math-pedagogy

[tawdry venture]

i see

[brazen pendant]

I wonder how lenient schools here are to things like this; I’ve seen teachers do their own thing to some extent but not entirely remodeling the grading system (in the end I’d have to give out a numerical grade in [1,6] rounded to the nearest 0.5 anyway)

luckily those are things I don’t have to worry about for a few more years ^^

also uh, if the content average cannot excede 1.0, why is there a 1.5 on the scale @turbid zenith?

here usually the grading system is just:
each exam gets a weight and a grade
your final score is the weighted average over all exam grades ± some flat bonus or malus of at most 0.25 based on participation or what not (if there is a malus, it must be clearly defined in the beginning of the year). average to nearest 0.5 and that’s your grade

importantly, it’s always only exams

I’ve never had graded homework

nor a participation grade, or anything like that

[turbid zenith]

@brazen pendant The average can't exceed 1.0, but you can have individual 1.5's

[brazen pendant]

ah so basically the 1.5s aren’t relevant for your final grade?

[turbid zenith]

Well they can boost some 0.5's

[brazen pendant]

but are nice to have for morale boost

ah, right

[turbid zenith]

So if you have all 1.0's except a 0.5 and a 1.5 that's the same as having all 1.0's

You showed partial understanding on one standard, but you maybe found some other standard really interesting so you asked to pursue it further

[brazen pendant]

yea my math teacher back in high school would often do a thing where he made exams have, say, 30 points for a perfect grade, but you could get 35 or so in total and he’d just give out grades higher than the maximum possible grade, but if your average exceded it he had to round down because you just … can’t have higher grades than max ^^

[turbid zenith]

The max possible overall grade I think comes out to 105

The only way to get it is to also get the CAP scores all to 1.5's, which I did to reward habits of good mathematicians

[brazen pendant]

I definitely see myself trying sth like this if:
a) I end up teaching and not pursuing academia
b) I can convince whatever bureocraty I need to convince

it’ll probably involve teaching “normally“ for a few years before I can try sth unconventional like that without them wanting to just fire me :P

[turbid zenith]

Yeah I gotcha

[brazen pendant]

I’m a bit confused tho

you say both that you average over the scores in content and that new scores overwrite old ones

isn’t that contradictory

[turbid zenith]

I average horizontally but I replace vertically

I average over your most recent scores in each standard

But if you get a new score on a standard it replaces the old one

[brazen pendant]

so if I really don't get solving systems of equations but trig makes sense, then depending on which order you teach these things in I could have a 0 or a 1?

[turbid zenith]

No.

You don't get systems of equations, you get a 0.0.
You get trig, you get a 1.0.
Currently your content average is a 0.5.
But now you worked on systems of equations, you get it, you retake that standard and get a 0.5.
That 0.5 replaces your old 0.0 in systems of equations.
Now your content average is 0.75.

[brazen pendant]

ah, you can redo things
how is this actually handled in terms of classroom time?

[turbid zenith]

"you can redo things"
That is the entire essence of it being called Mastery Grading. 🙂

To retake things students had to schedule it with me.

[brazen pendant]

how did you actually assess their proficiency? oral exams?

[turbid zenith]

I'd come up with a new problem on the spot.

[brazen pendant]

did you ever do classic exams, if only for the purpose of practicing the sadly necessary skill of test-taking?

(because they’ll have other teachers after you and you don’t want them to do badly in those classes just because they’re not used to math exams)

(idk if this would be a concern in the US but here we typically have the same math teacher for all four years of high school)

(the schools also mandate a big written final covering all materials of those four years)

(which is the same for all classes)

[turbid zenith]

Most of it was classic exams and quizzes

But the grading was just different -- I gave holistic grades based on how well they performed on the skills they needed rather than "points earned out of total points"

[turbid zenith]

https://www.mathvalues.org/masterblog/launchings201911

[brazen pendant]

“I cannot read your solution to exercise 3, but I know it is wrong because it is not solvable in one sentence”
-comment I just had to leave on a homework assignment

I still don’t know whether I love or hate the fact that my uni generally has about 90% of each class’ grade be the final exam

on one side, I abhor graded assignment, participation grades and what not

but a single exam is more representative of your mood on that day than your understanding of the subject tbh

at least we don’t have the time pressure as much, we’re given a long time to prepare for the exams

5 weeks in winter, 9 in summer

(the expected workload being 20 credits worth of exams in winter and 40 in summer)

[wispy slate]

I would hate if my final exams were 90% of the grade

[grand laurel]

i love what my uni does

you have to pass a written exam that is not graded

to take an oral exam

that you can schedule yourself

and which is graded

it produces overall better grades and higher understanding in students

[wispy slate]

For me there are 3 exams during the semester, the third one being most important. You also get points for homework and activity, but it doesnt matter that much. That said, I know people that had a really bad day on finals, (literally 1 person had 3/50 points) and still passed, just because they did really well in the first two exams.

I think its a good system, appreciates the hard work through the entire semester, not Just cramming for finals.

[lime echo]

The standard here is 80% exam, and 20% via whatever the means the professor likes (biweekly assignments, class test midway through the semester, a coursework project), however my GGT module is 100% examination x-x which I'm kinda scared about

[wispy slate]

I just finished my lecture about induction and proofs by contradiction I gave to 14 year olds. I feel like a lot of people didn't understand it (at least the induction part) and just feels kinda sad. Although I think many people got the contradiction thing, because there has been a lot of questions and ideas how to solve the problems, which is good since induction is less important for the kids olympiads. Kinda made me appreciate my lecturers more now haha, and I suppose giving a random answer no matter how stupid it may be will be much better than no answer at all!

[leaden rose]

Well Godel, mathematical induction and proofs tend to be a hard subject, however it can be fun. I know I'm not a teacher and I might not be allowed to talk here, but we're doing mathematical induction in university right now and a lot of people are still struggling with it. I'd recommend you to find a way to make it fun, especially if I look at the way I was when I was 14 @wispy slate

[wispy slate]

It isnt an obvious concept thats for sure, although it was aimed to kids who are supposed to do olympiad stuff, So I thought it would be a good idea.

[leaden rose]

Ah, united states?

Honestly I think this is way more fun than the calculus that was shoved down my throat in highschool for 7 years

Like mathematical induction, the fact that it has a default format, as a programmer attracts me a lot

All I can say is try n make it fun for kids, I've taught kids programming before and keeping it fun and having an obvious goal is very important

it's actually something i never had in math in highschool

[wispy slate]

Well, me neither, I got opprtunity to have one class for kids that as I said prepare to olympiads. So I chose contradiction proofs and induction as my topic, because I like it and think its useful, but this was probably the only class I was going to give, unless I have time in future I suppose.

[desert hinge]

How can I bring up the concept of symmetry groups to high schoolers after making them complete this activity:
So the example I will be using is having playdoh and dowels. In the example a group of kids gets 3 different colored dowels and have to rearrange them to make distinct squares,pentagons. What makes each shape distinct is if you can not rotate to get the other one. When they finish the examples we will make the symmetry reflection instead of rotation

[sick fractal]

@wispy slate as someone who has done quite a large amounts of olympiads during high school, I feel the most important aspect to introduce about induction is that it’s not strictly a x to x+1 and so on type thing. Rather, it’s really a proof by contradiction that there does not exist a predecessor statement that is false and that the existence of a predecessor implies a successive statement.. It is this understanding of what induction reallly is, that really pulled me through my uni years

And still are helping me

So as a tutor for some high school math kids, I would really focus on getting them to question why instead of how

Because induction etc can all be done with practice, but if they don’t understand how to apply induction because they do not recognise the inductive hypothesis as being a progression, it will be infinitely more problematic

[turbid zenith]

@desert hinge : I love that as a start!

Something I did once with high schoolers was to have them label a square, figure out all 8 symmetries, and then complete a Cayley table by physically playing with it

We did the rotation・rotation part of the table together, then I had the class split into three groups, one to do rotation・flip, one to do flip・rotation, and one to do flip・flip

And then each group had a representative come up to the board and fill in their part of the Cayley table, after which we made a whole bunch of observations

[wispy slate]

@sick fractal I disagree. You might wanna think about it this way, thats why I taught them contradiction proofs before I started induction BUT it really is the n to n+1 thing for me, as it makes much more sense when you think about why the induction really works. Its that jump. In other words, the inductive jump being the consequence of a well ordering and Id rather think about it that way.

[sick fractal]

@wispy slate what i meant is this. this was something i gave to a class about induction and no one could figure it out

I'm going to prove to you that all horses are of the same colour
Base case: In a set of only one horse, there is only one color.
Inductive step: Assume as induction hypothesis that within any set n horses, there is only one color. Now look at any set n+1 horses.
If we number them {1-n} and {2-n+1}, its obvious that both have a size of set n and all have the same colour
so there must only be 1 colour among n+ 1 horses
why doesnt this work?

the n to n+1 has 0 errors, the base case is right, but why is it wrong?

the reason is because at n=2, 1 + 1=2 has no overlap between the sets

which is why i do not like to think of induction as n to n+1

rather Statement n implies there is a sucessor that is true aka statement n+1

[wispy slate]

Thats what I said though about the successor

That +1 is the successor for naturals

[winged urchin]

That horse example feels flawed... I don't know how but I'm not convinced by your reasoning...

I'm just curious, why is there no overlap in this context but there is overlap in contexts where induction works?

