calc 1. Importantly, integration was not a part of this process until calc 2. But we would take elementary functions case-by-case, using the limit definition to compute the derivative of several examples before compiling thd derivative rules based on patterns. That was motivated by a general understanding of what a limit or a derivative even is. Yes, it was repetitive, but in the case, it only took one-and-a-half class days to derive these rules, leaving us with a deep understanding of patterns, and plenty of time to practice with more structurally complex functions

quasi maple
quasi maple
# dapper flume Besides pure speculation, what reasons might you have for shifting the perspecti...

As alluded to here in points 2 and 3, one major issue I have tutoring A Level maths [which essentially covers precalc and a lot of calculus to give a rough translation into US terms] is that a not insignificant number of students tend to not yet be sufficiently fluent/confident in their algebra; I would rather prefer the approach that allows students to keep checking that their algebra is in order instead of throwing what I'd argue is a plethora of functional operations (differentiating/integrating inc. techniques associated) at them

quasi musk
quasi musk
cosmic ibex
turbid zenith
quasi musk
#

Procedurally it gives students good practice with inequalities, graphically it's very intuitive, and it will clear common misconceptions about limits that students develop if repeated throughout the course intermittently

#

There's the triade to teaching limits: the graphical, the numeric, and the algebraic approach. Epsilon-Delta builds a concrete bridge between all three

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It also prepares students going into Linear Algebra to not solely rely on their intuition, but to work with a difficult definition

rain comet
#

(Not even sure if I’m asking in the right place since it’s less about the actual maths)
I’m explaining the theorem ‘there is always two opposite points on earths surface with the same condition X and Y’ and wanted to try and give an answer that’s more imaginable - if we let condition X be ‘brightness’ it’s easy to imagine the line of points on earth experiencing sunrise or sunset, but I can’t think of a second condition where the intersection is a well known concept. I could try explaining it entirely abstractly but I like examples like this

serene dew
# quasi musk I think epsilon-delta definition of limit should be mandatory in AP Calculus and...

Speaking of which, I wanted to ask you all a question (which proposes a different perspective).

Why isn’t the epsilon-delta definition taught only at the very end of calculus?

Understanding epsilon-delta was never particularly a problem — rather, the problem arose because it didn’t match the intuitive definition.

In short the epsilon-delta: we take increasingly smaller epsilons on the $y$-axis, and for each we find appropriately small deltas such that the $f(x)$ values belonging to the $x$-values within the delta fall into the epsilon — except for the center of the delta interval, which is usually denoted by $c$ or $a$.

The intuitive definition states that as we get closer to point $c$ with the point $x$, so does $f(x)$ to $L$. Whereas, the epsilon-delta intervals "trap" the point $L$ belonging to $c$ with increasingly narrow intervals.

The solution to the problem is to rethink the intuitive definition from a different perspective:
to $x$-values that are increasingly close to $c$ belong $f(x)$-values that are increasingly close to $L$ or, more precisely, $f(x)$-values that are arbitrarily close to $L$ come from $x$-values that are appropriately close to $c$.
<-- This is what the epsilon-delta proves, without stepping anywhere. It just examines the neighborhood of $c$ and $L$, about which it establishes that the arbitrarily close $f(x)$-values come from appropriately close $x$-values.
In general, increasingly close $f(x)$-values come from increasingly close $x$-values.

burnt vesselBOT
#

Mēdèn ágān

1236220929495924750.png
serene dew
# serene dew Speaking of which, I wanted to ask you all a question (which proposes a differen...

I think it should rather be done like this: first explain what the intuitive definition is, maybe temporarily introduce the expression infinitesimal. Once the intuition has deepened, the Heine definition of the limit should be introduced as a mathematical formalization. When integration and differentiation have been built on top of these, only then should the epsilon-delta be introduced, as an alternative but completely equivalent definition to Heine’s, which treats the limit statically — see what I wrote above.
It’s much, much harder to build concepts like differential quantity, integration, etc. on top of epsilon-delta, and without proper explanation it only leads to misunderstandings.

quasi musk
#

The issue with that is that derivatives are defined by way of limit, and so are integrals. I've experimented with doing epsilon-delta at the end of the class when I had extra time, and that's always been good

#

But I feel the payoff for the extra effort at the end isn't there. Whereas if it's baked into the course, you get so many more moments to tie things back to epsilon-delta

#

As far as you want to explain epsilon-delta, I'm sure everyone has their go to phrases

turbid zenith
#

When I teach Calculus I, I do derivatives first, then integrals, and finally limits.

#

And under those circumstances, I’m okay with presenting the epsilon delta definition because I can actually argue at that point why it matters.

quasi musk
#

Is that usually the business-calculus that you're doing that in?

