math-pedagogy

[shadow basalt]

Wrong channel

Do students (university aged) find historical motivations and stuff helpful? I was thinking of picking up a history of calc book or something to maybe find stories to share

[pastel horizon]

Stories could be a great way to break up material

[shadow basalt]

They would be very short but idk if it will just be annoying

[strange bronze]

i think theyre good

[pastel horizon]

Just be careful they don't remember the story instead of the material!

[strange bronze]

i dont think it hurts to make lectures entertaining

[pastel horizon]

The material is the most important thing

[brisk ruin]

Are your students math majors or other majors

[strange bronze]

like even if you take 2-3 minutes away from lecture time, theyll probably pay more attention (and find it easier to follow along pacing-wise) if you include historical anecdotes and quick jokes

[brisk ruin]

I feel like non-math majors would be less likely to care

[shadow basalt]

Probably not math majors

Well as a math major I also don’t care hahaha

But some people find things easier to remember if there’s a story attached

Maybe I’ll just sprinkle them into lecture as factoids

No need to make a big deal of it

[pastel horizon]

Yeah that's the best way to do it I think

If you overdo it the only thing they'll take from the lecture is the stories

[shadow basalt]

Yeah it would be like one or two sentences at most

But I’d need to go thru the effort of learning the history myself hahaha

[grim spindle]

I'd say so, my class really liked our linalg 2 prof sprinkling in historical facts, they were math majors tho so

[shy ice]

Anyone have good strategies for self studying new math stuff? I’m talking grad level, I have an MS but want to learn more to eventually get to algebraic geometry (hartshorne level)

Is just doing a shitload of exercises the best way? I’ve never really been good with taking notes, just scribbling on legal pads all the way through but I feel like if I want to learn on my own I should be more organized

[kindred stag]

Personally I think the relevant question is how well do you actually want to know the material. If you want to be able to use it then yeah, exercises probably help, but if you just want passive familiarity, just light notes + reading is typically enough for me.

[shy ice]

@kindred stag I’d like to have a pretty good idea what’s going on

I just see some people taking such nice notes it makes me feel bad all I do is scribble lmao

[winged urchin]

I think doing exercises is pretty much the only way to know if you have the right understanding

Taking notes can help me a little but honestly in a lecture environment I prefer to listen along and try to understand, if the notes are available afterwards perhaps

I think I find note taking to be more effective by myself actually, like making cheat sheets or summary sheets kind of to make the information more accessible to me after

More to the passive learning side too I do like to just listen to lectures or explanations on topics like on Youtube. I won't really get the picture completely but I find it is helpful to some degree

Also respecting your mental state is good, at least for me, in so far as I can kind of tell when I'm not going to absorb anymore info or I'm just falling into a mechanical process

Just my perspective

I'll also comment briefly on the neatness. I find personally I write nicer notes when I'm calm. And nice notes are better for reference later anyways

So if I notice my notes getting very scribbly it could be a sign that I'm burning out or getting frazzled and need to shift my mind back to a more thoughtful, calm place

[shadow basalt]

Yes doing exercises

Is the only way

Unless there are no exercises

In which case you have to invent them

[shy ice]

I guess my main thing is, should I bother taking a nice set of notes or just run through exercises and call it a day

[winged urchin]

Don't worry about 'practicing' note taking. Nice looking notes are better but not 'that' important

[shy ice]

Yeah that’s kinda what I figure, if I need a reference I can just use the book haha

[winged urchin]

I'd say, do exercises but make notes of things you find yourself referring to a lot perhaps? Maybe if you find some new insight or understanding of a certain kind of problem make a note of that to help cement it

The book is okay too, depending on the course

[shy ice]

That’s a good idea

In my case it’s not a specific course

Just to keep learning more algebra, which I enjoy quite a bit

[real mauve]

what i used to do was scribble a bunch of stuff without worrying about neatness, do a bunch of exercises, and about a month before the exam, start writing a nice and neat explanation of everything (the notes + tips and tricks one picks up thru exercises)

that explanation/summary was my study time

[shy ice]

Thanks everyone, nice to know y’all have similar methods

[chrome wind]

https://early.khanacademy.org/early-math/ does anybody know of any projects that have implemented ideas like this?

I think it sounds like a very interesting idea

[zealous edge]

ive always found rewriting notes slowly and neatly as if you're teaching it has been effective for me going thru grad level maths

only writing the step once youve got it

[dawn walrus]

I have no idea how similar this is to what's at khan academy, but have you heard of the Dragonbox games?

[chrome wind]

Oh yea, I grew up playing the Dragonbox games (mostly the algebra one) :P
I am not sure how much influence they have over my thinking today, but one problem is that when I started learning to solve basic equations, I would think of "dragging" each term over the equals sign and it becoming negative, instead of subtracting that number from two equal quantities. It's probably important to avoid things like that when trying to design an app like what Khan Academy suggests. On the other hand, it definitely gave me a good intuition for things like substitution, multiplication, distribution, etc.

