math-pedagogy

[quasi musk]

There's no way there isn't a trial and error process. It's hard to introduce things that will be useful in research without also being extremely difficult

Since most of math is streamlined

Even if the thing useful in research is comfortability with trial and error

[wispy slate]

In the realm of "how do we productively talk about math" and teaching people how to choose positive rhetoric towards math in general, I struggle a lot when trying to amend statements that might have been false or over-complicated.

I don't want to use that same performative self-hating language in front of the people I am tutoring because I know it will just influence them to do the same

[brisk ruin]

Do you have an example?

[wispy slate]

Yeah like say I explain something wrong and then follow up with "sorry I am a dumbass, this is the right thing"
Then they go on to do the same thing cause in a lot of uni environments making mistakes is shameful especially when people are trying to be perceived as having some kind of "prestige"

And that is not good, but I am definitely just bolstering the problem and want some better ways to say "oh that was wrong" without apologizing or shaming myself

So that they also do not feel like they have to do that to themselves

[brisk ruin]

Ah I see

What about something like "oops, that is wrong, this is right"

[wispy slate]

that's a good one

if anyone has any other things they use like that please do mention

[dawn walrus]

I think maybe in my case, I still want to apologize whether or not I'm going to say something self-deprecating.

[brisk ruin]

Oh hello ranyakumoschalkboard

Anyways

I think this is a good point

Certainly there is a culture of excessive self-depracation

And one would want to avoid perpetuating it

However, an apology is not necessarily an example of excessive self-depracation

[wispy slate]

tru

[brisk ruin]

Of course, it would depend on how you go about with the apology

[dawn walrus]

For me when tutoring it comes down to the student. If I feel that the student finds too much confidence intimidating and responds well to a funny self-deprecating joke, then I'll do that once in a while because it can help them to lighten up. But I always try to throw in a little quip about growth and how it's completely normal to make mistakes. Obviously I'm not like the best tutor ever or anything so maybe I could do better by being completely positive.

I also think there's an aspect to the authenticity which helps someone who already thinks this way to connect with you a little better.

But I can also see where it would be harmful.

[brisk ruin]

Yeah making mistakes and owning up to them in an honest manner can definitely help with approachability esp. with people who might be intimidated

But it seems to be quite nuanced

Like it might have the opposite effect of "Oh no [smart person] makes mistakes how could I possibly get it right?"

[dawn walrus]

Yeah, you need to be ready to u-turn it away from that and toward "they're just like me" which is sort of what the growth thing does.

For me, I think the toughest sort of mistake to tackle is not when I do something wrong solving a problem or presenting an idea, but when I do something wrong while formulating a problem.

Like I get halfway through the solution and go "wait this isn't right, I meant to add this condition"

And that seems to really throw people off. When it comes to other mistakes, at least the person you're working with might spot it before you which can be really good for their confidence.

(That's another thing: If THEY spot it before I do, I definitely don't call myself stupid, because that's removing the credit from them for spotting the mistake. Instead, I thank them and compliment their attentiveness.)

[shadow basalt]

for it is definitley inverse to the mistake

[brisk ruin]

Are you missing a pronoun

[shadow basalt]

like if I make a mistake that is really easy to make or not obvious

yes i am

then I will just be like

Oh whoops that was silly

I think framing it as "careless" is also better than "stupid"

lots of the mistakes math teachers make are do to a lack of being careful

[dawn walrus]

Ooh, I like that.

[shadow basalt]

and even the best mathematicians fail to be constantly careful

I think the carefulness focus strikes an importance balance between like

[dawn walrus]

We need more anecdotes of great mathematicians making silly errors. Can't use the Grothendieck prime one more than a dozen or so times on the same person.

[shadow basalt]

you shouldn't be hard on yourself for mistakes

but at the same time

you can try hard to avoid them

and you want to encourage that

[shadow basalt]

a lot of them are a lot harder to appreciate

which is why the groth example is infamous

[dawn walrus]

That's sensible

[shadow basalt]

bc even a young kid can see why its a mistake

When I tell you "wow John McCleary wrote surjective when he meant injective in a spectral sequence computation"

its the same level of carelessness, really just a typo

but its harder to see why that ruined my afternoon

lol

[brisk ruin]

[meager bronze]

during my 2nd year of grad school I "proved" that the only odd prime number was 3

I like to bring that up as an example of how everyone makes mistakes hahaha

[shadow basalt]

i mean

4 isnt prime

so looks okay to me?

[meager bronze]

it all checks out

[brisk ruin]

[kind salmon]

Hello fellow didacts and pedagogues!
I have a student whom I am 99.9% sure has synesthesia.
Now i struggle to aid her further and barely find useful scientific research on that topic. Since I am obviously not much qualified but hesitate to send her to a neurologist right away:
Do you know any reliable sources or social platforms to recommend to her (in english, french or german)?

[shadow basalt]

I would be very surprised if there were not a few discords about synesthesia and for people with it

Would there be a purpose to the neurologist? It's kinda cool to confirm the diagnosis I guess but I am not sure if there is anything a neurologist would actually do for her?

[kind salmon]

Ty, that's a good idea. Will check discord further.
And yeah, I don't think a neurologist would be a wise approach. It's just the only reliable I found yet.

[wispy slate]

Any advice for how to be a better tutor? I currently tutor CS courses at a community college.

It seems like students generally come to my university tutor department after they are already several weeks or a month behind, to give context.
A student will ask about something like iterating through an array that sort of reveals a lack of understanding about loops, but then they'll have some knowledge gaps with loops, and so we have to go into control flow, and then from control flow to variable assignment sometimes.

This happens quite often and it seems as though the students get pretty frustrated or it makes them even less confident than when they arrived. The actual subject can change but it's still kind of a one issue becomes a lack of understanding of another, and you have to trace it back, sometimes revealing a lack of understanding in multiple things.

I'm not sure if this is a bad approach. Students generally get frustrated when I do this but I don't think it's a good idea to just "fix" their code for them, I think tackling the understanding issues is more important if they exist.

At this point I generally give them a realistic time frame and study plan and ask them to come back if they need help during my office hours, but they sort of recoil at that time commitment. I.e. if a student is a month behind with no prior programming experience (common) I tell them that they're going to want to be studying 2-3 hours a day.

Any advice is appreciated.

[brisk ruin]

I definitely agree that tackling the underlying issues is the correct solution as opposed to just fixing their code

[wispy slate]

So like there will be a question that comprises multiple "primitive chunks" i.e. loops, control flow, variable assignment, and usually the problem is they don't know one of those "primitive chunks" for lack of a better word, a basic concept.

It's rarely a simple issue.

[brisk ruin]

Providing simple code fixes seems to be setting someone up for failure in the future

[austere inlet]

it's like a math tutor just giving students the answers or algorithmic solutions to some exercises

I think suggesting a study routine for them is good, often problems like that stem from people not having such a routine (it doesn't even need to be something set and rigid, just them dedicating some time each day to studying) or not expecting to study much or at all in their higher education (I think this is pretty common since you can pass in HS by barely doing anything)

[dawn walrus]

I think sometimes making people feel a little frustrated is not a bad thing, because it is indicative of a realization of what they don't understand. The first step to understanding something is realizing what precisely it is you don't understand, so that you can focus on it and not skip over it with an incomplete understanding. Maybe there's some way to get across to your students that their frustration about this is actually a healthy sign of progress, because it involves a sort of meta-realization with regard to their relationship with the material. (Sure, maybe not as great as a non-meta realization, but usually those come afterwards).

[brisk ruin]

[dawn walrus]

And once you get that across I guess it provides a gateway into grinding in some of those foundational concepts haha.

[lethal leaf]

I mean there are some hard truths you just gotta deal with

They're only coming to you as a reactive, not preventative measure

And like there's only so much time in a session

It's hard to fix every underlying issue in an hour or two

But yes you're right you shouldn't just feed answers

[winged urchin]

Yeah, as much as I try to lead them down the path to learn something, sometimes I admit I find I have to take the simpler path. Just showing how I might go through a practice problem for instance. And sometimes I do believe this is okay, sometimes students do just need to see how it's done and they don't have the time (or money) for the longer explanations

[dawn walrus]

I agree

I don't think that it's helpful to think about pedagogy in a "NEVER do this!!!" way, well besides the obvious things.

There are situations in which it will help someone tremendously to just see a problem done in front of them

With the thought process fully explained

[wise lodge]

This is just my two cents

Perhaps at the beginning of your first session, you could ask your tutee what they want out of the tutoring so you can adapt your teaching style. Sometimes tutees get frustrated because what they get out of the session is not what they expected. Some students just want to pass their course and want to learn math hacks. We may not agree with this strategy (they’re going to get screwed in later courses!) but as the saying goes, the customer is always right.

[wispy slate]

Well the issue is that it's programming so you can't just grind formulas the same way, even in an intro class, while in my uni's calc 1 course you certainly can.

I don't tutor mathematics, sorry if I gave that impression.

But I feel like they might be similar enough to where asking the question here is beneficial.

[wise lodge]

I brought up mathematics because of the nature of this server and my own personal experience. I never studied computer science, so I wouldn't know how applicable my advice would be, but I thought it was worth a shot.

