math-pedagogy

[autumn trellis]

He never pretends that he know the answer.

His lectures starts and ends with problems. Usually Teacher/Professors tend to start with theory.

[spark flare]

Hmm

I'll look at it when I'm back home

[turbid zenith]

Huh. I have AoPS vol 1 and 2, but I've never seen his videos.

[autumn trellis]

You can see them by clicking the above link.

[turbid zenith]

Yes, I figured that out.

I was just commenting that I was unaware that he had done videos.

[tawdry venture]

i'd like to share something with y'all that i just came up with

it is a prototype for classifying the various ways a proof can go wrong, which may perhaps be useful for grading purposes and whatnot

i do suspect however that it may be incomplete

basically this is a system of codes, in the range [0, 6] at present, with lower codes corresponding to less severe faults

i will now type out what the present meanings of the codes are

0: Impeccable. No comments to be made about wording or argumentation. This code is present for completeness.

1: Minor wording slip-up. This may be an innocent typo or a turn of phrase that conveys the intended meaning but is clunky.

2: Major wording slip-up. This may be a misuse of terminology, a confusing turn of phrase, or (in algebraic manipulations, if present) a mistake which irrevocably ruins whatever follows it (i.e. cannot be fixed, for example, by retroactively changing + to - in relevant places).

3: Minor argument hole. This is for argument holes which do not affect the conclusion and are readily patchable by inserting the relevant paragraph(s) at the appropriate place in the proof. This includes incomplete case analyses, errors in case delineation, and missing base cases in inductive proofs.

4: Major argument hole. This is for holes which affect the argument to a sufficient extent that a restructuring needs to take place (i.e. not patchable with the reinsertion of a missing piece). Examples include: arguments which apply to situations where they shouldn't, arguments which use extra assumptions without justification, arguments that do not lead to the appropriate conclusion.

5: Definitional error. Any proof making use of an incorrectly stated definition falls under this, except when it may be classified as code 1 instead.

6: Incomprehensible. Self-explanatory. If the proof is illegible or written in a way that makes it impossible to understand, this is the code it receives.

thoughts?

[wicked minnow]

I approve of zero-indexing

[tawdry venture]

ok but seriously

any criticisms or addenda?

[tawdry venture]

btw, it is my intention that codes 0 and 1 should signify proofs acceptable in an exam setting

[brazen pendant]

3 seems less bad than 2

Also, what about the good old "what you do is correct, but your justification is not rigorous enough for our grading standards"?

Also, while the scale itself seems good, I dont see the practical applications - when grading someone's homework, for example, you'd want to be much more specific than just giving a number

[tawdry venture]

well, presumably one would point out the point(s) at which the mistake is made

Also, what about the good old "what you do is correct, but your justification is not rigorous enough for our grading standards"?
i'd say that falls under 3

[turbid zenith]

Ooooh lemme read this

❤ rubrics

Okay so ... question.

Why numerical codes? Are these ordered?

[dusky scarab]

yeah shouldn't really be ordered imo

[tawdry venture]

intended to be ordered by severity

[brazen pendant]

well, presumably one would point out the point(s) at which the mistake is made

I feel like then it might be more instructive to make some acronym for it, like...

mw - minor wording (upper and lowercase w alone would be hard to distinguish)
MW - major wording
ma - minor argument
MA - major argument
D - definitional
X - incomprehensible (I is too nondistinct)

and a tick if it's good

[tawdry venture]

you could do "OK" for good then

[sharp kelp]

Hey guys, would it be possible for help? I’m unsure how to do this one, sorry :/

<@&286206848099549185>

[exotic frost]

Wrong channel

[scarlet perch]

wrong channel and you're supposed to wait 15minutes before ping

[wicked minnow]

[wispy slate]

What's 2×1? <@&286206848099549185>

[calm fjord]

this is the wrong channel, and are you serious about that question?

[wispy slate]

Yes

Why

Did I ask for help in the first place

[exotic frost]

No trolling

[hexed perch]

[empty badger]

hello

I am in need of math education

[limpid gust]

henlo @empty badger i believe you might be in the wrong channel

[empty badger]

tahnks

[turbid zenith]

Anyone in here a college-level instructor or professor?

I'm looking at creating a survey to send to college professors in my state -- I'm with my state's council of math teachers and we want to support post-secondary educators better. Could use someone to pick their brain a bit to help figure out what to ask.

[lament wraith]

I thought your website would be a crank website, but it's actually really informative and well written

Anyways I've TA'd for a couple semesters at a big state university so maybe I could help

[turbid zenith]

LOL @lament wraith Why did you think it would be a crank site 😛

Also is it okay if I PM you the stuff?

[lament wraith]

Lots of cranks feel like they've discovered something new about dividing by zero, saying it's infinity in some way or something

Yeah sounds good

[spark flare]

<@&268886789983436800> maybe the channel should be renamed "Education" to avoid confusion

[inland radish]

you'd have to ask woog to do that

mods can't rename channels

[spark flare]

People see math they post here

Oh

@weary ferry

[sterile rover]

there's a major renaming proposal in progress

see ivory pins

[brazen pendant]

well I’m overwhelmed with my next class

the material they’ve covered and need for next homework sheet is difficult stuff I don’t really understand myself

and on the ohter hand, not really something that’s a “main takeaway” from the class

[boreal sentinel]

What class

[brazen pendant]

numerical methods

broyden’s method (variant of newton’s method for root finding), rosenbrock-wanner methods (variant of runge-kutta methods for ODE) and then QR decomp, that last bit is the only thing I actually understand of these

the others are just algorithsm that are painful to implement and even harder to understand

idk what I can say about them

[gleaming glen]

Any tips for drawing math

Handwriting math

On paper especially

It looks like there is a general assumption that every mathematician went through college lectures and therefore knows all handwriting

I found isolated resources

But they are way too isolated and incomplete

[gleaming glen]

Pls

[scarlet perch]

do you mean like. for notes?

or for homework and the like

[gleaming glen]

When I'm doing math on my own

On a paper book

I want it to look pretty

But also sometimes I take pictures of it and publish it

Because I'm lazy to do latex

[scarlet perch]

ah, i was gonna suggest just using latex lmao

[gleaming glen]

I publish a lot of things

But it's github

Latex is unnecessary and slow

Because I keep updating it

[normal aurora]

latex is fast af

[gleaming glen]

And because I'm the only person who reads it already

...

[scarlet perch]

do you mean slow in the time it takes to write it?

[gleaming glen]

If there's no centralised resource in how to handwrite math I will make one

I'm just lazy

Tbh

Because I keep updating iy

It's not a paper it's a GitHub

So it's continuously updated

[scarlet perch]

i supposed you could look up some handwriting recognition stuff?

[gleaming glen]

Now that I'm doing machine learning

The equations are huge

I'm so lazy to keep updating that

I prefer to just erase and draw using pencil

I get it it's not the normal thing to do

I'm lazy and want to be fast

...

How to handwrite matg

There's a lot of hidden random tricks

Like drawing a complete bipartite graph with squiggly lines

Or the 4 new symbols used in reinforcement learning

[boreal sentinel]

Tbh people's handwriting are all generally pretty bad

Including mine

I would like to have some proper time spent on improving the way I write, especially mathematical symbols

[frail cypress]

Ikr, I defended a math report recently and the censor asked what symbol i had drawn on the blackboard (it was little alfa)

[sterile valley]

I have a professor who will just write smiley faces instead of your typical variables

[boreal sentinel]

I've heard of a math professor somewhere who drew animals to label equations and would refer to them as such (story courtesy to @junior roost)

Sorry for the ping but I remembered lel

[junior roost]

"as you can see, little fish definitely converges [...]"

he also used to put derivatives "into little houses"

[boreal sentinel]

Tf

That's a new one

[junior roost]

he also talked about his brother goats from time to time

[boreal sentinel]

This is #math-pedagogy at its finest for sure

[junior roost]

that was when he decided to label some function as goat
then he kept differentiating and pulled out a formula that converges to the number of possible partitions of a set

(Bell's numbers)

[boreal sentinel]

Wasn't it also the guy who smoked weed before lectures

[junior roost]

yeah

[boreal sentinel]

Explains a lot

[karmic socket]

Woow

Nice

[wispy slate]

lmao

[turbid zenith]

@sterile rover Where would I find said "ivory pins"

[sterile rover]

you wouldn't

[turbid zenith]

Well that's unfortunate

So what's the renaming plan?