[tawny slate]

@left meteor replying to your post about "pop math"

I used to hate these kinds of posts as well, but as I delved more deeply into the education side of math and matured I realized that I don't actually hate the "pop math" posts by themselves, what I really hated was the misrepresentation of math. recreational math and pop math serve a purpose in getting an audience who don't understand or appreciate math well enough to at least romanticize it, and in society we need a little bit of this. you can't teach people who don't care, and if they don't care, very few rational arguments will make them care. an emotional appeal is one of the best tools we have for getting a population who don't care to engage in it

example of "pop math" done right: mathologer

[left meteor]

To be fair, I don't really hate it, I kind of agree you. I just dislike it appearing on r/math. I'd hope that subreddit would be more focused on discussion of mathematics by mathematicians and other stem students or researchers, instead of things like "hey, look how math is cool" that is generally targetted to the rest of the population. But hey, that's just my selfish interest 😆 .

[strange bronze]

sadly /r/math, outside of the quick questions thread, doesnt really go beyond the early undergrad level and the top posts mostly hover around "look how cool this is" visualizations or "i used to hate math but fell in love with calculus" tier posts

this is kind of the path any unfocused math community takes on the internet, in fairness

the only reason this discord server has a higher math community at all is excessive compartmentalization + the honorable system

math.stackexchange tags topics and closes repeat questions aggressively and mathoverflow still exists as a separate website

(one thats very very picky about what fits on mathoverflow)

and let's not get started on facebook or quora...

[tawny slate]

It feels very difficult to sort through questions and such on math.se for your answers

You have a question about a function and don't realize that it's actually brouwer's fixed point theorem

You have a question about algebra but it's actually a combinatorics question in disguise

You have a concept for a question but dont really know how to describe it well, but we already have a definition for metric spaces like Hamming distance

How does one search for their questions when they lack the vocabulary to parse it in various equivalent forms?

[strange bronze]

not exactly sure how thats relevant to the convo at hand

[shadow basalt]

You have a question about stable homotopy but the only person willing to answer it is only on alg top discord

[lethal leaf]

there's an alg top discord?

[dawn walrus]

WHY THE HELL DO YOU DO ALG TOP?

[brisk ruin]

#chill

[shadow basalt]

Yes

[lethal leaf]

sounds cool but yes #chill

[shadow basalt]

Ange has literally no power in this channel

Ignore him

[lethal leaf]

ok

[shadow basalt]

This is a special economic zone under the ostensible but ineffective rule of the moderators

[lethal leaf]

one day I shall take a topology course lol

[civic tree]

what

[pastel horizon]

Mathologer is amazing actually

[patent beacon]

From a teacher's view, channels like Mathologer and Matt Parker have always been my go-tos for when I need to break things up with some other interesting maths. It gives the students a break from me 😅 But it also depends on the class/audience you have

[lethal leaf]

I like Michael Penn alot

3b1b is great for intuition type stuff also

[grim spindle]

Michael Penn does too much olympiad maths to my liking

But his serious maths vids are cool

Esp his series on his research on vertex operator algebras

[lethal leaf]

Yeaaaa I love that series

The olympiad and temple geometry I find nice also

[pastel horizon]

I liked his video on exploring the idea of calculus over the rational numbers and seeing why it doesn't quite work

One of the problems is you can get functions that are discontinous over the real numbers but actually continuous over rational numbers (let's say a function is undefined at √2)

[patent beacon]

I'm just learning of Michael Penn 👍 cheers

[regal wadi]

Could someone recommend me some themes for my undergraduste thesis in math education?

[left meteor]

investigate pop math

[patent beacon]

I have friends who decided not to go into the practice of teaching, but do more research education

By memory, there's always discussion around methods and curriculum, but I know that's super broad. My last semester was a research project into how to use technology effectively in mathematics education. That was 2019, so when 2020 came around and COVID first hit, I got to flex my muscles a bit while everyone else was trying to figure out how to start

In saying that though, I always found the maths education discussions always came back to the same principles of simplicity in term of lesson goals, focused discussion and participation, clear explanations and engagement. One thing I tend to disagree with is that differentiation isn't nearly as important as its put out to be, it's more about how the content is delivered. That's your explanation and questioning methods.