[visual breach]

because horses have more than one color 🐎

[brazen pendant]

the horse example works if and only if the case for n=2 holds (which it doesn’t)
if it was true for n=2 (i.e. whenever you pick a pair of horses, they both have the same color) then indeed all horses do have the same color

however, the proof starts at n=1 (for which the statement is trivially true) and as said, the flawed step is that the inductive step does not work for 1→2 (but it would work in all subsequent steps)

I’m not sure how exactly this example illustrates the point you’re making, but it’s a classic example in “watch out that you covered all your edge cases”

we got it as homework in second week at uni or so

most people couldn’t do it then, but looking back at it just a few weeks later it seemed trivial

[grand laurel]

we have like half a dozen variations of that exact same flawed reasoning

most people always answer that horses are not natural numbers, so you can't do induction on them

so i have this example induction problem

about tiling a "checkerboard" of dimension 2^n * 2^n with l shaped "tetris blocks"

always wondered if there is a better example to show that induction is not about natural numbers necessarily

(in the first week of a uni course, so i dont have access to graphs, which i would do instead)

[brazen pendant]

I think a visual proof on graphs could be done even if they aren’t part of class, you can just define it as “a thing that looks like this”

that’s good enough for week one

and you could e.g. prove that every tree with two or more vertices has at least two leaves

that’s a straightforward proof by induction on the number of vertices and the definitions needed are all “obvious” i.e. you can just illustrate what you mean and everyone should get it

[sick fractal]

anyway is there a method to derive the sum of 1/n^2 without going to go into too much depth?

[woeful folio]

https://en.wikipedia.org/wiki/Basel_problem#Euler's_approach

[toxic spire]

Ye

Well, infinite series of ln(1-x) and e^ix, you get cos(n)/n^2 evaluate at 0 rearrange the terms you have (1/2) times the series

You can derive ln(1-x) by integrating the geometric series

[vestal quiver]

There are ways to calculate that sum without committing intellectual dishonesty and that don't require a lot of knowledge: the picture below shows one. It's from an old maths test from my upperclassmen and it should be doable by early first year university students (as they were first years when they got it).

@sick fractal

[wispy slate]

Thats a nice test

[toxic spire]

Tuong, u accuse me of inteletctul dishonesty?

-.-

@vestal quiver

[vestal quiver]

I think I already explained in the past what I think of your "proof" of ζ(2)=π²/6

there was many unjustified steps, it was super super suspicious, it was more black magic than mathematics

you went full "I don't care" style even though there was obviously something complicated going on, and this is precisely intellectual dishonesty

this kind of frivolous behaviour may be common place in physics, but in mathematics it's not quite as welcome

[tawdry venture]

physicist dunk

[wispy slate]

Praise Tuong 🙏

[turbid zenith]

Wait, intellectual dishonesty?

I think that's a little harsh, depending on the audience

@woeful folio I do prefer Euler's way yes

I think that's probably the most accessible to, say, high school students who have taken AP Calculus BC.

And even if they haven't, if they've at least heard of infinite series, it's not too hard to get them up to speed with the idea that complicated functions can be represented as infinite polynomials

[woeful folio]

what is this ping

[turbid zenith]

Basel problem

[woeful folio]

what about it

I'm confused to why I was pinged

[turbid zenith]

Because I was agreeing with you 😛

[woeful folio]

about what

all I did was answer a question

[turbid zenith]

. . . okay, never mind then

[woeful folio]

what is this channel even

[turbid zenith]

It's supposed to be a place to discuss teaching mathematics.

@sick fractal I suppose I should ask what level of student you're looking to show the derivation to

[sick fractal]

its fine the test was sufficient enough

[tawdry venture]

sufficient enough

[civic tree]

lol

[sick fractal]

Wait @vestal quiver u know the sheet u sent? can u check that 2b was meant to be a_2n - a_2n+2

Wait nvm my careless error

[turbid zenith]

Of course!

[turbid zenith]

What course @wispy slate ?

[turbid zenith]

Very cool. I've taught students Python at the high school level before.

[lament wraith]

https://blogs.ams.org/matheducation/2019/11/21/the-dysfunction-of-functions-in-abstract-algebra/?fbclid=IwAR2UOpOeuMMxaahZCefjnx7GbYG_fZKfXdMy2lzylurGc8dF11Rsqm4sz1w

Interesting stuff

[left widget]

This seems strange to me. When students were introduced to functions as abstract concept between two sets in first semester in undergrad no one really had problems. It might be that giving too many examples that suggest "functions = some formulas" is counterproductive.

[turbid zenith]

Yeah, that's surprising sort of.

Though I've even seen students in calculus class have trouble getting past "functions = formulas"

They have trouble, for example, with the idea of working with a function purely based on a few values given in a table

[turbid zenith]

BTW, regarding infinitesimals in teaching of elementary calcuulus...

My Complex Analysis professor just linked me this: https://www.tandfonline.com/eprint/CUHXJUXFJDPPRKDSXIBY/full?target=10.1080/07468342.2019.1662706

Best quote:

[woeful folio]

introducing hyperreals to do calculus should be harshly punished

[turbid zenith]

By whom lol

[turbid zenith]

Ahahaha

It's throwing me for a loop though that this article is using ω as an infinitesimal

That screws with my head

It uses Ω = 1/ω, so I get what it's going for. But ow.

[lethal leaf]

For tutoring do y'all charge a cancellation fee if they don't cancel before a certain time?

Yesterday I had someone cancel 5 minutes after the tutoring session had started

[sick fractal]

@lethal leaf ok that shouldnt be allowed

u should just say "cancel with 24hr notice"

but regardless that the stuff about the function is interesting cuz when I learnt about functions, it was directly in the form of domain maps to codomain, and satisfies the vertical line test

and that was only in like year 8

[lethal leaf]

@sick fractal but like let's say they cancel less than 24h before the tutoring, should I charge a cancellation fee? Especially if I already travelled to the library they agreed to meet at (which is far from my house)

[sick fractal]

Then yes

That is my experience with tutors

[lethal leaf]

Cool

[left widget]

@sick fractal How would you define the vertical line test with two arbitrary sets? You can still do some demonstrative graphic where both sets are represented as 1-dimensional but that might be confusing too.

[sick fractal]

@left widget i started off with understanding the vertical line test to get introduced to the idea that we must know f(x) given any particular x

which was then extended to domains codomains injectivity surjectivity

etc

[left widget]

but the vertical line test is such a graphical description

[sick fractal]

well it worked for me

[left widget]

I actually never had the vertical line test I just encountered the definition written in logic

only heard about the vertical line test from others

[sick fractal]

oh right i was introduced to like is f(x)=x^2 a function obv yes but is f(x)=+ or - √x a function then no

[left widget]

I mean we introduced functions as relations

so a function was a triple $(X,Y,f)$ with $f\subset X\times Y$

[burnt vessel]

mophra:

[sick fractal]

oh interesting

[left widget]

and the special requirement

[sick fractal]

i just think of a function as any operator satisfying given x we know f(x)

[left widget]

I think about it in multiple ways

mostly like you do

but I've seen a pretty cool proof that used that linear operators are subvectorspaces of $X\times Y$ and then used that as a closed operator it was a Banachspace

[burnt vessel]

mophra:

[left widget]

so show that a sequence converges in the domain and co-domain at the same time

so that was pretty neat

The idea of a function being a Banachspace as itself was pretty funny

[civic tree]

what

[wispy slate]

Can anyone recommend books on teaching in general and teaching math?

I'm starting to read "Becoming the Math Teacher You Wish You Had" by Tracy Zager which is pretty nice so far actually, I was surprised the author cites Lockhart sometimes which is nice to see someone with a similar philosophy writing a popular education book

I've also been recommended "Elementary and Middle School Mathematics, Teaching Developmentally" by Van de Waal, Karp, and Bay. It seems to be very sanitary and unhuman in its pedagogy (really gives you a peek at current school math curricula! :/ ) but paired with the former these seem to have a lot of really useful content

Let me know if you have any insights for a practical newbie

I'm a 3rd year math undergrad so I'm new on the education ropes but Lockhart is what really lit a fire under my butt

[tender birch]

How do y'all manage to figure out how much time each thing takes? I've done some lesson planning before, but I always seem to underestimate the time something takes.

[shadow basalt]

I think it just takes practice

And like plan out lessons so that if you notice its going slower than you expected

You have natural cutoff points or things your can skip

[sick fractal]

@vestal quiver remember that sheet abbout integration you sent the other day?

do you have anything else similar to that

of the same type of difficulty

[vestal quiver]

I've got quite a big collection of exercises
unfortunately for you, it's all in French

[sick fractal]

ah ripperino

its time to brush up my high school french skills

!

[remote pumice]

Hmm, isnt this a hard question for being on the "no calculator" section on the SAT? Are there a shorter way to solve this one? https://youtu.be/S0qXkVjujAg

I guess "guessing" actually is the best way to approach this one... It would be to hard to solve if the answers were "big" huh?

[sick fractal]

well the heuristic approach would be to plug in some values to utilise the factor theorem. The coefficients of 1, -5,4 also remind us of (x-1)(x-4). The only difference is the that all the x terms are squared, which leads us to a nice factorisation. @remote pumice

(thats after u factor out the x)

aka x^5-5x^3 + 4x)=x(x^4 - 5x^2 + 4)=x(x^2-4)(x^2-1)

pretty easy question imo

[tender birch]

@remote pumice What's the solution you do? I can't be bothered to actually watch the video.

[sick fractal]

@tender birch he does a quadratic equation for no reason

[tender birch]

I mean, your method also uses a quadratic...

Letting z=x^2.

Or at least that's how I'd explain it.

