math-pedagogy

[tawny slate]

assuming you can just "feel out" the truth by just taking differing viewpoints at face value is also speaking from a position of privilege

[long pelican]

I don't read people's writings on the internet with the goal of finding the "truth", more like finding what the current state of societal beliefs and norms is. Truth is another matter

[tawny slate]

sure, my point about privilege still stands

[long pelican]

the fact that I can approach it like this?

Whereas people less privileged are faced with no choice but to ... approach it in a naive way?

[tawny slate]

yes, young people are becoming increasingly isolated and dependent on social media, and the problem is made worse by socioeconomic conditions

[long pelican]

wait am I in that category or not

I can't tell

increasingly isolated and dependent on social media sounds like this is meant to be an indictment on their ability to get good information, so I guess no

[tawny slate]

like to give one example of the impact social media has on gen z that i think millenials are out of touch with

https://knowyourmeme.com/memes/gen-z-stare

this source is cringe as hell but the underlying idea is that gen z consumes so much social media that they've forgotten how to be human

and the context is that there was a resurgence of this phenomenon as noticed in "workplace interactions" which i have witnessed personally to the point of searching online to find that this was in fact already a viral thing, which rarely ever happens

we really underestimate how much gen z consumes digital media and how easily and deeply they are affected by that information landscape

[long pelican]

I think there's more to the Gen Z stare than forgetting how to be human

[tawny slate]

i was being slightly hyperbolic there, but yes

[long pelican]

I can tell you don't like Gen Z 😛

Maybe you like Gen Alpha even less

[tawny slate]

no i hurt for gen z, i don't dislike them

that's why i went into education in the first place

[long pelican]

My assessment of Gen Z Is split on socioeconomic status I'd say. High SES Gen Z are a mirror of their boomer parents, middle to low SES gen Z are more like an actual new generation

[tawny slate]

i mean, yeah, the middle class is shrinking, which only exacerbates almost all of society's issues, yes

full agreement there

[tall bolt]

Generations are nothing but a marketing term, I don’t buy into it as an at all meaningful way to group people

[tawny slate]

i dont disagree, i am merely piggybacking on its convenience to refer to objective consequential differences between people of different time periods in a rapidly changing society, but i do want to reject it in the sense of the classification of people to the detriment of the group by stereotyping, and i realize that these labels come with that baggage, i apologize

[rain hawk]

If you're teaching a group theory intro course, how would you motivate the concept of homomorphisms

[tawny slate]

the high concept of group theory i describe as abstracting the "ways something can be symmetrical"

if we imagine rotations of 90 degree angles and integer addition mod 4, they are "symmetrical" in the same way, but how would you formally describe this in a way that makes it concrete? less obvious is that multiplication mod 5 (excluding 0) also is symmetrical in the same way, but how can show that it in fact is the same?

by formalizing the notion of a group, we can show the underlying structure of them is the same. at this point, i would add motivating examples for how this can be useful in problem solving applications but I don't have any examples off hand

[rapid tusk]

"how can two groups talk to each other?"

[lethal hornet]

How would a computer determine if two groups are the same?

The data of a finite group of order n can be encoded in a multiplication table, or a matrix M of size (n + 1) x (n + 1).
You don't have to explicitly state what the axioms are for this encoding, since writing them down is a pain, but maybe give a few examples.
In fact, an alternative definition of a group is a multiplication table satisfying some axioms. Just try to make this clear when explaining.

A large part of group theory is classifying and distinguishing them based on only their group theoretic properties.
Any property of a group that can be expressed referring to only the group laws should also be expressed in its multiplication table because of these equivalent definitions,
so two groups should represent the same group if and only if their multiplication tables are the same, up to reordering of the rows and columns, and relabeling of the group elements.

A great example of this is to display the multiplication tables of C_2 x C_2 and C_4, and ask them to tell you if the multiplication tables are the same or not.
This shows that we can have two different group structures on a set with the same number of elements, and the only way to tell them apart is to find some difference in their multiplication tables.

If two groups G and H have the same multiplication table up to reordering of the rows and columns and relabeling of the group elements, then what we have is a bijective correspondence
f : G -> H between the elements of the first group and the second, such that f(m_G(x,y)) = m_H(f(x),f(y)), that is, the product of two group elements x and y in G is mapped to the product of f(x) and f(y) in H by f,
or rephrased again, the multiplication table entry at (x,y) for G is sent to the multiplication table entry at (f(x),f(y)) for H by f.
Conversely, if there is such a bijective correspondence f : G -> H, then the multiplication tables of G and H will be the same, up to a reordering of the rows and columns, and relabeling of the group elements.
Such a function tells you that any property that G has in its multiplication table, H will also have in its multiplication table, and vice versa.

So an isomorphism of groups, or a function that tells you when two groups are the same, up to relabeling of the elements, is a bijective function that respects the group multiplication,
and this is the only way that a computer could tell you if two groups are the same: take two multiplication tables and produce a multiplication preserving bijection.

A function between groups that respects the group multiplication but fails to be bijective is still important: it is one that preserves only some of the properties of the domain group, and may not preserve others.
The most basic example of this is the zero function, which only preserves the group multiplication, and is not guaranteed to preserve any others.
Any group can be embedded into its symmetry group, and this is a way of extending all properties of some group into a larger one.
On the flip side, any arbitrary bijection isn't guaranteed to preserve the multiplication tables, i.e., the group structure, so there is no way of ensuring that an arbitrary bijecton will preserve any of the properties of a group,
and thus won't tell you if the two given groups represent the same group.

So a group homomorphism should be a function between groups that preserves the group structure, but isn't required to preserve all properties of the domain group.

[quasi musk]

I think it's simple, $e^{x+y} = e^x e^y$ or $ln(xy) = ln(x) + ln(y)$ is a natural entry point

[burnt vessel]

MoonBears-C-

[main gale]

a "method to go from one group to another:" a motivating example is often the determinant as a function on matrices, so this homomorphism takes all matrices over a field, then uses their determinants to end up in the field.
really, i feel like the motivation for a homomorphism comes from trying to find an isomorphism between groups. my intro course introduce hom's, then gave some "motivating examples," and revisited them with the iso theorems, etc.

[vagrant meadow]

personally i like the idea of even + even = even, odd + odd = even, even + odd = odd. pretty much anyone with knowledge of arithmetic can comprehend that. modular arithmetic is basically just a homomorphism from Z to Zn. the idea being that evenness and oddness "preserves" addition and multiplication. structure preserving is what homomorphisms are all about.

[boreal agate]

honestly motivating homomorphisms from isomorphisms is a really good way imo

its very natural to lool at a pair of groups and ask "if they are basically the same thing"

one could then try to formalize it as a pairing of elements in either group such that the multiplication tables translate accordingly

then one might look at this and notice that really, this is just like bijections, so we can encode this using some kind of pair of function, which are each others inverses

then the immediate next step would be figuring out what "correctly translating the multiplication tables" means in terms of these functions

which directly leads to the idea of an isomorphism
after that, a reasonable thing to try is removing the requirement for invertibility

which is now our beloved homomorphism

[tight star]

what most surprises people when they start teaching in person for the first time?

as someone who mostly does explanations in an informal setting, whether it's to friends or over discord, i'm curious as to what parts of math pedagogy i might be missing out on

[native iron]

I think the big thing is that teaching students involves a lot of managing motivation and emotions.

[tight star]

could you expand on this?

[native iron]

So when you're chatting about math with friends, they want to be there and they like talking to you about math. When students come to my office hours, it's often because they don't like calculus and they don't think they're good at it, but they need to get good grades on their next calculus quiz so that they can pass the class and complete the requirements for their major.

So some of my time is spent encouraging them and trying to get them to believe that if they put more into the homework, they'll get more out of it.

They can also get very upset because they really don't want to fail the class but they feel unable to succeed or they feel that what's being asked of them is unfair in some way.

[tight star]

gosh, right

[tight star]

and, how do you deal with that?

[clever girder]

i tutor ED students in math, and yes 100%

[clever girder]

while explaining, a lot is avoiding accuracy for "good enough". long-winded explanations full of jargon will often make things worse. repetition of simple rules for how to do a certain type of question are helpful, so are analogies

[tight star]

do you have an example in mind?

[clever girder]

and trying to show them what to expect. before a midterm or exam i'll just write out practice exams that are my guess at what will be on the test

[native iron]

It's very case by case because different people process that kind of thing differently. I always try to make my office hours largely student-directed, and I think it's really important when you're dealing with those negative emotions. Letting the students dictate what they work on helps them feel a bit more in control and distances me from the instructor (whom they may have negative feelings about) without having to explicitly criticize the instructor (which I don't do, as a rule).
Beyond that, I try to make students feel heard. I listen when they talk about feeling frustrated and I try to express sympathy. (Frankly, I don't feel very good at this—I struggled a lot with other parts of school, but I don't feel like that's carried over to being good at sympathizing with students.) I try to troubleshoot with them—for example, if they have a concern about how the class is run, I can help them politely reach out to the instructor. I try to steer them towards practice problems that are more manageable or do like X% of a practice problem for them so that they can just fill in the blanks that they feel comfortable with. From there, I can try to develop better capabilities.

[clever girder]

I say sometimes that <=> is basically an equals sign

[native iron]

Honestly I don't feel very good at the emotional management aspect of teaching, and I sometimes find teaching a bit emotionally difficult myself because I feel like I don't have the tools to get students the help they need.

[clever girder]

yes and i agree with the emotional management thing. i try really hard to connect with students as a peer and listen to their complaints about the class and the instructor

[native iron]

I went into math tutoring thinking "math is so lovely; students just need to be around someone who enjoys math and doesn't emphasize the boring computational aspects". Then I realized that some students are missing important skills or struggle to apply the skills they have and that there isn't really a way around that. It's not really clear to me how to get them to a good place.

[native iron]

What does "ED" mean in this context?

[tight star]

yeah sometimes they do need help with those computational aspects, right?

[clever girder]

education, specifically elementary education

[clever girder]

true and also knowing how to guide students to those resources. multiplying fractions was a struggle for some of the students i had

[native iron]

As in, students who are studying education at university?

[clever girder]

yes

they have to take one 'mathematical reasoning' course, which is mostly proofs

[clever girder]

they do, but most of the time that's the sort of thing that they can get on their own with practice

[tight star]

then, as a teacher, what should I be focusing on in terms of providing value?

[native iron]

Ah interesting. I worked with a few math/ed double majors who were doing AYA. I never met any of the elementary ed majors for whatever reason.

[clever girder]

70% of the effort is convincing them they're able to do the computations, 30% is showing them examples of how to do them

[clever girder]

what's AYA?

[tight star]

I’ve been feeling more and more of a drive to create expository math resources recently

but I also don’t want to make redundant stuff…

[native iron]

I might be misremembering the acronym but adult and young adult or something like that. Mostly prospective high school teachers but the way the major was structured learning to teach high schoolers was the same track as learning about adult education.

[clever girder]

and yeah. their attitude is 95% "i just need to pass and then i'll never have to do math again"

[tight star]

how do you convince someone they can do a computation?

in a way that’s separate to showing them how

[native iron]

I think being a stable, confident presence is valuable for a lot of those students.

[quartz oracle]

I'm in this exact boat right now

[clever girder]

ah cool

[native iron]

Frequently they say something useful then don't actually act on it. So I tell them to write it down and see if it makes them think of anything else.

"Maybe I could distribute this term"
"Yes, you could! Can you show me what that would look like?"

[tight star]

that’s actually happened to me so many times 😭

I’ll get stuck on a problem for ages because I keep trying different strategies in my head

but once I actually start writing stuff down I make more progress

[clever girder]

example:
doing geometry (in this class) is pretty much always

• radii are the same length
• angles add up to 180
• thales theorem
  if you get stuck, just keep hitting every problem with those three things wherever you can see them being applicable

if you get stuck, write down everything you know. and try to add more things

[vagrant meadow]

for tutoring at least, very often you end up not teaching the concept you expect to teach. ex. someone asks for help with sketching a function. they think that's what they want you to help them with. but it becomes quickly apparent they barely understand what the hell f(x) even means, let alone f(-2) and how to calculate that. you end up having to start much further back than expected. sometimes they can be frustrated by that. "i just want help with sketching functions" (often, the between the lines translation is "I just want you to do it for me"). but, sorry, you need to know what functions are if you want to be able to do it on your own.

[rapid tusk]

XY problem moment

[tight star]

mhm, I see…

[spiral elbow]

I've experienced this too, it can be so hard to convince them that I can't help them understand this particular topic right now, we need to take 4 steps back and understand the basics first

It's particularly frustrating if it's homework or something with a deadline

[quasi maple]

Weirdly, though maybe that's because I haven't been tutoring in-person for a few weeks now and mostly been active here for those weeks, I find the emotion part more predominant in the help-channels

[quasi maple]

"Adolescent to Young Adult"

In this context, Middle School (maybe) and High School (definitely) pedagogy

[native iron]

Ah interesting. My model for "explaining things casually" is "my friend wants to know what I've been up to" as I'm not active in the help channels here.

[quasi musk]

The biggest thing for me was saying "No" to questions that were outside of that day's objective, especially if they're very interesting questions that I'd love to discuss more

It is not a "We're exploring this together", especially at a university where people's grades matter, and you have set material to go over. I encourage those discussions to continue at office hours. 99% they never do, which is kind of sad, but you can't just willy nilly do stuff

The other thing that you might not have is a list of examples/counter-examples ready to go off the cuff in case someone asks "Isn't this obvious?" or "Isn't this just always true??"

[tawny slate]

or just in general questions you dont expect

sometimes you get real interesting insights that you didn't realize, sometimes you get questions about things you took for granted, and other times its just headscratchers

[tall bolt]

Yeah I’ve always found in tutoring or my week of actual classroom teaching, no matter how much I try to prepare for the confusions I expect to come up, someone always finds a new question or confusion I hadn’t considered

[turbid zenith]

Or a new "who would have even thought to do that" mistake

[turbid zenith]

Really basic but I think for me it was the realization that just being good at explaining math isn't enough.