[wispy slate]

I tutor in community college so yeah, we already have a 50% drop out rate or something, and then I assume worse for STEM

I just wanted to ask if I was doing anything wrong because I've yet to have a student actually go and, do the study plan that I've given them or return after 1-2 tutor sessions

so I assume they're just coming to me reflexively when they begin to panic, then withdrawing or dropping out

and I'd really prefer that at least some of them this doesn't occur, it's happened like four times

[wise lodge]

Is it free tutoring?

[wispy slate]

then they never come back for another session (which is free to them), I'm the only CS tutor rn oddly enough

yeah

included with the college

I get paid out of work study

But sounds like from what you guys have said I just have sort of limited influence on a student, I can only provide them materials and plans that, if they listen will rectify their situation, I have no real ability to make them do it

[wise lodge]

Its not your fault and if the drop out rate is that high, then I have a feeling the same reasons why the students are behind in their courses is the same reason why they do not follow the study plan.

[wispy slate]

Yeah, it's open enrollment

I just wanted to check with you guys and see if I'm objectively worsening their situation or doing things badly because I have no pedagogical training whatsoever.

[wise lodge]

I have some tutees that literally do not do any studying outside of the tutoring sessions.

[wispy slate]

Yeah, that is most of them

[wise lodge]

They end up doing well but not because of my teaching per say, its just a trick to get them to study.

[wispy slate]

I think CS might also be one of the majors where people seem to really uh

They get into it because of this promise of a career but have no idea what it is, get in, don't like it, they don't study, but are afraid to swap majors

So of course there is a natural attrition that occurs because people realize they simply do not enjoy it along with the base level attrition for STEM programs at my college

Well at this point I'm more sort of ranting, but thanks for the advice and thoughts.

I will continue doing what I am because I am entirely sure if people did what I said they'd easily pass the course with an A or at worst B

I think a lot of them just realize they don't want to study CS

[wise lodge]

haha not sure how strict the rules are about on-topic, but I would wager as long as its in the spirit of the channel, its gucci. It's a good idea to rant sometimes.

[wispy slate]

The same thing happened to me for chemical engineering

I thought I was having trouble in courses but really I just did not enjoy chemistry or chemical engineering in any way

[wise lodge]

They get into it because of this promise of a career but have no idea what it is, get in, don't like it, they don't study, but are afraid to swap majors

I think often times people like the idea of what they're studying more then actually doing that thing.

[winged urchin]

Also I mean, to be an academic I think you need to be comfortable with some length of time where you're not enjoying it

Like I don't necessarily enjoy doing long calculations that you don't necessarily know how to navigate

Its a lot of trying things, failing, then trying other things

And you only REALLY get that sense of joy and satisfaction when it (hopefully) finally clicks and you figure it out or get the result you were looking for

When the program finally compiles and does what you want

There's also a pretty huge disconnect between school and research in this way

In school you can be relatively sure that even if a problem is giving you trouble, there IS a relatively nice solution you just need to find how to get there

Sometimes in research you're simply asking the wrong question or thinking the wrong way. There is no guarantee of a nice solution. And furthermore you can't really determine whether you just haven't tried hard enough or whether your approach or question is fundamentally unsolvable

[shadow basalt]

Okay so unsurprisingly #chill was very cringe about this, but I had a question for (primarily) people educating highschool-early college students

Do you think including simple memes that convey somewhat complicated ideas can help students?

This was the image in question. Obviously tons of boring graphs and polygons can give you the same idea, but to me if a student was already bored and half paying attention, I could see a meme sort of "breaking through" that

But I also already understood the concept before seeing the meme, so its hard to know if it would actually make sense to someone learning it

[rapid kettle]

i guess it would make a lot of sense for students familiar with software development

or just familiar with graphics and rendering

[round robin]

i have no idea wtf the meme is trying to convey tbh

[dawn walrus]

yes but it can also be cringe

but like

that's probably fine

if you're confident enough in being cringe then you can gain the respect of highschool age students, but they will ignore the point of the meme if you're timid-cringe

[molten urchin]

I think I like this meme. It seems to convey the idea really well.

Although one would still need to explain the meme itself, it is catchy, and for a younger audience this seems to matter a lot.

[turbid zenith]

I love it

Same energy https://twitter.com/howie_hua/status/1246205490444431360

Howie Hua (@howie_hua)

I made a calc meme.

Retweets

3485

Likes

15547

Twitter · Apr 3, 2020, 10:38 PM

[dawn walrus]

wait i dont really get why sums are bumpy and integrals are smooth

oh i guess if you're thinking about riemann integrals of continuous functions

sure

[brisk ruin]

[real mauve]

me likey

[stark pine]

I think this photo is much clearer than the one max sent tbh

[real mauve]

it's not as funny tho, it kinda takes itself too seriously

i'd put the cat one on my slides when presenting stuff both to students and professors

[round robin]

tbh i feel this is clearer

[brisk ruin]

Meme war

[lethal leaf]

that's the exact context ya

[molten urchin]

Max's post above has more of a meme vibe; Ashura's post actually just seems to convey the idea. Depending on the audience and occassion, one may be preferred over the other.

[teal kelp]

i like the floppa one more

[turbid zenith]

I like both!

[mint lark]

hmm.

So I was recently grading a homework in an introductory analysis course (so they've taken linear algebra with proofs but no other proof courses) and someone wrote something like C_n = {V \ closed ball of radius 1/n about v} for some fixed v in V. They then claimed without proof that the union of all the C_n was V \ {v}. I counted off 1 pt/15 for this, because it was critical for the proof to work (it was a proof that compact => closed). The student submitted a regrade request and I'm trying to figure out like what people expect when they submit work

I try to grade with the intent that like, barring very hard problems, everyone in the class should be able to read the proof and understand it, and claiming set equalities like this without any proof or explanation seems to inhibit that. Do yall think this is a fair assessment? I did something similar when they claimed that a union from n = 1 to N of C_n would just be C_N

ofc this would be acceptable in further courses, but for now it seems like something they should verify

especially considering some students wrote false equalities or wrote things like C_n = {V \ open ball of redius 1/n about v}, which meant it wasn't an open cover, etc. etc.

[shadow basalt]

I would give the point honestly

i think its hard and a judgement call

and your point removal was very small

[mint lark]

it was 1pt for that and 1pt for the other thing

maybe should take 1.5pts overall

wait no it was 1.5 pts overall that I took off :)

[shadow basalt]

I would say like

if you believe the student understands what they are doing

then only take off points once

for effectively the same mistake

[mint lark]

ye

[shadow basalt]

bc you haven't given them a chance to correct it yet

[mint lark]

yeah

[shadow basalt]

I think as much as you want newer students to be more explicit

figuring out the correct level of explicitness is especialy hard for new students haha

and for some statements like the above are so obvious that it might not even occur to them

(fwiw, I don't plan on grading with your "other students" framework, but more of a "am I convinced you understand it" framework)

(this probably has its own disadvantages)

[mint lark]

The way I see it is that there are two equally important components that the homeworks test

1. Mathematical Understanding
2. Mathematical Communication

I'm convinced this student had 1

But for 2 I think there wasn't enough

Where we assume the target audience is the same level of student or slightly higher

[shadow basalt]

Well

Who does the student think they are communicating to

Have you told them that their audience is their peers rather than you?

You clearly understood their communication

[mint lark]

This is something I've /the professor has stated in class a few times. Though I don't think it's written down.

It's something I would include in a syllabus if I were writing it but I'm not lol

[shadow basalt]

Fair

Yeah idk it certainly isn't poor communication in my mind

just not detailed enough

for what the prof wants

you're like halfway through the sem right?

[mint lark]

yeah

but this was from an earlier homework

[shadow basalt]

Ah

[mint lark]

we're on homework like 10 and this was homework 6?

[shadow basalt]

Idk hot take

if they can explain the details when pushed for them

[mint lark]

and we're a few weeks from finishing semester

[shadow basalt]

i don't mind if people are concise

its all totally arbitrary

[brisk ruin]

I think I would agree with your initial assessment

Just based on my experiences grading intro real analysis

[shadow basalt]

idk its like

not a mistake

and I believe the student knows what they were doing

[brisk ruin]

Sure, but the point of proof is communication

[shadow basalt]

I could see rudin not justifying that statement

in his textbook

[brisk ruin]

Rudin is a pedagological disaster

[mint lark]

^^^

[brisk ruin]

And if the class has previously communicated

[shadow basalt]

idk it seems very harsh and arbitrary to me

to act like this student did anything wrong

[brisk ruin]

That proofs should be targeted for classmates

[shadow basalt]

Idk to me this is worth like

red-penning on the paper

without any reduction

but i also think a 1/15 reduction is fairly harmless

[mint lark]

the class is heavily curved. Like everyone gets an A in the end if they've been submitting homeworks essentially

so my grading is essentially

[brisk ruin]

No one pays attention to feedback if there isn't a point deduction

[mint lark]

red-penning without any reduction

at least in the end

[brisk ruin]

Like

I have graded multiple classes

[mint lark]

and more an indication of how serious I think the thing is

[brisk ruin]

Where people weren't stapling their homework together

There was no issue with the math

But literally pages were being lost

And regardless of how many times I asked for stapling and the instructor announced it in class, nothing changed

[shadow basalt]

Did you have a stapler in the room

otherwise thats a u problem to me lol

[brisk ruin]

Yes there were staplers

There are many staplers around campus

I wasn't losing papers

[shadow basalt]

in the room?