[remote pumice]

@sick fractal Thanks for your input buddy, appreciate it! 🙏 👍

I have come to the conclusion that the fastest way is "gussing". The reason guessing should be bulletproof here is that they would never put an equation with high x-values in the no calculator section

Now I know how to approach a similiar question if it appears again! 😊 👌

@tender birch And yeah, my method was to factor out x, then substitute x^2=t and solve the quadratic

[tender birch]

That's just about how Lacer does it.

At least the easier to explain way.

[sick fractal]

except factorise

but not solve with QR

@vestal quiver yes ivd decided i will take u up on ur offer of reading french math papers

/questions

[vestal quiver]

i guess you could start by downloading this:
http://www.concours-commun-inp.fr/_attachments/nouvel-accordeon-2/Banque_2020_corriges.pdf?download=true (with solutions)
http://www.concours-commun-inp.fr/_attachments/nouvel-accordeon-2/Banque_2020.pdf?download=true (without solutions)
those are exercises given during oral competitive entrance exams for some group of engineering schools

here, 21 years of maths exams from another group of engineering schools
https://www.concours-centrale-supelec.fr/CentraleSupelec

these are usually quite interesting, it's not just "calculate this, calculate that, is this true, is this false etc.", they all have a sort of goal, they make you discover cool stuff... but they're very long

it's really not hard to find maths ressources on the internet when you're french and when you are/have been a mole

and those who teach in CPGEs sure do make use of all that free stuff

and there's a lot more ...

I have yet to see to same thing in english maths

[sick fractal]

Ure not weong

Wrong

A lot of maths in England/ America causes many mathematicians to lose their ability to rely on intuition

Because they don’t have that idea of finding heuristics to simplify the goal

And they just want to jump straight to the answer, rather than trying to investigate more about the system of properties of a certain object

[boreal spire]

your are a CPGE prof ? @vestal quiver

[vestal quiver]

no, but that's my career plan for now

(j'suis juste un mec lambda au magistère d'Orsay faut pas en attendre trop de moi)

[boreal spire]

ah lel jsuis à Kchan

le dep de maths est nazissime

[sick fractal]

@vestal quiver yeah ok its actually pretty easy to follow

esp with a slight background in french

and since its maths anyway

[umbral saddle]

@remote pumice Could you not just pull it all together and solve it using the following
x^5-5x^3+4x=0
x(x^4-5x^2+4)=0
x^4-5x^2+4=0
x^2(x^2-5)=-4
x^2-5=-4
x^2=1
x=+-1?

That gives you 1 solution which is all it's asking for and you can plug it in and check

[sick fractal]

@vestal quiver the notation of ]a, b[ is our equivalent of [a, b] right?

[round robin]

i think its like (a,b)?

[sick fractal]

french notation o_O

gonna wait for him to confirm

[vestal quiver]

]a,b[ is open interval, [a,b] is closed interval

[boreal spire]

French notation is more natural than english notation ♿

🙃

[grand laurel]

I used to really like ]a, b[, but it changed over time

(a, b) is just more aesthetically pleasing

[trim violet]

the french notation is terrible to parse when there are a lot of them

[wispy slate]

(a, b) looks better but it might confuse a bit at some points, unlike] [

[boreal spire]

It's same shit. Don't say that's its terrible. It's not like we write o a,b o for (a,b) x)

[wispy slate]

?

What Im saying is, it might be confusing when working with like functions R to R2

Although not hard to figure it out.

[boreal spire]

Oh ok yes

[grand laurel]

i never confused (a, b) with a tuple or anything

[wispy slate]

I have, seen people do it as well.

[round robin]

I'm wondering, what exactly is making subjects like (basic) algebra or (basic point set) topology hard to understand for like a general high school student? i thinking like all the definitions and so on?

[trim violet]

because they think the mathematical world is made of algebraic formulas

[wispy slate]

Well, because it's hard to formulate the proofs and understand them even for undergrads who had some experience already

I guess if you do it non rigorous way they might understand it, just like with basic set stuff right?

[feral vector]

It is not hard to understand but it is that it is a completely different subject for them. Before, for them, mathematics was just manipulation of some laws which they kept in the back of their head. There is usually no emphasis on proofs so the kind of mindset required to understand what a proof is, what it should accomplish, when it finishes is lacking.
Consequently, it is not the content of these subjects that is hard. Just the framework in which they are taught, in my opinion.

[sick fractal]

@round robin https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/

this blurb captures what ure talking about

[thin aspen]

yeah a lot of things don't have formal prereqs but need mathematical maturity

[round robin]

ahh right proofs its hard to teach

[turbid zenith]

Just wanna share some links on mastery grading and mathematics teaching: https://rtalbert.org/tag/mastery-grading/

[wispy slate]

Hello. Does anybody in this channel use the Typoma Minion Math Font (Adobe Compatible)?

Pic related:

[sick fractal]

Nope sorry

[brazen pendant]

nope but it looks pretty. I like it

[turbid zenith]

I usually use the Euler font

[wispy slate]

Thanks for the responses!

[wispy slate]

Hello everyone, I am very happy to announce that I was able to virus-free and pay-free retrieve the minion math font. I would be more than happy to post the otf files in the server, but it may come up as spam, so if someone would like to save 600 euro, I would be happy to DM them to you unless I had the permission to upload them all here.

[wispy slate]

Wtf the font costs 600 euro?!?

[wispy slate]

Yes, but I did not dare pay that

[turbid zenith]

Why would anybody charge that much for a FONT

[wispy slate]

I have no idea. I think it's ridiculous. But no matter, I was able to find the files.

[round robin]

i would want it so i dont have to hunt in the future lol
the server should be cool with you sending i believe unless stalin exist

[shadow basalt]

wait

why would someone pay for a font for academic work

[vestal quiver]

¯_(ツ)_/¯

[wispy slate]

Well if you like the font and want to publish your work using it it might be cheaper to actually buy it if they see it lol

[zealous sky]

I can't see why, say, Latin Modern doesn't cut it

[obtuse widget]

@remote pumice I know I am late, but there is one more approach. Since x > 0, it means that RHS is negative--->LHS also must be negative. x^3 is positive, so x^2-5 must be negative. It limits possible value of x to (0; sqrt(5))

[ionic dagger]

@remote pumice Throwing my two cents into the ring. If x >0, you can multiply and divide by x.

So, let's divide both sides of the equation by x^3.

x^2 - 5 = -4/(x^2).

from here you can sub in w - 5 = -4/w, but it is "clear" x = 1 works.

[lethal leaf]

So like sometimes when I'm tutoring

I'll be explaining something and showing it on paper

But then like

I look up and see they're staring off into space

Not paying attention and looking

How do I deal with that?

[sage crag]

Well, here's one strategy that I use with the guys I'm tutoring while still in the military:

Make them work through the problem. So, do enough such that they can use what you've written. However, make them work through the rest themselves.

[sick fractal]

Another strategy is to keep asking them questions @lethal leaf

As long as u make them “stay active” and not listen to you drone on for more than 5 mins ure fine

[sage crag]

^This

[sick fractal]

Because I’m assuming these kids are slightly younger since most older kids don’t zone off

[lethal leaf]

These are high schoolers

[sick fractal]

Yeah under 16?

[lethal leaf]

And it's just me explaining for like 2 minutes max

[sick fractal]

Most mature by 16

[lethal leaf]

Under 16 yea

[sick fractal]

Ok then just prompt them step by step

Or straight up ask them whether anything is wrong/ if ure going to fast. Putting them on the spot tends to scare them a bit

[lethal leaf]

I don't want to scare them tho

[sick fractal]

Not scare well but make them realise they should pay attention

[sage crag]

But that's a good thing, though. It forces them to think. They'll remember things even better.

[lethal leaf]

Also holy shit teaching basic algebra is hella hard

Like I was helping someone solve 5x-4 = 16

And they didn't understand that to get the 4 on the other side you had to add 4

And to isolate then x you gotta divide by 5

Like I just know that's how you do algebra

Having to think of the why and the how is wack

[sick fractal]

yeah man

thats why i only help

anyone older than 16

/ decent at maths

LOL

@lethal leaf do you tutor as a side job? cuz thats what I do

[lethal leaf]

It's a side thing yea

I'm in HS

[sage crag]

The why and the how is what makes it matter, though. To be fair, math only makes sense when taught from the ground-up.

[sick fractal]

well to be technical, anyone can derive all of maths from axioms alone!!!!!!!!!

[sage crag]

😄 It's painful but it's fun. With my tuition kid, he prefers it when I derive results from the axioms.

[sick fractal]

@sage crag derive the quadratic formula from the field axioms of real numbers

!

[sage crag]

nah nah nah, you gotta create propositional logic first, then create the ZFC axioms and then create the Peano axioms before you move from there 🙂

[sick fractal]

nah nah nah

you gotta define the alphabet first

[sage crag]

what's an alphabet? 🙂

[round robin]

alphabet is just a random sequence of lines

[sick fractal]

Hmm what about lines

What are two points

[round robin]

a set of 2 ordered pairs

with the same finite cardinality

[pliant drum]

@lethal leaf so how do you explain the reason for adding 4 and then divide by 5 to solve for x?

[lethal leaf]

Well you have to isolate x

So first we'd add the 4 over

We add 4 because it's a -4 and addition undoes subtraction

And so we have to add 4 on both sides of maintain equality

Then we have 5x = 20

That 5x is 5*x so we have to undo the multiplication by 5

Division undoes multiplication so we divide both sides by 5

So we get x = 20/5 which simplifies to 4

[pliant drum]

and the student finally understood it when you explained it this way?

[ionic dagger]

I just play my own half-baked "chocolates" game with them except we start with positive chocolates. It allows for creativity and sometimes makes it obvious what methods are quicker.