[tall bolt]

Yeah for sure, I had that especially when I was teaching younger kids (10-11), I don’t know if it was actually worse than any other age group, or if it was just because it was on things I consider to be so rudimentary, but I remember quite a few times struggling to even work out where to begin explaining where they went wrong

[turbid zenith]

I used to tutor younger kids and I agree

It's a specific skill set to teach that age group

...and it's not really my skill set

[tall bolt]

Yeah… I can’t say I thrived

[turbid zenith]

Part of the issue was being at a tutoring center and having them at the same time as older students 😛

Trying to tutor an AP Calculus student while also wrangling a run-around-the-room kindergartener is not fun

[tall bolt]

I got to teach them in their classroom, I’m not sure having 30 of them together helps much to be honest

[turbid zenith]

Oh yeah I imagine

[tawny slate]

the trick to developing this skill is to spend less time doing research math and more time roleplaying annoying kids in your brain by continuously asking yourself "why" and break every rule because you feel like it

ok but serious answer, i think one thing i do that helps is that i treat solving easy math problems, like K-12 school math plus AMC style problems, as a learning experience for myself as well. after solving a problem, i reflect on what techniques i used, and try to come up with a list that is as exhaustive as possible, and then pretend someone asks me "why does that technique work, and how do i know to use this?" and do this ad nauseum

and i really mean ad nauseum, because the next step is now to not only compile interesting instances of this, but also pull examples from real life. i try to train myself to take notes every single time any math related thought pops into my head in my daily life, to use as possible things to talk about and reference and apply

one time i was driving and someone cut me off, and i had to break really hard to avoid a collision, and i thought about how much gas i was wasting from constant usage of hitting the brakes. so i thought about how to maximize fuel efficiency through driving habits and bam mental calculus

whenever someone mentions that its okay for rich people to be rich because they dont actually understand how much a billion dollars is i think about how poorly people understand exponential growth and ways to explain it, which also now motivates why we should care about exponential growth

student solves x^2 = 4 and forgets the plus/minus, imagine they ask "well how am i supposed to remember that" and you come up with 3-5 specific bullet points of intuition to drive the concept home

[quasi maple]

[insert "a million seconds is about 11 days, a billion seconds is about 32 years" here]

[turbid zenith]

I have good memories of those contests

Even though I was never particularly stellar

[turbid zenith]

I’m doing a lesson on that this semester again!

[turbid zenith]

So, sorta math education related I guess ... would anyone be up for giving constructive criticism of some of my math videos? I really want to start getting back into making them now that I've graduated, but I've just kind of been going off of vibes and feel like there are definitely things I could be doing better. Especially would appreciate feedback from anyone who does visual design and/or video production, but definitely not required.

[tawny slate]

willing to trade, feedback for feedback

[turbid zenith]

Sweet 😄

I can definitely do that

[winged maple]

Would it be acceptable to use this channel to post a lecture slide for feedback?

[spice scarab]

That's a lovely idea. いくぞ

[quasi maple]

I mean, that would be what this channel's for

Also, 日本語 jumpscare

[turbid zenith]

是非、どうぞ！

[lethal leaf]

How do y'all deal with emails in a manageable time?

Set hours of checking them?

[halcyon glade]

I just try to reply to emails as soon as I notice them. If I can't, I'll give it a little "Needs reply" tag in my inbox. Ideally, I just check my email once or twice a day and keep it closed for the rest of the day because checking email often can become a big waste of time/attention. (Ideally, I also have Discord closed all day...)

[quasi maple]

Sorry, what's a "Needs reply tag"?

[native iron]

Like a pin in Outlook or similar.

[quasi maple]

Oh I thought that was a "reply tag" called "Needs", instead of a tag called "Needs Reply"

[tight star]

To what extent have people found type-theoretic “sequent calculus” reasoning helpful (or not) for introductory proofs?

I was trained in the “Hilbert style” of formal proofs

[twin lichen]

Hilbert systems are basically unusable for formally proving theorems. Their main purpose is to prove high level metatheorems, such as

The set of theorems is computably enumerable

Natural deduction or sequent calculus are the way to go. I also don't understand or agree with this characterization of natural deduction or sequent calculus

You can have an intuitionistic sequent calculus where every line has exactly one asserted proposition but this is still not trivially isomorphic to a natural deduction system

[tight star]

hm i see

do you have a recommendation for a resource i can use to learn more about natural deduction and/or sequent calculus more properly?

[twin lichen]

I'll think and see if I can come up with a good example to illustrate of the difference

[twin lichen]

All of these systems are discussed in section 1.3 of "Basic Proof Theory" by Anne Troelstra

natural deduction systems contain "elimination" rules which allow you to go from more complex to less complex

like introduction elimination, aka modus ponens, is:

• Suppose we can prove A -> B, and suppose we can prove A. Then we can prove B.

This is going from "more complex" - the implication (->) - to less complex - the conclusion B.

[tight star]

mhm

[twin lichen]

Gentzen's sequent calculus doesn't contain elimination rules for the logical constants that let you go from more complex to less complex, instead the rules for deriving a new sequent always tend to increase the complexity of the sequent, I think, either the assumptions or the conclusion.

Suppose from assumptions Gamma we can prove A, and from assumptions Delta, B we can prove C; then from assumptions Gamma, Delta, A -> B, we can prove C.

So a natural deduction proof allows you to bury complexity (like A->B, which is more complex than A or B) in the middle of the proof even when the assumptions and conclusions are simple. A sequent calculus proof lets you do that too, but all the power to do that is concentrated in a single rule, the cut rule. If you remove the single cut rule, sequent calculus proofs contain no buried complexity, the final conclusion \Gamma => \Delta of the proof is the most complicated sequent occurring in the proof.

This is the cut rule, it has nothing to do with logical implication (->) or any other logical constant (and, or...) it simply says that proofs can be composed

and similarly that the identity proof exists, so proofs form some kind of multi-category

all the other rules of the sequent calculus deal with the logical constants (or are structural rules like permuting/duplication of variables) and these tend to increase the complexity of the sequent, either on the left (increasing the complexity of assumptions) or on the left (increasing the complexity of conclusions)

[cosmic ibex]

Being a computer scientist who learned type systems for programming languages before I learned formal logic, I find natural deduction (with explicit contexts) by far the most intuitive and natural of the trifecta. Sequent calculus is valuable from a theoretical point of view in the way it makes De Morgan duality very explicit in how the rules look. Hilbert systems are not really good for anything except definitional minimalism.

If you remove the single cut rule, sequent calculus proofs contain no buried complexity, the final conclusion \Gamma => \Delta of the proof is the most complicated sequent occurring in the proof.
... with the caveat that this is true for propositional sequent calculus. Once you want to treat quantifiers too, this nice property has to go away (since it would make provability decidable, and we know predicate calculus is not decidable).

[twin lichen]

Good point, thank you.

[cosmic ibex]

(Also your sequent calculus appears to be intuitionistic).

[twin lichen]

it's the "minimal" or implicational fragment introduced to motivate the sequent calculus

[boreal agate]

yeah i tend to agree with what has already been said here

hilbert calculus is very unusable

if youve ever tried to prove anything in it from axioms, you know how insanely hard it is without first providing a proof in natural deduction and then converting it to a proof in hilbert calculus

being able to provide context is both easier and more intuitive (and also helps with what context always does, it limits the scope of relevant arguments we could make)

as for sequent calculus vs natural deduction, natural deduction is probably easier to deal with for a human prover

[tight star]

#advanced-lounge message

based on this, i wanted to ask

for math students, how do you convince them of the value of concrete examples and computation?

as a physicist this always just came naturally to me, but i understand that it's not the case for everyone

and apparently the bias against computation is what learning cat theory early exacerbates, which is why it can be a cognitohazard until you have enough mathematical maturity

[boreal agate]

what do you mean by concrete examples and computations here?

[tight star]

I guess arguments for specific groups, for example

[boreal agate]

oh i see

while i can sympathize with disliking specific constructions and preferring abstract theory, i cannot recall a single time i properly umderstood something without seeing examples

whats very easy to miss is that math is taught in the opposite direction of what research might be

when everything is already worked out, of course its safe to start from the axioms and develop a theory while also explaining the general principles guiding us

but when the math isnt preexisting, figuring out good axioms starts to become a difficult goal

what i assume can be given as motivation for specific examples and computations, is that rarely we have a clear picture of what we want to achieve or explain when such a picture is not given to us

in that case, all we have to guide us are the examples and computations

it is then our job to take these examples and generalize them in a way that uncovers the patterns we see (but cant yet make precise)

i think this is pretty good motivation, especially if you get a math student that doesnt appreciate examples enough to try and figure something out on their own

[cloud zealot]

plenty of students do not need to be convinced; there are many who actually want to see more examples

[quasi musk]

Usually these are students that are very enamored with proofs. All you gotta do is ask open ended questions, try to prove something, realize there's a step that requires an assumption that doesn't hold in this example

Either that question can be answered minus the particular cases, or the particular case is so significant that the whole theorem makes no sense

ab = ba for numbers, does AB=BA for matrices? Well consist of numbers and each number commutes so it should be true

Wait, did we even use the definition of matrix multiplication?

etc.

[tall bolt]

I think giving more open ended questions which require you to make a conjecture and then a proof is good, because to actually make that conjecture you’ll need to do some computations to actually understand the behaviour of whatever it is that you’re working with

[tight star]

That makes sense

[tall bolt]

My LA/ring theory class did this through sage, we’d have some basic problems just teaching us how to use new syntax and functions etc, then we’d have to write a program to compute some sort of something or other, and as a final problem make a general conjecture and prove it

I also think it kinda comes naturally as you do harder stuff, I think the only people who don’t value computations are weird early UGs with a superiority complex from watching too much flammable maths

I constantly make computations to understand how things behave for proofs, it really helps me get a grasp on things and work out how the proof should go

[tight star]

Wait what does flammable maths do

[tall bolt]

He’s just a cringey YouTuber who enjoys making those engineers say pi=3!!! Videos

[tight star]

I see

[plain valve]

Doesnt flammable maths mostly do computations lol

[austere delta]

Yeah, I thought it was mostly like "heyyy boys!!! Today were gonna solve this CrAzY integral!!!!"

[tall bolt]

Possibly lol, i only really know him through Andrew Dotson skits and those were all “god you physicists and engineers are so dumb!! Do things right!!”

[plain valve]

oh i mean I agree with that, those things are silly

I remember being sad because uh

[tall bolt]

I always found him incredibly annoying I can’t say I’ve watched any of his solo stuff

[plain valve]

I had a hard (in my opinion) interview question for my undergrad physics application and like

afterwards a friend pointed out it was on flammable maths

Lol

[tall bolt]

Oh lol, could’ve been helpful

[rapid tusk]

average bprp video

[turbid zenith]

That holding two pens technique is useful though frfr

[rapid tusk]

wish he would do videos on more substantial stuff but unfortunately basic HS math + calc1 is what gets views so

[feral swan]

Any sources to help me through:

1. exercise checking (The job is to check, grade, and give feedback to exercise sheets for a course). I know how to check if something is correct, but i'm not quite sure about giving feedback.
2. supplementary TAing with small group. (The job is to help a group of ~10 university students in a year 1 course. Not much direction is given by the faculty.)?

Any source of things to expect, tips, guidelines, general philosophies, ways to deal with problematic situations would be helpful.

I had a few teaching jobs (a few private students, cramming sessions with ~10 students before an exam)
For which I showed up and did my best... I'd like to be more prepared this time.

[halcyon glade]

I mean I think his videos are quite good though and they teach more math to a wide audience

[tight star]

blackpenredpen essentially taught me calculus

[lethal hornet]

1. this server is a good spot to practice with low stakes and people to correct you if you’re wrong.

[feral swan]

For some reason it doesn't feel like real practice. Maybe not enough commitment?

[tall bolt]

I think helping here is quite different to actually tutoring a group. If you’re tutoring you’ll know the syllabus they’re following and what they’ve done week to week so you should have a better grasp on where the students are

My main advice is to make sure you’re fresh on the material they’re covering, and try to think about where people might get stuck or confused, and then how you might want to work through that confusion, preparation is key.

That being said without fail every time I’ve tutored people have hit me with a question I’ve never even considered or got stuck in a way I couldn’t even really conceive of and you just kinda need to know your stuff well enough to be able to roll wi the the punches (but you can still go think about it for a bit first! You don’t want to add to the confusion)

I think it mostly comes with practice though, I was pretty shit at it when I started tutoring, but you just kinda get better at it through time

That being said I do think helping here has value, it’s not a completely different skill, and people do occasionally post proofs looking for feedback and this can be a good way to practice giving feedback because there’s other people who can weigh in on your thoughts

[turbid zenith]

It's not going to be perfect but it'll still help. 🙂

[mossy hazel]

what is math pedagogy

[turbid zenith]

How to teach math effectively! 🙂

[mossy hazel]

ooo

[feral swan]

the thing I hate the most is when they ask me a question, and then I show the way to do it, and they give me this wide eyed look as if I came down from mount sinai.

I feel like im scamming them out of their time and money... I'd like to believe I do more than showing them recipes for solving questions, but that's what it boiled down to most of the time.

In 10-20 hours of instruction I felt I truly helped someone once, as in, touched something fundamental and corrected a misunderstanding. The rest feels as if I was used by the students to lower their anxiety and cram.

[vocal phoenix]

<@&268886789983436800> they're spamming here as well

[dapper flume]

This is part of why merely revealing the answer to a student's questions doesn't work. In order to learn a new concept, a person has to somehow fit that concept into their pre-conceived notions, either by seeing how it fits, or by altering their notions to accommodate a concept which contradicts their beliefs.

If someone is learning to solve an equation, like 2x-4=32, they might seem hopelessly hung up on trying to parse what "x" even means. They have not accommodated the concept of variables into their mental framework, so even if you perfectly explain how to solve an equation, they might be left extremely dissatisfied. In this specific concept, sometimes it helps to state the problem as 2(___)-4=32. The blank in place of the variable can sometimes short-circuit a mental blockage related to interpreting variables, allowing them to reason the problem on their own.

[tawny slate]

a tip here is that when a student asks you a question, dont jump to answer it immediately, ask a question back. have a conversation mostly in questions

this is not only to help build habits where youre listening to the student more, but often the student is trying to figure out what to even ask (you cant know what you dont know) and youre trying to figure out what they are really asking about. if a student says "help i dont understand sin and cos" its possible they don't even understand what a function is, and never thought to ask because they dont know that sin/cos are functions

additionally, sometimes you make reasonable assumptions about what they know already, and it turns out you have to keep adjusting to it, which is even more reason why you have to keep asking probing and checking questions to make sure the student can follow. I once helped someone trying to solve linear congruences and modular arithmetic and they didnt realize that multiples of 5 end in 0 or 5, they didnt know any divisibility rules whatsoever outside of bashing long division

[feral swan]

thank you both, ill try to keep it in mind.