#

I found business-calculus very difficult to TA for, and I've asked never to be assigned it again

turbid zenith
#

For the business calculus one we don’t do limits at all

quasi musk
#

Even at my own university now, we largely don't cover epsilon-delta. We're on the quarter system and it often feels like fast-food calculus

turbid zenith
#

I used to be in the “it doesn’t belong in calculus” camp

#

And I’m still skeptical, but I can get behind it if it’s woven into the narrative the right way

#

What I don’t like is “well derivatives are ✨DEFINED ✨ in terms of limits, and limits are ✨DEFINED ✨ in terms of epsilon-delta proofs, so clearly before the students see any kind of application of calculus we need to juggle Greek letters around”

steel vortex
#

I feel like you only need to put epsilon/delta in the course if there are math majors in it, otherwise it's a lot of work for not a lot of gain

#

Even then, you don't need them to be good at epsilon/delta, it's really just a preview for what they'll see in later courses

turbid zenith
#

Addition is ✨DEFINED ✨ in terms of Peano axioms but first graders get by without them just fine ;P

wind hemlock
#

the thing is defined after it's been created

#

the Peano axioms were invented in the late 19th century, but people already knew how to add before then

#

if you don't know why the thing matters, you don't have a reason to define it

turbid zenith
#

And limits were invented long after calculus was developed, largely to account for pathological technicalities that the vast majority of calculus students will never see

steel vortex
#

True, but having the limit definition of the derivative helps explain why the product and quotient rules are the way they are

#

It’s fine if students black box those proofs once they’ve seen them, but it’s nice to explain why those formulas aren’t as nice as we’d hope

#

(Even if the ‘limit’ in question isn’t a fully formal epsilon/delta)

turbid zenith
#

And the quotient rule comes from rearranging it.

steel vortex
#

I should include that as an extra proof, next time I do Calc I (whenever that may be)

tawny slate
#

i think epsilon-delta is one of those things that students need to do at least once

#

in the same category as like proving Pythagorean theorem or deriving the quadratic formula

#

like its core foundations, but its tedious and you can eventually do without them by operating at a higher level for the most part

#

order wise i dont really care when its introduced as long as it makes sense and fits the class

#

but i do like the idea of introducing it last

serene dew
#

On second thought, it is understandable that the formal definition with epsilon and delta may seem premature or even counter-intuitive when presented at the beginning of Calculus. However, maybe the decision to introduce it early on has some definite didactic reasons. From the very first lessons, actually, there is talk of limits, continuity and derivability: concepts that, without rigorous language, remain suspended between verbal images ("$x$ approaches $c$ and $f(x)$ approaches $L$") and practical calculations. The move to a definition specifying "for every epsilon greater than zero there is a corresponding delta" forces the student to distinguish clearly between what is arbitrarily small on the $y$-axis and what is close enough on the $x$-axis. In other words, it makes explicit the logical structure "for every ... there exists ..." that governs most of the proofs in Calculus: if this is practised early on, later on, when more complex theorems appear (such as the existence of the limit of the derivative or criteria of convergence of series), the student will find himself already accustomed to handling quantifiers and precise estimates.

What do you think?

burnt vesselBOT
#

Mēdèn ágān

1236220929495924750.png
turbid zenith
#

My view of this comes from a high school geometry book I’ve seen called Discovering Geometry.

Most geometry books want to introduce proof so they do it very early, and students have to flip back and forth between learning concepts and doing (two-column) proofs. So in practice most students rarely get a handle on proof.

What this book does, though, is it treats everything as a conjecture, where students discover the various properties as they go, and then at the very end of the course it introduces the idea of proof and the last two chapters are dedicated to proving everything.

steel vortex
#

Remembering back, I didn’t care for ε-δ the first time I saw it, but sequence converge came quite easily via the “ε-N game” (you pick epsilon, I pick N. Convergence means I have a winning strategy). After working with sequences, limits of functions came much more easily.

tepid smelt
turbid zenith
#

It certainly doesn’t work when entirely unguided

#

But entirely unguided discovery is a bit of a caricature of how pedagogies like IBL are usually implemented

warm valley
#

What are people's thoughts on books about learning proofs (not specific recommendations, but the concept)? In the pedagogical environment that I matured in, all the professors/mentors I respect thought they were pretty dumb (along the lines of, "to learn proof you need to do, not to read a book listing out proof techniques"), and a better approach is essentially a "playground course" where students see things they have seen before (in our case, elementary number theory) so that they have intuition beforehand, but are taught how to do it rigorously in this environment, and proofs really take the form of the next step of showing your work. This always seemed a bit dismissive of such books, but has really worked in practice in my experience

cosmic ibex
#

I think the situation is that ideally proofwriting should be taught together with some actual mathematics when the students first meet seriously proof-based math -- say, in beginning real analysis or linear algebra. And ideally there should be one textbook that covers both sides in a well-thought-our progression.
However, if you don't happen to be taught by such a book and a course that gives sufficient emphasis on the details of proofwriting, then a dedicated proof book can be the next best way to remedy that after the fact.