You are given 5 boxes of chocolates with the same number in each and also an extra 4 pieces. Your friend has 19 chocolates. The final fact is you both now have the same number chocolates. If you guess correctly the number of chocolates in each box, you both get a useless silver ticket.

Let O be a box and o be a chocolate.

OOOOOoooo

ooooooooooooooooooo

Canceling makes sense because you have the same number in both piles. Or ... "You both eat 1/2/3/4 more chocolates so the two piles are still the same." That kind of thing. Parentheses are gift baskets: "A T-shaped gift basket contains two boxes and an extra 3 chocolates."

So two (or more) T baskets is 4 boxes and 6 extra which can be subbed into an equation dealing with distribution.

I find it to be a nice tool.

You're likely going to get a smart-aleck student who wants to add 1000 first to both sides and then take away 1004, but that's kind of nice actually as further along there's expression manipulation where you want to add and subtract the same thing to get a particular pattern to show up.

[lethal leaf]

@pliant drum no the student was still struggling

And like idk how else to explain it or come up with a way to explain it on the spot

It's been so long since I learned basic algebra rules that I've forgotten how they're properly taught to new people

Like with calculus, I took it last year so I know how it's taught

[dusk linden]

anyone know Danish?

https://issuu.com/lr-undervisning/docs/matematrix_9 can you tell me if this book covers solving one equation in one unknown?

I dont know if the question asker is supposed to know it

[wispy slate]

it is not even there

let me try sending a pic

[dusk linden]

of the table of contents

[dusk linden]

go back to the questions chat for a sec

[feral vector]

Moore's Method? Any advocates?

[turbid zenith]

Hell no

[lament wraith]

Only really could possibly work for some classes, basically only the intro algebra, analysis and topology classes

[round robin]

feels likr it would only work for intro classes in a small group tbh

if its a big lecture hall most of the ppl are just gg sit there and do nothing

[tawdry venture]

what is moore's method

[wispy slate]

I think it sounds like a great motivation to skip class

[wraith shell]

^

[lament wraith]

Uh

Wouldn't it be the opposite since you can't get the material from class anywhere else but in class

[wispy slate]

Doesnt sound that bad tbh

[boreal sentinel]

This system will surely weed out anyone who's not motivated enough

[wispy slate]

But kinda feels like you would learn much less but much better

But if you wanna Just pass you can Google proofs and learn in advance so

[boreal sentinel]

Would it even be fair to have a measure of pass/fail for such a type of class

[tawdry venture]

yes

fail everyone

[boreal sentinel]

[wraith shell]

my prof did something similar to this but he didnt single out people, he just asked the class how we should progress on the proof

it was helpful for those who talked

everyone else just sorta chilled

[boreal sentinel]

I think it advocates favoritism

[wraith shell]

also people with anxiety will get pretty fucked

[round robin]

^this especially

[turbid zenith]

Here's another thing that loses me:

The students were forbidden to read any book or article about the subject. They were even forbidden to talk about it outside of class.

So if you're confused, you're shit outta luck.

My friend took a Moore Method linear algebra class once, and he showed me some of the work that was being done, and it was in the most bizarre order I had ever seen almost as if it was as a deliberate attempt to obfuscate it. Things that you think would be definitions of, say, linear dependence were instead given in an equivalent but unmotivatedly abstract form, and they would have to prove the thing that should have been the motivating definition in the first place.

And the Moore method restriction meant that he couldn't, say, open up a textbook or watch 3blue1brown's videos to get any intuitive idea about what all the stuff he was proving actually was supposed to be

[tawdry venture]

oh to add on to that

students are even forbidden from discussing it with each other like wtf

[lethal leaf]

this is like the more evil version of "how do I do this. Oh you should have learned this last year"

but instead of last year it would be last week

[wraith shell]

that just sounds like the equivalent of throwing a bunch of kids in a room and telling them to prove something and then leaving

like where is the teaching

[tawdry venture]

nowhere

[feral vector]

I mean I think he capitalizes a lot on the "competition"-al driving force which presumes enough motivation in the students themselves. But advocates have to say that while the material they learned was very jumbled up, the learning they got can never be matched by any lessons.

Like I am taking a class in which the professor is super lazy, so I do end up doing a lot of work on my own and then make him teach me stuff. Ofcourse I do not like the doing so much on my own part, but couple this with good teaching, and I think it is a very good way to teach

[turbid zenith]

Wow, that last story at the end

That competitive atmosphere is the exact opposite of what I want to foster in my class

[feral vector]

I guess it is more dependent on the student's motivation, but I will argue that competition fosters more will to improve than does cooperation. Just consider the simple example of class exams (individualistic) and take-home exams (cooperative). You would find a student preparing much harder for class exams as opposed to take-home exams.

[turbid zenith]

I think a big part of that is what you put on those exams

If you give the same test as a take home that you might give in class, yeah, people won't take it as seriously

But if you take advantage of the at-home nature of it by asking meatier questions you counteract that sort of

At least a bit

[feral vector]

Yeah, but the point is, that the above method is not as ridiculous as it was implied in the discussion above, as supported by another mathematician.

[turbid zenith]

I'm not speaking as a mathematician. I'm speaking as an educator. From an educator's point of view it's pretty ridiculous.

But, that's my opinion. 😛

[brazen pendant]

man, that method sounds horrible

[ionic dagger]

So (starting a separate topic though I'm relating it also to academic competition) ... I have, after being a tutor for years, found a few simple ways to increase an "average" kid's SAT: Math section score by about 100 points. The test taking strategies I use are easily explained in 15-20 minutes.

Some strategies are normal test tips: bubble in a few at a time, etc.

But the one that is most helpful is Show What You Know.

The test is generally arranged easy to hard (for each part of each section). You assess the student's strengths and don't let the student even read problems that are Hard if they generally get them wrong. It saves time and allows for checking answers on problems that they are likely able to handle and that they can correct. I mean yes guess randomly at the end because there is no penalty but reading an extra 5-10 problems just to try testing strategies on each answer is likely a waste of time.

My point here is ... a student can raise their Math section score 100 points in 20 minutes without really learning anything. Doesn't this lead to an admissions problem? Just presenting my thoughts.

[long pelican]

Yes it favors people who do SAT prep over people who don't do SAT prep

[tidal whale]

isn't there already a thing about MCQ-based tests being very weird in nature when it comes to assessment

I agree in that it heavily promotes the notion of prepping

[ionic dagger]

It's not even prep. It's 20 minutes.

Anyway I'm just very annoyed about this at this time. I wouldn't know what to do if I were trying to sort this out in an admissions office.

[tidal whale]

IIRC, they don't use it as a heavy assessment

more-so as a threshold

[lethal leaf]

Man in tutoring I'm really seeing how gaps in peoples knowledge form

Like this kid I was tutoring knew that for graphing in equalities dashed versus solid lines

But he couldn't graph the lines themselves

Like y = 4x - 3 was a struggle at first

Idk what the teachers are doing

[lament wraith]

their best

[turbid zenith]

Graphing lines is something that teachers seem to need to reteach every single year

[lofty mantle]

Hi, i'm looking for a teacher/consultant (will be paid, part-time) who can help us regarding some mathematical problems, mostly related to cubic curves, b-spline curves, 3d-geometry etc. Any programming experience would even more great.

[winged urchin]

I tutor university kids and they seem to forget the simplest things possible. Forgetting how fractions add or multiply. Not knowing how to graph a line. Not knowing function transformations. Just humans being human though.

Sometimes I think it has to do with how we compartmentalize the skill testing. We test them on hard(er) multiplication and division in elementary and then in highschool we setup easy integer solutions that require only the minimum of arithmetic skills

Then we do all the transformations of parabolas and radicals and all that in highschool. Then it barely comes up in university since we focus on higher level concepts

I mean sure, the odd question does appear but the gaps are far enough for us to forget. But there will always be inefficiencies I suppose. Only so much time can be devoted

[sick fractal]

It’s also a reflection of the education gap between private and state schools

[pure skiff]

I wonder if any of my math teachers are here...

[lethal leaf]

If any of my math teachers saw me here they'd probably be like "boy I just taught you this shit how are you already confused"

Huh?

[lethal leaf]

How do you know my name lol

Do I know you?

That's crazy lol

[sick fractal]

Are you guys joking around or is that for real

Wait nvm

@lethal leaf I just realised your reddit has your name

[lethal leaf]

Ya probably

[ionic dagger]

Time for me to sound a-holey. I'm defending teachers who DON'T cater to "every" learning style. My point is in real life, you don't generally get a choice as to how information is presented. Your job probably isn't going to cater to all learning types. It'd be nice in the real world if every job training video addressed the same info in various learning styles, but clearly that's not the case. It's also better for the student if they can learn in multiple styles. Teachers who cater to the top 2-3 learning styles are already amazing and near-perfect. I'd research more about this, but I'm just not a good learner when it comes to search engines. 🙂

[normal aurora]

appropriate name

[ionic dagger]

Yes and every teacher should be stressed out more.

[prime linden]

"learning styles" is nonsense similar to "multiple intelligences" and so on

[frosty pewter]

@prime linden que

Some people are just smarter than others lol. But I do agree learning style is a bit nebulous and in general it's the student's responsibility to resolve if a course isn't tailored to the way they like to learn

[sage crag]

Hmm, from the perspective of a teacher, what would you say is the most important thing that any math course should emphasize?

Like, just in terms of teaching alone? So, for example, high school math is very different from university math etc. What would you say is the most important thing that needs to be introduced in the high school curriculum for math?

[vestal quiver]

Highschoolers need to be taught discipline.