[elfin sleet]

someone I know, who passed high school math at a good school with grades around 70-80, just told me that she's concerned about taking calculus in university because "it sounds like something where you need to use f(x), and I've never understood f(x)"

...sometimes I'm hopeful about math education but then I hear things like this

[tawny slate]

to be fair, the jump from algebra to thinking in terms of functions is one of the most significant in high school

[elfin sleet]

yeah true. But you'd think that by the end of high school they'd at least know what that notation means

[mint lark]

Obligatory garrity moment

https://youtu.be/PAZTIAfaNr8?si=PvoG_ST-g-JA_Qaa

[tight star]

do you know what she’s found confusing about f(x)?

[tight star]

http://launchings.blogspot.com/2016_02_01_archive.html

reading through this, i wonder - how does one avoid structuring a lecture only for those who already understand a concept?

[long pelican]

In the introduction, this paper presents the example of the Feynman Lectures, widely considered to be some of the finest scientific expositions ever made. Yet, the fact is that when they were given at Cal Tech, “Many of the students dreaded the course, and as the course wore on, attendance by the registered students dropped alarmingly.” (Goodstein and Negebauer, 1995, p. xxii–xxiii).
I always like to be reminded about this bit

I wonder if the same can be said of Walter Lewin's physics lectures which similarly achieved fame on the internet?

[tight star]

This is the kind of thing I worry about regarding my own explanations

A disconnect between how students actually find the explanations, and how others “in the know” find them

[long pelican]

The truth can be quite ego-deflating

[tight star]

True, but it also means there’s so much more I can improve

[elfin sleet]

I don't really know for her specifically, I imagine it's mostly the same things that others find confusing with it. So like confusion about functions in general, confusion about what the notation f(x) means - is it a number? Is it a function? Is it an expression (by which I mean something like what we would call formal polynomials)?

But I'll be trying to help her with it soon so if you are interested in the specifics I can update you

[native iron]

My dad, who I think of as decent at math for a non-technical person, once told me that he never understood why we would write something like f(x) = x^2 + 3 rather than y = x^2 + 3. Something about graphing a relation made sense to him while writing down a function didn't.

[rigid reef]

i think of f kind of as an operator applied to x similar to how we can think of a matrix as a type of operator applied to a vector since Ax=b basically maps x to b. in this case the notation f(x) is just a way to say we are specifically mapping x to some other point. the graphing relation like what @native iron said might be the most clear

i think the least mathy way to maybe explain to them about f(x) is to imagine f is a machine with a slot where you put in a number. the x is the input you feed into the machine. the f(x) is is the output that comes out after the machine does its rule. so f(x) means what you get when you put x into the "rule" called f i.e f != f(x)

[elfin sleet]

Yeah, the person I was talking to also said she preferred having a y. I think it's because all those questions about what f(x) actually is don't apply - as long as you know (or think you know) what a variable is, you're fine. You're also not directly confronted with the concept of a function, you just have variables to push around

[elfin sleet]

Yeah but someone who is struggling with this doesn't understand what it means to map something to something else

[elfin sleet]

I agree, I'm planning on explaining it like this with her

[rigid reef]

glgl!! praying for their success

[cinder citrus]

im very interested in math pedagogy, what would you recommend for setting problems for revision for students

[tawny slate]

i like to use the square root as an example of a function being practical, compared to a relation

students often say that, for instance, the square root of 4 is 2 and -2

if that is the case, then the expression sqrt4 + sqrt1 has four possible values: 3, 1, -1, -3

this is ambiguous and not very useful

by forcing the square root to act as a function, which only has one output for any given input, we remove all ambiguity

if you wanted the -2, just do -sqrt4

compare this to the equation form using y

so functions differ from relations in that they only allow one output per input, which is very useful

[feral swan]

my fellow students really liked the professors that were well-presented, didn't stray from the material, followed their course notes, and didn't skep steps in the proof. It was quite unromantic.

the professors that made a show out of their lectures were either loved or hated depending on the students.

That's a very interesting blog...
Did anyone ever try a variant where the professor writes down the points he's trying to make?

[tight star]

One thing I’m curious about is whether looking at non-numeric functions could help for this

I feel like functions are a concept where looking at a more complicated example might actually help

In schools you usually only seem to deal with functions which take in a single number as input, and produce a single number as output

To me I think this restriction is artificial and actually unhelpful

I think it’d be nice to talk about “add” and “multiply” as functions

[tight star]

E.g. a “name” function which tells you the name of a person

Or a “reverse” function which reverses the contents of a list

[boreal agate]

probably actually

[tawny slate]

not a good example unless we are specifically referring to, for instance, full legal name. functions can only have one output so you might point a person to multiple names/nicknames

a bit pedantic but i got chewed out for this in a reddit comment once and i think its fair lol

[tight star]

Yes, there are ways you have to tweak this to make it precise

Though you could have a “names” function which outputs the collection of all names a person has

[boreal agate]

actually this can also be used to explain the axiom of choice

by asking whether a function mapping any person that has a name to one of their names exists

assuming infinite people ofc

[tawny slate]

actually this can be used to explain infinite ordinals, its not that we actually have omega/infinite objects, we are simply generalizing to allow an arbitrarily large finite case

[native iron]

My instinct is no. I think students get uncomfortable when concepts do not clearly connect to math. It's notable that part of my father's objection is that he doesn't understand why functions are a useful concept.
I think that functions with restricted domain are a bit better. For example, you can define a function mapping each of {1, 2, 3, 4, 5} to some arbitrary number via a table, and I think that's a little better.
To answer why we'd want to write f(x) = x^2 + 3 rather than y = x^2 + 3, my best answer is that we can then write things like f(2) or f(x-1), which makes functions a bit more flexible. However, I'm not convinced this is a fantastic answer.

Also, functions are much better for calculus, but I don't feel like the average person necessarily gets the "point" of calculus.

[halcyon glade]

I was taught functions using the vending machine analogy. You punch in a code into the vending machine and get a snack out of it. You'd be really disappointed if you got a different snack than what you were expecting, so every code has to correspond to only one snack. But multiple codes could be the same snack.

[native iron]

I like machine analogies but I seem to like them a lot more than my students.

[long pelican]

I’d say the worst outcome is an experiment where the subjects know the points by rote methods, because that would fool the researchers and the scientific community toward the wrong direction to pursue. Think of it this way: knowing the points is a good proxy for understanding the lecture but it is only a good proxy if it’s not made the target

[feral swan]

not that I disagree @icy but at some point you'll have to tell the students, for example "the triangle inequality is the technical tool used to prove ..." or "a cauchy sequence, intuitively, is a sequence where ..."

The professor in the experiment said those points aloud, and the students did not pick them up and instead thought the theorem he was writing on the board was important (it was a toy theorem which was used as an aid). Seems they got the wrong emphasis - that the things on the board are the important things.

[rain hawk]

A lot of students seem to have this intuition that 0 is "neither even nor odd". Anyone know why this would be?

[long pelican]

Yes, I'm bringing into consideration other concepts and pitfalls than what the researchers are studying

Other ways of looking at the problem of education

[halcyon glade]

Not an expert but probably a combination of

1. For the most part, you only ever talk in class about positive numbers being even or odd; even and odd as concepts are therefore not applicable to 0 and negative numbers
2. Confusion with the old adage that 1 is neither prime nor composite

[feral swan]

Im not sure, but i've found it is a common surprise that every number divides 0, as in, it's surprising that a|0 makes sense. So maybe they're confused about 2|0, reasoning that 0 is not even, but on the other hand Saying 0 is odd is absurd.

[halcyon glade]

It seems like one of those problems that's easily solved with more exposure time to 0 and negative numbers

[long pelican]

Maybe: "0 is nothing. Nothing cannot have any properties. Even and odd are properties. Therefore, 0 is neither even or odd"

[spiral elbow]

I had no idea people were confused by this, but apparently there's an entire wikipedia article about it: https://en.wikipedia.org/wiki/Parity_of_zero

It seems like a recurring theme, both for this case and other misconceptions, is the inability to use definitions. There's an interesting paper about this, where they find that (some) math undergrads have a tendency to argue based on an intuitive image they have, rather than the definion; and if the definition disagrees with their image, then they would rather disregard the definition: https://www.jstor.org/stable/4145268

[tawny slate]

which is sometimes a really good way to understand ideas in the real world, but a generally very bad practice when it comes to math

[austere delta]

I mean, I feel like it's good when it comes to math too.

If there were good reasons for 0 to not be even, then I would agree we should change the definition to 0 not being even by convention.

I think there are good reasons for 0 to be even though, and I don't see any argument against it.

The only important thing is to realize that it doesn't actually matter if 0 is even or not.

It's just a matter of convenience how many times you would need to say "if x is even or 0" or "if x is even, but not 0"

[spiral elbow]

Sure, but in this case they're already given a definition, and they're asked whether some object satisfies it. Here's also an interesting case study, where math majors are asked whether 0 is even: https://www.diva-portal.org/smash/get/diva2:542328/FULLTEXT01.pdf#page=197

[austere delta]

They've been given the definition, but that doesn't necessarily mean they've accepted that definition.

[tawny slate]

i mean, youre not wrong, this is why we no longer accept 1 as a prime number

but if you consciously decide to select mathematical definitions that satisfy your intuitions rather than the established conventional definitions without formally acknowledging and explicitly laying out the alternate definitions, there are several yellow flags

1. this usually demonstrates a lack of formal rigor
2. they probably dont understand philosophically how math is done
3. they probably dont respect how those definitions came to be, the design choices for these definitions

i am very much pro-"go against convention and make up definitions", but at least from what is linked and described in this channel, it seems there is a contradictory disconnect between what the subjects accept as the formal definition vs how they apply it, which is different from if you were to declare alternate definitions at the outset

[rain hawk]

I think this is what's interesting to me, not only does 0 satisfy the definition of even given to students, there's actually afaict basically no good reason to think zero wouldn't be even either, so it's unclear to me why the common convention is unintuitive to them

[spiral elbow]

I feel like not using a definition and not accepting a definition are two sides of the same coin in this scenario - they likely don't understand the point of definitions in the first place

[rain hawk]

Like at least 1 being prime makes sense (the definition usually supplied kinda does include 1)

[turbid zenith]

They may also be confusing it with the fact that zero is neither positive nor negative.

(Though in French it’s BOTH positif and négatif!)

[midnight scarab]

(The crazy part is how to interpret "non négatif", which is often taken to mean nonnegative (cause otherwise you'd say "strictement positif"), which is not the negation of "négatif"...)

[elfin sleet]

My guess is that it's mostly this. I think that one big hurdle is that people are very used to only having a|b when a < b, so 2|0 can seem very surprising

[quasi maple]

French when French, ig
-# Français quand français, je suppose

[long pelican]

This might help you get into the mindset of someone who has trouble believing 0 is even, instead of "neither": think about what goes through your head for the following question:

"Does the string '0' contain any leading zeros?"

[austere delta]

My thought process is

Yes, there is clearly one leading zero.

Now, the context where the concept of leading zeros might be used would be were you want a standard representation of a number. In which case '0' is the standard way to write the number zero. This is an argument for why it wouldn't have any leading 0s, but it would seem more natural to have an exception in "the standard representation of a number doesn't have leading 0s" then changing the definition of leading.

I'm not sure it's very analogous to how someone would come to think 0 isn't even

[tardy ember]

tbf the actual "natural" representation of the number 0 in positional notation is

it's just that that's kind of inconvenient to use in practice so we went with "0" instead

[austere delta]

I mean, is that "natural"? I would argue
...0000000
is even more natural, tough I guess even more inconvenient

[tardy ember]

i mean, it is if you think the number ||||| should be represented as 5 instead of ...00000005

[long pelican]

I asked ChatGPT and it prefers to change the definition of leading to say that single digit numbers don't have leading digits

[tardy ember]

the "infinitely many leading zeroes" notation does have some things going for it but i think it's meaningfully a different notion to the version where we don't write leading zeroes

[long pelican]

So obviously there's not universal agreement on what is to be done here, but notice your thinking process is less "apply formal definition, done" and more... neural-network-y, for lack of a better term

[tardy ember]

well then my next question would be, how many leading zeroes does 00 have?

[long pelican]

well not yours but more like someone who is waffling between your reasoning and the one ChatGPT gave

[long pelican]

I think ChatGPT would say 1 there

[tight star]

For me, i think the main reason functions are useful is that they can be composed with each other

You can build up a very complicated function from a bunch of simple ones

I do really think that having functions like “add” or “multiply” might help as well

Or just, any familiar functions from programming

You could do “remainder” as well

[austere delta]

I don't know what reasoning ChatGPT gave, but I can totally imagine a thought process leading to '0' having no leading zeros.

I can't quite see a reasoning leading to 0 not being even, other than "it doesn't feel even".

But I'd be happy to hear if someone had some reasoning

[tight star]

When you say “students get uncomfortable when concepts don’t clearly connect to math” - do you have an example of this?

Functions are extremely pervasive throughout math after all

[long pelican]

Here's one:

• Suppose the primary definition of even in your head is "can be divided into two equal groups"
• Can 0 be divided into two equal groups?
• Doesn't dividing have to make something smaller?
• Maybe yes, maybe no, I've never heard anyone deal with this case. So I have no idea

[tight star]

And, what do your students dislike about them?

[long pelican]

Continuation:

• Let me now resort to my example data bank of phrases I've heard before involving "even" to see if 0 was ever mentioned
• Nope, doesn't seem to exist

[austere delta]

Yeah, maybe if people are thinking about dividing things into groups. Dividing no objects into empty groups would seem like an exception type situation

[tight star]

I feel like this is a similar statement to “green is neither even nor odd”

I think from their pov it’s not a property that applies to 0

[long pelican]

o yea that's a good one

Based on examples (we train on examples, just like LLMs, after all) we have no data about even and 0, or odd and 0. All data is from 1 onwards

[tight star]

It’s not immediately intuitive that -6 should count as an even number

[long pelican]

("we" here means a 5 year old student)

[tight star]

Mhm

And then you could ask “is 3i even or odd”?

[long pelican]

4i instead of 3i creates a more contentious example >:)

Does x/2 have to be a member of Z? Is that part of the definition?