#

a "playground course" where students see things they have seen before (in our case, elementary number theory) so that they have intuition beforehand, but are taught how to do it rigorously in this environment
Isn't that essentially what dedicated proof books tend to do anyway?

tawny slate
#

are proofs at university+ level any different from proofs at a high school and below level

#

seems to me the same principles generally apply

lethal leaf
#

most peoples' view of proofs before undergrad (and thus for most people in general since most people don't major in math) is 2 column proofs in a geometry class

#

you ask any person who went to high school (at least in the US) this is what they think of proofs as:

two-column-proof.png
#

I went to a good public high school and even then this is all I knew of as proofs before my proofs course in undergrad

orchid steeple
#

i wanna get an honest (pedagogical) take from an educator on a math / theorem prover llm model made by a famous company called kimi moonshot from a specialized group called AI-MO that does a lot of lean4 automated theorem proving with a little bit of natural language mixed in , but i really dont want to like share unsolicited links... it's just a demo i'm hosting on an h200 on huggingface with freely accessible code, if anyone is curious at all just dm me and i'll send you a link to try it out

#

@lethal leaf if you've got a minute ... i'd love to share tbh , just want to see what professionals think about this

#

that's a no right ? i'm not an expert signal reader xD i wont be a bother about it , totally understand

lethal leaf
tawny slate
#

like yeah, those two column proofs in geometry aren't as rigorous as they could be, you can go further

#

you can explicitly lay out what axioms/theorems are allowed, you can be more precise in how they are applied

#

but how is it different?

lethal leaf
#

But my point is not really the rigor (in fact they're perfectly rigorous)

#

But rather the strict formatting

#

When I took such a geometry course, I was forced to use this format. Even if I wanted to just write sentences I couldn't

#

Versus in a university proofs course you write complete sentences

lethal leaf
tawny slate
#

....that's it?

#

the emoji reacts and your reply made it sound like there was something way more obtuse and deep to it

quasi musk
#

I always told my students that two column proofs are the training wheels to writing proofs

#

I had one middle schooler that I taught for 2 years that preferred writing complete sentences, and his proofs were very good

quasi maple
#

The main thing my students get wrong about proofs - though this might be because I'm a tutor and not a teacher, so my pool of students is decidedly biased - is that what they are meant to prove is something they think is a given

#

This is only amplified when I have to teach contradiction

tight star
quasi maple
#

So like, when you have to prove a certain quadratic expression is always positive, for instance

dapper flume
#

I'm an American Geometry teacher. The two-column proof method really only exists for two reasons. First, because that's how they're presented on standardized tests. And second, because they serve as a structural reminder to students that every single deduction in a proof needs to be justified.

Used right, they're plenty rigorous, even though the presentation is kinda ugly and I hate it lol.

Standardized tests will often leave parts of a two-column proof blank and have students determine how to justify certain steps. Which is neat i guess. But not very representative of how proving stuff actually happens.

When i teach proofs i like to have students read aloud the 2-column table as if it was sentences. Helps them internalize the language of deduction

quasi maple
tight star
quasi maple
#

It results in them having written a proof backwards, so to speak

tight star
#

Mhm, but what’s the issue with that?

quasi maple
#

Essentially, you end up perhaps with a proof of the form "A if B, if C, if D"; but students end up presenting it "A, meaning B, meaning C, ..."

#

i.e. the implications are in the wrong order

tight star
#

Yeah i think this is why explaining $\iff$ could help

burnt vesselBOT
#

Pseudo (Cat theory #1 Fan)

547515245258735616.png
quasi maple
tight star
#

I think there are times when doing a proof “forwards” makes it seem mysterious is the thing

#

Analysis comes to mind

quasi maple
tight star
#

Hm, i see

quasi maple
#

[I feel it's related to when they solve equations equal to 0, but only write an expression because they think it's the same thing]

dapper flume
#

Any experience with using the words (and, or, if... then, if and only if) before using the symbols?