[ionic dagger]

I am fully ancient here, but I remember the days when if you didn't know a vocabulary word or equation, you just wrote it over and over until you did. Now (and I appreciate the effort) math has to be sung or rhymed. I'm reminded of a Simpsons joke where Homer is caught in a fire and he can't remember the important part of the fire song.

There are not-so-difficult, tried-and-true ways of memorizing things. Yes, it's boring but being able to memorize boring things is what makes you amazing later.

[shadow basalt]

I disagree completely

High schoolers need to learn to enjoy learning

Nothing else they do really matters

They have to redo all of it in college anyway

[ionic dagger]

They'll have to redo undergrad topics in graduate school anyway ... jk

[grand laurel]

the way math especially is taught, is horrible

they either learn stuff that is completely useless to them later in life

or if they want to become an engineer or mathematician or whatever, they have to re-learn that stuff anyway

bcs what was done in school is not sufficient

[lethal leaf]

I think flipped classroom can work but really only in environments where people are willing to put in the work outside the school

We had flipped classroom for my AP calc BC class

And goddamn it was so much nicer than previous years

AND according to my teachers the over all grades and AP Pass Rates are better than non-flipped classroom years

I do think it will only work in a class where people are motivated though

Like my physics class wouldn't work because of the grading system in that class (which is a whole other topic)

But for my calc BC, everyone who took it was like "I'm going to grind to pass the test" so they put in the work outside of school

And having the class period open to asking questions and doing problems is really good

That's also how it kind of is in my calc 3 class

We have 1-2 days of lecture in class and then 1-2 weeks of in class work time on a bunch of problems

And we learn from the problems and have open access to asking questions

[obtuse widget]

as a, let's say good engineering student, I have to say that it really depends on class. I had something like flipped classroom during learning calculus and for me it worked out good, but I have seen huge struggle and lack of motivation in my schoolmates. I have heard a lot of argument "what's the point, in the future if I need this i will use wolfram anyway lol"

in such case I think classical approach would work better

maybe graded homework would also help, idk

[lethal leaf]

Grades do suck because not everything is about a score

But damn are they good for motivation

My physics class has that issue rn

[obtuse widget]

Like I said idk, but I have seen people forgetting what was learnt 2 weeks before

[lethal leaf]

The teacher wants to make grades less stressful

So the grading distribution is this:

60% learning and growth (which is really participation which is really BS because even kids who don't show up to class get 100% in this catagory)

And then 2 midterms 20% each

No other quizzes, tests, or homework is required

So the idea is you learn in class

Do homework

And ask questions

And the quizzes and tests identify gaps in knowledge

And come midterm time you're good

What actually happens is that no one does anything for 2 months

We have a midterm

That people cram for

[obtuse widget]

yes, exactly

[lethal leaf]

You get a "decent" grade

And move on

So like obviously relying on people to be motivated to do homework on their own volition isn't gonna happen

My calc 3 class has graded homework tho

Like graded for accuracy graded homework

And while that seemed bad at first

1: we get to check it before submitting it

2: it makes us actually do the damn homework to a high level

So people are actually learning shit

And that class still also has high grades

Now granted calc 3 is literally the highest math class in my HS and so we have the most motivated students so there's that bias

[obtuse widget]

one problem with graded homework is that people exchange solutions but there is nothing to be done about it, unless there is implemented some task generator

[lethal leaf]

But alot of those kids are also in the physics class

And are lazy in that class

So it looks like a product of the way the class is run

And I agree with what you're saying about people cheating (cause it does happen)

[grand laurel]

grading homework is a really bad idea

[lethal leaf]

But still people are actually working and learning

[grand laurel]

the best way to grade people is by just talking with them

[lethal leaf]

I think the homework should be required to do (and you get a grade for turning in and completion)

[grand laurel]

there should be some written tests, to test if they are able to write mathematics

and maybe some homework in addition

but to actually produce a grade, just do a 20 min oral exam

[lethal leaf]

And it should be marked up like it's graded

But no actual grading for accuracy

[obtuse widget]

talking has problem of favoritism and can create really bad atmosphere

[grand laurel]

bad atmosphere?

[lethal leaf]

Also it's really impractical in a high school situation

Idk how college is

[grand laurel]

its impractical, but producing grades shouldnt be the main objective of teachers

[lethal leaf]

Well like I'm pretty sure my teacher would be more lenient with me

For an oral exam

[grand laurel]

there are school system that dont even have grades

[lethal leaf]

Because he knows I know my stuff and I help people

[grand laurel]

and they work well

some studies show they work better than traditional schools

[lethal leaf]

Huh interesting, what motivates the students tho

[grand laurel]

the will to learn

[lethal leaf]

That's crazy

Wouldn't happen here lol

Sounds amazing tho

[grand laurel]

its also set up differently

there are no grades

if youa re good in math, you learn advances math

if you are bad, you study slower

older students have to help younger students

[obtuse widget]

in my class will to learn has like 15% of students

[grand laurel]

learning how to teach is part of the education

[obtuse widget]

so nope

[grand laurel]

it can create some problems

[lethal leaf]

^

[grand laurel]

but it works

[lethal leaf]

There is no will to learn here

[grand laurel]

then why go to school

[lethal leaf]

People focus on grades with the goal of making a top college

And then with that top college degree

[grand laurel]

to learn more stuff they dont want

[lethal leaf]

Get a high paying job

[grand laurel]

then school should focus on not making students have that attitude

[lethal leaf]

how tho

[obtuse widget]

public education with funds based on student number has its drawbacks

[grand laurel]

i mean it works here

in small privately run schools

but still

but well, i was the same in our equivalent of high school i guess

but people grinding through college just to get a degree is ugh

[obtuse widget]

welcome in my world

"whats the point of this stuff" "i will learn everything after getting hired" "i will never need this"

[grand laurel]

well, you shouldnt learn just because you "need" it either

you should want to learn, because it is fun

[lethal leaf]

people don't truely understand shit

Like yesterday in physics

we're starting cross products

and we're learning hte magnitude of the cross product is |r| |F| sin(theta)

and then we put the sign on our selves

using right hand rule

which is fine for physics

BUT

all the other calc 3 kids started losing their shit

because we talk about vectors as jst <1, 20, 3>

and using matricies and stuff

which is also fine

BUT DAMN people were losing their shit

[sage crag]

It's actually not fine for physics. It works for basic problems but not when things get complicated lel

[lethal leaf]

"how can two different operations get the same result" when we talk about vectors in fundamentally different ways

well it's only AP Physics

so like

it's not complicated problems lol

we still in HS

[unreal ledge]

No reason not to know how to easily get the magnitude of the cross product, but yeah you'll need to know what the difference is between these two answers

[lethal leaf]

yea and people didn't get why we're using different operations

[obtuse widget]

for me my main motivation is that the more I learn the easier it is to learn and I can learn more high level stuff. It is impossible to learn powers without knowing multiplication. I also see how much for example learning programming helped me with other courses. Not actual programming but just the way of thinking used during programming.

[sage crag]

I personally think that physics shouldn't even be taught until people have learned calculus fully

but whatever lol

[lethal leaf]

same here

but alot of schools require that people take chem and physics in HS

and then you run into the issue that not every is at a high enough math level to take calc in HS

or at least before their senior year

[unreal ledge]

I mean I can't imagine calculating the forces on a truss and actually working out the cross product every time I only need the magnitude

[sage crag]

I also feel that calculus shouldn't be taught until algebra, geometry and basic set theory are covered.

[unreal ledge]

But I need to know what I'm actually working with. That being said as a little engineer I'm sure I had no clue

People learn when they freak out, maybe this is natural

[sage crag]

I mean, you only know what you don't know when you've been pushed into a corner and you have to fight your way out.

Instead of 10 problems that look similar to those in the past year papers, just give 2 or 3 that are really hard.

[obtuse widget]

I'm meche student and we have like 5 projects during semester, 2 big one and 3 small one. But man, it is hard to work with people like this.

[unreal ledge]

I find arrow diagrams really help me understand shortcuts.
a,b → a×b → |a×b|
OR
a,b → |a×b|
Using the relevant formulas over the arrows

I'm not communicating this well

Hrmstv

[ionic dagger]

No worries. It's just a systemic problem: Discord doesn't easily allow for draw wings.

[tawdry venture]

do y'all think it's worthwhile to keep telling my tutee to avoid the excessive use of decimals? (such as writing 0.5 for 1/2)

[charred silo]

I think it depends on the grade they are working towards, what they intend to do in the future.

Like, if you know they want to take an engineering degree in the future, just let them be~

[lethal leaf]

I say yes

[wispy slate]

who cares? If it's easier for them to write is as decimals why bother changing? They will most likely switch later on anyways

[lethal leaf]

well are they working with decimals because they don't know how to work with fractions?

[wispy slate]

don't you learn fractions first?

[unreal ledge]

Most people no not really D: will just forget because the calculator doesn't work in fractions

[lethal leaf]

^

[grand laurel]

in real life fractions aren't really used because who really knows if 1/3 or 1/4 is bigger

real life uses decimals which leads students to believe they are somewhat more natural

[sick fractal]

I mean

Once again depends LOL

[tawdry venture]

3 < 4 so 1/3 < 1/4

[arctic rampart]

ordered fields

[charred silo]

With home delivary included.