I've given lessons that I do with my senior students to the juniors of the whole school. It takes them a little bit longer but with the right explanation they tend to understand it. It's all about that communication, which if you want to take the pop math route, you can look at the impact of how content in presented. Sounds trivial but there's a noticeable different with how younger teachers present compared to the older teachers

[pastel horizon]

It depends on what type of maths education you're interested in as well. Someone who is 5 will learn much different than someone who is 14 or even 20

[pastel horizon]

Differentiation for us is more like having an extension activity if pupils finish early or having varying difficulties of questions but yeah I agree you're basically teaching the same content for everyone

What I noticed today was the pupils (13-14) found rounding to a higher number decimal places (say 3 dp instead of 1) was harder even though for us there's no difference in difficulty so there's also kind of a psychological barrier what kids will find hard

[fluid storm]

Hey, this is not an exercise help, but could you guys give me tips to how create exercises to help students?
-Fractions (operations, simplification)
-Decimals (conversion, places after the dot)
-Powers and roots (not knowing the properties and when to use them)
-Equations (process to isolate the variable)
-Interpretation of the question

they don't have a good base of these and have difficult with basics concepts

[fluid storm]

here it's the list to the topics they have the most trouble

i'm thinking to mix the last one with the other but don't know how

[next relic]

I suggest checking out variationtheory.com to see if there's anything nice you can use.

If you want to craft them yourselves, I'd recommend writing questions that your student would find "huh?" (example: 3x . 1/4 vs 3 . x/4) or something that can start a discussion (perhaps draw attention to those features later once they've finished the task?)

[fluid storm]

Probably start by the simples one and then devolepd is the better way to gom i tried to craft one exercise, but some professor said to me that is very complicated

quation r above describes the amount of fish in an aquarium for a time X, it is such that the tangent of theta =2 and do what is asked:
The ration is measured in units, example 1 = 1 unit
A) Calculate the line r
B) Find the line S, perpendicular to r, when the time is one unit
C) Write the letter B, but generalizing to any X
D) Calculate the area formed by the Y axes and the r and S lines, with data from exercise B
E) Calculate the volume of the solid of revolution formed around the Y axis from the figure

[next relic]

When the basics aren't in place, I'd focus on getting the basics sorted first.

If you're teaching uni like most folks here, please feel free to disregard what I say.

[fluid storm]

I will help students that didn't have a good education and are sering classes of a popular "cursinho", a free teach that helps the student to pass by the entrance exam

[quasi musk]

I feel that clear explanations don't always help

Like if it's too smooth or too clear

Then nothing sticks

I often devise tricks in my lectures to try to purposefully trip up students, catch them off guard, or make them think "huh that's funny"

I also switch topics/type of problem we're solving every 5-10 minutes

Really keeps them on their toes, forcing them to pay attention

My idea is to make it fairly incomprehensible unless you pay attention

So if you tune out for longer then 2-3 minutes you'll be confused and have to ask me questions

[patent beacon]

That's how it looks in my classroom. Not sure about where you are in the world but Aussie kids aren't the most keen on maths so I have to disguise the extension question because if they see the word "extension" or "challenge" they don't do it. I've come to call my extension questions the What's Next? question because that's what they ask me, I can just point to the board and say, "that, that's what's next." It's also the time when I weave some number theory or other question that's interesting into the lesson.

[patent beacon]

What age groups are you working with? I'm at a high school level. And perhaps *clear *wasn't the right word, more the right explanation is a better way to put it. I'd be interested to see how my students go with the incomprehensible to comprehensible approach, I think for the most part I see students give up when they miss a key point, so I have to start with a go segway into the content or with the right explanation so they're set up to do well with the first few questions and after that I throw them a curve ball that forces them to extend what they're just learned a bit further or link what they've just learned to something we've done already

[pastel horizon]

We have the opposite here. Some of the higher ability students would enjoy a challenge and get bored if none existed but on the same token you can't make it too difficult or they give up

[quasi musk]

I'm teaching freshman at university taking remedial algebra

So most of them have seen the algebra before, I abuse that against them

And try to draw connections between disparate material

[turbid zenith]

These are always an interesting bunch

Something I've found seems to resonate is the "lemme show you what they never told you" route, with regard to how everything ties together

Like for my college algebra class I told them there's really only two ways to solve an equation: "undoing" and "splitting." Everything is either one of those or manipulating equations until you can do one of those.

[quasi musk]

That's exactly my approach

I draw a lot of pictures

[patent beacon]

Going to be honest, I'd never heard of remedial algebra before reading that

Is it just a basics sort of course you run?

[shadow basalt]

It’s common at large universities I think

I’m sure ucsd has one

[brisk ruin]

U Mich also has one

It's pretty large

[slate spruce]

hello, im new to this channel, im 17 in highschool but i tutor and think about pedagogy a lot, and enjoy learning from even the teachers that teach me math

I am in highschool, but in Uni at the same time, its a weird setup. I go to UNCSA

University of North Carolina School of the Arts
I am a music composition major

I tutor for students there

[slate spruce]

and hell yes this is the way to do it my experience too. cough cough mathematicians lament cough cough. I think people need to see the "real math" behind what they are doing, and the best way to give an intuition for that is to draw shit.

its funny how out of place my text formatting is in this channel lol, you guys are all adults (ew) i guess

perhaps you can learn what it is like to be a current day high school student through me