quasi maple
#

[e.g. and I find this a common trope at this point, when writing a solution

x^2 + 5x = 24
x^2 + 5x - 24 
(x + 8)(x - 3) 
x = -8 x = 3

which needs some proper formatting to be a clearer solution - the only problem being that at this point I'm talking about the difference between writing correctly/appropriately and _writing enough "to get the marks"]

quasi maple
#

i.e. write down A [what you want to prove]
(if) B
(if) C
(if) D - and you know D is true "by default"

#

And then for your proof
D (is true)
(which implies) C
(which implies) B
(which implies) A, QED

quasi maple
tight star
#

Yes I think students are often unsure what they’re allowed to assume

quasi maple
tight star
#

If I was tasked with proving associativity of real number addition I’d be quite lost

quasi maple
#

(even with using number lines)

tight star
#

As a student

quasi maple
#

Worse when you end up with "no" solutions, which still appears to be a foreign concept more often than not

tight star
#

I could totally see students being unsure of whether they have to prove x^2 >= 0

#

Or e.g. whether they have to prove “every natural is either even or odd”

quasi maple
#

Specifically I tutor A Level (UK) maths; I've been tempted to write a bunch of things from the GCSE that students can use as a refresher before starting out on the A Level because I feel sometimes they forget things that by this stage should be second nature

#

A not insignificant number of them still struggle with adding fractions, for instance

tight star
#

Yeah fractions are hard

#

They’re probably the first instance where students really have to grapple with an equivalence relation, even if only implicitly

dapper flume
# quasi maple Assuming this was directed at me, explicitly I can only think of `and` and `or`;...

Yeah that was for you.

It sounds like students have a lot of interesting misconceptions! I fully believe that these misconceptions are actually there as you say.

Your most crucial role as a tutor is to address misconceptions and show students how to break down those notions and reconstruct better notions.

What is your approach to showing students how and why they need to dismantle misconceptions?

quasi maple
#

["misconceptions" in every sentence there honestly comes off as the following comic: ]

why-did-robin-hurt-his-own-body-is-he-stupid-is-he-v0-s9d9xo118uec1.png
dapper flume
#

Lol yeah I am a bit verbose. Don't like juggling a million pronoun antecedents in a way that makes sense

quasi maple
#

Honestly if I realise they have a misconceived notion of something, I try to take a zero-prerequisite approach if I can, because it gives me the most freedom to do so

#

[I still don't know how to teach order of operations properly though girlbleak ]

#

The other issue is, ideally I don't want to overload a student with first-year-uni maths to explain a concept if I can avoid it, unless I'm sure they can handle it

#

I explained the logic behind the "r . n = d" equation for a plane using some lecture notes I found from a first-year physics course once lol, but the student in question was already flying through the relevant questions so I didn't have a problem with this

#

[and neither did my supervisor lmao, which was an interesting conversation]

cosmic ibex
quasi maple
#

...Are you sure you pinged the right message there?

cosmic ibex
#

It reminded me of this meme, which also uses badly drawn superhero panels in order to make a vocabulary explanation "relevant for the kids".

lex_luthor_40_cakes.png
dapper flume
# quasi maple Honestly if I realise they have a misconceived notion of something, I try to tak...

I like that your instinct is to remove all the preconceived notions. I think for students who struggle to "understand the assignment" (i.e. those who go on autopilot trying to factor every quadratic they see before reading the question), it's valuable to encourage them to forget math for just a second and digest what the problem is actually asking for.

It really is hard to go from the ground-up for students at the A-level who lack fundamental skills. It does seem to be a questionable investment of time and energy for some students.

There is an advantage to approaching fundamental concepts structurally. For example, for using the order of operations to solve equations, I use shoes as an analogy. Specifically, if you put on socks, then put on shoes, then tie the shoes, you have to do those steps in reverse to fully remove your footwear. The advantage of this approach is that it puts the ball in the student's court to make the logical connections clear using a general intuitive framework. It will help them understand the order of operations, but you collect interest on the added intuition the analogy gives for other areas of math.

For proofs, the logic of the analogy goes a long way to help students figure out what deductions are necessary to connect the givens to a conjecture. Logically, the layers could be like:

  1. Hmm... I need to prove that the diagram is a parallelogram. What would need to be true?
  2. (Next layer deep). The definition of the parallelogram says blah blah... how can I show that this is satisfied?
  3. (Next layer deep). Ah, analyze the triangles in the diagram! Could they be congruent...?
  4. (Next layer) Yes! By AAS!
  5. Now I will work backwards through this logic to write my proof from the given!