[brisk ruin]

The student was losing papers

[shadow basalt]

oh lol

[lethal leaf]

^^^^^

[brisk ruin]

But the point of this isn't about stapling

The point is to demonstrate that people don't listen to feedback if there isn't point deductions

On gradescope it's even worse

You open gradescope, see the grade, if it's a perfect score you don't even bother to open it and see if there's feedback

[shadow basalt]

in fairness that feedback will be useless pretty soon though

like the students argument would be perfectly fine in the future

I guess I am just not sure what one is trying to teach

by saying that argument wasn't good enough

[lethal leaf]

See here I think is the disconnect

IMO

they are both important

but not equally important

[shadow basalt]

Yeah I guess I am not convinced there was an issue in communication

[lethal leaf]

I think understanding > communication but some amount

and that getting better at communication

[shadow basalt]

This is the sort of thing an oral exam could immediately clear up

[lethal leaf]

will come with time

[shadow basalt]

or a course with corrections

[lethal leaf]

as you write more proofs

[shadow basalt]

Idk like consider how this would actually work in research

You would submit an article to an editor

and they would be like

I am not convinced by this proof

and you'd flesh it out and return it

There isn't any parallel in the grade once and done pset scheme

[mint lark]

research isn't the only thing mathematicians do though

[shadow basalt]

Replace article and editor with expository and audience

[mint lark]

they also teach and write expositories and communicate to the public (at least some do)

what about textbooks?

[shadow basalt]

Math that is intended to communicate

goes through countless revisions

[mint lark]

true

[shadow basalt]

because what is clear to one person

is not always to another

but psets don't

[mint lark]

mhm

[shadow basalt]

and it seems weird to me to take off points

when the correct solution in reality would be like

"why is this true" "oh heres the easy lemma" "okay great"

[mint lark]

I mean I have general problems with psets. But I can see why this would be a good thing

For now I think I'm going to keep it as is considering the context of how the class is graded and how few points it is

[shadow basalt]

The real issue is that grading-with-corrections is like

impossible to implement

bc of time constraints on graders

[mint lark]

but in the future what I might do is like. If I take off points for minor details, ask them to submit a regrade request for details

major details / big mistakes wouldn't get this treatement

[mint lark]

with a description of how to prove it in the regrade request

[shadow basalt]

I would just say as policy at the beginning like

If I mark off points for a lack of details

you can resubmit within idk 48hrs

and if you fix it

or just email me the details

ill give the regrade

something like that

dep. on how big the class is ig

[mint lark]

ye

this is a 20 student class or so

[brisk ruin]

At least at Berkeley, the students are not supposed to know who the grader is

[mint lark]

so that's not hard to do

[brisk ruin]

So there is no direct communication

[mint lark]

I don't just grade I also course assist

[brisk ruin]

Ah ok

[mint lark]

so that means like I'm in class 2/3 times a week

[brisk ruin]

That is sensible then

[mint lark]

and stuff

also gradescope has like regrade requests where you could put in details

if grader was anon

[brisk ruin]

Yeah this was back when I did paper grading

With physical paper homework

[dawn walrus]

it's so nice to be grading a class where the prof doesnt allow regrades

like just trust me i know what i'm doing

i have never gotten a legit regrade request

[stark pine]

at my school lower division courses don't do that, but small upper year classes will often list which TA graded which problem so you can just talk to them directly

[brisk ruin]

Oh

We have the instructors deal with regrade requests

However this is only for graders

[stark pine]

yeah some courses do that

[brisk ruin]

If the grader is also the course assistant then the students will know them

[stark pine]

not allowing regrades is kinda ass - what if there's legitimately a major grading mistake? I've had graders like... miss a page of solutions

[brisk ruin]

Yeah there are definitely times where regrade requests are proper

But I feel like Gradescope makes requesting a regrade request too easy

So you get buried in them

Where as when grading in person

I think regrade requests are more reasonable

[stark pine]

What I've heard has worked well is profs saying "We will grade more harshly and there is a good chance your mark goes down unless there was a major error"

[brisk ruin]

I feel like this can intimidate students, especially if they aren't super confident

[shadow basalt]

ugh thats so gross imo

a student asking for a regrade shouldnt be worried about reducing their own grade

might as well just not offer it

[shadow basalt]

i feel like even in a regrade free class you could email the prof about this an get an exception

[stark pine]

but the point is to prevent frivolous regrades

[shadow basalt]

I think its just a dick move hahaha

[stark pine]

I've also seen "we will regrade your entire paper not just one specific question, so if there were any errors in your favour they'll be fixed too"

same idea

[shadow basalt]

yeah those are all bad policies

i care a lot more about helping students w legit regrades

[dawn walrus]

I usually would allow regrades, but I dont set up the gradescope, the prof does

[shadow basalt]

than i care about trying to hamper students w frivolous regrades

[dawn walrus]

However if I make a mistake and someone contacts me some other way / mentions it to the prof I'll obviously fix it

It's not that the policy is that "grades are final" (which is stupid)

It's that the button is just disabled

Every regrade request I'd ever gotten prior to this was someone complaining that I took off points for an actual error, and trying to tell me the error was a mistake and that they didn't mean to make it

Which like... Yeah, that counts as an error! Sorry.

Obviously it's tougher to get to the prof about a grading mistake during covid (well, actually its probably easier due to virtual office hours, but my point is that it's different from usual)

[mint lark]

I've had students send me regrade requests just to vent

the context was like we had stated Heine-Borel at some point, because our progress in the worksheets / the homework can be different

[strange bronze]

regrade request: i couldve gotten a 100 on this if i just didnt make any mistakes

please revise

[mint lark]

and someone used Heine-Borel to prove compact => closed which was the whole problem

and I was like "this trivializes the problem and you weren't supposed to be able to do this"

and gave them 4/15 (4 points of completion on that problem since it was hard)

and they were just like "damn this is 1/4 of the homework grade...."

But like idk there was nothing else I could do

[turbid zenith]

Hot take: All of this stems from pointmongering and is why I'm against traditional grading in the first place 😛

[mint lark]

that is an extremely tepid take :P

[turbid zenith]

I mean

"Get rid of traditional grading" makes a bunch of people hiss

But in this company yeah thinking about it it's pretty tepid

Problem is incorrect? 0/1.

Fix it for full credit.

[mint lark]

I'm very glad that the course I do grading for is like

so curved

bc otherwise it would be depressing

feeling of: what right do I have to make these people stress out by hitting a button

[turbid zenith]

Yup

[shadow basalt]

I've seen a few problems on psets where even coming close to a solution is worth full credit

but honestly regrades can be logistically quite painful

[turbid zenith]

That's respectable depending on the problem I think

[shadow basalt]

I am not sure what the correct solution is

[turbid zenith]

I think with mastery grading you can hold them to a high standard without worrying their grade will tank

"This isn't right. Fix it."

I'm going to see how that works next semester when I teach intro proofs ... basically the way I was taught intro proofs a couple semesters ago (required course in my program but it was still good to see how it was done)

But logistically it would be difficult at large universities without the right structure

And I'm not 100% sure how they could do it

[shadow basalt]

This is probably intentionally terse but I would feel remiss if I didn't point out that you should probably explain why its wrong at least a little

[turbid zenith]

Of course.

On the actual feedback you'd point out what exactly is the issue -- give a counterexample to their statement, point out something they forgot to consider, etc

But "This isn't right. Fix it." is kind of the underlying mantra I guess

[dawn walrus]

In lieu of not having regrades, I have opted to try to explain somewhat to every student why I'm taking off points for any given thing and give them suggestions on how to remedy it

sometimes that's easy like "I think you copied this definition down incorrectly"

sometimes that's pretty tough to do if they show a conceptual misunderstanding

or if they don't "see" how to prove something and flounder around a lot

what i will say is that i did have one grad class where the grading was, hand in a homework, prof grades meticulously and asks for us to fix them and resubmit them, sometimes more than once if needed.

that was definitely my favorite style, but it was also a small class with like 6 psets

i can imagine it would be too much logistically to grade and regrade and regrade in a standard weekly pset class with like 30-40 students.