[brazen pendant]

not a fixed plan yet but some vague ideas are subdividing the topics into three groups (high school, early undergrad, advanced) rather than two. there’s disagreements over how the question channels should be handled and what the scope of the server is

[turbid zenith]

That's a good idea honestly.

It would make it a lot easier for people to know where they should go.

[brazen pendant]

the main reason why nothing’s happening is becasue there’s not really an agreement on the scope issue (e.g. how much do we think texbook questions are an acceptable thing? personally I think they’re fine as long as they don’t spill over into the advanced sections)

(and to that end I personally would prefer the questions channels to be at the top where they’re easily seen)

[tawdry venture]

this isn't a math help channel

go to #❓how-to-get-help

[pale steppe]

ok ty

[turbid zenith]

I guess that was something I didn't think of when I suggested a #math-pedagogy channel

The idea that people would come in and think "this is where I should ask for help"

[brazen pendant]

yep, that’s one thing I wanna get changed

I’d personally rename it to sth like TA-discussion or teachers-discussion or maybe teachers-lounge and put it into the most advanced section

[shell wharf]

Umm, i need help with my homework, i just need to understand how to do two-step equations:integer that has fractions in it

[proper sky]

This isn’t a Math help channel see #❓how-to-get-help or #prealg-and-algebra in your case @shell wharf

[shell wharf]

okay

[lament wraith]

This isn’t a Math help channel see #❓how-to-get-help

[fluid isle]

Wtf how is this happening to me guys

Ok

But why is it like that

I will transfer but I need a ans

[wicked minnow]

*lounging intensifies*

🚬

[sharp raft]

🚬

need a vaping emoji

[scarlet perch]

hm. im thinking about doin some tutoring on the side again, but ive always had trouble going through fundamental stuff when teaching

like. im not certain how basic id have to go to teach sometimes

cause ive had v intelligent students get stumped by stuff relatively simple

to their level

it's also bothersome bc i tend to teach people forced to learn by parents and sometimes don't care at all kek

[wispy slate]

are you a teacher by trade

[scarlet perch]

no, ive been a fairly consistent/long term tutor on a variety of subject tho

[wispy slate]

oh

[civic tree]

i like tutoring but lecturing w/ discussion sounds more fun to me

[scarlet perch]

i used to do some courses like that!

it was neat to review the concepts after presenting

v fun experience

shud i be a college prep shill?

[civic tree]

wowow

i wana do course like that

[wispy slate]

i feel very lost in lectures actually, i'm in an American high school student. It's hard to understand anything beyond memorization because that would be beyond the scope of the course

are you in college by any chance?

[scarlet perch]

yeah

do you mean hs lectures?

[wispy slate]

yes

[scarlet perch]

ah, those lack discussions which can help a good amount if the subject is nontrivial to you

and if the TA cares

[steady forum]

are people here actually teachers?

[meager bronze]

I teach at a university

[lament wraith]

I TA at a university

[pliant charm]

yes

[lyric osprey]

I'm a high schooler tutoring community college students

[tawdry venture]

@wicked minnow you aren't becoming IBT are you

[wicked minnow]

[unreal ledge]

@scarlet perch
How do you handle the business side of tutoring? Like, how do you find clients?

[scarlet perch]

uhhhh there's a couple o ways, sometimes i sign up to teach at a local college prep place

also theres uhhhhh librarys and the like

toss up an add

[strange bronze]

"like. im not certain how basic id have to go to teach sometimes"

my rule of thumb is:

however basic you think you should be

go 2 layers of basic deeper

worst case, you tell them something that they already knew, which gives them a confidence boost

like "ah, this is just rules i already know applied in a different way" or whatever

in my experience, most students are pretty bad at forming connections themselves, but once those connections are pointed out to them, they find it really easy to latch on to the concept

idk what it is - probably lack of confidence?

mind, i mostly tutor/TA high school/early calculus/LA students, so ymmv

theres a lot less mathematical maturity in that demographic than, say, advanced undergrad - and a lot less care

[scarlet perch]

ahh, i plan on tutoring from like

elementary to hs level peolly

*prolly

[wispy slate]

idk

George Polya said the same thing in his book "How to Solve it"

the teacher should aid, not give answers to his students

[wispy slate]

I will be a TA next Fall (Fall 2020)

[civic tree]

i would like to ta or something like that

i enjoy the process of lecturing w/discussion cos its like a presentation but its like involving the students with the process

i feel as if im inspired a lot by prof leonard method of explaining

but i need practice

and should spend more time preparing these

the most i do is make a quick writeup of the things i wanna do on a page and a back

how much time do yall prep for ur teaching stuffs?

[brazen pendant]

the largest factor for me is how much I have to research myself

research can take up to five hours (just relearning things I should know from when I took the class but don’t quite remember/didn’t really understand back then). if I wanna present specific examples or give exercises, those take me about an hour to two to prepare. and then preparing a general outline is another 1-2 hours

[lament wraith]

When I ta'd last, I basically ran problem sessions and those really didn't need much preparation. I would just work through the problems once by myself

[civic tree]

ah

[lament wraith]

I do wish I had more flexibility to run things though. I definitely would've done things like you described

[turbid zenith]

Hey this is a good way to name the channel

[hexed perch]

I wanna start tutoring math as a high school student to other highschoolers and stuff yeah

you guys think its a reasonable idea?

and if so how much do u think i should charge

[brazen pendant]

My high school encouraged that by having a sort of internal job board. If you're good at a subject, it's reasonable to do one on one tutoring to people a few years below you, I've done it before.

as for pricing, that would depend on where you live. Since you're likely only gonna be working 1-2h/week directly (plus prep work), you can definitely go a bit above a "low" wage. the parents of the student probably have an idea for reasonable wages, as might yours, or your school administration

[scarlet perch]

i think it varies depending on what you plan on tutoring and what degree of availability and the like you offer

if you're teaching something in greater depth than schools offer, you can charge a fairly high price

check with your school though

sometimes they have a pre-existing network like sascha mentioned

[brazen pendant]

oh shit I was just offered an assistance post for algorithms and complexity

that’s unexpected

I will definitely accept that, that was my favourite class last semester and I did not expect there to be an open position

[meager elk]

nice

[inland kayak]

Awesome

[brazen pendant]

the only drawback is that it’s supposedly only 9h/week instead of the current 15

so, less money

but also less work

(way less in fact cause for algorithms I had a much more thorough understanding than for numerics)

[meager elk]

Is this a job in your university? What are you supposed to do there?

[brazen pendant]

teaching assistants at my uni do primarily two things: give tutorials (usually 2h/week, but this one’s only 1h/week) and correct homework

tutorials being classes for groups of about 10-20 students that supplement the lecture

they include homework discussion, working through examples together or reviews of difficult topics

pretty much the only requirement for becoming a teaching assistant is good grades in that subject

[meager elk]

where I study, afaik you can get a job as a grade, not TA, so just grading homework, unfortunately but glad u got it, wish I could do that as well

[brazen pendant]

grading homework sucks

[lament wraith]

There are a couple online grading systems now though that make the process not as bad imo

[brazen pendant]

only if your school uses them

my school decidedly does not

homework is handwritten except for programming parts

(ofc sending a pdf is acceptable too)

[lament wraith]

Ours are handwritten too. We scan them into the online grading system and grade there

It's nice that I can like copy common comments over and don't have to pass papers back and forth

[brazen pendant]

what class do you grade for?

[lament wraith]

Multivariable calc

[brazen pendant]

so I assume your homework is mostly calculations based?

[lament wraith]

Grading for lower level math is easier in that you can just check the answers too

Yeah

And only look at the work if their answer is wrong

[brazen pendant]

even if the correct answer is 0? :P

there’s many ways to get a 0 the wrong way after all

but yea numerical methods is primarily code so that’s mostly the same, if the plots look alright then I won’t look too closely

algorithms is proof-based though

slash, well, finding and describing algorithms

and proving things about stuff

[lament wraith]

Yeah definitely and that makes grading way more painful

[meager elk]

I have this grader this year, linear algebra, the way she grades is so annoying - cutting points cuz I row reduced a matrix using online calculator, lol. Or like, cutting points because I wrote too much/not enough (cuz sth didn't seem obvious for her, even though we had that certain thing proved during classes). Don't be like her, lol

[brazen pendant]

cutting corners is fine… but you have to justify it

if you take a shortcut cause you proved you can do it, you should at least write a comment on why it’s correct, even if it’s just “as shown in class”

otherwise the TA will have to assume you just did something weird there

or cheated or what not

[meager elk]

true

[turbid zenith]

To be fair, there also should be a clear rubric from the professor so you know what constitutes an acceptable answer.