TLDR; a good logical analogy goes a long way getting bang for the buck with students who struggle with fundamentals

quasi maple
sick axle
#

Hello,
I'm a graduate student in France, and I'll be starting giving my first course next year. I'll be teaching first year undergraduates. I'll be teaching them while helping them solve problem sets. I've been reading a bit what is said here, but are there any resources/books I could read to help me ? I know practice will probably be a lot different than theory, but I wanna try reading up a bit, to do as best I can

#

Also if any of you know any books about the history of mathematical schools of thought, I'd be glad to know

quasi maple
#

<@&268886789983436800> yk the drill

zinc dove
sick axle
zinc dove
sick axle
zinc dove
# sick axle math

I think you can always ask your colleagues for suggestions, and you also have your experience from been a student, I tend to use both

vocal phoenix
#

<@&268886789983436800>

serene apex
#

what's up?

vocal phoenix
#

A scam link was, but it's gone now

serene apex
#

was there a spam bot that got smited already?

vocal phoenix
#

Yep

gray dagger
#

but its good at giving both theory and actionable advice that you can feasibly use

#

history of math I know less about sadly, though I plan on picking it up later

dusk linden
#

ill check, one moment

#

heres some that might ne relevant from his bibliography:

#

history of pi by Beckmann

#

Introduction to the Foundations of Mathematics, by Wilder

#

these are the two that seem relevant

turbid zenith
#

My dissertation has finally been posted! 🥳
https://scholarworks.gsu.edu/entities/publication/4e24d2d4-ab00-4f59-b86e-397b798209eb
It's at the intersection of combinatorial game theory and mathematics education!

Using Combinatorial Game Theory to Introduce the Concept of Mathema...

This dissertation investigates students' development of the notion of mathematical proof in a Mathematics for Liberal Arts course through the lens of Combinatorial Game Theory. There has been extensive research on the cognitive processes of mathematics majors as they learn about proof; however, much less research has been done on non-mathematics...

frank shell
quasi maple
#

"Using casinos to teach kids about gambling"

#

Seriously though, major congratulations

turbid zenith
quasi maple
#

I WAS RIGHT!?

turbid zenith
#

Pretty much lol

#

But the big takeaway is that expected value is always negative 😛

vocal phoenix
#

the correct takeaway for them should be that they ought to be running the casinos

zinc dove
cosmic ibex
#

I'm not sure that's all that counts against poker.
There's also the game-theoretic complication that there's hidden information in addition to randomness. You'll want to optimize your betting behavior based on which cards you have, but to the extent your behavior reflects your cards, that will also reveal information to your opponent. It's not a priori clear how to find a fixpoint for that dilemma even if you can compute all the probabilities flawlessly.

mossy mulch
#

How are you supposed to dress as a TA?

#

Like should I make an effort to look more professional?

cosmic ibex
#

In most places I think TAs dress like students do. Perhaps dial it up just a smidge so you're above the median level of formality in the room.

vocal phoenix
#

I'm a teacher and I just wear Jeans and a T-shirt (or a flannel shirt if it's colder), as to other teachers at my uni, no matter the rank

next sinew
#

I wore sweatpants and a hoodie because it was comfortable

valid patrol
#

Anyone on here ever think of not dismissing radical foundationalism as waste of thought? and actually trying to find meaning at the deepest levels of logic rather than dismissing it as nihilism? It seems like the deepest assumption is basically thought of well if we don't assume coherence then nothing works. Somehow theres deep respect for acknowledign all paradoxes and issues yet this is conveniently ignored the most basic assumption that you have to presume coherence and if not its nihilism, why does almost no one question this furthere other than as a dead end?

mossy mulch
#

what on earth are you talking about sir

valid patrol
#

we all take for granted say 2+2=4, its provable from axioms. Im saying why do people never try to inquire about the base assumptions? its circular logic. What im saying is people seem to dismiss deep assumptions about coherence identity and persistence.

#

isnt possible that we miss new foundations in mathmatics by dismissing the one giant base assumption about ontology? and instead say oh your a nihilist

tawny slate
mossy mulch
#

what do you mean by coherence identity and persistence 🤔

valid patrol
#

Coherence being the idea that parts of a system behave consistently over time and dont arbitrarily shift meanings. Identity being that an object like an electron can be treated as the same thing accross time or interactions. without this we couldnt even say what were adding and persistence being that the objects we're counting dont just vanish or transform unpredictably before we finish measuring them.

mossy mulch
#

I think those are all fairly obvious assumptions no?

#

since human thought satisfies all three of those conditions

valid patrol
#

thats literally the point, what makes those valid in the first place?

mossy mulch
#

well if you couldn't assume those to be true of your own thought processes then your brain might as well be soup

valid patrol
#

if all of math and physics quietly depends on those three - maybe they're not just background conditions and we should be asking why they hold at all

mossy mulch
#

i dunno sounds like a non starter to me

#

like the old "what if my mom was a bicycle"

uneven kayak
#

It’s just turtles all the way down

valid patrol
#

so your response is that oh lets ignore this blarring gap in mathmatics in mathmatics pedagogy chat?

valid patrol
mossy mulch
#

I just explained why i don't think its a gap?