[brisk ruin]

That does seem very ideal

[shadow basalt]

i also really think a good class has weekly psets

esp if it moves quickly in material

[brisk ruin]

Weekly homework is fairly standard

[shadow basalt]

well

6 psets

is a lot fewer

[brisk ruin]

I don't think I've had a class that assigns homework less frequently than that

Except grad classes I guess

[shadow basalt]

this is what i was talking about

[brisk ruin]

Ah

Most of the grad classes I've been in assigned much less homework

[dawn walrus]

these psets were biweekly and usually had one or two problems which took the whole 14 days, so i'm kind of glad we had those tougher problems as opposed to more easy problems

i also don't think there are that many reasonable but also unique questions you can ask a late undergrad/early grad student when it comes to certain topics

[brisk ruin]

This is also true

[dawn walrus]

it's kind of this tough transition phase which lies between the early undergrad "still has difficulty doing proofs" and the mid grad "can handle larger scale problems"

[brisk ruin]

Simple problems have probably been seen before and aren't very insightful

Hard problems easily become research level

[dawn walrus]

which i found that almost all the "second year" grad courses I took at berkeley suffered from

yeah

[brisk ruin]

Yeah

[dawn walrus]

258 harmonic analysis definitely suffered from this

[brisk ruin]

That homework was a mess

[dawn walrus]

a lot of the problems were either trivial with the notes, or way way too hard

[brisk ruin]

Yeah

[dawn walrus]

christ's strategy was clearly "throw everything at the wall and see what sticks"

[brisk ruin]

Hopefully the notes undergo a few series of revisions

Before they make it into a book

[dawn walrus]

i was referring to C* algebras with rieffel when i was talking about biweekly psets

two of those psets were just one problem each lol

terribly hard problems

oh well

[brisk ruin]

Yeah

[dawn walrus]

278 noncommutative euler equations was the worst with this

[brisk ruin]

[dawn walrus]

because voiculescu literally only gave us trivial problems

[brisk ruin]

I mean

[dawn walrus]

and they were weekly

[brisk ruin]

Why are topics courses giving homework

I'm not sure they should be

Maybe they should

[dawn walrus]

but like wow. they were essentially just verifying proofs /computations from class in 2 or 3 dimensions.

[brisk ruin]

Oh

That's boring/not insightful

[dawn walrus]

i mean

it was insightful

[brisk ruin]

Oh well that's fine then

[dawn walrus]

in the sense that you don't have any clue what's going on when he's babbling about infinite dimensional lie algebras

but once you get your hands on a nice one and run the calculations for it you "see" why the noncommutativity matters more directly

like on the ax+b group and stuff like that

[brisk ruin]

Yeah

[dawn walrus]

i dont mind the problems being easy for sure, i didn't need them to be hard for them to do their job which was to illuminate the material

which WAS hard was

his way of grading them

[brisk ruin]

[dawn walrus]

which was that every week someone would have to volunteer to present their homework

and he would critique you while you did so in front of the class

which was honestly very healthy but still was stressful to no end, especially since the volunteer process was literally us sitting there awkwardly each week until someone said they'd do it

and we assumed that we had to somehow give everyone a go in this way

then he collected the homeworks and looked at them but didn't mark them at all, and then when covid started he just gave up and the only person whose homework he saw was the presenters

that was funny

214 differentiable manifolds had weekly homeworks and they were so hard

oh god

that class i spent 20 hours per week on the psets

[brisk ruin]

Oh my

Presentations sound very stressful

[dawn walrus]

i guess 10-15 some weeks

[brisk ruin]

214 homework sounds like my 118 homework

[dawn walrus]

yeah

[brisk ruin]

Asking undergrads right out of multivar/lin alg to compute distributional derivatives

Is something

[dawn walrus]

the worst was when he decided he thought the homework problems one week were too easy and decided to add a 6th one

[brisk ruin]

[dawn walrus]

and none of them were easy!

the next week he made up for it by only assigning 4

and one of them was like 4 parts

but whatever

anyway: in conclusion, i think biweekly problem sets for a suitably high level class are often better if not almost necessary.

[brisk ruin]

Yes

[dawn walrus]

OTOH my problem sets for undergrad real analysis / undergrad diff top / undergrad fourier analysis were all very reasonable in length and just difficult enough to make them worth doing, and those were classes where I think the weekly format was good.

[stark pine]

yeah, this is one of my problems with my course that only posts notes. There are only 5 assignments in the whole term, and no other assessments. It's really hard to stay on top of the material

and the assignments are pairwise disjoint in content

[shadow basalt]

yeah thats rough

[dawn walrus]

i feel like as long as i take physical notes and fill in any skipped steps later i tend to stay on top of material for a course without needing as many exercises

definitely when i get lazy and don't take notes, then i need more consistent exercises to keep me on top of things

[real mauve]

sounds about right. as much as people love the convenience of having recordings and taking screen shots, actually writing stuff by hand helps you retain the information more easily

[dawn walrus]

I also find that TeXing during class is not nearly as helpful for me.

And yet writing notes on a drawing tablet is totally fine.

There's got to be something about the hand movement there. Indeed, I wonder if young kids who grow up on keyboards will have them inextricably linked to learning in their minds in the way I have pen and paper linked. Perhaps liveTeXing will become commonplace in a decade.

[austere inlet]

I definitely grew up on keyboard and yet I feel the same way, I take notes on my tablet too. I feel like it gives me greater liberties and it's much easier to sketch stuff or draw diagrams

[stray crane]

I also tend to understand concepts better when writing them on paper. It gives me the freedom of creating a few quick sketches, and visualise. That just isn't possible with LaTeX.

[vagrant sundial]

Does anyone know how to troubleshoot LaTeX on overleaf I am having an issue where I cannot compile two pieces of JavaScript

@wispy slate do you have a link?

[molten urchin]

I can DM an invite if you'd like.

Or maybe share it here, if that's okay with the server rules.

[vagrant sundial]

@molten urchin just dm me

[molten urchin]

Sure.

[wise lodge]

Could you please DM me for the LaTex Server as well?

[molten urchin]

Sure.

[wintry ravine]

Has anyone had any experience in homeschooling? Perhaps knows of a curriculum that is accessible online that can be shared?

[wispy slate]

Current 4th year math undergrad, very passionate
I was homeschooled for a semester of high school in a small group of other kids by a tutor, and one thing I can tell you is that the "sit down and calculate" approach made me hate what I thought math was. Eventually got into a transfer school and a collaborative environment with lots of project-and-inquiry-based learning had me falling in love with math. Just one person's experience tho @wintry ravine

[shadow basalt]

Thoughts on giving students (a small amount of) bonus points for writing their problem sets in latex, w plenty of support for learning how to do that if they wanted? Let's assume that I implement the bonus properly so it doesn't affect curves etc. and is actually a bonus.

Twofold objectives for this

[brisk ruin]

Yeah sure

[shadow basalt]

1. Getting kids to learn latex w for stem majors is essential

[brisk ruin]

This sounds pretty reasonable

[shadow basalt]

2. I hate handwritten stuff

[brisk ruin]

Well

It depends on what level the class is at

[shadow basalt]

Yeah see

It's calc

[brisk ruin]

I wouldn't do this for an intro calculus class

I would for a real analysis class

[shadow basalt]

I really really really hate

reading handwriting

and this seems like the most innocent way to reduce it

[brisk ruin]

Are these calc students stem majors?

[shadow basalt]

Not necessarily

[dawn walrus]

my current class does this

that i'm grading for

i think it's warranted

[brisk ruin]

Your current class is also a real analysis class for math majors

[dawn walrus]

we advertised a latex learning workshop to them

and one person came

[shadow basalt]

Okay so

Let's say

[dawn walrus]

out of the 35 person class

[shadow basalt]

for sake of argument

we are in a world where someone else getting bonus points

[dawn walrus]

most of them still don't know how the fuck to use latex

[shadow basalt]

doesn't hurt you at all

is there any real downside to my idea

[dawn walrus]

no

[brisk ruin]

No

[shadow basalt]

Okay so the question ig is how I can make that assumption true

[dawn walrus]

huh

[brisk ruin]

[tame lion]

I would fully support that

[brisk ruin]

Seems difficult

[dawn walrus]

that depends pretty deeply on the expectations of the school

[shadow basalt]

As long as you apply the bonus points post curve

I feel like it would be fine?

[brisk ruin]

How is it curved

[shadow basalt]

Does it matter?

[brisk ruin]

I think so

Does it not?

[dawn walrus]

you could do something different with it than just applying bonus points, like give the students that latex all their homeworks an extra homework drop

hmm

that's kind of weird

[tame lion]

I think that might be too much

being able to drop a whole hw

can be big

[shadow basalt]

Well of course how I curve matters in the sense that like

i should choose a decent curving scheme

[dawn walrus]

i mean if you're not already doing one homework drop

[shadow basalt]

but if you apply bonus post curve

it shouldn't hurt any other students

[brisk ruin]

Hmmmm

[shadow basalt]

it might help them less

[brisk ruin]

What if

[shadow basalt]

but im ok w that

[brisk ruin]

LaTeX

[shadow basalt]

if you latex all ur psets you get a free A

[tame lion]

max's take is colder than my take in discussion general a while back

that latex should be mandatory

[brisk ruin]

Meant that you would alter the grade breakdown from homework/tests/other stuff in a favorable manner

[tame lion]

for certain courses

[shadow basalt]

I think this is true above calculus (i.e. analysis)

But im gonna be grading/teaching calc

most likely

[tame lion]

true, there are a lot of calc students who're not in STEM

I think a fixed post-curve boost is fine

[dawn walrus]

well

latex is not mandatory for the core upper divs at berkeley usually

i guess sometimes a prof decides to require it

(analysis, abstract algebra, complex analysis, etc)

[tame lion]

if I were a prof for basically any math course other than a general first calc course

I'd mandate latex

ngl

[dawn walrus]

most of the time for upper div electives, it's highly encouraged

i had grad classes where it wasn't mandatory

but that was a terrible decision on the prof's part when that happened

and anyone who wasn't latexing, let's be honest

wasn't doing too well in the class i bet

for other reasons, but still

[tame lion]

most of the people I know latex basically everything, because it just works well

I know some google docs users, but it's pretty fringe

[lethal leaf]

I'd mandate latex in any proof based course

Cause as someone with shit handwriting I'd rather die than read my own handwriting

But also like my probability class is mostly numerical, not proof based and I'd want latex for that

But I think expecting people in a calc class to use latex is overboard

Especially since most of those classes use software like cengage and stuff

[dawn walrus]

ok

here's a take that should be ice cold

if you make your students pay for software to submit their work, you should be fired and forced to go find a job in industry. bye!