[turbid zenith]

Also something interesting ... an AP Calc group I'm in on Facebook was talking about the possibility of teaching derivatives before limits in a calculus class.

What do y'all think of that?

[lost raptor]

I think it's certainly interesting, ||though I prefer limits first||

[strange bronze]

@hexed perch just for comparison's sake

as a high schooler in an upper-middle-class suburb

i charged $25 an hour

which was about $13 above minimum wage at that time

this is in canadian so a bit above american dollars (take off 25% or so)

i probably could've charged more, though, honestly

as an undergrad, i jacked up the prices a fair bit

since having "math honours major at the best school in the province" sounds pretty good to parents (even if its practically kinda meaningless)

and, yknow, i had a few years of experience at that point

but yes, id strongly recommend trying out tutoring if you have the time + skill

[pliant swan]

I think teaching derivatives without teaching limits is the wrong approach. Are you going to use infinitesimals instead?

[strange bronze]

its a great source of money - one of the highest-paying jobs a high schooler can get

and looks amazing on a uni/scholarship app

pretty much everyone who won a math scholarship at my UG tutored in high school, so take that as you may.

[turbid zenith]

KD, I'm not talking about how I'd do it personally, because I know I'd use infinitesimals if I got to teach it at the college level 😛 But we were talking about at the high school level I believe

As it is, most books structure it as all-of-limits then all-of-differentiation and then oh hey btw L'Hopital's Rule is a thing.

But a textbook author I'm a fan of actually starts with differential equations (he just calls them "rate equations") and uses them to motivate needing to talk about rates of change more precisely, then after some numerical computation introduces limits as a tool to be able to find derivatives

[pliant swan]

Do you plan to use derivatives and a context to justify limits? Since I think limits are more interesting than derivatives, I think it's better to start with them.

[turbid zenith]

More interesting how?

[pliant swan]

Because we think about limits a lot, and formalizing them allows us to do that with confidence.

For example, there are many times we might think "what if we did it forever", and limits basically allow you to address that question properly

It also let's us understand what indeterminate forms are.

I suppose what I really meant by more interesting when I typed it, though, is that if you understand limits, it's not that hard to understand derivatives/integrals. On the other hand, getting from algebra to limits seems like a much more difficult step. Limits are a very intuitive concept but formally somewhat challenging. I think the opposite is true of derivatives and integrals.

[turbid zenith]

I see. I suppose I've found quite the opposite from having taught it.

A lot of my students struggle with limits but then find derivatives very intuitive. Integrals seem to give people trouble no matter what.

As for "thinking about limits a lot", I think limits of sequences are something we think about pretty often, but limits of functions in general give students a lot of trouble. They get so lost in the formalism -- even without the epsilon-delta gobbledygook -- that they fail to grasp the intuition.

But derivatives are rooted in things students can intuitively understand. Number of miles per hour you're traveling. Number of gallons per minute when a tank gets filled. Number of people born per year.

[brazen pendant]

@turbid zenith our solution sheets are often really bad as the main assistants can be somewhat lazy in writing them. not rare they're even just wrong. so correcting for me boils down to "do it yourself and then compare with the students"

(the professor isn't even involved in homework/tutorials)

[turbid zenith]

@brazen pendant How are students graded/given feedback?

[brazen pendant]

not graded
red pen on paper?

[turbid zenith]

So there aren't points assigned for homework?

[brazen pendant]

no

[turbid zenith]

Awesome. That's how it should be. 😃

[brazen pendant]

homework is optional

[turbid zenith]

But graders do give feedback on how students did?

[brazen pendant]

yes

[turbid zenith]

Even better.

That's formative assessment done right, right there.

I do imagine it takes a while though.

[brazen pendant]

yes

I get paid 15h/week and only teach 2h/week

[turbid zenith]

I'm working on a project that will make giving feedback easier for teachers/TA's/etc. I might contact you for input.

If you're interested.

[brazen pendant]

sure, I'll look at it

foe now I gotta leave bed and go shower or I'll be late for class tho :P

[turbid zenith]

Okay, have fun

[hexed perch]

i just read all ur responses

thanks for the tips

[civic tree]

this channel is really nice 👍

[turbid zenith]

I really do like it. 😄

The entire server is a very welcome change from the IRC channel I came from.

[small musk]

I enjoy teaching students when they behave, not chew bubbly gum

[turbid zenith]

hah

Chewing gum was the least of my worries when I was teaching 😛

[scarlet perch]

chewing gum is fine, considering it can actually help with memory. The issue arises when they start putting it wherever they wish

[iron halo]

Any teachers here willing to help a student making his small english paragraph look better? idk where else to ask and dont know about other good discord help servers

this is the paragraph

[scarlet perch]

this is not relevant to the discussion meant for this channel

[iron halo]

oh

where can i ask

[scarlet perch]

asking for english help in a mathematics server most likely is a wrong start

[iron halo]

true

[pliant charm]

this channel needs to be nuked @wooden agate

[turbid zenith]

No it doesn't, @pliant charm . You don't even post in this channel.

[pliant charm]

I would if it was on topic

probably

[turbid zenith]

I see plenty of on topic discussion above.

In fact, maybe you have something to say on what I was about to ask. Has anyone here taught graph theory?

[pliant charm]

why

[turbid zenith]

I'm going to be teaching it this summer at a program for gifted/motivated high school students, and I'm looking for ideas for activities and other ways of teaching that don't involve just me standing at the board and lecturing.

[lament wraith]

You could build the seven bridge of Konisberg in someway if you have space for it

Then ask the kids if they can cross every bridge

[pliant charm]

oh a lot of people have ideas on that

google "graph theory for high school"

you immediately come across high quality material

[turbid zenith]

I actually imagine a number of the students will have seen the Konigsberg problem, so I'm going to start off with something different -- sort of a "draw this picture without picking up your pencil" challenge

[vestal quiver]

You could make them code some interesting algorithm in whatever programming language they're used to

[lament wraith]

Yeah that's actually a fair assumption if they're honor students

[turbid zenith]

@vestal quiver , actually I'm adapting some of this from the Algorithms course I taught last summer! This year I'm not going to assume coding knowledge, but I'll probably encourage any students who want to implement some of these algorithms to do so

The way I was planning on doing the algorithms this year was to present a problem to the students and let them hack at --- say, finding a minimum spanning tree --- and seeing what they come up with before going into the standard algorithms like Prim and Kruskal

Very much a "here play with this in a group and see what you come up with" approach

I really like the idea of them creating something though

[lament wraith]

There are also a couple of adversarial graph theory games that are actively being researched

[turbid zenith]

Funny you should say that. I'm teaching three different courses, and my second is "Let's Play a Game", about game theory. 😅

[lament wraith]

One of the cool ones is adversarial graph coloring where you basically have two people and the first is trying to color the graph in as little colors as possible and the second one is trying to force him to use colors and they alternate turns coloring a vertex

[turbid zenith]

Ooooo. That's really cool.

[lament wraith]

So if you bring up graph coloring, where you can also bring up the four color theorem and stuff, you can bring this up too as a game

[turbid zenith]

Yeah that's amazing, I love it.

Do you have any links about that?

Here are the topics I plan on looking at so far:

• Eulerian/Hamiltonian paths/circuits
• Adjacency matrices
• Markov chains
• Google's PageRank algorithm
• Tree searching (BFS, DFS)
• Minimum spanning trees and Prim's/Kruskal's algorithm
• Shortest paths and Dijkstra's algorithm
• Directed acyclic graphs
• Flow networks maybe?
• Graph coloring, 5-color and 4-color theorems
• Ramsey theory and Graham's number
• P vs NP problem maybe?

It's a 4 week program and we'll likely have about 14 classes, 45-50 min each

[brazen pendant]

that seems like a lot for 14h unless you wanna only somewhat briefly cover every topic

also the order’s a bit all over the place but I assume that’s just cause it’s an unordered list?

[turbid zenith]

Yeah its unordered

And probably longer than what we'll actually cover

[brazen pendant]

is this going to be pure lecture time or also workshops n stuff?