[dawn walrus]

Perhaps we can

[weak bane]

The best student is the one willing to learn

[pastel horizon]

Intuition is kind of a weird one. I observed two lessons on ratios with a higher level and a lower level secondary school class. The higher level class understood the more abstract concept of ratios whereas the lower level understand the physical meaning of a ratio (e.g. 2 paper clips for every 3 pencils) but when it came to the actual symbol 2:3 they got confused

It seems like intuitively they both understand the concept of a ratio but the lower ability don't know how to abstract it to pure numbers

[slate spruce]

that helps
i think it should always start physical then go visual then go abstract

in that order

of several days

[slate spruce]

teach me 😳

[weak bane]

If a student is determined enough, he may learn without a teacher 😌

[pastel horizon]

But they may also learn misconceptions ;)

[slate spruce]

thats a good and a bad thing from a person perspective

[pastel horizon]

That is actually one of the major drawbacks of lessons where you let students explore for themselves, they may end up discovering 'new' circle theories (which are wrong) and then that gets ingrained in their mind

[slate spruce]

i learn the right way so much better if i had that misconception

something something veritasium had a video discussing it

[pastel horizon]

Because they feel like it should be right since they discovered it on their own

[slate spruce]

i feel like one should let it happen and never tell them they are wrong

until its nessesary to prove to them that they are

[pastel horizon]

The earlier you can catch them out the better

[slate spruce]

i am biased towards my own experience i guess

[quasi musk]

I think UCs only go down to Pre-Calc

[pastel horizon]

Otherwise you end up with 16 year olds that still have bad habits when it comes to adding fractions

[quasi musk]

CSUs only go down to college algebra

CCs go all the way down

[pastel horizon]

Imagine if nobody ever told them you can't have 1/2 + 1/5 = 2/7

[slate spruce]

i have learned how to check my own theorys on my own and through my grandfather

so maybe you teach how to prove something works

[pastel horizon]

But when they're 16 it's almost too late to get rid of that misconception

[slate spruce]

maybe you teach how to not hold on to your special idea

of course
of course
start in kindergarden

which is hard i guess

[pastel horizon]

Yeah kindergarten don't learn fractions

[slate spruce]

yeah

[pastel horizon]

Or I don't think they do

[slate spruce]

i think the best way is to guide them, even if they are already having a misconception

just nudge them along until they contradict themselves, then point it out

[pastel horizon]

If you want to show them something it's usually better to do it yourself

[slate spruce]

yup

[pastel horizon]

You show examples and non examples and soon they figure it out

[slate spruce]

split the time between telling and guiding

oooohh now i know what you mean
that works for me too\

[pastel horizon]

But non examples can be dangerous if they don't pay attention

Like they might think "oh sir showed me a triangle can be a quadrilateral it was on the board" even though it was a non example

[slate spruce]

have a reward for whoever figues out the the pattern
that worked in a summer camp class thing i did

[pastel horizon]

What I found was whiteboards are underrated when it comes to teaching, they can answer questions quickly and you can see them quickly

But the drawback is it's a potential distraction

[slate spruce]

YES

and i guess?
ive never been distracted myself when someone uses the whiteboard that way

[pastel horizon]

Make sure when you're finished they put them away or they'll be drawing on them

[slate spruce]

ooooo thatttts what you mean

ok

i was thinking group white boards

[pastel horizon]

Ohhh nah

Like the individual ones

[slate spruce]

yeah i know

[pastel horizon]

They are a really good tool if you use them right

[slate spruce]

i dont know thats always felt clunky to me

id like to see them used by someone who knows how too;)

[pastel horizon]

Well you can have multiple choice questions on a bigger screen and then they just vote

That's one way, it's very quick

And then you can ask people "why do you think it's C?" Or "why is B wrong?"

[slate spruce]

i always like that

but its better if you can get us students in groups where we like the people we are playing with

then students get have a chance to explaain to each other why its wrong

[pastel horizon]

Groups are ones you have to pick carefully

[slate spruce]

yup

[pastel horizon]

Not just behaviour wise either

[slate spruce]

groups are hard

hard hard hard

[pastel horizon]

You don't want all the smart kids in one group

You want mixed ability

[slate spruce]

like open mathematics in graph theory hard

except the nodes are students and they have oh so many properties

and then you have the smart kids that over power and the ones who are good at now

not

and what are you going to do pull those students that ruin it aside?

maybe?

i dont know

[pastel horizon]

Or having all shy kids in one group for example too

You want at least one person to take on the leader role but not too many leaders or it gets imbalanced

[slate spruce]

when i say figure out the pattern

i mean find the pattern

as in describe it

and you dont ever actually share the solution when someone gets it

[pastel horizon]

We have rewards if a student actively helps others I think it's a good idea

[shadow basalt]