[lethal leaf]

facts

[dawn walrus]

i was never forced to do this dumb shit

but if i ever was

[lethal leaf]

I'm so glad I did calc in HS

cause we didn't use that software

[dawn walrus]

i don't think i'd be able to stop myself from giving the prof a piece of my mind

[lethal leaf]

I don't think the profs enjoy using it either

it's deals with the university

[mint lark]

this is bad imo

lots of students don't have the time during semester to devote to learning latex

[lethal leaf]

hm

fair

[dawn walrus]

i mean like

if you're taking proof based math

then using latex is an invaluable skill

[lethal leaf]

but I mean plenty of courses here mandate it

[mint lark]

mmm

[lethal leaf]

even the CS Algo class mandates it

[mint lark]

lots of people at my school

[dawn walrus]

that you will need whether in industry or academia

[shadow basalt]

latex is easy to learn

[mint lark]

have to take 217, which is proof based linear algebra

you underestimate how much people have trouble with it

[shadow basalt]

i honestly am not that sympathetic

maybe

[mint lark]

I have tried to teach many people latex

[lethal leaf]

I can see how it's hard

[mint lark]

and they have struggled

[shadow basalt]

ive never met anyone who struggled w latex

[mint lark]

very hard

[lethal leaf]

I know people who struggle

[dawn walrus]

i would say latex should be mandated in any class that is only a requirement of math majors

[lethal leaf]

but no one who never gets it eventually with a little work

[shadow basalt]

i can't imagine learning latex would take like

more than one or two psets

to get the hand of

for basic math?

[dawn walrus]

well like theres a difference between learning and typing fluently

[lethal leaf]

it doesn't but I wouldn't throw them full in the deep end

[shadow basalt]

not asking for tikz lol

[lethal leaf]

give them a reference, a template

[dawn walrus]

for a year or so it took me a lot longer to type my hws

[shadow basalt]

hell make them learn Lyx

for all i care

[dawn walrus]

than to write them

[lethal leaf]

wtf is Lyx

[shadow basalt]

WYSIWYG

latex

[dawn walrus]

but the problem is that if you don't learn it in undergrad

then are you really gonna have more time to learn it wherever you are next?

[shadow basalt]

lmfao

the thought of not learning latex in ug

is so cringe i cant imagine being a grad student

and not knowing it

that would suck

[dawn walrus]

ok i think the argument here isn't so much about learning in ug vs not

i think it's more about what year of ug

[shadow basalt]

asap

[dawn walrus]

so i'm being misrepresentative

[shadow basalt]

college only gets harder

[dawn walrus]

sorry

[shadow basalt]

imo

[dawn walrus]

agreed (for the most part)

[shadow basalt]

first year you have more time than any other year to learn latex

for sure

[dawn walrus]

if there was some way to mandate every math major learn it in year 1 or 2

that would be nice

[shadow basalt]

ive seen people suggest a class

but honestly

latex isnt hard enough

[lethal leaf]

you do not need a whole class

[shadow basalt]

to justify that

latex is very straightforward with some practice

[dawn walrus]

well you know how some schools have a class about intro to proofs

[lethal leaf]

just record some videos using pirated bandicam

[dawn walrus]

shouldn't that also be about writing mathematics

[shadow basalt]

it should

rarely is

[lethal leaf]

I think an intro to proofs class is a perfect place to require latex

[dawn walrus]

i think we were talking about a proofwriting class before that wasn't just about how to think about proofs

i agree that a proof-based linear algebra class isn't necessarily a great place to introduce people to latex (though the people outside math taking that are usually EE/CS who should also know latex)

[shadow basalt]

idk if ur writing proofs

they should be in latex

i think i would need to really see some firsthand examples of ppl struggling w latex

to change that opinion

[stark pine]

a friend of mine has dyslexia, and it took her a long time to learn LaTeX properly, and she still often struggles to use it well. She sometimes gets someone to transcribe what she says, and doing that with LaTeX is impossible. Handwriting works much better. If you go through with this, I would check how it affects students with accessibility needs

That concern aside I like the policy, but it's a pretty big concern

[shadow basalt]

That hadn't occured to me, why is transcribing harder with latex out of curiosity? Is the dyslexia worse too? I guess it would be a big issue with easily messed up markup langs

[stark pine]

she's said that taking the commands and turning them into words is hard. She mostly only transcribes actual essays, no math, so she can just like give a speech. With latex, you need to articulate how to spell the commands because you can't just say the word, since not all commands are phonetic. I think she's said even when dictating mathy stuff for typing in word or whatever, you can just say the plain words for the mathematical expression. "the sum of a sub n from 0 to 10" is easier to say out loud than "backslash sum underscore open curl brace n = 0 close curl brace up arrow open curl brace 10 close curl brace a underscore n", and when the commands are weird things that aren't words, spelling them is also hard

if you were transcribing to someone who knew latex I guess it may work since you could say plain english, but during the pandemic she's been using friends and parents, so that's not a guarantee. If your uni provides transcribes who do know latex this may not be an issue

@shadow basalt I hope that's sufficiently clear, if you want more details I can ask her sometime

[dawn walrus]

i do think it's important to account for accessibility accommodations with this

and also for the possibility of limited technology access, though that is less of a question nowadays

[austere inlet]

definitely

my own experience as a student required to hand over (and now grade) TeX-ed assignments is that pretty much anyone can learn enough to be able to type their work, but you often see a ton of hack-ish workarounds and bad typesetting, and sometimes even glaring compile errors which are ignored by e.g. Overleaf. I've even seen this in notes written by my professors. Guess it's harder to get proficiency than learning the basics. I don't think it's a huge issue though

[tawny slate]

I feel like somewhere there exists an app in which you could scan in a picture of a handwritten math thing and it would spit out the latex transcription

Could've sworn I've seen this before

[real mauve]

by latex transcription you mean latex code or the typeset output?

[wispy slate]

http://detexify.kirelabs.org/classify.html

https://mathpix.com/

[tawny slate]

yep that's it

[alpine wyvern]

God mathpix has saved me so much time 🙏

[craggy atlas]

has anyone take the math content exam (TeXes) from grade 7-12

[brisk ruin]

Why do you ask

[craggy atlas]

just want to hear their experience

[austere inlet]

and why here

[turbid zenith]

I assume because it's a test for teachers, @austere inlet

Unfortunately no I have not @craggy atlas

I took Georgia's version, GACE

[craggy atlas]

yeah test for teachers

[molten urchin]

How do you motivate the idea of irrational numbers to someone who has very basic knowledge of algebra(~at the level of American Algebra 1, and knows what the rationals are)?

[brisk ruin]

We want polynomials to have solutions

sqrt(2) is a solution to x^2-2=0

sqrt(2) is not rational

Thus, we need more numbers

[molten urchin]

The problem is Ange, it is not immediately obvious why sqrt(2) shouldn't be rational. To patch this up, I presented a proof of this, but I still feel I somehow didn't do justice to the idea of irrationals-I remember the first time I was given a proof of this, it felt like symbol pushing.

Is there not an alternate path to talk about irrationals in general, without diving into too much technicality?

[brisk ruin]

Hmmmmm

[molten urchin]

Is the only half-convincing way to achieve this, at this stage of education, to say that rationals have terminating or eventually repeating decimal expansions, while it is not so for irrationals?

[molten urchin]

In a way, yeah. It's hard to tell someone why something like \pi can't be the ratio of two integers, but I can probably circumvent to say that \pi is known to have non-recurring non-terminating decimal expansions or sth. Not the best way to put it, but it probably captures the intuition of someone at that stage better.

Hmmmm

The problem with "not rational" characterisation is that it is definitional and not constructive. If I'm showing someone there exist numbers which are not rational, then definitions don't help.

[clever jetty]

Well,Irrational numbers are supposed to be annoying

You have to live with them because lub

[molten urchin]

I agree, irrationals do not have as many interesting properties.

[molten urchin]

I'm addressing a Grade 10 student here lol.

Even building up to the machinery for something like that would take a year or more.

[clever jetty]

idk,Took me about 3 days

[molten urchin]

[clever jetty]

I think Pugh is kinda good at that section

[molten urchin]

Drake, this is literally a grade 10 student who still doesn't know how to solve a system of two linear equations at this stage.

[clever jetty]

[molten urchin]

If I could go into that knee-deep technicality, then it would be a non-issue obviously.

[clever jetty]

What does he know

[molten urchin]

Suffice it to say he knows what naturals, integers and rationals are and understands the motivation for them.

[worthy latch]

maybe use geometry?

[molten urchin]

Has little knowledge of algebra(solving linear equations in one variable, or quadratic equations)

[worthy latch]

length of diagonal of a unit square?