#

Math is just structured logical thought

valid patrol
#

your response was I cant answer

mossy mulch
valid patrol
vocal phoenix
#

Yep, but it's a delicious and occasionally useful soup (also I don't see how any of this has much to do with math pedagogy)

#

Unless you suggest we invite math students to delve into the depths of foundational underpinnings of coherent thought

quasi maple
#

Suppose a priori that it doesn't, and then you don't really have anything to stand on

quasi maple
turbid zenith
#

OH

#

YES

#

That video should be required watching

tawny slate
# valid patrol so your response is that oh lets ignore this blarring gap in mathmatics in mathm...

math is a formal study, meaning that it is the study of logic in and of itself. it happens to be useful in the real world, but the math is self-contained no matter what happens in science or in measurable or observational ways

in other words, under the assumption of such consistency, you can do math. you are totally free to assume that the world does not operate under such rules, but if you do, now math is no longer relevant. it's not that math depends on such rules, it's that such rules are the basis for how math is defined

#

this is why this is metamath and philosophy, not pedagogy, so not relevant to this channel

#

it's like saying you can study bananas all you want, and you find out that bananas are a fruit, and then someone goes "but what if you also studied earthworms?" okay, you could include earthworms into your dataset, but now none of the conclusions you made about bananas are now true for both of them, but it doesn't falsify what you learned about bananas, it doesn't change the fact that bananas are still fruit

valid patrol
# tawny slate math is a formal study, meaning that it is the study of logic in and of itself. ...

It’s easy to call this metamath or philosophy and wave it off as irrelevant—but doing so glosses over the single most fragile assumption in the entire mathematical enterprise: that coherence simply is. Before any axiom, before even the law of identity, there must be a medium that permits stable distinction, recursion, and aggregation. You can say “math is self-contained,” but self-containment still requires a vessel—it requires that the rules not only hold internally, but hold at all. That’s not metaphysical noise; it’s the silent precondition of all formalism.

Dismissing foundational inquiry as if it's a category error is convenient—especially when it protects the illusion that formalism is insulated from ontological commitments. But the truth is: you can't escape those commitments. You inherit them the moment you allow anything to “count.” If students never confront that, pedagogy becomes rote repetition built on unearned ground. This isn’t about mixing bananas and earthworms—it’s about realizing the soil came first, and pretending it didn’t makes for shallow roots.

tardy ember
#

...in what sense is that an assumption that is necessary for mathematics

#

as far as i can tell the only assumptions you need in order to do mathematics are that you are capable of performing extremely basic computable operations like concatenating strings

#

and that the outcomes of those operations doesn't depend on who is doing them

#

and i think doubting those is really getting into the "pointless philosophical excessive doubt" side of things, it's not that far off from "well maybe i can't do mathematics because i don't actually exist! have you considered that?"

uneven kayak
valid patrol
# tardy ember and that the outcomes of those operations doesn't depend on who is doing them

The irony here is that you’re affirming my point without realizing it. You say all that’s needed is “basic computable operations” and “consistent outcomes,” but those are assumptions. They're assumptions about distinction, determinacy, and coherence across time and context. To say they’re obvious or practical doesn’t make them any less foundational; it just means you’ve inherited the structure without questioning the scaffolding.

Calling deeper doubt pointless only works if your goal is comfort, not clarity. No one’s claiming you can’t do math without metaphysics but if you think formal systems float free of ontological ground, you’re confusing usability with absoluteness. That’s not rigor. That’s faith in disguise. And pretending I can concatenate strings is the foundation of mathematics is like saying architecture begins with drywall. Sure, you can build without asking why gravity holds or space has dimensionality. But don’t confuse that with depth.

tawny slate
#

fyi we all understand you, but you're not understanding us, this is not a profound or difficult or even novel concept

mossy mulch
#

Every time I open up an algebra book the words on the page are the exact same. Every time I read the word homotopy it means the same thing. Even if somehow these things did change I have no way of measuring or perceiving that fact, so its not scientific in any way to consider whether or not coherence, persistence, and identity are not true.