[civic tree]

isnt that what happened to some burger making company who tried to compete with mcdonalds quarter pounder

by selling a third pounder for less

[lethal leaf]

who really knows if 1/3 or 1/4 is bigger

[wispy slate]

it really matters what kind of math you're working with

in calculus, fractions are more important

in finances, decimals and percentages are more important

[lethal leaf]

Find out why they prefer decimals to fractions

because also while you're working through a problem

fractions are WAY easier to work through

and then final answer you can put in decimal form if you fancy

but if they prefer to work with decimals because they don't get how fractions work

then you should force them to work with fractions and get used to them

[wispy slate]

decimals are easier because you know 35 is bigger than 25, but its harder to tell whether 13/25 or 8/15 is bigger on a first glance without doing extra calculations

yes you may say its easy to determine but once you git gud and it still takes some thinking, and there are examples which are not as easy as this one, just threw some random fractions

[lethal leaf]

If you're solving a system of equations

IMO

it's easier to use fractions if they come up in your work

and then if you want at the end, go for a decimal answer

but I don't think that's what Ann was asking about lol

[winged urchin]

I mean it is all perspective and based on the context you're working in.

In my mind fractions are far superior to decimals. And just multiplying by denominators to avoid fractions altogether until the end is the best.

Granted my context is school-based questions and in those cases the numbers usually stay nice enough.

[tepid citrus]

i am just in hs, but decimals are more convenient for chemistry and physics and fractions for math in my experience

[tawdry venture]

LEVEL 0: Not knowing the concept.
LEVEL 1: Knowing the concept well enough to apply it.
LEVEL 2: Knowing why the concept works the way it does, and being able to rederive it should one forget.
LEVEL 3: Knowing the concept well enough to prove theorems involving it.

can someone suggest any modifications to this scale of understanding i just made up in 5 minutes and which could probably be more precise

[round robin]

whats the diff between 1 and 3 lol

[tawdry venture]

knowing how to differentiate arctan(7 sin(x) + 55sqrt(cosh(e^2x)) - 7) vs being able to prove MVT on the spot, intuiting exactly why it's true, and applying it to further results

[round robin]

ah so 1 is just knowing how to plop things into it directly

[tawdry venture]

yes

[brazen pendant]

being able to prove MVT on the spot, intuiting exactly why it's true, and applying it to further results
that seems to me like it’s a few steps above what you described as 3

in particular “on the spot”

[tawdry venture]

then what'd you describe as my level 3 for derivatives

[charred silo]

I'm.. not a fan of putting arbitrary numbers on these things.

[tawdry venture]

i mean if you want names you can make names

the numbers as i used them are just labels purely for convenience

[turbid zenith]

arctan(7 sin(x) + 55sqrt(cosh(e^2x)) - 7)
Ew

[normal aurora]

bruh give me MVT on the spot over that tbh

[winged urchin]

I think as a grading system it's not so good. But as a knowledge assessment tool I think it's good

Like level 3 is the kind of knowledge I'd expect from someone directly working with that particular idea or field

Level 3 is something I wouldn't expect someone who hasn't dealt with the topic for awhile to achieve

I think your grading scale roughly translates to F/B-/A+/Gonna go far kid grades

In that if I ever saw a student with level 3 understanding Id put good money on them doing very well in life

[tawdry venture]

a knowledge assessment tool is exactly what i'm intending it as @winged urchin

[feral vector]

Level 4 : Know the concept well enough to create hilarious memes involving the concept

[jaunty echo]

Level 5: Know the concept well enough to become the concept and be used.

[lethal leaf]

Level 6: thesis

[wispy slate]

Level 8: skipping Level 7

[modern solstice]

Level 100: having orange username on Math discord

[toxic spire]

Slurp

[stone tusk]

I thought level 2 and level 3 are similar, until... I thought... category theory for me is still level 2

[cedar lotus]

Level 9:Engineer

[loud dirge]

Level 100: Boss

[ornate swift]

I'm level 3 in MVT.

If you're able to understand why rolles theorem works, you should be level 3.

[round robin]

lol idek whats mvt anymore
or rolles theorem

[tidal whale]

Rolle's Theorem: If a function f over [a, b] has f(a)=f(b) then there's at least one stationary point in (a, b)

MVT basically implies that for a function f over [a, b], there is some c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a)

[prime socket]

And f should be continuous

[tidal whale]

Oh right, I leave that bit out a lot

[tawdry venture]

f should be continuous on the interval and differentiable on its interior

[ornate swift]

Also, if f is differentiable over the [ , ]

Closed interval.

Because differentiability involves continuibility

[tawdry venture]

implies*

and that's a stronger requirement than the statement of the mvt

[tidal whale]

Level π: Prove CMVT, then use that to prove MVT

[charred silo]

Or knowing how to prove MVT, but doesn't remember what MVT is.

[tidal whale]

Level e: Just write The proof is trivial and left as an exercise to the examiber

[charred silo]

Please refer back to the notes you gave us.

[lethal leaf]

Hey do you guys have any experience with tutoring online?

Like through video?

How hard is it to setup and teach through video?

As compared to in person

[mild void]

@lethal leaf only ever taught younger kids like this, but losing some of your drawing freedom hurts. If you expect to need a drawing that’s intricate then prepare it imo

Definitely try to screenshare if you can

[lethal leaf]

See when I tutor it's mainly like "what do you need help with in math this week"

So I can't really prepare

That's what I was worded about tho, using a whiteboard

Or like not being able to draw

[charred silo]

Having the student show you their notes, homework and what they have tried might also be an issue?

[lethal leaf]

Yea that too

Idk when I go to college I'd like to keep tutoring to make $

And I'd like to keep my current client base I have at home

[obtuse widget]

I can recommend www.bitpaper.io as an online whiteboard

[lethal leaf]

yea but drawing with a mouse 🤢

[obtuse widget]

you can get graphical tablet

or get some cheap normal tablet

but idk how bitpaper.io works on android or with graphical tablets

other option is using skype with screen capture

yet another one use streaming software and organize like 1-to-1 streams

but for drawings you want graphics tablet

[modern solstice]

@obtuse widget put dollar signs around your name

[obtuse widget]

Why?

[loud dirge]

@$a^b+a^c=a^{b+c}$

[burnt vessel]

EpicGuy4227:

[obtuse widget]

Ohhh

[lofty lantern]

any other teachers struggling to get back into the swing of things in the classroom? break is the best but it’s also the worst lol.

[hoary steeple]

Wheres the student lounge lunchroom

[rigid thistle]

#chill

[fair girder]

Hi I’m not a teacher but is it normal that u people to give quizzes everyday

I have a 4.0 gpa

I skipped 2 years of math and have an A

Btw for drawing my teacher uses a Microsoft tablet or something

[boreal sentinel]

I think your question belongs more nicely in #math-discussion

[fair girder]

OK THANK U

it’s 1:30 am

[boreal sentinel]

I'm not a teacher but it's advisable to go sleep

[fair girder]

cough

No not until I finish my 3 ixl assignments

This new week is going to be all khan academy

My teacher assigned 125 assignments

[rigid thistle]

not a teacher here:
weird flex but ok

• not a teacher

[fair girder]

Yes flex

[turbid zenith]

So I was a teacher for four years ... I do like the idea of small frequent quizzes

Just to get an idea of where people are at the moment

Rather than having no idea where you're at until the day of the test and you bomb it

And only then do you know what you're not getting

[tawdry venture]

short five-question quiz every other lesson or so sounds like a good idea to me

[round robin]

yea

[shadow basalt]

depends on the age group

i feel like forcing ppl to attend

is not a good idea unless you are very sure you are a good prof teaching at the right pace

bio classes here have weekend check ins tho

where you can skip lecture if you want and just come in for a 5 minute quiz on the weekend

[indigo shale]

and imo there's very rarely a single pace that is right for all students

[turbid zenith]

So, wanna throw this out there and see what y'all think.

This year at my local enrichment summer program for high schoolers, I'm going to be teaching a new course called "Close Enough", which is going to be basically numerical analysis with perhaps a smidge of other related topics like dynamical systems. My plan was to make it be for students who have already had some programming experience (we always have some of those), and thus for it to be as hands-on as possible — not just people sitting and listening to lectures and taking notes, but actually coding algorithms, testing them out, analyzing when they do and don't work and how reliable they are, etc.

Do y'all have any suggestions for what sorts of student-centered activities might work for something like this?

[turbid zenith]

Topics I imagine we could hit are floating point numbers, error propagation, interval arithmetic, root-finding (bisection method, secant method, regula falsi, Newton-Raphson method), interpolation (Lagrange, Newton), optimization, dynamical systems (some DiffEQ, logistic model, chaos theory, that sort of thing). Probably will also look at Taylor series, Fourier series, Weierstrass factorization, as different ways to approximate functions.

[winged urchin]

Off the top of my head, maybe computational complexity?

You covered a lot of the bases I think

[turbid zenith]

What I'm looking for is less what topics to cover

And more what kinds of activities to do

How to make sure students are really involved

[tidal whale]

What you've mentioned sounds incredibly thought out

[boreal sentinel]

Once my transport phenomena lecturer used the attendees of the lecture as nodes in an algorithm to solve the Laplace equation

or something like that

[winged urchin]

Ohh that sounds kinda neat

How did that work though?

Just like.. "alright everyone! look at number your cardinal neighbours have and perform this operation"

?

or whatever stencil is being used

[tawdry venture]

[wispy slate]

can any teachers here help with my resume? It's my first gig and they said I should improve on my resume before showing up to the interview
I'm super nervouse kek

If you can help pls pm me so I can send you a link for editing

[calm turtle]

@vestal quiver

[wispy slate]

@vestal quiver

[rigid thistle]

@vestal quiver

[charred silo]

Perfect examples of how teachers should behave!