[turbid zenith]

But these students often surprise me with how fast they go

[brazen pendant]

in the 14h

[turbid zenith]

So I'm obviously trying to over plan!

I definitely do not want it to be just pure lecture a lot of it I imagine is going to be them tackling a problem to see what happens with it, and then I can briefly explain the algorithms involved and then find a way to go further with them. I also plan on giving them something that they can work on on their own afterward.

[brazen pendant]

I wonder if perhaps instead of trying to cover such a wide base, what if you picked something like 3-5 topics to go more in-depth about

[turbid zenith]

So some of the topics may be extensions they can choose to tackle if they want.

[brazen pendant]

seing as from what I gather it should be more of an expository thing than a thorough course

[turbid zenith]

Also, something I'm going to try this year is to have each group tackle a different problem in depth, and then present their findings to the other groups.

[brazen pendant]

could even let them vote on the last topic in the beginning or sth^^

oh, or that

of course presentations take a ton of time, but I assume they’ll have time to prepare outside the classes, right?

[turbid zenith]

Like for example, maybe one of the probably 3 or 4 groups tackles the question of minimum spanning trees, another 1 looks at shortest paths, and so on

[brazen pendant]

let someone find an algorithm for second shortest paths

[turbid zenith]

Yes I would wanna make sure they have ample time

[brazen pendant]

the second shortest paths problem was a fun challenge problem but ofc way too hard (took me a few hours to solve)

[turbid zenith]

One thing I'm also not sure of because my own background in graph theory has been very algorithms based is, are there certain fundamentals I need to be teaching them?

I mean most of what I can think of in terms of fundamentals seems to just be vocabulary. You need to have the language to describe what you're looking at, but after that point you can really start right in playing with these questions.

[brazen pendant]

(the statement was sth like, you have a graph and two distinguished nodes s and t, and for every node on the graph there exists a shortest s-t path between them going through that node. find an algorithm in O(|V||E|) which finds a second shortest s-t path)

(it was fun and won me a book so yay)

(but it’s quite tricky)

[turbid zenith]

Awesome!

I suppose I tend to like the idea of breadth over depth because I want to expose these students to as many interesting problems and concept as possible, and give them the tools to go further in depth where they want

So many of these kids, all they ever see in math is algebra and geometry and calculus

But I want them to play with it, rather than just sit and take notes.

[brazen pendant]

and I feel like going into breadth will make tha tharder. but if you offer a breadth of topics for a group project and let them choose what sounds most interesting they’ll at least get to choose something fun

[turbid zenith]

I see what you mean yeah

That's why I came here to talk about it XD

it's a weird balance.

And I'm still trying to figure it out.

[brazen pendant]

but yea you’ll have to spend a bit of time explaining basic graph concepts I suppose

and the associated notations and vocabulary

luckily it’s all pretty easy stuff

[turbid zenith]

So in a typical graph theory course, what other things are gone over? I somehow doubt it's all applications.

[brazen pendant]

I haven’t had one

my experience in graphs is from an algorithms class only :P

[turbid zenith]

Oh. XD

[brazen pendant]

which featured them pretty heavily, but still

[lament wraith]

Vaguely what you listed is what I went over in my discrete math course

[turbid zenith]

Gotcha.

As for order ... I actually don't know what a good order would be. <_<;

[scarlet perch]

are you assuming they've had prior experience with graph theory and graphs as a whole to a degree?

it sounds like it from what you've said

[turbid zenith]

@scarlet perch No -- I'm assuming we're starting from scratch

But these students usually pick things up quite quickly.

[scarlet perch]

i mean i'm certain you can explain graphs in something like. 5-10 minutes and it'll only become more clear as the course progresses

[turbid zenith]

Yeah, explaining what they are should be really easy 😛

[scarlet perch]

i'd say something like. adjacency matrices, DAG, eulerian/hamiltonian paths/circuits, then BFS/DFS

i feel like these are relatively related topics that would definitely help explaining the rest

[turbid zenith]

I hadn't think of putting DAG's there, hmm.

I had been planning on putting DAG's (and PERT) closer to Dijkstra, Prim/Kruskal, and Ford-Fulkerson, as a part where each of four groups investigates a different thing you can do with weighted graphs and then presents

[scarlet perch]

ahhh, well that does make lots of sense, i think when you get into weighted graphs it would be a good counterpoint as to the limits of things like BFS/DFS

[turbid zenith]

Limits how?

[scarlet perch]

BFS shortest path

[turbid zenith]

Ahhh I think I see what you mean 😛

The rest makes sense though

Would it make sense for coloring and then Ramsey stuff to come at the end?

[severe cedar]

This is maybe a controversial topic, but do you have any suggestions for "fixing" office hours in the context of a huge class (250-300+)

[brazen pendant]

is this a class where you can expect that students will actually make use of them? at my university, I don’t think I’ve ever heard of anyone actually using them

[severe cedar]

Right now we have a staff of 35+ TAs and OH are going on like 6 hours/day and basically students line up (queue gets massive before homework is due) and do their best to get answers from TAs

[brazen pendant]

okay that sounds like “yes”

[severe cedar]

It's a very hard class and I'm not saying students shouldn't get help on the homework, but it just feels "broken" in some way

[brazen pendant]

mandatory homework certainly causes some… effects

for the better or worse

[severe cedar]

That said I guess students never really go to office hours just for pedagogical reasons as you noted

you have intro classes with non-mandatory homeworks? 😮

[brazen pendant]

they got rid of mandatory homework at my uni a few years ago

across the board

decided it was too much handholding and it’s the student’s problem for not doing it

before that you needed to have handed in 50% of your assignments to get approved for the exam

[severe cedar]

wow

could be wrong, but I'd guess this is in the UK maybe?

[brazen pendant]

you are wrong, but if the guess was between US and UK then you’re closer

^^

it’s ETH Zürich

[severe cedar]

ah ok, I mostly knew this would never fly in US

[brazen pendant]

ETH has a very… non-handholdy attitude

it’s the students‘ problem if they don’t pass, you know?

the only mandatory thing is the final exam

everything else is either purely for the learning experience, or at best a minor grade bonus, that, most of the time, won’t even matter

but going back to your issue… I have no idea, sorry

sounds like a big hassle

[severe cedar]

hmm , wow... at my uni students stretch themselves notoriously thin and I can't imagine how responsible students would be without mandatory homework given the culture is currently the opposite

[brazen pendant]

you could refuse to give homework help altogether, but then I guarantee you, plagiarism will spike up

[severe cedar]

but yeah the current system is OK and we have made reactive fixes, but something feels off about the entire way it works

[brazen pendant]

I can't imagine how responsible students would be without mandatory homework
on average, not very. pass rate in the first year is about 60% per semester, meaning a sold 40% make it through year one at all

after that it levels off quickly though. most students who make it to semester three make it to the end

[severe cedar]

I had been wondering how TAs might be able to help many students at once which could make the help less specific to a specific student's homework, but that might require changes to the collaboration policy (in theory it's pretty strict right now)

[brazen pendant]

…except in my year where there were two profs in third semester who decided to go over-board with material and so we had an unprecedented fail rate for that semester

[severe cedar]

And yeah that is sort of insane

[brazen pendant]

it’s part of the “we let almost everyone in but only those who are actually capable graduate” policy

[severe cedar]

Here it's usually hard to fail as long as you mostly keep up, but people can get decently low grades

The intro courses are honestly much "harder" in many ways than the upper level courses

Much more work and usually larger learning curve for students

[brazen pendant]

what do your TAs do, generally?

only grading, or do they hold proper classes too?

a lot of our TAs basically just give tips for upcoming homework and do some example on the blackboard or with the class

[severe cedar]

We have a bunch of small sections in addition to the large lecture (think 15-20 each), each led by a TA (or a pair sometimes)

which meet once a week

But it's mostly review and worked examples, not new material

[brazen pendant]

and those sections are held like classes or more like “you can come here to solve exercises together and a TA will be present for questions”?