#

Also this has nothing to do with pedagogy

#

this is like, philosophy of philosophy if anything

sacred wave
#

Ts dude is just not understanding this simple fact that in order to establish a committee, you need some founding members, the one who'll help you to grow the committee, they aren't invited by anyone, they're the one who invites the other members. I'll start from here. As the dude says, consider a committee without founding members. Say X is in the committee. then X might have formed that committee, then he's the founding member. A contradiction. Hence X must be invited by Y. Hence Y is a founding member. A contradiction.
This exhausts all the cases of X existing in the committee. Hence the committee is null, that is , it doesn't even exist.
I'll conclude that, any rigorous approach to a logical subject needs some basic observatory pillars, those, upon denial, will lead to a contradiction, which contradicts the existence of the subject.

quasi maple
#

I don't know, is Architect arguing that metamathematics be taught alongside mathematics? i.e. that students learning maths should question the assumptions that are made along with the maths involved, such as why we have the axioms we assume a priori?

#

That would be an argument worth discussing here (even if it's a roundabout way of approaching the subject...)

tawny slate
#

i dont think he is actively making that general point

#

i think hes fixated on the consistency thing

#

i think he would have mentioned something in response to multiple people saying this is philosophy not pedagogy

rancid minnow
#

SInce y'all were discussing the structure of mathematics, I thought I'd share something I've been working on. We're building a fully formalized graph-database of mathematics. The alpha-signup link is in my bio. We're opening the open-access beta on August 1 🙂

valid patrol
# tawny slate i think hes fixated on the consistency thing

Lol, you’re all acting like I’m here to reinvent counting. Enjoy teaching your algebraic topology without ever mentioning why your spaces are Hausdorff or why your morphisms commute. If foundational questions aren’t relevant to pedagogy, then I guess you’ll be teaching your committees to “just trust me, bro” whenever students ask uncomfortable questions. Sounds rigorous.

cosmic ibex
#

You mean you're not here to reinvent counting? It was honestly difficult to tell, with the super-abstract language you were using.

#

And you started by apparently saying we shouldn't take for granted that 2+2=4. That seems to be a proposal to radically revamp elementary-school teaching to be based on .... um, something heavily philosophical.

valid patrol
#

I never questioned that 2+2=4. I questioned why that expression retains meaning at all, why identity, coherence, and recursion are so foundational we no longer see them. That you equate this with teaching arithmetic metaphysics only proves the point: the deepest assumptions are invisible to those who’ve never had to justify them. You’re not defending rigor, you’re defending inherited blindness.

cosmic ibex
#

I'm not even defending anything; I don't get what your point is at all.

#

Do you agree that the meaning of 2+2=4 is typically something that is taught in elementary school?

#

So if you want to change (in some direction I have no idea what is yet) how 2+2=4 is taught, you're therefore proposing some change to elementary-school teaching?

valid patrol
#

Yes, 2+2=4 is taught in elementary school. That’s precisely the level you're operating at, confusing the product of coherence with the premise of coherence. The point isn't the equation, it's the metaphysical scaffolding that makes any equation possible. If you can't distinguish between a symbol and the ontological stability that lets it mean the same thing twice, then of course this sounds abstract to you. You're not engaging with the question because you're still asking where the chalkboard is.

cosmic ibex
#

Okay, here is where I conclude you're more interested in spewing insults than actually explaining a point.
Apologies for wasting your time.

valid patrol
rapid tusk
#

joins server

#

immediately starts yapping incoherently

uneven kayak
valid patrol
# rapid tusk joins server

Calling it "yapping" is a useful defense mechanism when ideas exceed your parsing depth but don’t worry epistemology isn't contagious. You're safe.

queen sand
#

@valid patrol don't you think these questions are better for a channel like #foundations ? and not pedagogy? especially considering you used the example of 2 + 2 = 4, something that is indeed taught at the elementary level — where it is quite unreasonable to discuss such meta-mathematics in this setting?

valid patrol
#

thanks chat gpt

queen sand
#

lol I swear I didn't use chatGPT to write my answer, I just try to be as articulate as possible and encode a lot of detail in my sentences 😛

quasi maple
#

You can't just 💖 someone directly after calling them ChatGPT

valid patrol
vocal phoenix
#

em-dash is a fine symbol and I will not let the chatbots claim it as their own

queen sand
#

@valid patrol I am not very advanced in my mathematics studies, but I think you are (possibly) asking great questions that are more suitable for a channel like foundations, and I hope — if you are genuinely interested in a good faith discussion on these topics — you will go to that channel to discuss there.

queen sand
valid patrol
astral sinew
# valid patrol Coherence being the idea that parts of a system behave consistently over time an...

I think there is a confusion between ontological questions, i.e. your questions that relate to how come is it possible the scaffolding works, which are not mathematical questions.
To these questions, you might look for similar questions on Philosophical Stack Exchange, the Standford Encyclopedia of philosophy as an introduction.