Be careful lol

you will have your strongest student

[slate spruce]

yeah

[shadow basalt]

strongarm the rest of the class

[slate spruce]

i know

[shadow basalt]

into letting them help

[slate spruce]

i know

[shadow basalt]

and reap rewards

[slate spruce]

i am that strongest student or was rather

[shadow basalt]

very cool

[slate spruce]

but there is no first prize

[shadow basalt]

i wasnt talking to you

lol

[slate spruce]

the prizes are equal

[shadow basalt]

I was responding to this

[slate spruce]

cool, my bad

[shadow basalt]

Most young students don't understand pedagogy well enough to be actually helpful

they will probably just tell the other student the answer

not calling young students dumb pedagogy is hard

[pastel horizon]

At the end of the day though the teacher decides

[shadow basalt]

i still have no clue what i am actually doing for this teaching position

[pastel horizon]

If they're just telling the answers that's not going to get a reward they'd have to go beyond

[slate spruce]

i recomend forcing them to learn
while teach them

i once devised a system which is just an extreme version of what one teacher did one time

You teach ONE person. then they teach, then their students teach, etc.

dont pick the same starting person everytime

always watch and correct

[pastel horizon]

I don't know if I like that idea

[slate spruce]

never saw it happen
dont think i ever will

but my teacher at the time did something that worked and it sparked my imagination

[pastel horizon]

It's like Chinese whispers

[shadow basalt]

that sounds insanely slow

and prone to fail

[pastel horizon]

Yup

[slate spruce]

then scale it down

[shadow basalt]

you should probably just like

be the teacher

[pastel horizon]

Chinese whispers but with teaching maths it's a recipe for disaster

[shadow basalt]

theres no reason to teach young students how to teach that early

[slate spruce]

i think it could work

[slate spruce]

and i think there is every reason to do so

[pastel horizon]

There's no reason at all really

[shadow basalt]

they can wait until they have to start teaching in 15 days and their department hasn't given them any training or information as they start grad school

[pastel horizon]

They are there to learn maths

[slate spruce]

look look
wether you know it or not
its been shown that kids are rather good at teaching other kids

and that kids are good at learning from other kids
and when i say kids i mean less that 3rd grade or so

[pastel horizon]

What I think can work is if you do group presentations but that's not the same as teaching each other

That's them showing they can research a topic and present it

[slate spruce]

yeah
yeah

i feel like the reasearch topic thing is pulls the joy out of everything when its employed

[pastel horizon]

You'd be surprised

If it's a computer activity and they make posters, it's almost a guaranteed win

[slate spruce]

i guess its form over method most of the time isnt it

what age kids to you teach

[pastel horizon]

Secondary school

[slate spruce]

oh yeah
never mind
i thought you meant in highschool

[pastel horizon]

Any lesson involving computers is usually guaranteed to work with them

Secondary school as in 11-16

[slate spruce]

we have it 2 years less but sure

[pastel horizon]

I think once they're a bit older they see through it

[slate spruce]

yup

and it gets repetitive

[pastel horizon]

You basically just disguise computer work as a reward

[slate spruce]

teaching people never gets repetitive thats why i like it

?

ive never felt that way about computer work and i dont think most people have....... except for one time

where do you teach

[pastel horizon]

Like you book a room out and say "Because you've all behaved well this term and done some hard work, today we're going to do an activity on the computers"

UK

[slate spruce]

ok might be a different student culture i dont know

[shadow basalt]

I did love a good computer activity as a kid

[slate spruce]

i think it is

im in america, school is looked upone culturally as a damn hell scape

(exageration but you get it)

[pastel horizon]

Teenagers don't really change wherever you go though

[slate spruce]

the problem is most of the time it is that damn hell scape

the structures of school really does

[pastel horizon]

Yeah but the last place they want to be really is in school tbh 😂 if they had a choice they wouldn't turn up

[slate spruce]

which does change how the students regardless of age, see it

[pastel horizon]

Half the battle teaching them is getting them to focus

[slate spruce]

thats so much more true (just from stories from some student british immagrants ive met) in america

[slate spruce]

this is true no matter what

[pastel horizon]

I feel like when you're older it's more on the student to focus

[slate spruce]

yeah

[pastel horizon]

If they don't the teacher/lecturer doesn't really care they still get paid

[slate spruce]

you were saying about the students not wanting to be there

in amerca they often just dont come

[pastel horizon]

But when they're younger someone not focusing ends up distracting everyone else then it becomes your problem

[slate spruce]

like they are supposed to come

but they dont
you get it

[pastel horizon]

Tbh there's probably a couple of students that teachers here wish wouldn't turn up 😂

[slate spruce]

damn

yeah same is true anywhere

anyways nice discussin with ya
gonna go figure out lunch

[pastel horizon]

I mean do the parents get fined if their kids don't show?

[slate spruce]

hahahahaha

hahahhhahahahaaaa

maybe?