[severe cedar]

Like classes, they're "mandatory" and we have pretty structured plans

So each TA does more or less the same 3 problems in a week

(full disclosure: I'm not the prof, I'm one of the TAs "in charge" who's been here for a while)

[brazen pendant]

ugh mandatory attendance?

my soul bleeds from thinking about that

[severe cedar]

no mandatory lecture attendance, but these are

although "mandatory" doesn't mean that much, it's worth a small grade percentage

[brazen pendant]

man my tutorials would be so much worse if people were forced to be there. the few who come here voluntarily are already passive enough

[severe cedar]

It can be hit or miss - the CS dept at my school has this system for most of the big intro classes because it's very easy for students to fall behind

And in the US they do try to keep everyone above a baseline

[brazen pendant]

yea as said, my school doesn’t care

you’re offered plenty of help to stay caught up, but in the end it’s you who has to put in the work

[severe cedar]

We have 2 profs who rotate fall/spring, and one of them is a much better lecture and makes the course very fast/rigorous

So usually the students who take the course in the fall are much more motivated

And I've had great experiences with holding the small section with them and participation is actually pretty good

Sometimes I cold call to make sure it's not the same couple students answering everything

Our fall prof is beloved because he somehow manages to learn all 250+ students' names... to cold call them in the lecture

[brazen pendant]

I assume he has access to some database :P

[severe cedar]

He does, but he mostly uses it to learn a dozen or so students names so he can cold call them by name on the first day to be intimidating

Then he also cold "points" at random students and asks them their name and then for the answer

And then he knows it for the rest of the semester

[brazen pendant]

I was semester spokesperson in analysis II
I don’t think my prof remembers who I am

some of the other profs do remember me though since I’m always a rather active person in class

[severe cedar]

haha one of my friends was the head TA for a prof in an intro course (last semester), and then took an upper level course with him (this semester!) and the prof did not know who he was

[brazen pendant]

my linalg prof once called me to her in the break… to solve a rubik’s cube for her because she saw me solve one before class

and I’ve since had some chats with her when I met her in the corridor

[turbid zenith]

Anyone up for taking a look at a lesson plan about Lebesgue measure for that high school summer program?

[brazen pendant]

oof that’s a topic alright

sure I’ll take a look

[turbid zenith]

For a bit more context, on Day 1 after introductions, I'm going to have the students debate "which is larger, [0,1] or [0,2]", and the answer of "it depends what you mean by 'larger'" will springboard into us talking first about cardinality and then measure

[brazen pendant]

I suppose that is a reasonable plan actually. are you gonna go into how the lebesgue measure respects countable unions but not uncountable ones?

(sigma-additivity)

[turbid zenith]

I didn't think of that actually.

I had only considered finite unions. :V

Well, explicitly

Though I guess for the Vitali set thing you are using an uncountable union

[brazen pendant]

you could showcase it by first chopping up [0,1] into e.g. (1/n, 1/(n-1)] (and {0}), if they’re comfortable with series

to show with an example at least that even infinite sums can work out

but then let them consider real quick if uncountable sums could work too

the conclusion being: no, because μ({x}) = 0, but [0,1] is the union of all {x} in it

[turbid zenith]

I can't take credit for the colors thing btw -- that's from Dave Kung

Oh yeah that is an easy example

[brazen pendant]

again, all depends on how comfortable they are with things

I don’t like your background color :P too gradienty

[turbid zenith]

😛 I was trying to still keep it pretty minimal

But yeah I've been meaning to revisit that

[brazen pendant]

hang on, so, I’m not sure if I remember the construction of the vitali set perfectly, but you’d have to add an uncountable number of colors, right?

[turbid zenith]

Yeah

It mentions that.

[brazen pendant]

then, this is an uncountable union an dan uncountable sum

so that’s throwing rigor way out

(to the point of being wrong)

[turbid zenith]

Right, which is why I didn't put something like $\bigcup_{i=1}^\infty$

[burnt vessel]

DMAshura:

[turbid zenith]

But left it as $\bigcup^\infty$

[burnt vessel]

DMAshura:

[brazen pendant]

yea, but that equality in the second line doesn’t hold

because μ is only additive on countable unions

[turbid zenith]

Then that means the original proof I saw of this was wrong

So I guess I need to look it up

[brazen pendant]

lemme take a look at how we did it in our lecture notes

non-measurability in general is kinda hard to even describe

[turbid zenith]

It was one of the last lectures from Dave Kung's course on The Great Courses

[brazen pendant]

so the way we showed it apparently was to show that all measurable subsets of both V and its complements are nullsets

which, if V itself was measurable, would imply that both V and its complement are nullsets

which is a contradiction

with the fact that [0,1] is the union of those two

but the proof is nasty

oh okay, so, it’s actually almost right

the way you have it

[brazen pendant]

yea, the way you have it is actually right, I misinterpreted what it said

you actually do have a countable union

[turbid zenith]

OH, you're right

There are uncountably many color classes

[brazen pendant]

because you’re rotating V

[turbid zenith]

But countably many rotations under consideration

[brazen pendant]

V is uncountable, but countably many copies of it cover the circle

yea, sorry, I confused myself.

[turbid zenith]

So maybe it should be "all the disjoint rational rotations"

[brazen pendant]

yea okay, it’s fine the way it is

[turbid zenith]

How's that

[brazen pendant]

more importantly, imo, fix the notation on the sums

maybe you could also actually write explicitly something like V+qᵢ

where qᵢ are the rationals

[turbid zenith]

I originally had that 😛

[brazen pendant]

like, maybe that and then “a rotation of V by qᵢ” next to it

[turbid zenith]

But I figured that being descriptive would be better

[brazen pendant]

my main complaint is that, just from looking at the slides I’ve gotten wrong ideas more than once because stuff was perhaps a bti too wishy-washy. now I understand you have to walk a fine line with not being too formal cause that’s scary and boring

[turbid zenith]

Yeah, it is a fine line ... it's something I'm always trying to fiddle with

Your point about sigma-additivity helped me patch something about that up

And I probably should mention it elsewhere as well

[brazen pendant]

but I must say, all in all your lecture captures the most important ideas really well

I like it

not a fan of your powerpoint styling tho

[turbid zenith]

LOL 😛

Is it just the gradient or something else?

[brazen pendant]

it’s also something about the font that I can’t quite put my finger on

I think it’s just a bit of a clash all in all

[turbid zenith]

Ahh. Yeah the math font is serif, the English font isn't.

[brazen pendant]

I like my slides to look very clean (and have stuck with that since early high school)

[turbid zenith]

Same, honestly. It's a work in progress though.

[brazen pendant]

I think I generally just dislike that particular english font… is that arial?

[turbid zenith]

It's called Century Gothic

[brazen pendant]

try linux binolium for a good-looking sans font

[turbid zenith]

Recently I've been using ... Georgia I think, or maybe it was Book Antiqua

https://twitter.com/mathsubgre All these are made in PPT as well

mathsub.com (@mathsubgre)

Preparing for the GRE Mathematics Subject Test has never been easier.

Tweets

47

Followers

48

Twitter

[brazen pendant]

http://libertine-fonts.org/ the downloads are linked somewhere here. libertine is a serif font, binolium is sans

[turbid zenith]

Awesome, thanks 😮

[brazen pendant]

they’re my favourite fonts. yes I have favourite fonts

[turbid zenith]

Oh I get you 😛

[brazen pendant]

it’s just… clean, you know?

hm wait a sec

something doesn’t look right

yea, thought so

that wasn’t binolium

I didn’t have binolium installed

on this pc

[turbid zenith]

Oh I thought you meant something else on my PPT XD

[brazen pendant]

there we go, same example but with the font fixed

[turbid zenith]

That is a nice sans font.

[brazen pendant]

it’s so nice it actually has notions of serifs on some of the letters :P

(if you look very closely, T actually has serifs, but they’re tiny)

(same with H)

[turbid zenith]

Yeah that's kinda interesting

My PowerPoints used to be much worse, heh 😛 I've done my best to keep making them sleeker when I can

So I'll try to take the font stuff into account

Why does writing on the board have to take so long :<

[brazen pendant]

for real

I have a messy blackboard way too often because writing stuff in detail takes too long

it’s one of the things I have to work on

the other major thing being to sound more enthusiastic

friend of mine sat in one of my classes and his feedback was basically “you stuctured the class really well and the presentation was spot on, but your voice is way too monotonous and it ends up feeling boring without actually being boring”

[turbid zenith]

D'oh

Yeah -- overacting a bit can help honestly XD

I have a tendency for theatrics sometimes

Once when I was in my infinity class I shut off all the lights in the classroom and turned the screen blank, just to have a "void" out of which the von Neumann construction of the naturals could appear

[brazen pendant]

I guess my goal for next semester is improving on that

it being a shorter class (45min instead of 2×45min) will make it easier I suppose. plus the fact that I’m genuinely more excited about the topics

like, numerics can be fun but it’s not really something I love

algorithms was really nice

[turbid zenith]

etc.