[grand laurel]

when correcting homework, how do you deal with people who clearly put in (almost) no work

i usually like writing detailed comments, but when someone clearly shows no effort i kinda cba

but just commenting "wrong" or "nothing shown" feels rude

[tawdry venture]

what extent of "no work" are we talking

if it's just a blank sheet then "nothing shown" is appropriate imo

[grand laurel]

blank page i just mark as missing

but im talking about stuff that is "unmotivated rambling" for lack of a better word

often misinterpreting the question

and bad handwriting with much stuff striked through

[tawdry venture]

mark bad handwriting explicitly as such

unmotivated rambles: underline and comment "off-topic" or "irrelevant"

[winged urchin]

Hopefully your handwriting is good during class else they might think you hypocritical :-P

But Ann is basically correct. Just hopefully it's communicated at the beginning of the year what is expected in terms of neatness and assignment submission

If you have the time and are feeling generous then try to weed through their answer for any pearls within the muck. Sometimes students have any number of reasons why they might not have the best writing or whatnot

[lethal leaf]

Make em use LaTeX

[meager girder]

that's what i do

[winged urchin]

LaTeX is actually great for assignments once you get used to it

Instead of rewriting a big long disgusting equation to the next line and changing one or two things... Just copy paste and change what you need

[grand laurel]

i tried to make them use latex

they are also CS students

i gave them a template, but nobody cares

this one year i had a group of 3 people who did their homework on git

that was sick

[round robin]

why will cs students what to write on paper thats dumb

[lethal leaf]

I mean the only way to make them do LaTeX would be to go "yea from now on if the homework isn't typed up from LaTeX I'm not grading it"

[tawdry venture]

"non-LaTeX homeworks get an auto 0"

[grand laurel]

that's a nice way to lose my job

[lethal leaf]

I mean my math class is based in mathmatica right now

And the teacher was like "yea good luck doing the assignments without it and also even if you do I'm not grading it if you don't turn it in through the software"

[grand laurel]

I'm just a TA, I can't just not correct stuff

[fading dust]

I usually show my students my own notes, which are usually in two piles - here's the scratch paper, here's the transcribed pretty notes. and they always say that the transcribed ones are reasonable and legible, but the scratch notes aren't

it's reasonable to tell your students that if you can't read it, you can't grade it. they need to take the time to transcribe their homework into something legible, whether it's latex or handwritten. I think it's ok to stand your ground on it if they complain, too, because again, if you can't read it, you can't grade it

[winged urchin]

I will say that while 'most' of the time messy writing is a sign that they have nowhere near the right answer, a few times I've marked questions that seemed rambly but actually approached the question in a completely different way than intended

It would take me a little while to dissect what they mean but it makes me happy when I do see these little pearls of wisdom

[granite nacelle]

Hello! I was wondering if I could get some help from you folks. Me and a couple of my fellow TAs want to buy a birthday gift for one of the Professors we work under, but are kind of lost on what to get her. We want to get her something somewhat personalized to either her field (she teaches Geometry, Number Theory, Trigonometry, etc.) or her interests (basically just math and crotcheting hyperbolic stuff).

Any suggestions? As a teacher, what is a gift you would like to get?

[wispy slate]

Eulers Disc

[plush elbow]

Well guys, I have wrote on a paper in a CS course :p

At least you are sure that deadass students gave back something (in exam/test case) instead of y’all forgetting about it and being late and shit or even just copy-paste from wherever

[storm tangle]

@granite nacelle Give her good research output. On a more serious note; even though most professors in mathematics tend to know about the history of mathematics, and geometry being one which is rich of history perhaps get her an old work in geometry? Euclid's Elements is probably something she's seen, but maybe some work from someone else that's influential?

[silk moon]

Texit has stopped working

$ proof $

[spark flare]

Get her a copy of Archimedes Palimpsest, he basically invented some sort of Riemann sums in it @granite nacelle

Could be interesting to read

Tho I dunno if there's enough to read in it as not much of the book was recovered

[wispy slate]

The summary of all this discord server

[round robin]

[rigid thistle]

ok why is number theory such a small chunk and vector calc takes up a shit ton of space

[civic tree]

i dont like this map

lol

[meager bronze]

map of mathematics drawn by someone who doesn't really know mathematics

[wispy slate]

i identify with self.repair_brain()

[round robin]

thats a map of like

high school math

[feral cradle]

^

[wispy slate]

Bruh what high school you went to

[round robin]

you can teach those surface level to hs kids

if they can use it or not is another qn

(they cant)

[radiant pollen]

my math prof showed that map today lol

[round robin]

[rigid thistle]

BS

[wispy slate]

vector calc is an engineering scam

[astral marsh]

yeah vector calc is an engineering scam

[kindred ferry]

Why is discrete math on the opposite end of computer science?

[charred silo]

It is trying to be discrete.

[brazen pendant]

what the hell is measure theory doing where it is

[rustic junco]

Measure theory is an extension of probability theory

[wispy slate]

Measure theory is just using rulers on steroids

[boreal sentinel]

All physical measurements are just using rulers on steroids

except for counting

[wispy slate]

Maybe you, I count by imagining a Real Line and a ruler and measuring the intervals

[brazen pendant]

I count by assigning a ruler of fixed length to each object, measuring the total length of the rulers, then dividing by their individual length

[civic tree]

yknow what is a real scam

electromagnetism

pretty sure i heard someone say that it was just

vector calculus but with extra steps

[feral vector]

All physics is just vector calculus or differential equations with less rigour and bad notation

oh and some prob stats

but thats not maths

[lament wraith]

This is possibly the worst take I've ever heard in my life, and I've been in this server for a while

[civic tree]

lol

[turbid zenith]

Anyone in here happen to be a non tenure track lecturer?

I'm applying for a job as one

[violet isle]

@turbid zenith I was a part time instructor at a university, that might be the same position.

[turbid zenith]

Ahh, what I'm looking at is full time

I'm trying to get an idea of what's a reasonable salary

I've done some reading online but figure asking people is a good idea as well.

[violet isle]

I don't think I would go lower than 50K a year.

You can look up what teachers/professors are paid at public universities

I know some experienced lectures at the university I'm at are paid about 75K

Did they make you an offer?

[turbid zenith]

Nope, I haven't had an interview yet but I believe I'm on their short list

[violet isle]

Yeah, wish I could give better advice on this subject. When I taught part time, they basically just paid me an allotted amount per class, which came out to be a little bit more than a graduate students' stipend.

[violet isle]

So are there actually any full time lecturers here?

[charred silo]

I think the pay is fairly standard in the UK.

[marsh cliff]

[meager bronze]

@turbid zenith what does "non-tenure-track" mean exactly

there are institutions that have "teaching-track" jobs

that aren't tenured but have a promotional ladder with some amount of job security

there are also positions that are basically adjunct positions with no real job security

and both of them can be called "lecturer"

in terms of salary it really depends on the institution. if you're at a small branch campus for a large public institution, you should expect 45-55

similar or worse if you're adjuncting

teaching-track at a top institution is probably 60-70

[robust mirage]

I’m currently 16 and I have 7 free periods during the week where I don’t have any lessons so I decided to teach some of the other students (which are also free) voluntarily.

They seemed pretty interested in it and I honestly only started it as a joke to pass time for like 2hours when we don’t have lesson but since our exams are coming up we’ve gotten a bit more serious.

I try to have a lesson plan on a sheet of paper I prepare the night before. We use an empty room in the school then I try my best to present the lesson.

My main issue is I always get lost like halfway through teaching or I don’t know the order to present the content. The content is only A-Level Further maths.

This for me, is the most effective way to learn. What do you guys think?

Managing the class is easy since most of us are friends trying to just effectively study but even so, trying to engage everyone for 2 hours is exhausting

Opinions, tips and advice would be highly appreciated! Just trying to maximise my time at school for the best results

[meager bronze]

have the "students" work on lots of problems during this time

[wispy slate]

sup teachers

[south bridge]

@robust mirage you'll get it eventually, its just practice. I take fluid machinery lab and teach autocad to kids, not as intensive as teaching maths but i think teaching is teaching so... I take 1 topic and discuss how we can modify a question and then how I would tackle it (after asking how they would approach it). You can give 1 or 2 examples at the end of each section so that you have time to see what is next and how to present it. Also 2 hours is a long session(in one go), you can split a session in 50-10-50 which is 50 mins of study and 10 mins of break where they can discuss problems among themselves or get a snack etc. etc.

[turbid zenith]

So I'm pondering things

Things are going well so far with the process of trying to get this lecturer position at a school by me ... I'm apparently on their short list and have a campus visit + interview + guest lesson next month

I'm going to be done with my master's in math in two years, and I'm definitely eventually going to go for the PhD

But I'm wondering if maybe it might be a good idea to just teach there for a couple years (if I get it) rather than jumping straight into the PhD

If anyone happens to have any remotely relevant advice I'm more than up for it

[round robin]

wew jus gave a 2hr lesson on intro top my entire body and voice is now super tired lol and one marker died

im now super impressed by teachers who can survive long days of teaching

time to ask to be paid

and rush lesson notes lol

[pliant yacht]

Is anyone here familiar with Ergodic theory? I’m looking for a tutor

[tepid citrus]

@round robin teach me topo pls

[round robin]

read munkres

im kinda teaching like a stripped down munkres with like less stuff and exercises but in the end still better to go through munkres imo

[meager bronze]

@turbid zenith Are you planning on finishing the masters at the same time as you're teaching? (Also, I'm a little surprised they're offering this to you since most institutions require a masters degree to teach in the first place. Do you have a masters in something else already?)

In terms of the PhD, it depends on what field you want to go in. If you want a PhD in math, sooner is likely better. If you come back in 5 years and say "I haven't done any math in 5 years, I've just been teaching" I think it'll be hard. But if you want a PhD in math education, then teaching might not be as bad. Is there any reason you would want to wait?