[molten urchin]

The problem is still the same det

[brisk ruin]

This is the first thing I brought up

[molten urchin]

It doesn't address why sqrt(2) can't be rational

You still have to prove that

And the proof seems rather artificial at first glance

[clever jetty]

Why not do rational roots theorem

Let's suppose √2 is rational

√2 is a rational root of x^2-2

Which means √2=a/b(a,b integers) would imply a divides 2 and b divides 1

But clearly √2 is not 2

[worthy latch]

maybe look at some visual proofs?

i remember this guy mathologer on youtube doing a lot of visual proofs

[molten urchin]

Hmmm, rational root theorem would be a major digression. Would have to take up a treatment of polynomials in greater generality.

[molten urchin]

That's what I was hoping to get too

I suppose I should look up for visual proofs

[worthy latch]

we want to look for some naturals such that a^2 = b^2 + b^2
So we want to cover the area of a square with 2 smaller squares

place these two smaller squares diagonally in the larger square

so that they overlap in center

[molten urchin]

This is slick!

[worthy latch]

but then the section left out are 2 smaller squares and we have an overlap of a square

so you get a smaller solution

[molten urchin]

So we either get an overlap, or we don't get a complete covering for integer values

This is neat

I'll be using this tomorrow. Thanks a ton!

[worthy latch]

algebraiclly,
a^2 = 2b^2
(2b-a)^2 = 2(a-b)^2

[molten urchin]

Gotcha

[worthy latch]

we can't keep on getting smaller and smaller solutions because the lengths are supposed to stay integers.

[molten urchin]

[tidal topaz]

It's definitely a technical subject, I think a good way to introduce it is a historic one
Show why the Pythagoreans liked rational numbers so much, and why the existence of irrationals came out as such a shock to them

[mint lark]

I agree with the historical approach

[molten urchin]

Okay, I'll look more into the history. It is likely that the Greeks had discovered some visually convincing argument about existence of numbers other than rationals, which might help.

[sinful thistle]

Oh, this proof's pretty

[tawny slate]

An interesting observation I made a while ago but couldn't figure out how to really explain the difference (which could also mean that I don't fully understand the difference):

In modal logic, by default you assume things are true until they are shown to be false

For instance the statement "If A, then B." This statement makes no claim about what happens if A is false, and yet if A is false we just say this is vacuously true, not vacuously false.

And yet, in epistemic logic, we assume things are false until proven true, because of the null hypothesis

I feel like the difference here is that the point of modal logic is to check whether something is consistent, if something is reasonable, or valid, while the point of epistemic logic is to make sure that all of the premises or axioms are also true, that the argument is sound

But that doesnt really explain why we assume modal logic in say digital logic design, which doesn't have anything to do with logical arguments, but as simply operations on boolean algebra

I would think this is based on convention, that either one would be valid, but is there any basis for that convention then?

[brisk ruin]

I think this is better in #foundations

[ocean hinge]

Agreed

[spark thorn]

Hey there
Guess what this is

For math history pedagogy

[lethal leaf]

Oooo

[brisk ruin]

Ah

Pascal's triangle

Dating from 1303

The characters used for the numbers are counting rod numerals

[round robin]

:o

is this old chinese numbers

if so then anything above 4 must have changed so much

[brisk ruin]

Not necessarily old

These rod numerals come from counting rods which were sort of abacus-like counting/calculation tools

Which is why all the strokes are straight lines

[round robin]

:o cool

[modern solstice]

I think the distinction you're making is rather between procedural processes and proofs

If A then B is a command

[tawny slate]

how exactly does that explain the truth tables for conditionals then?

[modern solstice]

Its just a different framework with different end goals

[spark thorn]

in fact 1261

[brisk ruin]

https://commons.wikimedia.org/wiki/File:Yanghui_triangle.gif

Wikimedia says 1303 so that's what I was working off of

[spark thorn]

from wiki article

The earliest extant Chinese illustration of 'Pascal's Triangle' is from Yang's book Xiangjie Jiuzhang Suanfa (詳解九章算法)[1] of 1261 AD, in which Yang acknowledged that his method of finding square roots and cubic roots using "Yang Hui's Triangle" was invented by mathematician Jia Xian who expounded it around 1100 AD, about 500 years before Pascal

[brisk ruin]

Sure but is the image you sent the earliest one?

[spark thorn]

I think so

[turbid zenith]

https://twitter.com/solidangles/status/1389194128832016389

Bill Shillito (@solidangles)

So I just realized last night that the AP Calculus test is tomorrow.

Then I realized how blissful it was to not have the looming pressure of a test on which the entirety of my teaching effectiveness would soon be based.

I do NOT miss teaching AP. (1/n)

Twitter · May 3, 2021, 12:24 PM

[civic tree]

ap calculus

[trim violet]

what do the a and p mean ?

[potent ether]

algebraic projective calculus

[mint lark]

Interesting article on teaching calculus

http://rtalbert.org/the-semester-that-was-a-3x3x3-reflection-on-calculus/

[pastel epoch]

Hi, I hope this is the right place to ask (if it isn’t please direct me to the proper channel 😅):
What should someone who wants to teach themselves something be doing in order to properly develop intuition and skills in said subject?
I have been blessed with incredible teachers most of my years, and I spent some time mentoring students in calculus for a while. And I noticed that a lot of the times a student relies on the teacher/professor/mentor to guide the development of intuition.
I am planning on attempting some self-study soon, and wanted to get some tips on how to properly approach this in a rigorous way without taking a formal class on it.
Any thoughts on the philosophy of teaching that apply to this are also very welcome.
Thank you

[dawn walrus]

The most important thing is always to do exercises

the second most important thing is to try to explain / talk about what you're learning with other people

[austere inlet]

I'm not sure I entirely agree with

If you do a problem asking to find and classify the critical numbers of a function, and you are allowed to check this with Desmos by throwing up a quick graph to see whether the local extreme values are where you say they are, to me it seems like manna from heaven — a pathway to mistake-free work.
since a graph is not proof, bluntly put. But a "conceptual" rather than "algorithmic" calculus is a very attractive idea and one that should be put into practice more often.

[winged urchin]

Does anyone ever think that algebra gets a bad rap? Like I just mean, 'pushing symbols' kinda thing of just manipulating an equation or expression to see how else you can express it. Just sorta leapfrogging with the current discussion

I do find value in students seeing graphs and ultimately having more interactivity

But I also feel like sometimes we demonize just... playing with equations to see what we can turn them into

Also @pastel epoch

If you're by yourself I'd say make sure you approach every topic very multiple angles. You shouldn't be happy with just one way to verify a claim until you're much more comfortable with the material. Since you have no teacher or resource to bounce back off of other than possibly answer keys

But ultimately I think approaching problems from different angles can be very enlightening overall

[round robin]

intuition develops overtime as you see and use it more often

if you think of something hard enuf, for long enuf, you'll likely get an intuition

[round robin]

i feel like main issue is that most people just learn how to do it and then just practice it, to the point they probably think math is all about increasingly complicated symbol pushing

[round robin]

agreed, less mundane practice more intuition = better

[pastel epoch]

Thanks for all the input everyone, appreciate it 🙂 I’ll try it out!

[strong carbon]

This is what happens when the people push too much on numbers and symbols but not for the insight itself.

[turbid zenith]

DONE TEACHING

Also here, have a thing, related!

https://twitter.com/solidangles/status/1391851225734385674

Bill Shillito (@solidangles)

It's time for the COR 314 Meme Review!

My Mathematics and Human Nature classes made memes related to the course for our last day today, and BOY DID THEY DELIVER.

Here are some of the memes, along with what we studied that prompted them.

Twitter · May 10, 2021, 8:22 PM

[balmy iris]

Guys, do you think average highschool students, that are not going to mainly study theoretical math, need to know strict definitions and be strict in their work? For example i gave 2 of my student a task to make presentation on translation rotation and 2 symetries and one of the studends found a strict definition of one of them in terms of maps, with right notation and everything, propaply they found some properties that are not tought in high school too. So i told them not to learn that since in real world they dont need it and that intuition of how that all looks in practice is just much more important than formal stuff. What do you think about it?

[strange bronze]

well, i certainly wish more students came into calc 1 knowing how functions work

the amount of students who see notation like $(f \circ g)' = (f' \circ g) \cdot g'$ and just have no clue how to parse it

[burnt vessel]

Namington

[strange bronze]

is really concerning

and the idea that we typically represent mathematical "processes" and "constructions" as functions is a good one to teach, even for very basic applied stuff

a proper formal definition, though, is almost certainly unnecessary

i wouldnt dissuade a student from pursuing that though

although id perhaps recommend they leave it out of a presentation

but i mean... the formal construction here isnt very complicated

unless i misunderstand what you mean

[balmy iris]

Oh, I didnt mean only in this case, but in high school math in general. I was tought a lot in high school becouse i wanted to know those stuff, but for students that doesnt want to do math i think there is really no need to preasure them.

I agree with you, when students come to calculus or even subject called "Elementary math" in my uni ( Mostly high school.math in there just more strict in solving equations and proofs) they need to know some stuff and have some solid foundations but for those who doesnt go for math... i mean just learn the basic intuition that you may need sometimes and dont bother with math more than you need to.

Its just the same when some like biology teacher force us to learn some latin names and memorize becouse "you need to know this" :/ i am honest with my students, in real life, you can live without knowing trigonometry

[wise onyx]

Dont discourage anyone from learning anything. You could recommend the student to omit technical details in their presentation. That's fine. But the student should still be free to learn those details if they want to.