Algorithms is cool stuff btw!

where do you teach if you don't mind my asking

[brazen pendant]

ETH Zürich as a TA

they let undergrads TA, which is really nice

gonna catch up with game of thrones now

before the next episode drops

[turbid zenith]

very cool

Have fun :3

[wispy slate]

@brazen pendant you're in the mf ETH?

Bruh

Are you Swiss?

[brazen pendant]

yes to both

[wispy slate]

That's epic

Is Figalli a cool dud?

[brazen pendant]

never met him

heard the name tho

[wispy slate]

Bruh

He won a fields medal last year

[brazen pendant]

but he’s not one of my lecturers :P

[wispy slate]

Sad 😔

Is it cool living in Switzerland?

[brazen pendant]

lots of people win a fields medal every year. like, almost ¼

[wispy slate]

[dusky shell]

wot

@brazen pendant but field medals are only given every four years

[normal aurora]

that's the joke

[dusky shell]

[wispy slate]

1/4 of a medal per year

[fading cliff]

[quartz steeple]

I'm doing my PhD in Math and I need to practice my English, I was thinking about tutoring in English (maybe freely). Does anyone know where I can do it?

[turbid zenith]

Another day

Another student who learned about complex numbers completely devoid of any geometric representation

This makes me more annoyed than it really should

[lost raptor]

i learned complex without the geometric stuff and I turned out fine

[turbid zenith]

Lots of people do, sure

But a lot of students I've taught have trouble with it and think it's stupid and arbitrary

It's just more arbitrary, unmotivated, and meaningless rules to memorize

[lost raptor]

true

[turbid zenith]

Why can't we mandate that all math teachers do things my way the best way

[lost raptor]

true

start teaching them right from the beginning

smh when you don't start by teaching them groups and rings

[lament wraith]

smh when you don't start by teaching them ZFC

[lost raptor]

true

[turbid zenith]

I do understand the absurdity of starting from first principles at the lowest level 😛

But how hard is it to say "hey what if there were up-and-down numbers, and they could rotate"

And let students play with that

[lost raptor]

true

but then again
why do that when you can do graph theory

[turbid zenith]

...truth

I would be a fan of alternative non-calculus pathways in high school

[lost raptor]

Same, but calculus is still rather imporant

since like everything in existence likes it

can't replace it too easily with a different path, but having a more algebra focused path (like say linalg or something at the end, idk what level hs kids should be at) in addition would be nice if schools could allow it

[unreal ledge]

One way I finally really clicked with complex numbers was associating them with vectors, but my prof made specific note that they aren't. I find thinking of them as a vector space with multiplication is ez

[lost raptor]

technically they do form a real vector space alongside a bilinear product so it's a real algebra

[unreal ledge]

I have no idea why my prof said that

[lost raptor]

To emphasize that they aren't just vectors

They have their own field structure, which makes them interesting

Though, yes, they are technically a commutative associative unital algebra over $\bbR$ or someshit like taht

[burnt vessel]

Darkrifts:

[unreal ledge]

Yes, they are more. But my little brain at the time found the simple explanation useful

[lost raptor]

I forget what all descriptors imply each other and stick

It's a good explanation, just a bit of a forgetful functor so to speak

[turbid zenith]

@lost raptor how do you know all this despite being a HS student ;P

[lost raptor]

I'm a lazy asshole

so i just read number theory and algebra

[turbid zenith]

I don't think I've ever had a student who even knows what a "functor" is.

[lost raptor]

I'm rather hazy on my cat

I'm working on anal and top atm so I can further my NT

@turbid zenith what can I say except
t h e n w o r d

in all honesty I don't know why I know what I know, just ended up learning it at some point and kept going

[wispy slate]

t h e n w o r d

[turbid zenith]

What's y'all's take on take-home exams?

[brazen pendant]

I find the concept really really weird

I’ve never had one and idk if I should be happy about it or not, but I think I am cause I’d be way too paranoid about doing something I’m not allowed to do

[severe cedar]

Eh, usually if you give a take-home exam you should fully expect students to google and collaborate

so you need to make the problems original and so difficult that it doesn't matter

[brazen pendant]

my point is even if they allow it I’d be too paranoid to do it

[lost raptor]

seems like a weird idea of an exam, but every case I've seen of it went well

[severe cedar]

one of the hardest exams i've ever taken was like a cross between take-home and a standard exam

[brazen pendant]

they’re extremely not a thing here (even open book exams are rare enough)

[severe cedar]

the prof released 24 problems and said i'm going to make 75% of your exam from these problems

[brazen pendant]

at my uni the usual practice is “bring 10 pages of notes”

[severe cedar]

my friend and i spent 2 weeks working like 8hr a day on those problems and couldn't solve them all

[brazen pendant]

oof

[lost raptor]

"lmao i'm not making an exam, just don't screw up these problems btw one of them is proving RH and the other is on the Collatz conjecture"

[brazen pendant]

my analysis exam was ⅓ replicating proofs from the lecture and we got a list of 40 possible proofs that could come up 10 days before the exam

well you didn’t have to do it as it was in the lecture ofc

but it had to be a valid proof

[severe cedar]

actually I can share the problems with you at the risk of slightly doxxing myself

https://www.dropbox.com/s/jobjmc84ifss7xu/Final Problems.pdf?dl=0

actually that doesnt doxx me at all

they start out pretty easy and then get very hard

[brazen pendant]

the idea being you should have studied them all (800 pages of lecture notes!) already and then in the last week you could make sure to fill any gaps in those 40 you might still hvae

the list contained such gems as the lebesgue-criterion for riemann-integrability (this actually came up on the exam, as a guided proof tho)

I think the implicit function theorem was also on the list

banach’s fixed point definitely was but that one’s ez

oh yea the proof that riemann integral ⇔ darboux integral was another one. but that proof is so ugly that I just ignored it

oh and “let X be a metric space. then X is compact if any of the following 5 equivalent statements hold:”

stokes apparently could’ve come up too

and picard-lindelöf

[meager elk]

My brother had this kinda exam before - there werent many students taking this course, So the teacher sent them email at 6 pm with problems and they had to email him the answers back till 9 am the next Day - Yes, students enjoyed it, even though it wasnt allowed to communicate, noones gonna prove you did it

[severe cedar]

6pm - 9am? thats seriously fucked up

[meager elk]

But it was a high level class So googling wasnt really helpful

Why fucked up?

Its a lot of Time, if it was in class he wouldve only had like 3-4 hours

[severe cedar]

well assuming these were sufficiently hard problems, then students have to basically pull an all nighter

[meager elk]

I dont think the problems were much harder then they wouldve gotten in a normal exams, at least some of them, So idk

[brazen pendant]

it’s like the worst possible time for an exam

[meager elk]

Why?

[brazen pendant]

ideally you’d want to wake up like 2-3 hours before the exam so you’re well-rested

have fun doing that if the exam’s at 6pm

[meager elk]

I mean, I usually study at night So idk

[severe cedar]

My ideal exam time is like... 3pm maybe

[meager elk]

Yeah same for me

[brazen pendant]

mine’s like 10am till lunch

or 9am till lunch if it has to be 3h

or 9am till late lunch if it has to be 4h

[severe cedar]

All my exams this semester were 9am and I wanted to cry

[brazen pendant]

I think analysis was 9-13

[severe cedar]

My average bedtime was like 5am that entire week for only partially related reasons so that was awful

[meager elk]

Lol, before exams I need to sleep ubtil like 11am at least

But sometimes they are at 8:30 which is not cool

Yeah jdk I feel you, fortunately I fixed my sleep schedule to some degree

[brazen pendant]

the worst one probably was the one at 9am when I was still on a 2h commute - 2h30 to the exam location - and it had freshly snowed

[meager elk]

Oof

2h commute wtf

Thats a commitment

[brazen pendant]

couldn’t find an apartment

until late in my first year

had to get up an hour earlier just in case something went wrong with my connection because of the snow

almost died too, slipped at the top of a long stair when I was rushing at the train station

managed to catch myself, but it would’ve been a great start to the exam session lemme tell ya

dying, I mean

[meager elk]

Lol

I know some people that have to wake up like 3 hours before lessons start, I feel like Id miss most of my classes if that was me, especially since I miss some of the early ones although I have a 20 min commute

[brazen pendant]

I’m constantly a bit late

I have two options

a tram at 7:46, which gets me there in time

or a tram at 7:54, which should get me there just in time (arrives 8:13 by plan, class starts 8:15) if everything went by plan, but it’s usually at least a minute or two late

[meager elk]

What % of lectures do you usually attend?