[pastel horizon]

That's a law we have in the UK

[slate spruce]

hahahahahahahaaaaaa

we have a similar law but

still

hahahahaaaaa

[pastel horizon]

So yeah a big incentive for UK parents to make sure their kids show up

[slate spruce]

yup

our law is often just ignored

uk governmental function is tighter

see you

[dawn walrus]

I'm the strongest student around these parts. There isn't room enough in this town for the two of us.

[slate spruce]

oh no!

[winged urchin]

I think that sounds nice to say but it is much more integral for a student to be willing to learn. Certainly a stagnant educator will eventually lose focus on the current world but I'd hazard to guess 'willingness to learn' isnt the primary identifier of great educators

Of course this is just small beans =p

[weak bane]

I was just trying to annoy ryc please stop tagging me here

[blissful latch]

The best troll is the one who is willing to learn

[weak bane]

[shy ice]

Question for self study people, when you are reading through a textbook do you solve every exercise before moving on?

[molten urchin]

This question is probably better suited to one of the general discussion channels; although I feel one would generally not do each and every exercise unless the book itself is very compact on exercises.

[pastel horizon]

The point of exercises in textbooks is to give the teacher a set of standard problems and a set of challenging problems as an extension activity. you would kind of just pick the set of questions that you feel are appropriate for yourself

[winged urchin]

If you can be honest with yourself then you can skip the questions you know how to do

Like if I'm doing problems and I can see the path from start to end and I've done these kind of problems before then I might skip it

There is always a chance that you're kinda lying to yourself or not seeing some hitch in the process you imagine however

[pastel horizon]

What would your thoughts be on a "find the error" exercise? Like if I show an example where someone has fell for a classic trap and ask them if they can see what went wrong?

[shadow basalt]

Generally good

Ideally designed so a student who would never make that mistake or understands it well will see it instantly

It shouldn’t be about “hiding” the error in a clever way

It should just be about identifying it

[dawn walrus]

yeah any time i've had a problem like this it's either been on the easy side, or has been a pain that i learn nothing from

and the ones that are on the easy side are actually good

cause they make me go, oh, i'm not gonna fuck that up now

[pastel horizon]

Yeah definitely, you need the students to have a bank of knowledge they can't screw up so that they can concentrate on the trickier questions

One of the ones I've seen is in prime factorisation, they'll write something like 6 as 3*3

Probably because they think "oh it's just two threes" idk it's a weird mistake though

[winged urchin]

Some mistakes have nothing to do with the students knowledge base and are just human failings though lol. Like the 3*3 = 6 thing

There's nothing to identify there really. They already know what you're trying to show them. So really it's a mentality thing in that case. Students not checking over their work in a calm, clear way

[quasi musk]

I think these are good for classroom discussion problems

I don't find them very good for exam questions

I did the classic 2 = 1 "proof"

and it got the students interested

[strange bronze]

which one

dividing by 0?

[quasi musk]

a = b

a^2 = ab

a^2 - b^2 = ab - b^2

It was good because we were also covering a^2 - b^2

and overall distributing/factoring

[strange bronze]

yeah okay

[quasi musk]

So it tied into the course in different ways

Than just "oh you can't divide by zero"

[twin lichen]

lmfaooooooooo ok i had this teacher in 8th grade that tried to do this

but instead of showing us the proof

she just wrote 2=1 on the board and asked us to prove it

[shadow basalt]

woke

[pastel horizon]

I'm not too sure about that, if students aren't paying attention they might actually get the misconception that it should be correct

[pastel horizon]

Was a guy here who posted about getting pupils to teach. One thing I found great success with for younger kids was getting them to show their work on the big board up front, you can ask the class if they agree (and correct mistakes if needed) but give them big praise and achievement points. Even the other pupils gave praise via round of applause

Obviously first you have to kind of drill the method in first and get them feeling confident

[shadow basalt]

This but for undergrads but you still give them way too much praise

Make them incredibly uncomfortable

[pastel horizon]

Why not? They may have to stand up and present/defend their ideas either in a PhD or their job

For that scenario I'd probably put more emphasis on the other students asking questions

And like I said if they make any mistakes you always have the option of stepping in

Another thing my lecturer used to do was make the female engineering students speak up louder because they are (unfairly) underrepresented in STEM so it's about giving them the confidence to be assertive in their careers

[shadow basalt]

oh no i agree that undergrads should present

I just know they would be made terribly uncomfortable with the same praise one might give to a young student

[lethal leaf]

^

praise is good

but should be in a different form

Like in office hours or on the class forum I'll say "oh this student asked a good question / pointed out a mistake with my original answer"

[lethal leaf]

this is also a great thing to do yea

[austere inlet]

in private during office hours might be better, you don't want to e.g. make the shy students feel worse for not asking or contributing

I've had a prof assign extra marks for participation in my TA sessions and even though contributions increased exponentially it did seem a bit forced at times, especially for those that naturally didn't ask questions in front of people

[pastel horizon]

Yeah with a young student you'd probably even praise them for actually sitting still and listening 🤣 definitely not what you'd tell a 20 year old undergrad

[shy fulcrum]

first

[winged urchin]

Zeroth!