[turbid zenith]

@meager bronze I have a Master of Arts in Teaching Mathematics

I'm going for my second Master's, this time in mathematics itself

Main reason I'd want to wait is to have a year or two break from splitting myself between taking classes and teaching classes

[rustic junco]

@turbid zenith I am not sure, but depending on your situation i will probably suggest you go work on your master and then if you will like to do a phd in mathematics it might be a good idea to pursue your interest in that particular field of mathematics before thinking of taking a teaching job.(Gaining some experience with problems and learning new things in the mathematics field that you find interesting).
Good Luck!!

[silver marsh]

For someone like you guys, who teach in the field, how well would you honestly say you know the material you’re lecturing about? I’ve been trying to poke around at grad level material and I’m struggling a little with retaining everything in. At that level, like say something like diff topology, how well do you know it off the top of your head? And say that vs something along the lines of undergraduate analysis in comparison?

I guess what I’m trying to say is: how well is someone who’s grad level or higher to know that material as a second instinct? How well would I be expected to know everything?

[winged urchin]

Honestly I think you'll find a lot of variability

There's probably lecturers who have to take on a certain course because of any number of reasons and may have to pick up the material just ahead of the students in order to teach it even reasonably well

Of course ideally it's second nature. Like I'm a tutor personally and because of how much calculus I see Id say I could teach any part of calculus off the top of my head (though I would still look up definitions of the more technical bits just to make sure it's 100% right)

But I tutor people in courses I haven't seen, taught, or even thought about for a long time

In those cases I do more prep to refresh myself before classes and honestly, those students probably are getting a worse service than the students getting tutoring for calculus purely based on my experience levels with either

I also would say about picking up and retaining information at higher level studies... I personally found that things don't 'click' nearly as quickly as lower level studies, naturally

I'd learn how to do something but not be fully convinced or have tiny little questions regarding some of the technical bits but just have to carry on due to time pressure or not being the best student I could be

But later on, usually after completing a big assignment or possibly after learning some later concept that ties it together in a neater group... Then I would be like AH HAH! THATS WHY THAT WAS LIKE THAT!!

[silver marsh]

That’s a relief

I’m trying to expose myself to it at least, some of the hard stuff. And it feels like my problem solving has gotten way better since I’ve started getting proactive about this, but it’s still just a lot to chew on. I’ve got time, but part of me also was getting worried that I was reaching the ceiling of what I could do in terms of mental fortitude

[winged urchin]

I've often heard teachers say students just need to stare at the problem long enough to 'get it'

And to an extent that's true

Some students do just look at the problem, think it's hopeless, and give up without trying anything

And sometimes even 'good' students just need to think about the problem a little more

But I like to say you shouldn't be stressing or 'force' yourself into that mode

You should stare softly at the problem, try not to fixate, try to look at the problem as a whole and any avenue you have to understanding any piece of the problem

It's good that you push yourself to consider things more deeply but if you find yourself stressing and really fixated on a problem with that mentality of "why cant I see the thing? am i not smart enough to see the thing?!" then you should step away for a bit

Come back and approach it on a different angle

Maybe dont attempt the problem at its fullest

Reduce the problem to a simpler one that's related or ask yourself about a tiny part of the problem

You should hear what others have to say on it too though of course

I'm interested to hear anyone else's opinions on this too

The whole learning process, especially at higher levels, is really interesting to me

[silver marsh]

For material I know should be hard for me and it’s new, I’m cool with getting bruised up and willing to take it slow. But that toxic mentality you mentioned started manifesting today when I was doing something on a mastering physics module

A super easy question I could derive backwards and forwards but it wouldn’t take my input and it gnawed at me badly because Idon’t have a lot of time to do the hw because of work and school. I’m taking 30 min rereading the chapter and just doubting myself super hard

Only to realize I flubbed an extra number when I was putting the answer in

But now I’m thinking that my cognitive faculties as declining

So I don’t know what’s better: accepting I’m exhausted and making mistakes or how close I am to falling apart right now

Way too close for comfort

I’ve been struggling with this software since the start of this semester

Any I don’t like getting tripped up by what I should be comfy with

Am I not paying enough attention? Am I spreading myself out too thin? Am I weak?

That’s what I was dealing with today

[winged urchin]

That question there "Am I not paying enough attention?" is exactly what I mean when you should look 'softly' at the problem

[silver marsh]

But it was a problem that I knew the answer to. So I need to be more aware that I can make a small mistake

[winged urchin]

I highly doubt it's a lack of care. Of course I don't know you more, but I would wager a good amount your errors are not coming from a lack of care

[silver marsh]

Check the obvious, is the number crunch wrong?

[winged urchin]

You will always make some errors, sign mistakes, off-by-1 errors, these will happen. I don't believe you can reasonably expect these to go away

[silver marsh]

It was such an innocuous mistake but because I didn’t account for it I made myself miserable

[winged urchin]

Right, I'd suggest you can either:

• Check your answer if you can. (Plug it in and see if left-hand side is equal to right-hand side or whatever your problem is) Granted if this just boils down to doing the problem again then it's not really a great method

• Think if your answer makes sense within the context of the problem. Are you getting a negative slope when you expect a positive slope? Is it an unrealistic number for whatever the variable represents?

[silver marsh]

The way I do my math it’s all set up for one calculation

[winged urchin]

• Or try to do the problem an entirely different way and see if you get the same answer. This is at the core of science in fact. Using multiple approaches and getting the same answer improves your confidence with the answer

Okay thats good, collect everything first

[silver marsh]

Looking back at it, it’s kind of funny that this was what I ran into

An honest mistake to remind me im human and definitely got a good bit left to learn in terms of wisdom

[winged urchin]

Yes that's right. You can never escape the tiny mistakes. And you will mix up even simple things, elementary things. It doesn't necessarily mean you lack care, understanding, or knowledge

[silver marsh]

It’s a relief, honestly

And about the pushing myself too hard to study: it’s more like playing a difficult game. There could be a learning curve, but the pay off is the experience of being a part of this giant fantasy world and really feeling the anguish in it all that makes the beautiful and elegant stand out

And the best part about math and science is that it isn’t some fantasy. It’s real, it’s there

And to me that goes from math being an escape to my way out to something better

[winged urchin]

Well haha. Lets not get into a debate about the nature of maths right here. Is all of maths real? Or is it a fabricated tool to see things through? But I'm being a bit silly anyway

Of course it's real in the sense that it has applications and use in the real world and indeed, you can 'see math and science' in the real world

I'm glad you find joy out of it though. Just make sure you dont push yourself too much. Check in with yourself

You don't want to end up snuffing the flame of curiosity

[silver marsh]

I guess a better way to mention it would be it’s a productive investment of my time and energy

And it’s something that makes me happy

It’s not a vice and the better I get at it, the more I can help others

Thanks for giving my plight a listen to

My headache has started going down and I feel way more relieved now

[winged urchin]

Absolutely. First and foremost you should enjoy it. And it's just an additional boon that it's a valuable skill to have in our world right now haha

No problems friend. Anytime. I hope you give yourself a bit of rest and come back to your studies with fresh eyes 🙂

[cold sorrel]

This is gonna seem specific but what's the best way to explain the formula for cumulative average? Because I have my personal favorite explanation but I keep sensing that I'm losing my pupils when I go through with it.

engaging ways of deriving $ \frac{1}{b-a} \int_a^b f(x) dx$? I'm a tutor

[burnt vessel]

datorangeguy:

[cold sorrel]

I will also gladly take this to #calculus if it fits there more

[tawdry venture]

i'd go geometric

imagine you have a metal thing shaped like the region under the graph of f

then you melt it

it turns into a rectangle

then it cools back down

its width is b-a and its height is the average value of f

[cold sorrel]

I also do geometric but I use the fundamental theorem to pull it together

People remember things like "average velocity" over an interval, if s is displacement

you just take (s(b) - s(a) ) / (b-a)

but $s(b) - s(a)$ is exactly $\int_a^b v(t)dt$ by the fundamental theorem

[burnt vessel]

datorangeguy:

[cold sorrel]

therefore ``average of $v$" is given by $$\frac{s(b)-s(a)}{b-a} = \frac{1}{b-a} \int_a^b v(t)dt$$

[burnt vessel]

datorangeguy:

[cold sorrel]

You can also do the converse and derive the fundamental theorem from the geometry and 'melting' argument

and if not for losing my audience I'd do this all in one

[lethal leaf]

That's the way I teach it

Through the lens of average velocity/acceleration

[obtuse widget]

Never seen this general formula before so I can give my feedback. Geometric made me instantly understand. In case of difficulties I would start with y=x and show moving the cut triangle to create rectangle from 0 to some value, then proceed to trapezoid from a to b and then show some symmetric functions so that one cut and piece transport still works. At the end I would introduce the general explanation like Ann said.

the velocity explanation seems to me like some black magic and is difficult to understand even now, but maybe for some one it's easier

[turbid zenith]

@cold sorrel I've got another way you can think of it

So the way you usually find an average is to "add up all the stuff" and "divide by the amount of stuff" right?

When you're dealing with continuous functions, your "add up all the stuff" becomes integration over an interval [a,b], and your "amount of stuff" becomes the length of the interval, b-a.

I've found that often resonates with students because it directly reconciles with their already existing definition of average rather than seeming like something completely different.

And it helps reinforce the important idea that integration is often a continuous analogue to discrete summation, which is an intuition students need to have in other places in calculus.

It helps if students understand that the "∫" symbol is literally an "S" for "Summation" as a continuous analogue to "Σ", akin to how "d" is a continuous analogue for "Δ".