And like, what time do they usually start, early morning or rather afternoon?

[brazen pendant]

most, but I’ve been a bit more lazy the past week

I visit everything blue (lectures) except numerical analysis regularly, green stuff (tutorials) when I feel like I could use some more assistance, and red stuff I’m paid to do so I kinda have to be there

next semester is gonna get weird

(grey one’s an alternative time slot. yellow ones are electives)

[lost raptor]

do i see information theory

[brazen pendant]

nah, you see theoretical computer science, in german

[lost raptor]

ah damn

So it's all german, and then you just see "Algebraic Topology"

[brazen pendant]

well that one is in english

[lost raptor]

smh not even consistently german

[brazen pendant]

as is differential geometry

[lost raptor]

and num anal

[meager elk]

All my lectures are at tuesday and thursady form 8:30 to 12, so I miss a lot, but thats also bcuz I feel like I dont gain much from going to them - this semseter Ive been Just Reading the script of the course by myself

[lost raptor]

seems like you've got a nice free day on Thursday, so that's good maybe

[brazen pendant]

except for the one hour I absolutely cannot miss on that day, ofc :P

[meager elk]

Is geometry fun?

[brazen pendant]

I hope so

[lost raptor]

well yeah, but only 1 hour

Geometry, smh just use law of sines

[brazen pendant]

cause I’m kinda going a bit all-in in that vague direction

well, geometry and topology

I guess I just like manifolds

the geometry course is actually mandatory tho, and could be taken in your first year

I just didn’t

and it didn’t fit into my schedule last semester

[meager elk]

Ive never really had Such courses, next year Ill start topology, but I feel like its gonna mess with my head, since Ive been having trouble with imagining thing on 2nd linear algebra course

[brazen pendant]

it’s not always about the imagining

[meager elk]

So im a bit scared but we'll see - many friends like topo

[brazen pendant]

topo is fun

[meager elk]

Yeah its not always about imagining but for some problems I feel like its essential to understand

[brazen pendant]

by far my favourite class this semester

and really, this year, it only has to contend with complex analysis. which was also fun tho, hard to pick a fav

coma was harder for me tho

[meager elk]

Ye sounds hard

[brazen pendant]

oh wait algorithms existed

[meager elk]

Although topology as a Word sounds scary itself

[brazen pendant]

that class was fun, but also a bit forgettable cause it was only 2h/week

[lost raptor]

topology s p o o k

[brazen pendant]

just replace it with “open sets” in your head

whenever you see it

[meager elk]

Oh i like sets

But dont know if open ones

[brazen pendant]

have you taken analysis at all

[meager elk]

Ye, 2 courses, 2 more next year

But its like calculus and analysis combined

[brazen pendant]

did you do metric spaces there

[meager elk]

Kinda, we'll do those on algebra and next anal course

[lost raptor]

metric spaces are nice

[brazen pendant]

aight. topology generalizes what you do there

also, topology is really just weird geometry

like this

[normal aurora]

[meager elk]

Ye thats what I know about topology hhaha

[brazen pendant]

in (euclidean) geometry, we consider two shapes equivalent (similar) if you can get from one to the other by a magnification and a rigid motion

right?

like, you can make it larger (uniformly) and then move it on top of the other

then they’re “the same shape”

[meager elk]

Yep, thats what I was told during explanation of a donut joke

[lost raptor]

oh boy smooth deformations and continuous mappings for homeomorphisms

[brazen pendant]

uh, we’ve not gotten to topology yes

no donuts yet

in geometry, a donut is only equal to another donut, but it can be a bit larger or smaller

in topology then we just say “fuck it, let’s allow all bijective continuous functions, not only rigid motions and zooming”

and then it kinda boils down to trying to figure out which things are not equal

[turbid zenith]

Wow, so I posted my question and then left and the talking happened after XD

I've been thinking about how I want to do tests when one of these days I teach undergrad

I've always hated how timed, in-class tests often require so much "sanitation" of the problems so that they can be done within the time limit

(Either that or you're a teacher who tests for cleverness, which I'm not a fan of)

My abstract algebra class had just about everything boil down to three in-class exams, and it often seemed the problems required you have some kind of "clever insight" to be able to make any headway in them, and we were never given any kind of help building up that insight :/

So I've been considering a model I saw another professor does (I wish I could find his blog post), where all his tests have two parts. The first part is in class, and it primarily consists of definitions, examples, counterexamples, etc. -- the sorts of things you should have in working memory to be able to do the problems. The second part is take-home and has the proofs and other meaty problems that you're realistically supposed to be able to bang your head against for a while until something works.

[brazen pendant]

I actually quite liked the structure of our analysis exam

it was a 4 hour long exam (appropriate as it was the final of a one year course) with three equally weighted parts.
one part calculations - things like finding extreme values on some manifold, computing an integral that needed some trick of sorts…
one part proofs - these were in line with homework problems and generally only required about one insight, though there was one problem which was harder
one part theory - replicating things from class, from definitions to proofs we did there; he handed out a list of possible topics 10 days before the exam

I did really badly on part 2 (I think I got about 2/20 points…), did well but not perfect on the calculations, and aced the last part. was enough for a decent grade

probably something like a B in the american system

[turbid zenith]

What's the use of replicating proofs from class?

[brazen pendant]

forcing the students to study them

[turbid zenith]

...well that's a tautology

[brazen pendant]

is it?

e.g. in my probability class, we’re doing proofs of theorems in class, but in the end we only have to be able to use the results

so while the proofs are there to convince us we’re doing math and not pseudoscience, we don’t ahve to actually study them

[proven cape]

What is everyone's opinions on the dreaded "common core" method of teaching?

I specifically ask because I was talking to some friends and we all agreed that our Uni's Calc 1 course was easier than our former high school's Calc 1 class, and we thought it had to do with the fact that the university course taught a lot more of the reasoning behind derivative rules, the fundamental theory of calculus, etc, making those easier to remember.

[meager bronze]

one of the most important things is having what's called "constructive alignment"

(this is to ashura and sascha)

basically your primary concern should be coming up with your learning objectives for the entire course. then build assessments that will assess those learning goals (so tests, homeworks, etc) and then you think about the activities that will actually teach the goals

and all 3 of those should be aligned

so if "memorizing proofs" is not a learning objective, then it shouldnt be on a test

@proven cape common core is not a method of teaching, it's a set of standards

and calculus isn't even covered under the common core

[proven cape]

Really?

[meager bronze]

you can look here: http://www.corestandards.org/Math/

[proven cape]

Huh

Isn't it a curriculum?

[meager bronze]

it's literally just a set of standards

[proven cape]

Oh, ok

Then that's probably more a problem with my state or local district's curriculum

[meager bronze]

yeah

the other issue is that often times high school teachers are more likely to not really be as expert in the field

so in college you are going to be taught by someone who really does know the "why" behind things

[proven cape]

What do you think about AP math curriculums?

[meager bronze]

whereas in high school you might just have an engineering person who only knows the rules and not the why

hmm it's been a long time since I thought about AP stuff

[proven cape]

In my freshman year of high school I had a teacher's assistant who had just graduated with a PhD in math or something like that, he was amazingly knowledgeable.

[meager bronze]

but I think AP emphasizes "here is how to apply the method" than "why does teh method work/how is it derived"

oh that's super awesome

[proven cape]

He did square roots of 4-digit numbers in his head, so he wasn't just theoretically knowledgeable but was also just a savant at the arithmetic.

[meager bronze]

i can do square roots of some 4-digit numbers

like 1600 :)

[proven cape]

LOL

[drowsy bear]

what about a prime number

[turbid zenith]

Wow I missed common core discussion.