[frosty flame]

This is horrifying. Trust me, your students already think math is useless and they're bad at it so they just need to pass. You don't need to encourage that.

[mint lark]

how do you teach topology to someone with aphantasia (an inability to visualize aka form mental pictures)

wikipedia for reference https://en.wikipedia.org/wiki/Aphantasia

[austere inlet]

some links that may be relevant:
https://mathoverflow.net/questions/237243/mathematicians-with-aphantasia-inability-to-visualize-things-in-ones-mind
https://carma.newcastle.edu.au/brailey/pdf/apantasia.pdf
interestingly both seem to mention geometry and topology, aphantasia doesn't seem to be an impediment at all. It even eases the process of abstraction for some, it seems

[frosty flame]

Fun fact: I used to think I have aphantasia, lol.

[turbid zenith]

Going back to this, I think formalism is eventually necessary, but I think it's best if it's saved until later when the student has a good "feel" for the topics. Nothing makes a topic inaccessible like an overabundance of formal notations.

I understand that as mathematicians we often think in terms of "definitions first, and the examples will follow", but pedagogically I'd argue it usually needs to be the reverse if you want it to stick.

[real mauve]

especially when introducing topics in lower level maths, i think this goes a long way

[weak bane]

How is this server so based

[stark pine]

And I mean when doing research math, you don't pull definitions out of your ass, they show up because of examples/patterns you've seen (at least that's what I've heard). Doing examples then definitions can lead students to feeling like they could discover the definition or come up with it themselves, which is really helpful for conceptual understanding

[brisk ruin]

I feel like in the context given though when students are exploring on their own, they shouldn't be discouraged from presenting the formal definitions though?

[wispy slate]

(A+B)² = A²+B² over any ring of characteristic 2

[stark pine]

I think that a formal definition might be a natural culmination to such a presentation? like once the intuition/exploration is done you can talk about how it's formally defined, or introduce the formal definition at the start and work towards it making sense based on exploration/intuition

[brisk ruin]

Yes this is very sensible

I guess what I'm trying to say is that if the students want to present the formal definition, I don't see anything wrong with it, especially if they have examples to build intuition

[stark pine]

oh wait I read "shouldn't they be" not "they shouldn't be"

I am much more literate with food in myself

[brisk ruin]

[balmy iris]

Agreed 100%

[strong carbon]

I imagine a toddler seeing 3 animals of different breeds. The child notices that all of the animals have similar snouts, have 4 feet, hanging their tongues out, barks, and wagging their tails. The parent points to one of the 3 animals and says "dog", and the child repeats "dog"... and they do it three times.

[turbid zenith]

TOPICAL.

https://mathwithbaddrawings.com/2021/05/12/against-mathematical-proof/

[austere inlet]

provocative title huh. I do agree with the sentiment that proofs are best left at the end of a lecture/chapter, after exploration and motivating examples have been presented

[stark pine]

I think there often isn't time to do as many examples as are needed, and that's left to the students. My first year math courses and profs focused a lot on how useful examples were in forming conjectures and developing proofs, and it's something that profs have consistently encouraged we do on our own, before lecture/when reading the notes/after lecture/when studying, I just don't think there's always time

[inner turtle]

Apart from the time factor which I think is important, I'd argue that it is quite an important skill to be able to navigate a „dry“ text that doesn't start with examples.
Due to the time factor, there's always gonna be material that isn't pedagogically polished, so it's important to be able to construct examples and ask questions yourself even if the text/presentation doesn't provide that.

And at least for me, being confronted with a lack of examples in lectures has driven me to develop that skill.

[tawny slate]

In regards to teaching about the formal definition of transformations, it is also much more practical in the digital age, as are many other math concepts. Whether you're working with image processing, coding, coordinate geometry, games, etc. they could all use these concepts. Just because it's may not be useful to everyone or may not be useful now doesn't mean it doesn't have value. Imagine if we were talking about calculus instead

Also, isn't there also something said to just be learning math for the sake of learning math? I get that there are a lot of students that may not find it practical and useful, but what if they just find it beautiful?

When I teach combinatorics, there are 3 main epiphany moments my students have during teaching:

• bijections are not only a powerful problem solving tool, they are necessarily required to count, by definition
• the relationship between the "choose" numbers and pascal's triangle and why
• after introducing and proving many different combinatorial identities, show that a large subset of them can be proven in one step using the binomial theorem
  Each of those "aha" moments are checkpoints which motivate the students to explore more and learn more, out of sheer curiosity and intrigue. Prior to introducing these, they find combinatorics very dry because they primarily have two mindsets: "I know how to count already" and "it's still just adding, subtracting, multiplying, dividing"

Why do we learn math? Because it's practical and useful. But being interested or passionate about it motivates that learning

[wise onyx]

abstractions without examples are like souls without bodies

[austere inlet]

bourbaki:

https://tenor.com/view/alex-jones-tantrum-butthurt-rampage-angry-gif-13615981

[frigid crest]

https://www.youtube.com/watch?v=rU-u2yjFncc

https://www.youtube.com/watch?v=KpR7Ekb6k5o

[frigid crest]

https://www.youtube.com/watch?v=z4c9Hg-iLnk

[brisk ruin]

Are you going to continue posting these videos

[inner turtle]

“Hmmm, the first two unsolicited clips didn't spark any discussion, so a third one will do the trick for sure!”

[brisk ruin]

I mean like

I'm sure this videos are relevant

But

If you want to have a discussion about them, it may be helpful to briefly summarize the content contained in the video

As well as express a viewpoint about them

Otherwise, you're just hoping that somebody else will be interested enough to watch a random video

Like

Lots of people may be interested in discussing ideas contained in these videos

And they might not necessarily want to watch a video in order to discuss them

[wise lodge]

I for one I'm glad that they shared these videos despite having no discussion about it.

I am fan of both Lex and Po-Shen Loh, and had no idea that they collaborated. Thank you for sharing.

Actually maybe posting three is excessive and having a small summary would definitely be appreciated.

[tawny slate]

another one of the disadvantages to discord channels is that without pins things just get buried over time

[acoustic wadi]

https://youtu.be/_zCWzPS-i4I

https://twitter.com/mathequalslove/status/1394839699303411713?s=21

Sarah Carter (@mathequalslove)

While searching for a conic sections GIF the other day to show my students, I ran across this awesome video (https://t.co/n6Erqr8j4H) for visualizing conic sections from CutOutFoldUp.

I decided to blow up the free template and make a giant version. LOVE it!

#mtbos #iteachmath

Likes

104

Twitter · May 19, 2021, 2:17 AM

[strange bronze]

does visualizing conic sections as cut from a cone actually help pedagogically?

genuine question since my high school (canadian) never covered conic sections

so idk how theyre typically presented

[civic tree]

im not sure cus like

while visually it makes sense

the algebra doesn't seem to follow?

like im remembering the kinds of questions i used to do with conic sections and i never really had to think about the cone much

it's nice to know though i guess

[wise lodge]

I also study in Canada, and I was not formally introduced to conic sections until Advanced Calculus I (multivariable calculus) in University. The paper model is neat, especially since its easily craft-able, but in a lecture setting (which is often when conics are introduced), I feel like computer 3D graphics will go a long way and be much more useful. Something akin to the 3B1B video on ellipses IMO is a lot more helpful.

[tawny slate]

does anybody here have a framework for how to teach ratios/proportions? I don't have any problems teaching the concepts themselves, but for me it feels a bit unmotivated and uninspired compared to other topics. obviously there are lots of practical applications and such, but I haven't figured out many approaches that aren't very dry, like just setting up an equation and solving

the only interesting things I can come up with that are somewhat interesting are as follows:

• percent being literally "per-cent" or "divide by 100"
• concept behind conversion factors/dimensional analysis
• constant of proportionality (for direct, inverse, joint proportions)
• rate problems and common traps (and by extension, arithmetic mean vs harmonic mean)
• relation between rate and slope

does anyone have comments or anything to add to this?

I also appreciate problems which challenge students to not fall into traps such as "if 5 students finish 5 problems in 5 min, how long does it take 100 students to finish 100 problems?"

[brisk ruin]

Ratios and odds in probability are also interesting I guess

[acoustic wadi]

it nurtures curiosity? i was mystified by that pattern and it made dig deeper. learned about dandeline spheres.

[acoustic wadi]

if you want to measure the height of a flag pole but can only measure its shadow. take a ruler and measure its shadow. the solution comes down to solving a proportion.

also certain basic stoichiometric calculations in chemistry (or baking) requires ratios to solve more easily.

in cooking and in baking, halving or doubling recipes are very common.

[cedar garden]

would it make sense to restructure the math ladder so that students learn linear and matrix algebra before taking calculus?

[wise lodge]

I think it would. At my school, you can learn calculus and linear algebra in any order you like (even concurrently).

[turbid zenith]

Throwing an idea out there to see what y'all think. Say you wanted high school students to get their feet wet with SageMath, plotting implicit and parametric curves in 2D and 3D. What kind of activity might you do?

My idea so far has been to give a few examples to have them try out and turn it into a matching activity, then give some room for them to try tweaking them to make their own

[brisk ruin]

Any reason for sage in particular?