1. It is now!
2. No, let's use the ambient ring

• 2a. Is the ambient ring supposed to be understood as Z[i] here? Can't we cause chaos by saying we can use Z[4i] as the ambient ring instead

[tight star]

I think this could be a good opportunity to teach students some math

Rather than saying “0 is even, end of story”

[long pelican]

Yeah

[tawny slate]

what about defining even as just double an integer?

i dont find this as a definition offensive

[long pelican]

What's this in response to, in the context of this conversation so far?

[tawny slate]

just if people in general have issues with this as a definition for even

because if there is no objection to that, it trivializes the even issue

its now obvious that negative integers and zero can all be even

pedagogically it might be useful just to sidestep all of the complicated stuff with division and fields and whatever

makes it far more accessible to the layman

[long pelican]

I think this

And then you could ask “is 3i even or odd”?
was targeted at us, not the hypothetical student

even this

It’s not immediately intuitive that -6 should count as an even number

Or this

green is neither even nor odd
That's a philosophical statement meant to remind us that answering whether something is even or odd, people naturally grasp for relevant training data in their brain

[boreal agate]

honestly getting the student to think about what it means for a number to be divisible by 2 is a good way of approaching this imo

because ofc its easy to answer whether something is even if we have our definitions in order

and getting to these definitions is too hard

its just that trying to figure out what numbers are even before figuring out the definition is difficult

[long pelican]

Also when I asid

Suppose the primary definition of even in your head is "can be divided into two equal groups"
Here I'm referring to the definition the student has inferred from their training data, not the one we chose

[boreal agate]

tfw aleph0 is even

[tawny slate]

i mean i think both lines of inquiry are important

like just because a student thinks about how a definition doesn't mean they will come up with good/useful ones

[native iron]

I don't know! It's not that they tell me they dislike it, it's that when I say things like that I get blank stares.

This is one of the biggest challenges in teaching for me. I know I find math more enjoyable and intuitive than the average person, but they don't even want to talk about it, so I can't sus out what they get or don't get/like or don't like.

[true granite]

Hi! Has anyone participated in organizing a math Olympiad? I've been contacted to help create the problem sets (for high-schoolers), and I'd like some "life pro tips" if anyone has any

Mostly, besides choosing what the problems shall be about, what do I have to keep in mind? Pedagogically-speaking, what are things that don't work and should be avoided?

[rapid tusk]

i've been writing problems for a contest at my uni for 3 yrs now uh

biggest things are

dont overestimate the level of your contestants; it's happened a ton to me where i look at a problem and go "oh this should be fine" only for it to have a super low solve rate on contest day, etc

by default contestants will tend to struggle more on topics that aren't in the standard curriculum (NT, combo, etc), include problems of those types in moderation

by design you'll want to have some especially hard ones at the end but be careful not to overdo it

[halcyon glade]

You might want to check out AMC problems on artofproblemsolving for ideas

[rapid tusk]

^if you do decide to borrow problem ideas from past contest problems try not to be too obvious about it

though this is less of a concern in lower level contests where most of the problem archetypes are super well known anyway

[true granite]

I see; that is indeed something that would be terrible!

Thanks to both of you for your output! I think I'm safe if I look into AoPS, since it's in English and high-schoolers don't typically look into foreign language stuff here

[cinder citrus]

I participate in quite a few olympiads (in fact i have a very important one coming up) and from the perspective of a high school student, even some of the easiest problems in hindsight can be hard in an olympiad environment, so really try and place yourself in the shoes of the test takers and dont pick problems which are really far out of their natural thought bias

[true granite]

Gotcha! Would you happen to know how to evaluate that? Cuz I find it pretty hard to make sure a problem is fine, despite trying my best at it (it's something I tend to struggle slightly with for my classes already, altho I've become better at it with experience)

[cinder citrus]

Hmmm well this is the most difficult part of finding problems. Even test makers at the highest level often misjudge the difficulty of problems but there are some strategies you can use. If this is just for your class of students, then I would assume that you know how you taught them material and also what you've taught them. This can help you when considering a problem. With Olympiad problems the difficulty lies in challenging your thought process to go off on an adjacent path and find different links in the problem. If you can find a problem where upon a little consideration (from the perspective of your students) you can gauge some ideas to attempt the problem then this is good because it will allow for your students to challenge themselves and reach out for the next step in solving the problem. Easier problems tend to be more simple and have less steps as well as smaller jumps in thought processes whilst harder problems are more complex, having many steps and often requiring large jumps as well as recognition over multiple areas of knowledge. I am not super knowledgable on math pedagogy as of now so this is more of a surface level review that im suggesting (im trying to learn more every day :)) but even if you just look at how many mental jumps you have to make in solving the problem helps, and also looking at how bizzare or intuitive these jumps would seem if you attempt to strip away the bias you have from experience (and obviously from knowing the solution). Hope this helps 🙂

[true granite]

That's some valuable insight! I'll keep that in mind, it sounds like a good thought process thank you very much!!

[formal ibex]

Has anyone experimented with Gemini for learning from math papers or plotting their own course? While I despise the idea of fixing a curriculum, I appreciate that it can be very helpful for others to know exactly where to go next and exactly why to go there (for psychological reasons, I assume) and I respect the usefulness of the metaphor in its capacity to act as an intuition pump for unknown domains (although sometimes it seems like the pump works in reverse...)

I'm really looking for an automated process that ramps up my intuition past 11 instead of glomming onto it like a leech, which is how structured curriculum feels to me.

Ideally, I would find a powerful AI tool that augmented my learning process and allowed me to generalize the tool beyond my own experience, but I have yet to find something that doesnt suffer from arbitrary bottlenecks at random intervals, although that could be a personal limitation and not a technical one.

[rapid tusk]

none of these models actually know what they're doing, just read the papers yourself

[halcyon glade]

I think it's very useful for logistical stuff like taking a list of topics and planning a learning schedule in a couple seconds which you can then ask it to modify in various ways to your liking

[rain hawk]

These LLMs are really bad at actually doing the hard part (understanding the paper) but they can be pretty good at finding stuff if you're like "hey what technique is this and are there any expository papers about it". In the past few months they've gotten very good at stuff like that.

[turbid zenith]

So I’m trying to come up with a lesson for my liberal arts class. I know I want to start with having them try to come up with a definition for “number” and then stress testing this definition by debating whether things like negatives, imaginaries, and infinity are numbers. And later they’ll look at the history later in the week but what to do after that debate, I’m lost.

Am up for any suggestions.

[native iron]

I'm not sure if this is quite the thing you're looking for, but I remember enjoying this when I saw it a few years ago.

https://www.youtube.com/watch?v=uo1xDbsYAcU

I especially think the Julius Caesar problem is potentially relevant.

[turbid zenith]

Have a timestamp for that part?

One thing is I definitely don't want to lecture at the students for an hour...

[quasi musk]

You should definitely give them some grounding. Maybe you start with one, ask them how to get to two, etc. Ask them how to add and subtract

Have some structure ready to go. Ask questions about multiplication, division

[turbid zenith]

Yeah, the idea of numbers as something you can do numbery things to is a direction I'd love for us to end up

Where things are defined in terms of how we use them

Though that has its limitations

[quasi musk]

I've been teaching this to middle school students learning algebra for a few years now

I just do one or two lessons on it

[turbid zenith]

How do you structure your lessons on this?

[quasi musk]

Since they were learning algebra, I started with the naturals

and I asked them to think of what 0 should be, and how it should behave with the four operations (+, -, mult., divide)

Or what should -1 be

[native iron]

I'm relistening to the lecture rn; I'll let you know when I get to it.

[quasi musk]

I had to give a lot more structure since they had little experience, so I spent most of the time lecturing and just answering their questions

[native iron]

Also of note is his distinction at the beginning between the representation of a number (say the numerals 57) and the concept represented by the representation.

[quasi musk]

"Why can't we divide by zero?"
the answers would be something liike "Because zero's not a number" or "You can't take objects and divide them into no groups"

So they had linguistic answers, but very little "math-y answers"

My primary goal was to get them to think in a more mathematical way, rather than this languagey way

[turbid zenith]

To give you an idea where we're at, we spent last week talking about the idea of abstraction and numeration systems (comparing and contrasting Hindu-Arabic, Egyptian, Babylonian, Iñupiaq, etc), and now we're about to get into historical struggles with particular types of numbers such as negatives, complex, also throwing in quaternions. And next week we'll also get into zero and infinity and how those concepts were developed.

[turbid zenith]

Mhm this makes sense

Honestly getting them to try to define a number precisely is probably going to take longer than I might think

[quasi musk]

The way I introduced zero was "It is the solution to x + 1 = 1"

Which gave naturally do defining 1/2 as the solution to 2x - 1 = 0

Then, when I later taught those students pre-calc, I introduced i as one of the solutions to x^2 + 1 = 0

The same students would ask questions like "How can a square be negative?" to which I replied "Remember when I taught you algebra and I said 'How can zero be a number? Zero represents nothing, but zero is something. Something can't be nothing; nothing can't be something'. This is nonsense, tell me how to add, subtract, multiply, and divide"

[spiral elbow]

how do you justify being able to introduce solutions to equations like x^2 + 1 = 0, but not to equations like 1/x = 0?

or is that out of scope for your target audience?

This was one of the things I was confused about the first time I learned about complex numbers. When I learned about them more formally in uni, it was just as a special operation on R^2, then a bit later I learned that you can adjoin roots of any polynomial to a field

but I guess you need to handwave some stuff on a first introduction

[quasi musk]

Multiply both sides by x. You get 1 = 0

[spiral elbow]

Well of course that particular case gives a contradiction. But how do you know the same isn't true for x^2 = -1?

[quasi musk]

Because if we do it, then we get a consistent system where we can add, subtract, multiply, and divide

[spiral elbow]

hmm, okay. Do you think your students understand that you can't just take any equation with no solutions, and claim it has solutions in a bigger number system? Like in a sense, I guess you're trying to teach some parts of field theory without the formalism, right?

[turbid zenith]

This assumes that's an allowed operation with whatever number system contains a solution to that equation.

But I could go on all day on division by zero. 😛

[quasi musk]

Right, but we're talking about plausible explanations for higher schoolers to follow

In a somewhat math-y way

[turbid zenith]

I still teach high schoolers how to divide by zero every summer XD

Good times

[quasi musk]

Let's all teach 5th graders peano axioms, and construct the rationals from there

They'll surely know what fractions are then!

[turbid zenith]

I don't think teaching how to divide by zero is any more difficult to understand than teaching how to square root a negative number, personally

[spiral elbow]

You wouldn't need to talk about field theory in any detail, but I think it would be good to emphasize that this doesn't work for any equation, but it works for any polynomial

[spiral elbow]

I'm just basing this on my experience with the amount of people coming to #math-discussion and claiming they have found a way of dividing by zero. I think it stems from a misunderstanding of how we construct the complex numbers

[turbid zenith]

Is this a common occurrence?

[spiral elbow]

people claiming to be able to divide by zero? yes, more than you'd think

[turbid zenith]

I mean specifically in #math-discussion

[spiral elbow]

I think it's among top 3 crank topics in #math-discussion

[turbid zenith]

...I have a whole website on it 😛

[spiral elbow]

it's been a while since last time though, but there have been periods where everyone was dividing by zero in #math-discussion

[turbid zenith]

Did anyone ever mention 1dividedby0.com? 😛

[spiral elbow]

hmm, not that I can remember but I'll link it next time!

[turbid zenith]

I do it in terms of projective geometry and the real projective line, but with the caveat that you don't have a field anymore.

Single point compactification and all that.

How do they usually do it in #math-discussion ?

[spiral elbow]

uh, you don't wanna know 💀

it's definitely not in terms of projective geometry

it's usually written by people who think algebraic geometry is about calculating areas and angles, but with x and y instead of actual numbers

[quasi musk]

I didn't know that you had this up

Very interesting

[turbid zenith]

It never got the kind of traction I wanted it to 😛

But I still teach students about it decently often when it's relevant

[quasi musk]

I did once say the zeroth commandment is thou shalt not divide by zero

And a guy stared at me in disgust

[turbid zenith]

lol

[quasi musk]

I was in a job interview for teaching, and I kept writing $+\infty$ or $-\infty$. The interviewer (an engineering prof.) asked why Kept writing it with signs

[burnt vessel]

MoonBears-C-

[quasi musk]

and I didn't want to get into it, so I just said "That's how I learned it"

"Well actually if you take the one point compactification, you want a pt at infinity, but it can't be signed"

Isn't a real good thing to say

[boreal agate]

it also shows mathematical inexperience lol

there are so many ways of treating infinity

[fiery quiver]

I start tutoring tonight for calc 2 and 3. What do yall do when you dont know an answer? Im particularly afraid of having to do a word problem as its been a bit since ive taken either course

[lethal hornet]

typically if i can’t see the answer right away, i’ll ask for a few moments to think about it and/or i’ll be transparent and say that we are going to solve it together

it’s nice to be inclusive and involve them; sometimes the students do have partial solutions or good insights if they have been working the problem for a bit, and that can speed/smooth the process while building confidence

in the case that you don’t end up solving a problem or something like that, direct them to someone who may know it, e.g., a professor or a different tutor. sometimes i would ask to exchange emails and i would let them know if i thought of something later

[fiery quiver]

👍 thanks!

[quasi musk]

"Oh man this seems tough ~ are there any examples like this in the book or notes? What did you learn about this in class"

[native iron]

Often, you should even do that if you do know the answer (IMO). It teaches students that there are answers in their book and notes.

[turbid zenith]

It’s also useful for students to see what the problem solving process is actually like

[halcyon glade]

Also useful for students to see that being stuck ≠ being stupid, it's just part of the problem solving process

[fiery quiver]

true true. it was kinda fun getting back into a calc 2 mindset, tho i definitely made some mistakes. def was rewarding

[tight star]

I’ve been thinking about this a little, and I wonder whether there’s something to be said about variables being what you need to “manage infinity”, so to speak

Since as far as I can tell, variables are most useful when they’re ranging over something infinite

(Or maybe something very large, even if finite)

[spare void]

one way I ease high school students out of their math anxiety is by straight up saying a problem looks hard if I myself find it hard lol

[quasi maple]

I do the same; I did the same yesterday in fact

One of the questions in the workbooks my students go through involves factorising 35x^2 +2x -48

And I try to walk through what my intuition (in the form of prompting(adj.) questions) is to figure this out

[spare void]

I tend to discuss in a way where I verbalize my thought processes live and if I screw up then I say something like "ok that's not right. so we should try something else." helps kids internalize that solutions are not supposed to be flawless and perfect in one try

but tutoring college calc and approaching behavior of adults are way diff from high school math + teenage behavior

[long pelican]

That's a very interesting insight

[tight star]

i tried it out earlier today if you're curious

#proofs-and-logic message

[quasi maple]

It's more difficult for my specific case where I'm tutoring, not teaching

Because essentially I have to cover the same amount of content but with 4 straight-hours a week

(well, there's a 15min break in the middle)

But it's still quite a long time to stay focused on stuff for that long

I'm literally procrastinating my current work, incidentally

[warm valley]

I agree with what others said, I also think a lot of my students have gotten value from me saying something along the lines of "Well, my first instinct to try is <insert> because <reason>, let's try it together". Often it fails, but making it clear how the instinct was wrong and how we can learn from this is very useful

[fiery quiver]

I like that

[winged maple]

I'm planning on trying some more interactive lectures/seminars to go alongside some regular ones, where I take more of a step back than usual and focus on guiding the class as they build whatever I'm trying to show them (such as a proof or derivation).