[pastel horizon]

Last (theorem)

[molten urchin]

Can we keep this channel on-topic

[south kernel]

No

[pastel horizon]

How would you guys approach teaching exact trig values for trigonometry (low ability)? At the moment we're kind of just resorting to rote learn

Just to be clear as well, it means 0, 30, 45, 60, and 90 degrees for sin and cos,
0, 30, 45, and 60 for tan

[kindred stag]

Well if you can remember sin and cos you don't need to remember tan

And personally I found visualizing the unit circle helped with remembering the values of sin and cos

[real mauve]

how about using the special triangles?

unit hypotenuse, and then you specify the sides and give them names

the classic "30 60 90"

[pastel horizon]

The circle is going to be a bit too complicated with these kids

Yeah maybe pictures of special triangles might be a good idea

Ultimately it seems like it's still down to memory

[plain valve]

I had a trick I used initially which worked quite effectively, if you'd like me to show you it, lol in a second

[real mauve]

pretty much

you can also try grinding it out mixing special triangles with pythagoras. like give 2 sides and have them find the length of the other. use laws of sines and cosines to find the angles

[pastel horizon]

My other idea was if you know say sin 30 = 1/2 you can draw a triangle and use Pythagoras to work out the other side but it might be a step too far for low ability

[real mauve]

kinda similar to my idea, yeah, but you have to guide them through a few examples

[pastel horizon]

This is 15-16 GCSE foundation tier

[real mauve]

mix it up by multiplying the sides by some scaling factor

[plain valve]

Lmao

[pastel horizon]

I think what's also kind of rough is them having trig questions on non calculator

[real mauve]

use as many of the things learned so far with it? recalling soh cah toa, pythagoras, etc

[plain valve]

this is what I used for gcses initially lol

[pastel horizon]

Yeah that seems quite intuitive

[plain valve]

I don't think it's intuitive at all why it works, just easy to remember

[pastel horizon]

I mean it's a pattern they can memorise

[real mauve]

try all of these and see, probably different techs will stick with different people

[pastel horizon]

Yeah, it's a bridge we'll cross when we get to it but it is a nasty surprise it used to be a level and they moved it down to GCSE

[plain valve]

Also er not to be that guy but this isn't really what this channel is for lol

'how would you teach x' is just paraphrased asking a normal maths question i guess for stuff like this

[real mauve]

idk, this sounds kinda exactly like what's written in the channel description

[pastel horizon]

It was specifically for lower ability though

[plain valve]

oh sorry i misread

[real mauve]

"teaching techniques"

[plain valve]

sorry lol i misunderstood

ignore me entirely

oop

[pastel horizon]

For higher ability obviously it seems a bit simpler you can explain it more intuitively with unit triangles

[shadow basalt]

This might be totally beyond your control but like

Can I ask why you want to do this in the first place

If math is clearly not someone’s forte I cannot imagine a less useful thing to teach them than special trig values

[pastel horizon]

Because it's on the curriculum

[shadow basalt]

I see

[pastel horizon]

I agree it's kind of unfair

[shadow basalt]

Yeah memorization for sure

I’m on team memorization

I think sometimes rote memorization is just the right answer haha

[pastel horizon]

Seems like it here

They should have saved it for a level

[kindred stag]

Is this like 15 and 16 years old?

[pastel horizon]

Yeah

[stark pine]

the sin and cos have a really nice pattern; for those angles, sin goes $\frac{\sqrt{0}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{4}}{2}$ and cos goes $\frac{\sqrt{4}}{2}, \frac{\sqrt{3}}{2}, \frac{\sqrt{2}}{2}, \frac{\sqrt{1}}{2}, \frac{\sqrt{0}}{2}$

[burnt vessel]

Nicholas

[stark pine]

This pattern is super easy to remember

and might make the rote memorization less painful

[quasi musk]

I remember when I interviewed for a math education company, and she asked me how to get one of these things and I derived it from first principles in trig

And she was like "Wow ok, but what if a student said that's too much I need to know it"

and I was like "just do that enough times till you know it"

Then she showed me that and the first thing I thought was "That's dumb"

[shadow basalt]

What

That’s amazing

I had no idea @stark pine

That’s so sick

[spark flare]

Just draw a circle and see where the cos and sin stand approximately

But ye that pattern is sick

[twin lichen]

Hahaha

[stark pine]

Haha ty! I saw this in grade 10 and the teacher said he’d start teaching it

Oh and tan goes

[shadow basalt]

I mean on the one hand it’s like

Idk if it’s something I would want to teach

But for people who already understand the stuff conceptually

It’s a nice tool to never think about it again

[winged urchin]