[turbid zenith]

Because it's free

And it's what I used in my Computational Algebraic Geometry course, so that's what I'm gonna use for my own CAG-for-high-schoolers course

[tawny slate]

i don't see math topics as a ladder, i see it as a graph, and in that sense I think they can be learned in parallel, until around calc 3 where I think both intro to calc and lin alg are both prerequisites

[tulip badge]

I would probably just show the students a similar video or maybe pre-build a few and pass them around for a few minutes with some prepared explanatory notes. (To me, the above video wasn't clear enough on the distinction between a parabola and hyperbola.) Anything more than that seems like an ineffective use of class time in my milieus.

[acoustic wadi]

the video itself was actually kinda lackluster. but it gives links to the templates for the cut outs.

[wise onyx]

I first used Sage to play around with partitions

You could have them try to verify known facts about partitions using sage to compute/test small examples

For example, you could have them verify/test Euler's partition theorem for specific values of n:

How would you write a computer program to compute these two sets of partitions? That's an activity that would get students to learn how sage works

Writing a general function to test this for arbitrary n, and the letting the computation run for all n up to 100 would build confidence in the fact that this statement is actually true

And that's how a lot of conjectures are actually made: you test out examples, and these days a lot of examples of are computed with computers, and then formulate conjectures based off those computations

[round robin]

feels more like learning how python works than sage ngl

but just throw any project euler problem and add additional stuff should suffice

[wise onyx]

Yeah thats basically it

If you know python you know sage pretty much

[charred dust]

Hi guys, wondering if any of you could provide me with some numbers games to create a 25 minute lesson for a 6 y.o

shes currently ok with 2 digit + 2 digit addition, the parent wants emphasis on partitioning past the tens threshold and wants to see some work around 3 digit + 2 digit addition

Also, any general advice for teaching children will be super super welcome.

[stone tusk]

exploding dots?

https://www.explodingdots.org/
might be useful to consider

[charred dust]

oh that looks so good!

[magic minnow]

Hi! I wanted to share my software for making interactive educational videos: https://ractive-player.org/math/

Here are some full-length examples of things made with this: https://epiplexis.xyz/m/9s7/elementary_functions

and a couple days ago I added the ability to embed Desmos in videos; here's an example (with source!) explaining function composition/chain rule https://ractive-player.org/blog/2021/06/01/desmos-react/

[shadow basalt]

can it do latex

[magic minnow]

yes, via either KaTeX or MathJax. The first link demonstrates this functionality

anything you can put in a webpage, this lets you put in a video

e.g. THREE.js for 3d graphics, d3 for graphs/charts, CodeMirror for coding tutorials, etc

[turbid zenith]

@magic minnow Do you mind if I PM you asking more questions about RactivePlayer?

[magic minnow]

not at all!

[tawny slate]

are there any plans for performance improvements in the future?

the more intensive parts lag really badly on my computer, and my computer isn't that weak 😦

[small quartz]

Woah, very cool

I submitted this 'graded' math portfolio while I was in Grad school to get my NCTM certification 🙂 I want to share it with y'all.^

[round robin]

I'm wondering anyone has a decent setup for like 1-1 teaching over zoom? Particularly how should one check the student's understanding other than like "could you take pic/point camera at work or smt"

[shadow basalt]

You could always ask them to prepare some of their work to present to you / walk through with you? I think working together on problems can be helpful? My gf had a one-on-one course this semester and she just did work during the week, emailed it to the prof a few days before the meeting or something, and then they met to talk about what was hard, what she understood etc

[quasi musk]

It's much more difficult to look at procedural understanding/mastery over zoom

It's easier to assess conceptual questions, or have students talk through the set up of the question, and how to work through it

e.g. "In this problem, what is the relevant information? How does this relate to the things we've learned lately?"

And so on

[round robin]

ohh thats possible yea jus like take pic send solutions before meeting

[round robin]

mm yea thats true maybe should focus more on this type of qns for now

[round robin]

the idea of taking pics before class sounds good tho

prob would do that

[lethal leaf]

The hardest part is definitely seeing their work yea

Especially if you give them a problem during the session

It's bad

Well not bad

Just hard to deal with

[dawn walrus]

My method when doing physics tutoring was to have them share screen with me with their problems/work, and I would use the Annotate feature to draw on their screen with my input. Of course, if they're not doing their work electronically that makes things tough.
Another way I did this was to share my own screen, and have them dictate their ideas to me. I'd write what they were saying (say, in onenote), and draw pictures of it. This was good because when they'd get stuck I'd throw in a suggestive picture that might get them to the next step. It also helped me to focus on conceptual understanding of how to solve problems (instead of wasting time waiting for them to do a bunch of arithmetic), they have plenty of homework to go through the details on their own but my goal was trying to help them to understand intuitively what they were actually doing when they'd solve problems.

[topaz scarab]

why not just do the work on the computer?

• ms paint / any kind of online whiteboard
• I like one note
• ms word equations
• overleaf latex

plus anotate like what RYC said

i find anotating with mouse is not too hard

LOL.
lower your mouse sensitivity, that helps a lot

lower it way down than what's normal

[round robin]

^ this is my top concern when trying to like ask hi could present work

ouchhhhhh

[round robin]

oo the second is not too bad actly

[molten urchin]

I personally use the ask-along-as-I-do strategy, since it is not feasible for the person I'm tutoring to share their work. (I do think I could ask them to submit the stuff I ask them to do).

[severe kelp]

yeah thats kind of where I am, depending on the student i ask them what to do or other probing questions while sharing the screen of my tablet

if they have a tablet or w/e i make them do the problems with like either probing questions or pushes in the right direction

[winged urchin]

Also another useful thing I find sometimes is changing up the current question they're working on

Like maybe they're solving sinx = .5

Once they seem to get that I might ask to solve sinx=1.5

Usually it's those kind of... Can they see the problem with this new question?

Kind of vibe

And it can test or reinforce their understanding

[round robin]

til teaching live is so different from typing notes

was so disordered rip

[hard badger]

I don’t know if this is an appropriate place to ask this, but:

I just graduated undergrad with degrees in math and physics, and while I’m in grad school I want to be a private tutor to make some money on the side. I have experience being a TA, being a volunteer tutor for my undergrad physics department, and minimal experience as a private tutor. I was planning on tutoring math and physics, and was wondering how to figure out an hourly rate that is fair to both myself and clients.

Things that seem like good ideas (I could be wrong though)

• Charge slightly more for GRE prep (or other standardized tests)
• If I’m not finding clients, reduce the price

[austere inlet]

sounds good to me, these are common sense rules really, supply and demand

try looking into the market as if you were a client, i.e. find out what the usual rates are for the topics you want to offer

[wise lodge]

I would not charge too low because that could have the opposite effect. Potential customers might assume that cheap rates equals not great service.

[hard badger]

That’s a fair point, thanks

[quasi musk]

You charge more for k-12

Charge less for college/grad students

They are broke

You can charge the most for AP Calc/Physics/SAT/ACT since the value they get out of it is better than other htings

[hard badger]

Ah, okay
So in terms of how much to charge:
1.) AP Exams, SAT, ACT, SAT Subject Tests
2,) K-12 Classes
3.) College/Grad Classes

Where would GRE prep go?

[quasi musk]

The highest are number 1

the second highest is probably GRE prep, third highest is k-12

Last is the college material

[hard badger]

Okay, cool! I appreciate the feedback
Do you know of any good places to look online to see what others are charging?

[quasi musk]

It'd have to be your area

[hard badger]

I’ll be in the NYC metropolitan area

[quasi musk]

ok for AP exams just go for $50/hr

Anything AP = $50 starting

[hard badger]

But also i imagine most things will be remote anyways

[quasi musk]

If you don't get hits, just lower by $5 till you get hits

k-12, you can probably go for like $35-40 easy

[hard badger]

Makes sense
So the highest I should reasonably be charging in your eyes is like $50/hour

[quasi musk]

SAT ACT you should be gunning $65

[hard badger]

Ah ok

[quasi musk]

College just go for like $25

they are broke

anything higher and you get nothing

[hard badger]

Relatable

Thanks for the feedback and actual concrete recommendations. I’ll also be looking at other things and whatnot, but this is a great start 🙂

[lethal leaf]

What what it's worth I've done HS and College tutoring for 30 an hour just when I was in high school

It all depends on the surrounding area

There were some richer areas around me and I could charge more

[round robin]

Any tips as to how does one like structure a class in the sense like
How does one figure how much content is "an hour worth of content" instead of jus running through everything lol

[real mauve]

i have not yet found an effective system for that. even when stopping and asking straight up if anyone has any questions, i usually get no response and jsut keep going. at the end, students just say their brains are fried and it went too fast

maybe having frequent stops where the student has to try using the stuff they just learned might help

[round robin]

i usually get no response and jsut keep going. at the end, students just say their brains are fried and it went too fast
this tbh

[round robin]

hmm perhaps a small "exercise break" could help 🤔

[grand laurel]

yes, ask frequent questions
although this has to be taught to a class especially if they are "early" in their education

i also try to be honest with my students and tell them that it's not a problem if they ask "dumb" questions and that it's hard for me to tell if i am being too fast

to make this more easy you can have small 5 min breaks with a question and have them discuss it with other students first and then continue with an answer from a student