Does anyone have any experience with something similar? Any advice or things I should be cautious about?

To clarify, I usually ask questions anyway and reason with students about their responses. The plan for these sessions is to take it to the extreme where I basically only guide them.

[dapper flume]

This is the inquiry model, in a nutshell, in case you need a search term to learn more.

Inquiry, done right, is indeed very good. Some things to note:

• Inquiry is a higher cognitive load on students. This simultaneously leads to more learning, but also more stress and anxiety. Be prepared to encourage students to persevere.
• Inquiry needs a strong set-up in order to achieve the expected payoff. Students need access to well-placed information to help them progress. One way I do this for adolescents is to have students work in small groups on sections of vertical whiteboards around the room. If a group gets stuck, they can look to other groups for inspiration.
• Think about the intended questions that you hope students will ask. Design your instruction around eliciting those questions.

[tepid smelt]

Building thinking classrooms is a good book on inquiry based teaching. I personally have observed the vast majority still find the 'classic' direct instruction to just work best. I think more research is showing that also.

[dapper flume]

^ my current fave

[warm maple]

anyone here a teacher Im looking for one on one teaching willing to pay

[halcyon glade]

this server's not the place for that, sorry, you can ask questions in the help channels though #❓how-to-get-help

[tight star]

How relevant would people say the psychological difficulty in learning new math is? Compared to, say, its conceptual difficulty

[halcyon glade]

What age of student are you thinking of

I feel like it will always depend a lot on the individual and their conceptions about their relationship to math

[tight star]

I think this is a phenomenon more prevalent pre-uni

But I have personal experiences with “psychological” difficulties in learning uni math, particularly category theory and algebra

[spare void]

psychological difficulty, are you referring to the negative pre-existing attitudes and feelings re: math as a subject for young students? because that lack of confidence definitely gets in the way of being able to study

just not being able to get out of their own head and giving up before even attempting. math anxiety, etc

[tight star]

yes that's a big part of it

[spare void]

incredibly relevant imo

the importance of studying education specifically and not just pure math is because we're taking the chance to learn how to dismantle those kinds of behaviors

content area mastery is the ideal in learning but building positive attitudes and cultivating good habits, growth mindset, curiosity, failure is part of the process, is equally important in learning

earlier our professor was demonstrating pmi. and she was walking us through like the "wrong" solutions that would lead us nowhere. essentially it went like:

"my first instinct upon seeing w is to try x" and she really solves the entire thing on the board and it's just totally wrong -> "but that didn't work, even though it looked like it should be in the x way. so let's recalibrate and consider y and z"

I thought it was fascinating!

[tight star]

for me i've found that, if i don't believe i can understand a piece of maths, it becomes a self-fulfilling prophecy

[tawny slate]

it actually goes further than that iirc, i think studies show that if you think you can become smarter, you will become smarter

one of the best pieces of advice you can give an aspiring artist is "fake it until you make it"

so i think psychology is not only important acutely, I think it's critical in the long term as well

[tight star]

an example that comes to mind for me recently is algebraic geometry

i've understood the definition of the spectrum of a ring for more than a year at this point

but that was only conceptually

because i didn't understand the significance of the definition and/or why i'd care about it, i had a block that meant i was unable to really understand AG much further

[tawny slate]

i was going to make comments about this kind of thing, but i wasnt sure if it counted moreso as conceptual than psychological, this feels like a weird grey area in between

motivation is a form of direction, so that can also inform what kinds of ideas you prioritize and therefore what results you find meaningful and what to pursue, which could be conceptual too

[glossy mortar]

I know this was asked a while ago, but I have seen students (k-12) who when told that they can do something, and told you believe in them suddenly able to do something they thought they couldn't do, become able to do it. So while it probably hasn't been studied extensively (idk I'm just guessing because it's hard to say for sure) I think that in practice it does hold some merit to consider.

[regal portal]

im not sure if this is the right channel but:
what do people generally do as a TA if there is a strict no AI policy in the course but the student blatantly used it to generate solutions for a very unserious assignment and the consequences (if i report this to the instructor) could be dire since they are a scholarship student

i am leaning towards reporting it but advising the prof to not report this to the honor council

[vivid coral]

An assignment is an assignment. If you're at the level of university, everything you put your name on and hand in should be expected to follow the rules set out. You can voice your opinion to the professors, but I would personally trust that they would be capable of making such a decision themself

[twin lichen]

I think given that you don't want to endanger their scholarship you can just email them and be like

Hi <student>,

It looks like you used AI to generate the answers to this assignment. I will remind you that there is a strict no AI policy in this class. The consequences of violating the policy are spelled out in the syllabus.

I would imagine that letting them know their AI usage is obvious might make them think twice about it.
If it were me I might also include a personal comment about the time I as a TA invest into grading their work and providing feedback, and point out that if they use AI on assignments they are wasting my time which is nominally spent for their benefit.

but college students are adults too

[regal portal]

yeah i have emailed the student something along those lines

[regal portal]

they are just freshmen taking their first ever course though (calculus 1) and the prof can be quite ruthless

ill voice my opinion to the prof then with a strong rec to let them go with a warning

[quiet gale]

So I'm tutoring this student on AP statistics.
But the problem is that he acts like a given exercise is easy for him but when I ask him to do it, he fails to do that almost 50% of the time. He lacks basic logic knowledge (which is not that bad, I can teach him slowly) but he just acts way smarter than he is.

Have you faced anything like that before?

[tawny slate]

unfortunately yes, and this was when i was fresh out of college so my approach to dealing with this was to flip a switch and crush the student's ego before flipping back to nice teacher

i have regrets, and i still haven't figured out the best way to handle this

[quiet gale]

There's so much joy in teaching open minded students.

[quasi musk]

If he says something like "This is easy" you can ask "Why do you think this is easy" and let them try. When they fail you can say "Looks like it's harder than we both thought"

Teens are very sensitive to things, no need to come down on this too hard. Kids will be kids. After building a relationship for a while you can tell them directly "Maybe you don't mean to come off this way, but this is how you come off"

To be honest about my own life, I realized my senior year of HS that I was smarter than most people around me, and became absolutely insufferable my freshman year of college. After (many) people told me how I came off, and many years of working in mathematics, and teaching people math, I finally found a balance of my own confidence and having kindness/patience with other people I'm working with

[serene dew]

Hi everyone. I have a question.

Sequences and series are related concepts but differ extremely from one another. I feel that students in integral calculus frequently mix them up. Part of the problem is that:

1. Sequences are usually taught only briefly before moving onto series.
2. The definition of a series involves two related sequences (terms and partial sums).
3. Both have operations that take in a sequence and output a number (the limit or the sum)
4. Both have convergence tests for convergence (monotone convergence and squeeze theorem vs. root test, ratio test, etc.)
   What methods can we use to teach students to distinguish between sequences and series? Specifically, methods that adress the above concerns. I would appreciate also appropriate references.

Thank you.

[turbid zenith]

You have to realize you're fighting against the English language. In English, "series" essentially often means the same thing as "sequence." A TV series really should be called a TV sequence for example.

[quasi maple]

one does not simply fight the English language

[turbid zenith]

So part of it is that you have to emphasize that the word "series" in mathematics has a precise meaning that specifically involves adding the terms of a sequence.

[quasi maple]

Ultimately it falls on Step 1 - you have to explicitly teach this^

[turbid zenith]

I'll be honest — as an educator, I trip up and will say series when I mean sequence or vice versa, and I have to correct myself.

[quasi maple]

It's not enough to say "series are like sequences"; you have to specifically define them

[quasi maple]

skill issue

[turbid zenith]

😛

Look I became a math teacher so I don't have to speak English okaaaay

I just speak in arcane symbols and Zalgo text

[serene dew]

Mhm, I don't think I understand...

[marsh compass]

series are sequences

[crystal spruce]

CAP.

holy cap

bro is so wrong and he thinks he is right

series are a sum of sequences with certain properties

they are not the same or Series -> Sequences

Is not true.

they are differeent

[cloud zealot]

every series is a sequence, not every sequence is a series

[cloud zealot]

series are just sequences of partial sums. the terms being added do come from a sequence, but that doesn't mean series aren't sequences themselves

[tawny slate]

technically as long as subtraction is defined on the elements any sequence can still be a series if you just take a finite difference

[tawny slate]

the way i like to think about it is:
when we say we are summing a sequence, we dont have to declare how many terms to add, because any number we pick is still a series, so let's call it n, and then write out a sequence of all possible sums indexed by n and voila also sequence

[marsh knot]

I do apologize, since I'm not a professor or teacher or anything of the sort, but how do you go about walking through problems and connecting them to other things in a way that 'clicks' with someone. Obviously, this is different on a person-to-person basis, but I would love to know more about it. Even some reference material would be great! Specifically, I want to know how to do this in a way that builds comprehension and understanding, so not just teaching them how to do something, but how that is connected to everything else. Thanks in advance, haha!

[tight star]

Some advice I’ve been given is to try asking them questions

You’d be surprised how often students can answer a Q just with the right prompting

[marsh knot]

Oh, okay! Thanks. That's a great start to try and build a model of.

[feral swan]

I've gone through undergraduate treating the two as the same thing (I think of a series as a special case of a sequence) and it never hurt me... You should probably find a way to show the downsides of treating them as the same or they will think it's an esoteric obsession, tradition, or historical artifact and go about the easiest way they find to do things.

The attitude that I had and indeed to this day I have is:

So, given a sequence a_n, you can define the series ($ \sum (a_i-a_{i-1} $). Given a series $ \sum a_n $, you can define the sequence of the partial sums.

They're basically the same concept if you squint on it the right way, and the sequence concept is more geometric and (to me) more intuitive.

Why should I think of them your way? My way is comfortable and works. Your way is confusing, for not much gain. Maybe my way is wrong, but It's sufficient to my purposes.

So I think you should address that.

[turbid zenith]

I disagree — I would say that a series is the limit of a sequence of partial sums

Not the sequence of partial sums itself

[tight star]

i think sometimes it is useful to view series as a kind of separate concept to series

there are things like absolute vs conditional convergence which don't really have an analog for sequences

[tardy ember]

...honestly this conversation is making me realise that i just, don't have a definition of the word "series" that's this precise

like i can tell you exactly what the notation $\sum_{n=0}^\infty \frac{1}{2^n}$ means, but which exact part of it is a ``series''? ...idk, who cares

[burnt vessel]

bee [it/its]

[tight star]

"series" to me is "infinite sum"

[tardy ember]

...yeah i think that's also roughly my definition, actually

so like, the difference between the sequence $a_n = \frac{1}{2^n}$ and the series $\sum_{n=0}^\infty \frac{1}{2^n}$ is that with the series, you're adding it together, and with the sequence, you're not

[burnt vessel]

bee [it/its]

[turbid zenith]

I’m just gonna leave this here and wait

[spiral elbow]

I'm definitely gonna use that when I need to compute the decivative of +² = 4" + 3

[compact isle]

this is hilarious

[uneven bay]

when i was in an airbnb they hung up shitty AI art on the walls and it was genuinely a pain to look it

[turbid zenith]

Here's another one!

[vocal phoenix]

To be fair, that graph is perfect in its way

Especially accompanied by the rest of the ad; it's a form of art

[feral swan]

fantastic ad.
If you checked the contents you're not the audience anyhow.

[quasi maple]

I love how this somehow both relies BOTH on the ignorance of what LLMs are combined with the ignorance of mathematics

This is double crazy

[plain valve]

"Unlock the unfair advantage"

[tight star]

It’s like how phishing ads purposefully misspell things to filter out people who would notice that

[feral swan]

yeah exactly

[twin lichen]

!nogpt

[coral copper]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[turbid zenith]

lol I didn’t know this is a command in here

[coral crest]

-Quadratic equation
-Look inside
-Cubic

[midnight scarab]

I think the meaning of "series" is really sentence dependent.
In "the two series are equal/the series equals 1", "series" means the limit
But when we say "the series is convergent/divergent/..." we mean the partial sums. As Pseudo meantions, there's also the subtlety where we talk about e.g. absolute cvgce, where we're referring to a different series: that of the partial sums

To say all we say about series, a series must contain the knowledge of its terms

[quasi maple]

This reads sorta like the introduction to the episodes of Full Metal Achemist

"If one wishes to obtain something, something of equal value must be given. This is the law of equivalent exchange: the basis of all alchemy" etc

[gray smelt]

Also like. Maybe he just is consistently wrong about how easy he'll find something. I've definitely been wrong about how easy I'll find something! Or thinking I understand something when I don't!

[halcyon glade]

You might also be able to goad him, like "Oh yeah, because you just use [X wrong technique], right?" Either he corrects you and is forced to do the problem, or he agrees with you and you reveal that you were just making stuff up

[lone nymph]

any here have experience getting math bs and ms not in teaching and have never tutored and still get a job teaching math? like middle or high school

[gray smelt]

yeah i'd start by first gestly pointing out that he predicted wrongly the last dozen times, and then if that doesn't work more absurd approaches like these are appropriate

[halcyon glade]

That depends a lot on the country

And in the US it depends on the state

In the US it's common to have alternative teacher certification pathways because of chronic teacher shortages

I would look to see if there's a local department of education that will help you through this process

[rich summit]

$\lim_{x\to{3}}x^2-9, \lim_{x\to{2}}\frac{x}{x^2+1}, \lim_{x\to{a}}x^4$ are all ones I remember doing

[burnt vessel]

alligator6

[rich summit]

spivak has a ton of practice problems, which is the book i used

I'd also recommend having the people in your group doing a lot of work with the definition (making sure they can write & negate it, understanding the individual parts)

[lethal hornet]

sequential limits or functional limits

[native iron]

delta suggests functional

Popcorn function if you're fancy

https://en.wikipedia.org/wiki/Thomae's_function

[rich summit]

oooh that reminded me of some of the limits with werid combinations of floor and ceiling

[turbid zenith]

So I'm wondering what y'all think. I'm designing a curriculum for my school's new version of the intro to proofs course, and we're supposed to teach about functions (proving one to one and onto etc). And I'm just finding myself wondering "what's the narrative here? How should I motivate even caring about that?"