@proven cape I'll agree with what was said -- Common Core isn't saying HOW things need to be taught, but just what students need to be able to do/understand in each course

But the big thing is it is about conceptual understanding as well as (but not instead of) procedural fluency

Also ... AP Calculus curriculum also does emphasize why things work, I disagree with @meager bronze on that

There are plenty of questions where you really need a solid conceptual understanding to answer them

Only thing it doesn't really do is introduce limits with the full rigor of epsilons and deltas

. . . which I applaud

[remote vine]

First of all, sorry if I say something that makes no sense since I'm not American and I don't know your system. Here in my country the mandatory education lasts until 16yo and then there are two years of "preparation for university", which are just an extension of what you were studying before (as in you still have to take things from sciences and liberal arts) but things are more focused towards the national university access exam

In this two last years (which I'm not a huge fan of) you are taught what universities supposedly think that you need as a previous knowledge

In particular this means that you are going to be exposed to "what a mathematician does" or "what a historian does" and so on

In the case of science class I guess we all would agree that the lab is fundamental, as a physicist/chemist/geologist or whatever is going to have labs in their education (and some of them in their whole career if they pursue research)

Not having labs would feel like lying to the students, and at the same time I think that it would make no sense to be on the lab all day, because the theory is important

I feel that the same is true for the math class

Avoiding proofs altogether for this two years is straight up lying to the students. I have had several people in my class at uni who left because the degree is not what they expected

So while applications are very important and the math class has to be taught for all, even for those who will not go on to studying math as their degree, I feel that some amount of proofs is necessary

Not all year long with proofs, but some part at least

From what I've read in here and Reddit and so on it seems that in the American system the proofs in highschool are these weird geometry proofs

[meager elk]

I don't think I've had even one 'real' proof in high school

[remote vine]

In here we do proofs in what I think you call "calculus": we give the rigurous definition of limit, continuity... (as well as intuition, of course) and prove things like Rolle's theorem, Bolzano's theorem, IVT...

All of this was to say that I think that teaching something rigurously in highschool is a positive

[brazen pendant]

completing the square to derive the midnight formula is a real proof I suppose

[remote vine]

😆

[brazen pendant]

we did that in high school

and I’m pretty sure I saw a proof of thales’ theorem in elementary school geometry

[meager elk]

@remote vine bolzanos theorem, where are you from? 😛

[brazen pendant]

and a teacher certainly proved pythagoras at some point

[meager elk]

I respect you calling it Bolzano, if you are talking about the commonly known Darboux

[remote vine]

I don't know, Bolzano is the one that says that if f(a)f(b)<0, then there is a point c in the middle where f(c)=0

[meager elk]

yeah, most sources call it Darboux theorem, although it was stated by Bolzano first

[remote vine]

We prove that and then conclude the IVT as a corollary

https://en.m.wikipedia.org/wiki/Darboux's_theorem_(analysis)

This seems to be a different one

[meager bronze]

@turbid zenith we must have just had different instructors then. In my class we got no explanation of why anything worked, no proofs or anything. just "here is the method, now do it"

[turbid zenith]

That makes me sad @meager bronze

@remote vine as for introducing what proofs are like, I really don't think epsilon-juggling is the right place for it. I actually would advocate proof being a bigger part of high school mathematics, though perhaps not at 100% level of formality.

There needs to be more emphasis on "how do you know this is true?" and not just "show you memorized this formula"

[meager bronze]

epsilon-delta proofs don't belong in a first-year calculus course unless the course is for advanced students

i'm not disagreeing with you ashura

[turbid zenith]

Yes this exactly

[meager bronze]

i'm just saying that my experience (and a lot of people's experience, given the number of students I get in my calc 101 class who have taken AP calc but have never seen a proof)

[turbid zenith]

Do you teach calc?

At the university level?

[meager bronze]

yes

[turbid zenith]

Awesome!

[meager bronze]

i'm a phd student

[turbid zenith]

I can't speak for everyone who teaches AP but I know what's in the curriculum framework

[meager bronze]

and i've taught the lowest level of calc at my university (which doesn't offer precalc but forces all students to take calc lol) and also the highest

so a pretty wide range of students

[turbid zenith]

The AP test has questions that test your understanding of when the various theorems do and dont apply, and what they do and dont say

Like the distractors on the multiple choice might have misuses of the IVT, for example saying that a continuous function on [a,b] never goes outside the range from f(a) to f(b)

They do their best to make it very difficult if all you learned how to do is take derivatives and integrals ;P

[meager bronze]

oh I mean I was taught all of that stuff

just no proofs

also gotta run, see ya

[turbid zenith]

I wonder if we might differ in what constitutes a "proof"

For anyone who's curious, here are the newly released AP Calc guidelines

https://apcentral.collegeboard.org/pdf/ap-calculus-ab-bc-course-and-exam-description.pdf?course=ap-calculus-bc

[brazen pendant]

a proof of the IVT is actually pretty annoying - you either need to introduce some topology, or do a super ugly thing

I would not want to prove IVT to anyone in a class before real analysis

[turbid zenith]

Okay usually you dont see proof OF the IVT

[brazen pendant]

yea

IVT is so intuitively clear that I think it’s perfeclty reasonable not to prove it until you do real analysis

[turbid zenith]

But you may USE the IVT to prove a function has a zero, or explain why the IVT doesn't apply in some situation

Kind of get used to the framework of "show the conditions are met so that you can make your conclusion"

[brazen pendant]

I think usually when people talk about proofs in a school setting they mean more like proofs of theorems

and not using theorems to show some particular statement

that’d be more “using the theorems”

of course those are still proofs

but they’re on a different cognitive level imo

more akin to calculations

[turbid zenith]

Sure, but I would say the latter are more developmentally appropriate

[brazen pendant]

oh, definitely

as said, I would not want to prove IVT to a high school student

[turbid zenith]

They're in kind of a transition

[brazen pendant]

that’s just bad times

but I do think proofs still get the short end of the stick anyway; we never learned even basic propositional logic in high school for example

[turbid zenith]

I think the classes before calculus are also great places to teach how to think about proofs

And we only maybe do it in geometry

I mean if it were up to me... high school math would look very different ;P

[brazen pendant]

same… and maybe it will be up to me at some point (and then they kick me out)

[turbid zenith]

I once came up with an idealized list if how I would structure courses

[brazen pendant]

I wanna see that

if you still have it

[turbid zenith]

It was all one-semester courses instead of full years

And one course you might do as a senior was called "Logic, Proof, and Limits"

[brazen pendant]

hm here the school year is pretty unimportant already

yea that kinda restructuring wouldn’t be possible here

probably the largest difference between my school and typical american ones is that there’s no such things as courses here

there’s just a class called “mathematics”, and it’s mandatory for everyone

and then there’s a core subject “physics and applications of mathematics”, in which there’s some more advanced topics

[turbid zenith]

Okay so for context

[brazen pendant]

(every student has to choose one core subject)

[turbid zenith]

At the high school where I was teaching, the new head rabbi said we should "reimagine" the school and dream big

So I did lol

https://docs.google.com/document/d/1rT41dO83r8r2SmqynYZaBAxxlwwcPNN09ASlBF7yhTc/edit?usp=drivesdk

[scarlet perch]

ah, tutoring for ap courses is probably one of my least favorite things to do

because usually the goal is to force feed a bunch of formulas into their mind and teach them a bunch of strategies for the test more than teaching itself

it's one of my least favorite things to do ill be honest

pay is good though

[turbid zenith]

I do agree that a lot of the time you have to teach to the test and that utterly sucks

Because all that seems to matter is the score on the test

Not how much they learned

Though to be fair the test is a decent proxy for that

If you dont know calculus well, you won't do well ;P

[civic tree]

yeah

the ap test is pretty good in my opinion

but lowkey really rushed

question quality is ok

[turbid zenith]

Rushed how

Like as in you dont get enough time?

Or the questions seem hastily written?

[lyric osprey]

The ap exam for AB and BC is as basic as it could get, since most of the material is just a recall. The FRQs, to an extent, is where it's better quality, requiring them to actually think lol

[tidal whale]

Kinda curious, how would you folks compare - at least in terms of math - the APs to its equivalent systems (e.g IB, AICE)?

[lament wraith]

In terms of the material covered, IB and AP are pretty similar

But having taken both tests, the IB HL math test is significantly harder than the AP BC test

Not to mention that the IB is on a 7 point scale which makes it much harder to score perfect on compared to the AP one

[trim mango]

IB is so underrated in Canada compared to AP

[remote vine]

Omg so many acronyms 😫