The second kind of questions are related to mathematical practice, be it pedagogy, mathematical fields, metamathematics.
This second kind of question is inherently instrumentalist in mind, with no questioning on the implicit ontological commitments one makes when actually doing mathematics.

By the way, your (philosophical) questions seem weird to be asked so vigourously by a "postgraduate", but I'll take the benefit of the doubt.

queen sand
uneven kayak
queen sand
valid patrol
#

well in fact, its been argued that numbers were first convieved of in reverse order, rather from splitting a whole, so this gets further down the rabbit hole

#

how many pieces of a hole not addition in the traditional sense

astral sinew
# queen sand At the risk of straying more off topic: Is the quesition, "What is a number?", n...

The question is worth an essay, but I guess it depends on the context: is the highschooler asking what defines syntactically a number? what is its fundamental essence, if it exists (independently of our minds)? how does it originate from an evolutionary standpoint? Notice how these questions relate to your same question, there is an issue of non-uniqueness already (issue if the highschoolers were to accept a single sharp answer). Anyway, I think only the syntactic definition can be the best approximation to what we would usually call a mathematical definition (when I say synctatic, I do not mean in the formal language theory sense, but in via language in general). But in the end, this choice of definition might be arbitrary (there are quasi-empiricist interpretations of mathematics like there are cognitivist, structuralist, intuitionistic, even game theoretic ones I believe, the justification of these interpretations are philosophical ofc, but internally they give rise to different definitions to the nature of mathematical objects).

cosmic ibex
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It's a more practical question that it sounds. I have a feeling many school students end up with a vague notion that "number" means "string of decimal digits", no more no less -- and the fact you can use them to count is a useful application of numbers but not really their intrinsic essence.
(And that's then what leads to 0.999...=1 resistance a bit further down the line).

astral sinew
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I remember my highschool teacher was as his introductory course asking us what 2 was? was it the symbol? was it the idea? Little did I know at this time I was exposed to a rabbit-hole question opencry I thought he was kinda like trolling and luring us to continue studying the field, I guess it worked for me lol.

quasi maple
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"Hey, Vsauce! Michael here!"

astral sinew
# cosmic ibex It's a more practical question that it sounds. I have a feeling many school stud...

I might see where you're coming from. I think my operationalism and physicalism are "biasing" my naive perspective.
From my point of view, some similar remarks may be done when asking, "what is a chair", or what is a "hole" ? It seems I have no trouble to have zero access to the original/essential concept of a chair (if it supposedly exist somehow, whether it is in my mind or in cognition or whatever).
Why would it be different for numbers? Ofc, some might say that numerals are not numbers. I guess there are different schools of thought, and that's ok.

I am interested though, do you think we can have access to this instrinsic essence of numbers you mentioned?

quasi maple
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well shit it's still metaphysics

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astral sinew
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Yeah it seems to always come back to the same ancient debate loop

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This is why I tried to say the original question was not very mathematical.
Perhaps I should stop.

cosmic ibex
# astral sinew I might see where you're coming from. I think my operationalism and physicalism ...

I think I might be able to accept identifying the "intrinsic essence" with string of tally marks (or, in more dignified words, "unary numbers"). Even then, though, we need to be able to say that the string of tally marks here is the same string of tally marks as the one we've written on the other blackboard over there. So that's still an essentially Platonic conception of "sameness", I suppose.
For general real numbers I'm much more prepared to say that they are just convenient fictions of the mind in the first place ...

astral sinew
cosmic ibex
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I suppose I'm partially using "Platonic" as code for something like "I don't think I can reduce this intuition to more fundamental concepts, and it doesn't really worry me that I can't".

astral sinew
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Ahh ok, so more like a primitive notion left undefined, but that will remain fundamental for (future) more complex definitions?

Thank you for answering btw

tawny slate
vocal phoenix
tawny slate
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yeah i figured as much

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basically im calling this person a liar

quasi maple
tawny slate
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well they must be so happy to continue to have people in that channel discussing nonmath and other members baited into engaging with them

uneven kayak
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FWIW it was nice not to feel like the silliest person in the room.

tawny slate
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oh god i just skimmed through that convo

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it's got big sye ten vibes and the only difference is he doesn't chalk it up to god

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"how do you know" ad infinum

astral sinew
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I am trying my best, but I have 9% left.

tawny slate
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just stop engaging, he clearly isn't listening to anyone, i legit think it's a troll

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he even labeled himself postgrad

astral sinew
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yeah, I guess I'll give up.

tawny slate
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better for your mental health that way

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spend that energy on someone that'll appreciate it

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you'd get more bang for your buck assisting the high schoolers in the help channels

astral sinew
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heck I forgot those existed lmao, thank you