Like I guess it's a nice way to demonstrate proofs involving quantifiers

But it still leaves me thinking "okay now what?"

[compact isle]

I think it can be fun if you tie it into discussions about cardinality.

Then you can talk about unintuitive results involving infinite sets and introduce things like bijections from N to Q and such

I assume that's already part of the course anyway, but I think its interesting at least

as a framing device

[pure bison]

You could do some bijective proofs
https://math.mit.edu/~rstan/bij.pdf

[lethal hornet]

math is about relations between mathematical objects blah blah blah

one basic question is, how do i tell if two sets have the same number of elements? if one set has more elements than another?

i think this entry point could be good lead into why you should care about injectivity and surjectivity. they are the two ingredients you need to answer the above questions.

[quasi musk]

I'm assuming that many students have somewhat that mathematics is interesting if they're taking intro to proofs. Galileo famously gave up on comparing the sizes of sets because of the bijection between the perfect squares and the naturals

There are the famous examples of removing a point from an integrals domain doesn't change the integrals value. How many values can you change? What does this say about length? How many pts can be removed from a line segment before we change the measure?

[tawny slate]

why are all these suggestions about cardinality lmao

i feel like while it may be philosophically interesting, its not particularly useful or meaningful in the grand scheme of things

my favorite application of why we care about functions being one to one or onto is how we solve equations

2^(3x+4) = 2^(7x-11)
3x+4 = 7x-11

why is it that we can do this step? how can we be sure that we arent missing solutions or getting extraneous solutions

the function we are applying to both sides of the equation is log_2

we can guarantee no solutions lost and no extraneous solutions because the function is one to one and onto

we can use these ideas to explain why x^2 = 4 will cause you to miss a solution if you square root both sides and why squaring both sides of sqrt(x-2) = x can result in extraneous solutions

[tardy ember]

i feel like maybe part of why this is hard is that there isn't really... one obvious thing that you use these notions for

like there isn't some field of "injective function theory" that just looks at the properties of injective functions, because there isn't much that's interesting that you can really say about them and most of the meaningful results are very elementary

they're just basic enough to show up all over the place when you're studying other things, which is why it's useful to know what they are

you might end up trying to count how many injective functions there are, or considering injective homomorphisms of some kind, or just, using them for things, trying to track down every use of the notion of injectivity would be like trying to find every example where the notion of implication is useful

cardinality is the closest you really get to injective functions being the primary object of study, even though that isn't most of what you do with them in practice

[tawny slate]

i agree, which is why my approach to finding motivational examples for things like this is to actually crawl up the curriculum from kindergarten and try to find places where i am implicitly applying these ideas or taking them for granted

i think we miss so much of this when we learn math for the first time, and just accept the intuition behind it

i think when it comes to motivational examples, too often people find math that is more advanced, not less, to motivate what is being learned, and this usually doesn't feel very strong, because the examples start to get abstract and it begins to feel like learning math for the sake of learning more math, which is fine if you already like math but for most people they need something more relatable, not less

[quasi musk]

Because that's how, historically, cardinality was developed. There are narratives about Galileo, Cantor, and other mathematicians that developed these notions to solve these problems

[tawny slate]

unfortunate timing because my last message would be my reply to that lol

[quasi musk]

I think the examples you provide are good at giving a practical problem, but completely miss the very interesting ideas embodied in the history. The problem of cardinality, the different sizes of infinity can be a lot of fun

[tawny slate]

uh yeah i sorta acknowledged this already

i phrased it very differently but i get it, i just think this just further makes my point, that the historical ideas, while fun, aren't that useful

[quasi musk]

Useful with regards to what?

[tawny slate]

its not like actual history class where the point is to gain some understanding of the superstructure to society, that which we live in and interact with

what does the layman gain from learning these fun math history stories beyond inspirational fun and general cognitive exercise

[quasi musk]

If the goal is to create a lesson centered around 1-1 and onto functions with a narrative, then a historical approach is a very natural one to take.

Not everything in math class is designed for a carpenter to use. Not everything in math class has a direct reference to a job

Even the questions that you propose don't have much for a layman to worry about

[tawny slate]

i actually agree with this too! i just think the "historical approach" is very dry and overdone, and that most people would prefer something like i described and it would more directly benefit them a deeper understanding of math

[quasi musk]

It's hard for something to be dry and overdone for someone that hasn't seen it before

[tawny slate]

learning math for a job is also part of the thing i am rejecting, like if you asked why we are motivated to learn partial differential equations and someone said "so we can do engineering"

most people arent going to be engineers so they dont care

im saying that slogging through historical examples is dry in the same way people dont like history class in general, and its overdone in the sense that everyone does this "historical approach", which was my original point

saying the "historical approach" is natural and we should do it is basically the same as saying "we do it because we have always done it this way" and im not convinced

[quasi musk]

I'm not convinced that most people teaching these things take a historical approach. Most of my lessons certainly didn't

[quasi musk]

The condition being "creating a narrative". Of course you can come up with some other motivation to investigate cardinality. However, there's already the real history that played out a very real narrative. That's what makes it a "natural" choice

[tawny slate]

ok sure

at this point idk what we even disagree on so lets agree to agree

[feral swan]

This is not directly related to your question but I found that questions around "composition of injectivity and surjectivity" like "If you compose surejctivity and something random do you get surjectivity?" are great for explaining how to work with counter-examples.

We would give variants of those questions as True/False, and if you shuffle the terms around enough you get statements that are either, so the students are open to methods that give them a heuristic to which is which.

Then in the counter-examples itself, a lot of the students give idealistic examples (for example, they use sets of the same finite size) where the statement is true (but it is actually false), so it's a great time to talk about using examples which are pathological, because the pathologies are very simple.

I had a very bright student who was bored to tears in my tutorial and this thing is the only thing that helped him throughout 10 hours of tutorial, so I feel rather pleased with it.

[tight star]

I think a version of the isomorphism theorem could be nice?

Every function between sets can be decomposed as a quotient map (surjection), followed by an isomorphism (bijection), followed by a subset inclusion (injection)

The first map is “abstraction”, since you’re giving the same name to different things

The second map is “relabelling”

The third map is “modelling” something within a larger context

So in a sense, studying surjections and injections helps you understand arbitrary functions

And the “standard” surjection is a quotient map, while the “standard” injection is a subset inclusion; arbitrary surjections and injections differ from these standard ones by a isomorphism

There are also more “equational” ways to discuss these concepts

Injective functions are those which have a left inverse (provided the domain is nonempty)

Any function with a right inverse is surjective - the reverse direction is equivalent to the axiom of choice

A bijection is equivalently a function with a two-sided inverse

Another thing that could be useful is emphasising the distinction between functions and formulas - there’s some nice history here where mathematicians of old didn’t see the difference

E.g. historically Lagrange assumed every function had a Taylor series, and used this to define the higher derivatives of f

And indeed most functions given by “formulas” do have a Taylor series, and are even analytic

The same function can have different formulae, like f(x) = (x + 1)(x - 1) and f(x) = x^2 - 1

The same formula can also describe different functions - “x^2” can refer to squaring naturals, or squaring integers, or rationals, or reals, or complex numbers, or polynomials… all of these are different functions, since their domains are different

I’d also be curious as to whether the curry-Howard correspondence could be useful pedagogically - viewing “proofs as programs” gives a pretty visceral way to connect functions to proof-writing

You also have Garrity’s famous quote “FUNCTIONS RULE THE WORLD”

There are also some fun examples of functions like space-filling curves or Cantor’s base 13 function

[tight star]

invertibility?

[tight star]

is it related to what i said here?

[spiral elbow]

This is a very good idea I think; in my experience, a lot of people who are new to proof-based math don't really understand that you can have a function without being able to write down a formula for it. If you define f(x) to be the unique y such that blah blah for example, they don't believe that f is actually a function, unless you can write down an explicit expression for y

[tight star]

:D

yeah i think pointing out the history could help too - even mathematicians didn't think that functions without formulas made sense

the way i like to think about it is that math studies functions "agnostic of their implementation"

you don't care how f(3) gets computed, you just care that f(3) = 5, for example

whereas the "formula" perspective is much more about how the function actually gets implemented

in the example i mentioned, (x + 1)(x - 1) and x^2 - 1 are two different implementations of the same function

[spiral elbow]

Yep 👍 another thing I've seen is that beginners often think that piecewise defined functions are less "legit" than ones defined by a single expression. Like if you define a function as a counter example to something, but it's piecewise defined, they think "it doesn't count", like it's not a real function

[tight star]

mhm, and historically people viewed piecewise functions as really two functions rather than one

david bressoud goes over a lot of this in "A Radical Approach to Real Analysis"

would highly recommend

i think it's important to point out that the student's concerns aren't wholly invalid, though

if you wanted to describe an actual function $\mathbb{N} \to \mathbb{N}$, it's hard to do so without some kind of formula

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

the domain is infinite after all, and there's not enough ink in the universe to actually explicitly write down what f(1), f(2), f(3), ... are

variables and formulas are what allow us to "tame infinity" in this sense - it's possible to describe infinity with a finite amount of information so long as the function behaves uniformly in some way

like f(n) = 2n - you haven't actually used an infinite amount of ink to write down all the values of f, instead you've noticed they follow a uniform "pattern" that requires only finite information to describe

one reason we need the axiom of choice is that, in reality, humans can't actually make infinitely many choices - there isn't enough energy in the universe to do so

[tight star]

that's why i like this perspective - it recognises that, in practice, many functions you meet are given by some kind of formula, since that's the only way to "tame infinity". but you often don't need to concern yourself with the specific implementation - instead, you focus on what the function does, the relevant properties that it has, which is that it has a unique output for every input

even in programming it's sometimes helpful to consider functions in an "implementation-agnostic" way - you don't necessarily need to know how a HTTP request is actually implemented at the physical level, just what output you expect to get from an input

[spiral elbow]

oh yeah? watch THIS

[tight star]

unironically i think i understood proof-writing significantly better the day i realised "proofs are finite"

so whenever you want to prove some kind of infinitary statement, some sort of "uniformity" is unavoidable

-# incidentally this kind of "uniformity" also crops up very often in category theory, where it takes the form of "functoriality" and "naturality"

[tight star]

this is what makes $\forall$ different than "AND" - forall proofs require a kind of "uniformity" to make the proof fit on a page, in a way that a finite number of statements "AND"-ed together don't

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[midnight scarab]

Maybe start with manipulating equations? I think that's a very concrete motivation, esp for injectivity. The whole "squaring the BHS" story

(Sorry scrolled through the convo too quickly and missed Cozmo saying the same thing)

And then I'd move on to cardinality, pigeonhole, how equal cardinality buys you injectivity from surjectivity (and vice-versa for free) and maybe say a word about how you can replace cardinality by other notions like dimension (of vector spaces)

And yeah Galielo, Cantor and co

It's a shame there's no "simple" proof of Cantor-Bernstein

Though we did do it in my 1st year

[tight star]

oh yeah cantor-bernstein is quite interesting to me

it's one of the significant set-theoretic results without choice that comes to mind

[midnight scarab]

(In fact one of the problems on test 2 was an alternative proof)

[tight star]

do you have a favourite proof of cantor-bernstein?

[midnight scarab]

Wait I just realised there are 2 C-B thms lol

I mean injections both way imply existence bijection

[tight star]

yes i mean the same one

[midnight scarab]

Good 😂

I like the Knaster-Tarski (fixed point) proof because I like fixed-point arguments (they're everywhere and "systematic" in a sense)

[tight star]

oh i haven't seen that one

[midnight scarab]

Though the König proof with sequences is cool

[midnight scarab]

The fixed-point one?

[tight star]

yeah

[midnight scarab]

If you can guess-read French it's proof n°2 https://fr.m.wikipedia.org/wiki/Théorème_de_Cantor-Bernstein

But the TLDR is that the monotonous map (for inclusion) on subsets of E is
G : A |--> E\g(F\f(A))

And you use the fixed point set M to decide whether you use f(x) or g-1(x) depending on whether x is in M or not

This can be motivated if you try to build the bijection in this way and you see what M must verify

[turbid zenith]

These are really helpful answers, thank you all!

I think I can make a narrative work.

[feral swan]

A logic professor tried to insert that (a variant, actually) in the early logic course for cs students. It doesn't work out in practice, unfortunately... It's too abstract and the proof is tedious and the students can't see past the proof to the meaning.

[tight star]

I see

[feral swan]

In the exam he gave 30 points proving part of this statement and he got an average 30-40 on the whole exam and probably an average of 5 points on this portion.

Truth be told there were other problems (it was the last material of the course and students were not ready for it) but that's telling.

[tight star]

What was the specific result he used?

[feral swan]

something like "any functions that agrees... decomposes through the qoutient"

but in a more elementary language.

Teaching the students qoutient sets was already a bit past their limit of their understanding and accomodation of abstraction, so that was far too much for them. The limit was somewhere around eqvuilance relation and dividing by eqvuilances.

Since im already talking my head off to stave off the point of studying I'll note that they found great difficulty in the mechanics of dealing with the fact that there are 2-4 concepts and notations around classes, eqvuilances, and the like, and they were confused.

Like the fact that the class [a] is the class of the representative a and that [a] = [b] if a, b are in the same class was a step most could do, but then that X/~ is composed of classes, and that calling the elements of X/~ the set {[a_i]} and working with that implies some form of choice of the representative and that that has consequences really tested them.

We had a lot of fun talking about well defined functions here.

[tight star]

Mhm I get this

I wonder how they’d find kernel pairs

But yeah X / ~ is an interesting construction

Maybe you could view it as a specific subset of P(X)

[worldly sigil]

I've found that a good amount of students generally struggle whenever having to work with families of sets.
I can't explain what exactly they seem to struggle with but a lot of them seem to "get lost in the sauce" so to say when working with sets like X/∼

[tight star]

like having a set whose elements are sets, you mean?

what trips me up about that is set-theoretic $\in$ is not transitive, whereas irl $\in$ kind of is

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

if a ball is in a bag and that bag is in a box, i think i'd say the ball is in the box

but that doesn't actually work with sets :P

[worldly sigil]

Yeah. It seems like a lot of them also struggle why we choose to construct the quotient in terms of equivalence classes rather than via representatives for example.
I think some students would actually benefit from being taught to think of the quotient only up to iso as a pair (Q, [-]) with [-] a surjection X -> Q such that x ~ y iff [x] = [y]. Then the construction in terms of equivalence classes becomes just one implementation of such an interface rather than the definition.

[tight star]

yeah that seems nice, feels like it dovetails with kernel pairs too

issues about imposing surjectivity aside, which i know we've disagreed with in the past before

[tight star]

do they have the same struggles with the power set, then?

[worldly sigil]

I'd say yes. From TAing an elementary set theory course it seems that some students kinda blank out as soon as a question looks sufficiently convoluted (even if the solution isn't actually anything complicated).
For a concrete example from the class, when asked to prove that {Px | x \in X} is a set via comprehension we often even give them the desired superset to separate from (namely PPUX). But when asked to verify that Px \in PPUX whenever x \in X, a lot of them blank completely or make uncharacteristic mistakes, seemingly forgetting how they've proved (much more difficult) theorems in the past by unfolding definitions etc.

[tight star]

wait what's PPUX

[worldly sigil]

power set of power set of union

[tight star]

hm...

what's Px

[worldly sigil]

power set

[tight star]

x is a set as well?

[worldly sigil]

this was in the context of an introductory set theory course so yeah

[tight star]

hm...

i guess it's like..

$Px \in PPUX \iff Px \subseteq PUX$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

hm..

so if i take $y \in Px$, so $y \subseteq x$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

then $x \subseteq UX$ i think, so $y \subseteq UX$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

so $y \in PUX$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

that does feel a little weird honestly

[worldly sigil]

the 'elegant' proof they're supposed to come up with (using some earlier exercises) is that since x ∈ X, x ⊆ UX. So Px ⊆ PUX, so Px ∈ PPUX.

but it would be no problem at all if they gave a "nuts and bolts" solution that works just with the basic definitions

but i think some students struggle quite a bit with questions that they can't visualize that well, even if they do have the ability to prove it if they sorta followed their nose so to say, without trying to paint a clear mental picture of the problem

[tight star]

i can sympathise

i'm honestly still not quite sure what i just did

[tight star]

how do you actually use comprehension here

[worldly sigil]

{Px | x \in X} is a subset of PPUX and the latter is a set by union and power set, so {Px | x \in X} is a set as well (since {Px \in PPUX | x \in X} is a set but {Px \in PPUX | x \in X} = {Px | x \in X})

[feral swan]

These questions trip the students so much.
I don't know how to explain them because for me it was always intuitive how to keep "track" of where I am in an abstraction and how to go up and down definitions.

[tight star]

interestingly enough this isn't quite intuitive for me

though doing the calculation did illuminate some things for me

like $x \in PY \iff x \subseteq Y$

[burnt vessel]

Pseudo (Cat theory #1 Fan)

[tight star]

which ig acts as a definition of powerset

but this is more about a technical understanding than a conceptual one, right?

like there's understanding powerset conceptually, and there's being able to work with powerset in a proof

these aren't quite the same skill, and i don't even think being good at one necessarily means you'll automatically be good at the other

[tight star]

i'm curious, what sorts of mistakes do they make?

[worldly sigil]

Most commonly they start out correctly and then once they're knee deep in unpacked definitions they seem to lose track of what their current goal even is
At that point some give up and others start throwing an assortment of random proof techniques at the problem, hoping that something will stick

[tight star]

interesting

i'm trying to think about how i'd assist students with that

[worldly sigil]

If you play the role of a proof assistant and constantly remind them at each step of what their "givens" and "goals" are that seems to work for some

[tight star]

yeah, that's one thing i considered

[winged maple]

I struggle with this in general. A solution can be incredibly simple and elegant, but just requires a little more thought than 'apply definitions', yet students just... won't get it.

I really don't know how to push that problem solving thinking.

[tight star]

any examples of this come to mind?

[winged maple]

Yes, however give me some time to respond as I'm a little busy right now.

...yes this is what I'm talking about.

[tight star]

that seems like a sensible strategy

[lone nymph]

hey, if another channel is better to ask this, lmk. when a math teaching job at a community college is paid per hour, what does that usually mean? I see some part time instructor positions with an hourly rate. is there a standard amount of hours they include per credit hour of the course? they can't possibly just mean during class time, right?

[spare void]

problem solving thinking is built through gradual and repeated exposure. i firmly believe the sort of creativity required for math is a muscle that can be trained over time. scaffolding through different levels of difficulty works, especially in high school but even more so in college-level math courses. even in abstract mathematics and proving

[winged maple]

I guess so, but the students don't seem to want to try. For many, any resistance is met by either shutting down or getting tools to do it for them (often incorrectly).
My struggle is that I don't know how to motivate them to try. I do what I can, I'm always around to help and work through it with them.

(These are undergrads and master's degree students, by the way)

[dapper flume]

Yeah the big advice i always got in teacher training was to offer "low floor, high ceiling" tasks. They should be simple enough that people with weaker ability could understand the task and make progress, while being rich enough for even advanced students to have a lot to unpack

The facilitation and debrief of this kind of activity is where the magic happens. If the instructor pays attention during the activity, they can sequentially select who they call on to offer ideas so that as many perspectives as possible can build on each other. Really great for exposing struggling students to the role of pattern recognition in problem solving, and for giving advanced students some alternate approaches to think about

[left pumice]

#❓how-to-get-help what is the remainder when X6 is divided by (x2 +1)?

[rapid tusk]

wrong channel.

[spare void]

math anxiety and challenging zero motivation students are separate beasts on their own.

only very tangentially related, but once, my professor challenged us: if a student asks you, a math major, why they, a high school student, HAVE to study math, what would you say? and every reply we tried to offer got shut down (you'll need this in the future: what if they're not even going to pursue anything math related in college; it's good to be knowledgeable: a non-answer that evades their lack of interest; math is used in real life: trigonometry etc? be serious; and much more)

[spare void]

anyway, I'm sorry to hear that :( tbh I am a little surprised to hear of that kind of demotivation/fear from masters' students. I imagine they have more agency in the courses they take? that is, they definitely wouldn't be taking more difficult math classes without having chosen it for themselves. I hope you find a way through it; fwiw, making yourself available and open for help is, i think, one of the best approaches you can take as of now

[dapper flume]

If I had the time to monologue in my class, I'd say this:

"I'm sorry, I don't get the power to answer your question in a satisfying way. It's like asking ne whether you'll ever use a hammer in real life. I could teach you how to use one, and that would surely open up possibilities for you to build things on your own someday, but a tattoo artist may never use a hammer. Sure, math gives me the best possible tools for prediction i could ask for, but you're a unique individual and I have no idea where or how math will show up in your life. You'll know it when it happens. But I'll say that even though my job is to be a math teacher, I don't directly use the formulas from all my college math classes in my career. I'm not solving any differential equations to input grades. I'm not normalizing vectors when having meetings with the principal. Yet I do not regret a single second of my math education. Each thing i learned shows up in my mind all the time as a lens through which to see the world. These days the world is scary and confusing, and it is becoming more and more clear that that is by design. It feels hard to know the truth even when it's put right in front of me. But with math, at least I know that if I'm skeptical that something is true, I have a rock solid way to prove it myself. I don't have to trust the world to tell me the truth anymore. I can check that the earth is round. I can check that what the scientists are doing at the Blue Brain Project is actually interesting. I can check that the meteorologists and the doctors and the sociologists aren't just making stuff up. And that ability to be a part of making truth visible makes me feel more powerful in every single part of my life."

[feral swan]

... one day you'll go down to the shop and the grocer will ask you how to prove <obscure theorem> and you'll be so embarrassed that you forgot.

[native iron]

I don't have a great way to articulate it, but to me the reason I want high school students to learn math is that a little numeracy can help you better understand current events. I think there should be way more emphasis on statistics in high school math and that statistics should emphasize interpreting statistics in the context of the actual domain we're collecting data about. However, I'm not very knowledgeable about statistics or education, so I'm not confident in this assessment. I'm curious if my position has been critically treated by people that think about this sort of thing.

I'm not sure that's what I'd emphasize. In particular, that seems replaceable with communicating the broader idea that the casino wouldn't sell you a bet that isn't profitable, which is more economics or history and less math.

I think probability is one of the harder parts of stats for non-mathy people so I'd prefer to spend a lot of time looking at histograms and talking about how medical trials use statistics.

This ties into my feeling that high school teacher should show students published scientific papers way way more than they currently do.

[quasi musk]

Whatever reason you provide, there are good counters for why high-school students shouldn't study mathematics. I'm actually ok with creating tracks in late middle school to high school. One for trades, one for office work, and one for academics. Funnily enough, the workers in trades have to do the most algebra, geometry, and trig.

My friend is a general contractor and he has to do so many calculations on floor tilings

Another difficult part is that, most people can live good meaningful lives without knowing mathematics

[native iron]

I'm also broadly fine with tracking.

[quasi musk]

The best reason that I can provide is that "If you don't put in the years for math knowledge now, and later you realize you do want to go into science or engineering, and you don't have the background, then you will never put in the 4-6 years of study necessary just to get to Calculus"

[native iron]

People can live good meaningful lives without learning anything after fifth grade. I think the question is whether there's some universal curriculum that improves most peoples' lives (or improves society or something).

[quasi musk]

So we subjugate most students to a math curriculum that won't be enjoyable, and won't be that useful, for the 10-20% of students it will have an impact for. The issue with tracking is you don't really know who is who, and it can be difficult to jump tracks

[native iron]

What about the more liberal arts argument for math?

"Like English class, math class doesn't teach you directly useful things. However, it introduces you to a way of thinking that conditions your mind in a useful way. Additionally, it might train you to have some aesthetic appreciation for one of the great achievements of human civilization."

I think that's how someone like my dad (high school English teacher) would argue for math education.

[quasi musk]

The problem is that this doesn't provide an answer for why math. This provides an answer for deductive reasoning and logic, and some quantitative skills. But why Algebra 1, Geometry, Algebra 2, and Trig?

Why not Stats & Data Science?

[native iron]

Well I'm a big fan of teaching data science more widely, and I think that's compatible with tracking.

[quasi musk]

Why not logic games? Why not puzzles? Why does it have to be this way

[native iron]

Why read and analyze the Odyssey instead of the Bhagavad Gita? Either one is a fine way to cultivate that state of mind, but we need to choose some substrate for developing that knowledge/style of thinking.

[quasi musk]

At least, I've summarized my issues with this perspective. Although, everyone will partially agree with at least the sentiment. Others will argue that this has no bearing on the real world consequences, who cares about the quadratic formula or learning about the Odyssey. I don't need that to run my business

[native iron]

So, your issue is primarily that even if we concede that math cultivates a useful style of thinking, it's not clearly the best way to cultivate that style of thinking and it imposes an unpleasant, time consuming experience on most students? (Just checking that I understand correctly.)

[quasi musk]

So in essence there's no real way to fully defeat the most disengaged student, the most hypercritical person that doesn't want to do this or learn. My answer says "Ok, you can go that route. We teach this to most people because about 15% of people will need this in their careers, and it takes years to learn. Tracking has its issues, so right now we're stuck with this. We don't know which 15% of people will use this in their careers and lives. For now you can think of it as a broad education, but this is really a cop-out we educators tell ourselves"

[quasi musk]

Yes, that's one argument. It provides why we need quantitative skills. But to that end, a study of finance is vastly more important than the quadratic formula. So while it answers "Why do we need quantitative skills, or skills in arithmetic" it does provide an answer to "Why is math on a track to either Calculus, or more recently, Statistics?"

There's a good ted-talk by a math teacher who became very disillusioned with teaching math, and instead uses games for deductive reasoning skills

[quasi musk]

The other thing to add, ok Math is important to learn. How many years of it?

How many years to cultivate it? What's the answer? (Aside from what topics/curriculum)

[native iron]

I don't think "Math is important to learn" is a good description of any view I've presented, but I might just be reacting to the vagueness.

(It's okay if it's not intended to be.)

[quasi musk]

I'm writing this from the perspective of a skeptic that is somewhat conceding

But not entirely conceding the point. Ok math is important to learn to some extent. Why do we need to teach it every year k-12?

(I know some schools stop at 10th or 11th grade, for requirement purposes). But when your primary argument is something along the lines of "It's part of a broad education of being a human, to engage with humanities greatest ideas" or whatever else, it's very unconvincing. More important than being unconvincing, it doesn't provide answers to "What curriculum, how long should this take, and how long on each topic?"

[long pelican]

One approach I don't see used very often is answering "Why is algebra 2/etc. mandatory" from a historical lens

[quasi musk]

I've learned to accept it just sucks to learn math, it's difficult, and it's only useful for about 10-15% of students once you get to Algebra 1, Geometry, Algebra2/Trig, Pre-Calc, Calc. Draw the cut off based on what they wanna do

Even for students that like math, they're always that "year" or that "teacher" that made me dislike/hate math. And from there, no recovery. Even math majors have professors that make them question whether or not they actually like Math. It's a brutal subject to learn, and has this air of objectivity that seems very daunting.

One of the most eye-opening things in my MS program is when I shared classes with people in the math-ed department that couldn't solve problems beyond Calculus 2

[long pelican]

Based on cursory research and my existing background knowledge: Leaders sold Algebra II and trig as a way to beat foreign competitors in the economy and as a signal of workforce readiness

starting in the 1980s (this is when leaders pushed for Algebra II and trig to be made mandatory for everyone in high school to learn)

[quasi musk]

I also feel like the math is fun trope, or the "3blue1brown fallacy" of "Wow look at the pretty visuals, I get it now! I understand Linear Algebra" without ever row-reducing, drawing out the vectors transforming. I feel that these things are there to help them delude themselves into thinking that math isn't a painful subject to learn

Which isn't necessarily a bad thing, but it (to me, anyways), provides insight as to why students find math such a painful subject to learn, to do, and to use

It's because it just sucks

It sucks be to told you're objectively wrong, it sucks to correct yourself, and it sucks to sit there staring at a problem for any amount of time feeling like a complete idiot because for everyone else it's so easy, but for me it's not

I don't wanna be too harsh here, obviously I have a deep love of mathematics, and I try to make it more fun and engaging when working with students

[tight star]

This seems a little strange to me, I don’t think there’s something fallacious or misleading about what 3b1b does

Math can totally be fun

And sure there are painful parts to it, but there are painful parts to every subject

I don’t think math is significantly more painful in that respect

I’m reminded of Lockhart’s essay

Which I don’t necessarily agree with all of, but I’d certainly agree that while music or sports have painful aspects to them, people still enjoy them greatly

I don’t really buy that this is something intrinsic to math, as opposed to being a function of how math pedagogy works

[tacit spruce]

its not the what 3b1b does is intentionally misleading

but some people could fall into the trap that math is always this experience of like having a revelation about a new topic

without recognizing the hours upon hours of nothing coming to you that come with those revelations

[tight star]

hm, but i would think people have enough personal experience with this?

everyone gets stuck on maths, after all

maybe i'm misinterpreting, but i don't see how people would believe both "doing math means you're often stuck" and "doing math means you're always having revelations"

[tacit spruce]

a lot of people still in middle or high school dont have that personal experience to recognize thats what its like

i remember i didnt

of course the issue doesnt apply to people actively studying advanced math and knowing what its actually like

but the amount of realistic communication about studying math at universities and knowing if its for you is very poor in my opinion

[tight star]

hm i certainly did, i thought getting stuck on maths is a fairly universal experience

[quasi musk]

It's not what 3b1b does. It's about people watching and interpreting it, obtaining a false sense of understanding. Grant Sanderson is incredibly talented, and I don't want to subtract from that. I'm primarily talking about people "understanding Calculus" without solving many, many problems

[tight star]

Sure, but I think there are definitely different kinds/levels of understanding one can have

There’s a difference between conceptual and computational/technical understanding

[quasi musk]

Yeah, I'd say you need to be able to compute, and interpret those computations to get either form of understanding

I once had a first grader come into my classroom and write $\frac{d}{dx} kx = k$. I'd say that first grader had neither a conceptual nor computational understanding of what they wrote

[burnt vessel]

MoonBears-C-

[tight star]

Ok this I don’t understand

I think you can definitely have a conceptual understanding without a computational/technical one

[quasi musk]

So if you ask someone "What is the definition of a derivative" and they say "Slope of the tangent line at that point", and then you ask them to write it down. Let's say a general formula and they can't do it, and they can't do it for a specific function, maybe f(x) = x^3

I'd say that person doesn't know what a derivative is

[tight star]

That’s probably where we differ

[quasi musk]

Much like the first grader that could write $\frac{d}{dx} kx = k$, but there's no understanding of what the symbols mean

[burnt vessel]

MoonBears-C-

[tight star]

If someone understood determinant as “volume scaling factor” but didn’t know the formula for a determinant nor how to get the determinant of a specific matrix, I think there’s still some understanding there

[quasi musk]

Yeah, I'd say they have a very surface level phrase that's parrot'd, rather than understood. For me an understanding needs to be able to connect definitions to concrete examples, to general pictures. You can have a stronger understanding of concrete examples than abstractions, or a stronger understanding of definitions/theorems, and proofs than knowing how to calculate

But if you cannot calculate simple examples, or make simple arguments, then I'd say you have neither a conceptual nor a computational understanding

It's fine to disagree on this issue, ofc

[tight star]

yeah i think we have different notions of understanding

there's certainly lots of concepts in physics i think you can appreciate at a conceptual level without understanding them at a computational/technical level

general relativity as the warping of spacetime, for example

and that is pretty helpful for building intuition, whether or not you actually get to learning GR formally

[quasi musk]

I understand when I turn the key in my car, that it will start. I understand how to drive my car. But I don't understand how cars work on an internal level. I don't understand the design of cars, and there are certainly weather conditions that I don't know how to drive in (icy/snowy conditions, for example).

I can't say I have an understanding of cars

Other than they can move, and I am licensed to drive basic cars

Just as I know how to type on a computer, but I don't know how computers work. Basically no understanding

[tight star]

yeah i just don't understand how "i understand how to drive my car" and "i can't say i have an understanding of cars" are simultaneously true

[quasi musk]

Yeah, in my world, or definition, you'd need both a practical understanding of how driving works, and a technical understanding of how the internal parts of a car impact that driving (or other functions)

[tight star]

But then isn’t practical understanding alone an understanding of cars

It literally has the word “understanding” in it

[quasi musk]

I'd say you have an understanding of how cars move, but not of cars in general. It's like saying "I know how to calculate derivatives, so I understand Calculus" but maybe the student doesn't know limits or integrals

My perspective is this: If the bar is so low to have a valid claim to understand a topic, then everyone has a reasonable claim to make that they have an understanding of everything (at least, in concept).

This isn't the case, so I reserve an understanding of a topic to mean both a conceptual and technical ability in that subject

[tight star]

I’m fine with the bar being low

[quasi maple]

To what extent "understanding" a car is true is vague, essentially

If you drive one of those new Nissan Leafs with that one-pedal BS, then it's literally a matter of "press pedal to go fast, release to go slow"

That doesn't mean you understand e.g. what a clutch is

~~insert comment about Americans and automatic gearboxes ~~

[tight star]

Sure, but do you need to?

[quasi maple]

Well that's the thing

Suppose all you wanted to do was drive Nissan Leafs; then no

[tight star]

I’m happy saying that there’s different levels of understanding

[quasi maple]

But if you wanted to then learn how to drive a manual, you need to learn what a clutch does

[tight star]

What I don’t understand is picking a specific level as the “canonical” one and gatekeeping understanding of a subject behind that

[quasi maple]

(I think my point is more that there're stages to understanding something)

[tight star]

Yes, and I agree

[tall bolt]

This is very much not true, I bet if you asked the majority of the UK population what the clutch does the answer would be “let’s you change gears” which while strictly speaking isn’t untrue, it’s not what a clutch does or how it works

[quasi maple]

Sure, but that's at least some understanding of it

[tawny slate]

this is a bit of a weird question, and is probably more math or math philosophy than pedagogy but i maybe you guys are still best equipped to answer it? idk

are there any use cases in math specifically where we create a label to group or classify math objects, but this label is NOT based on a property or pattern or concept, but kind of just vibes or an arbitrary set of objects?

I don't think so because i feel like then it wouldn't be math, almost by definition, right? like suppose you had a set containing all of the serial numbers of a product line, this fits the question im asking, but this isn't purely math anymore

[austere delta]

Prime knots are labeled by two numbers, the crossing number and a numbering of knots with the same crossing number.

As far as I'm aware the numbering is pretty much arbitrary though for example 5_1 is vaugely more similar to 6_1 then 6_2.

The exceptional Dynkin diagrams are E6, E7, E8, F4 and G2.

The Es look similar, but they could might as well be labeled by three different letters. And I guess this is supported by their extended Dynkin diagrams extending in completely different ways.

There's also no reason for the order (as in why E isn't called F etc)

[tawny slate]

ah it appears i wasn't clear enough in my question, this is my bad

I meant not in terms of the names given to the objects, but rather that we don't just arbitrarily "culturally" decide, for instance, what numbers are or aren't prime. the number itself will determine whether that number is prime or not

for context, this was in regards to a previous discussion, where we were talking about how when people have a definition and an intuitive image, when these things come into conflict, most people will prioritize their intuitive image and reject or rewrite the definition, which is generally not a good idea in math. I'm wondering if there are any such cases in math, where when the definition and the intuitive image conflict, we prioritize the intuitive image

obviously, if we find that changing the definition so that it's more useful, we will do so, such as defining 1 to not be a prime number, but this is a refinement of a definition that will eventually calcify. I'm wondering if there are cases where this definition really is just freely changing and arbitrary based on vibes, we say this number is <label> cause we feel like it

like suppose we defined "purple" numbers to be numbers which are prime. then i took a look at 6 and went "shrug you know what i have decided it's also a purple number". and we just keep doing this whenever we feel like, but not because we keep "refining" the definition of purple, we just do this just because we feel like it for whatever reason

[austere delta]

Well prime numbers is kinda such an example.

If the definition is "number divisible by itself and 1" then 1 is prime, but we decided 1 shouldn't be prime (and then ammended the definition appropriately)

[tawny slate]

yes but once we amended the definition, we found it was better, and kept it, and it pretty much doesn't change anymore. are there any examples where we aggressively redefine a label, or at least much more often?

one that isn't simply an abstraction

[tacit spruce]

I see redefining due to discomfort a lot in philosophy but unsurprisingly it usually doesn't have good results

Not saying i have a definitive answer but my experience with vibes defining is telling me that I doubt math does such a thing

[austere delta]

I mean, there's a lot of disagreement around the definition of ring or algebra for example.

Though it's not like everyone changes their mind periodically in tune

But people (often) do change the definition of ring depending on what kind of math they're doing

[tacit spruce]

Natural numbers is another good example of that

[austere delta]

Indeed

[tacit spruce]

Though usually those definitions change because the fickleness in their definitions isnt majorly problematic

[austere delta]

Yeah, definitions are really about convenience.

If someone has to say "nonunital ring" every time they have to refer to rings they might just change the definition of rings to be nonunital.

[tawny slate]

yeah that part i totally get

[tacit spruce]

You could technically argue that all definitions are based on vibes but I think thats underselling it a lot

Mathematicians with an interest in defining something know what they need that thing to do and they give it the properties it needs for it to do so

Like the set theoretic definition of an ordered pair comes to mind

[tawny slate]

yeah that much is pretty clear, which is why i was mostly just sanity checking this question, because it's a cultural one and not one i can easily verify myself

[tacit spruce]

Yeah i understand

Its an inherently philosophical question so theres no way to know for sure

But every mathematician I know would be frustrated at the lack of a good definition

And imo having good definitions is maths biggest selling point

[feral swan]

forgetful functors is a definition-less concept, but i don't think it's what you're asking about.

You can look up the history of what "compactness" used to mean which might be closer to what you're looking for. It changed a lot, and there's disciplines that mean compact + hausdorff when they say compact.

[abstract grove]

perhaps what qualifies as “geometrical”. Once meant something quite specifically down to earth. Has gradually been softened based on vibes beyond all recognition

[tawny slate]

oooooooh ok that's actually a really good example

dang, nice one

[abstract grove]

it’s not really arbitrary though, I was being slightly tongue in cheek. But now it takes an expert to explain why something like a scheme has anything to do with geometry

for how much freedom you ostensibly get with the study of arbitrary collections of axioms and arbitrary definitions, surprisingly little math that's been taken at all seriously, is arbitrary

Saunders Mac Lane (one of the inventors of category theory) wrote a really interesting book called Mathematics, Form and Function, and there's this great table in it:

[turbid zenith]

Does anyone have experience writing lecture notes from which to teach a course, rather than using a textbook?

[zinc dove]

At this stage, my lecture notes are the slides. I take pieces and bites from various books, including those from the main bibliographic sources provided by the university. (I teach CS btw)

[turbid zenith]

Do you typeset it yourself? Or are you grabbing stuff from PDF's etc?

I'm due to teach our Proofs course soon, and I can't find a book that works with how I'm planning to do it

[rapid tusk]

what little pedagogical material I’ve written I usually make a point of typesetting myself

[zinc dove]

Beamer for the slides

I grab examples from books and create my own

[zinc dove]

You could mix from various books and make your own course, keeping it consistent

[tall bolt]

Reaching out to UK academics if you know any may be a good move, I’ve pretty much exclusively used specially written lecture notes here and afaik that’s fairly standard across the UK

I think I’ve taken 3 courses which followed a book as opposed to specially written notes

I’ve always been a big fan of it, I guess it depends on how good and organised your lecturer is

I’ve had one course where I didn’t enjoy it but I just generally think that lecturer shouldn’t be allowed anywhere near a classroom

[turbid zenith]

This is probably going to be what I’ll do.

We have such a weird idiosyncratic split anyway.

We started a new sequence of two courses, each of which is half a semester

MAT 195 Mathematical Thinking (2 hours)
This course serves as an initial introduction to the mathematical reasoning and communication techniques required for advanced mathematics. Proof techniques include direct proof, proof by cases, proof by induction, and Boolean logic. Topics are drawn from set theory and number theory as needed. Offered every Spring semester. Prerequisite: None.

MAT 196 Proof Techniques (2 hours)
Building on MAT-195, this course serves as further development of the mathematical reasoning and communication techniques required for advanced mathematics. Topics include proof by contradiction, quantifiers and negations, equivalence relations, functions, set theory, and number theory. Offered every Spring semester. Prerequisite: MAT-195 with grade of “C-” or higher.

I helped design the sample syllabus but now that I’m about to teach it in like “man I should’ve put this topic in that course”

[halcyon glade]

Huh you don't teach proof by contradiction or quantifiers until the second semester?

I don't really understand that division

[turbid zenith]

I think the original idea was that the second half of the semester would be more “advanced” proof techniques

Now that I’m planning it that doesn’t make sense

So I’m going to introduce it in the first part and just reinforce it in the second half

The first half was supposed to mostly be about attacking a problem and just a basic introduction to proof without going heavily into specific techniques

…except induction I guess IDK

[tall bolt]

I’m generally not 100% convinced of intro to proof courses in the first place but having a 2 semester sequence of intro to proofs seems like just a flat out bad idea to be honest

Is the 2 hours thing saying that there’s 4 hours of content total? Or are they both 2 hours a week? If it’s the latter I’m not sure how you even fill the time

[halcyon glade]

Actually is there research on student outcomes for students who took an analysis class where intro to proof techniques was integrated into the curriculum vs students who took an intro to proofs class and then a separate analysis class?

[turbid zenith]

It’s a one semester sequence of two half semester courses

Mathematical Thinking is the first half of the semester, Proof Techniques is the second half

Meeting twice per week for 90 min each

[tall bolt]

There’s a few IIRC, this has been a topic of discussion in education for a while, I read a couple for my education class iirc

[tall bolt]

That seems less egregious but still rather odd

[turbid zenith]

It’s supposedly to make it more modular?

[tall bolt]

I mean it does, but for what purpose lol

Who needs direct proofs but not contradiction

[turbid zenith]

I’m gonna make it work fine, the big thing was to make it so that it wasn’t a 200 level class anymore and help students get into major classes earlier rather than having to wait to take everything

[tall bolt]

It just makes discussing any actual content in either half of the course harder because you may need a method you don’t discuss for another few weeks in the “other course”

[turbid zenith]

Once I teach it I can most likely request the catalog be amended

Based on having taught it an seen what’s needed