I personally don't think the circle is too difficult for any student who would be learning special angles anyway

And that's always my go to for describing the values of sin and cos

[stark pine]

$\frac{\sqrt{0}}{3}, \frac{\sqrt{3}}{3}, \frac{\sqrt{9}}{3}, \frac{\sqrt{27}}{3}$ which is a less clear pattern but I always remember the three powers of 3 in a row, and all the denominators being 3

[burnt vessel]

Nicholas

[stark pine]

Yeah

[shadow basalt]

I mean

I’m tempted to say that teaching trig special values at all is just an insanely antiquated idea

I can’t think of any good reason to do it

[winged urchin]

Do you mean you cant think of any good reason if you don't have a calculator?

I think it is a decent way of getting an idea of a function, having a list of function values I mean

[shadow basalt]

I cannot think of any reason why a person in the 21st century would not have a calculator either lol

[shadow basalt]

I mean, even if you don’t have a calculator, what is the argument in favor?

[winged urchin]

If I don't have a calculator and don't remember the values from experience then let's see...

I can try to draw a sin or cos graph but the shape is rather particular and I wouldn't assume my sketch of those are accurate enough to get nice values from

I guess what I'd do is use a compass to draw a circle?

[strange bronze]

why would you do that rather than just constructing the triangles?

[shadow basalt]

Okay but why would you want to draw that graph lol

Like my point is

In what situation would you need to know cos(45)

And also not have a calculator

Other than like a poorly designed math quiz

[winged urchin]

Constructing the triangles is better for the special angles, yeah

I think it's reasonable to want semi-accurate values of functions

[shadow basalt]

Why

Like why would you want that and not have a calculator

[winged urchin]

Like when I'm thinking something out in my head I can think, oh this term is approximately this value and then that means... Etc etc

[shadow basalt]

I mean I’m all for mental math but mental trigonometry sounds incredibly niche

Like I guess my point is

I can’t think of a time when this information would be useful to really anyone in any serious way

I don’t think I’ve used it since highschool algebra when I learned it

[winged urchin]

Ah for like the random person? I mean a lot in highschool isn't used if that's the case

[shadow basalt]

I don't even mean for a random person

like I don't think it is that useful to anyone

like even if you taugh high schoolers infinity category theory

it would be useful to the future math majors

but i dont know if any profession or field of study would really benefit from this

and theres a lot of important functions for which no one memorizes many if any special values

I guess I think that back in the day, finding values of trig functions was useful for some like physics problems and stuff and the problem designers used the special angles and taught those angles

but now physicists can use whatever angles and a calculator

[winged urchin]

I mean, algebra and that gets considerably messier if you don't recognize special values sometimes

[shadow basalt]

But you can put them into a calculator

[winged urchin]

Like students who leave log(1) in their work not realizing what it is

[shadow basalt]

I think that is more arguable since it is a unique special value common to all logs

and it is like

very conceptual

I don't think the special trig values are. If you want people to memorize like sin(0) and cos(90) or something thats a little different

But even then like

[winged urchin]

Well the cardinal directions on the unit circle I think are similarly conceptual but thats besides the point

[shadow basalt]

Like even with the log thing

that was less emphasized in my education at least than cos(45) or whatever

[winged urchin]

I think in response to calculator use... It's like the difference between memorizing something vs. looking it up. I personally believe it impairs you overall thought process if you constantly have to look stuff up. That's probably my best argument against just using a calculator but it's not as solid

[shadow basalt]

sure but for almost every possible angle

you have to use a calculator anyway

[winged urchin]

Well you don't 'have to'. If all technology broke we could still figure out what sin56 was to a reasonably accurate degree if you're prepared to do the calculation

But I'm being a touch ridiculous

You can certainly catch errors more often when you have memorized some values, no?

Like if a student had the calculator in the wrong mode or mixed up the syntax and gets some value

And then they don't recognize that it's wrong because they have little to no idea of the values of sin or cos

Perhaps is a point

[shadow basalt]

I feel like we are okay with this problem in every similar situation though

like students don't have intuition for like, sqrt(432345324^3)

and could easily mistype something like that

[twin lichen]

this is a good point. 'number sense' captures some of this but not all

[pastel horizon]

I had $\frac{0}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{\sqrt{3}}{\sqrt{3}}, \frac{\sqrt{3}}{1}$ for tan

[burnt vessel]

Gaunter

[wispy slate]

Yall thinks its a good idea to teach abstract algebra to high school kids?

[strange bronze]

no

maybe in a fantasyland where its possible to run classes that cater to <1% of the student body

but this isnt that world

if you mean outside of school (like in tutoring), sure if theyre up for it - why not?

i think the farthest you could realistically justify is literally just defining a ring/field and proving week 1 facts

because that can be justified as aiding in linear algebra

even thats a stretch, but the french do it (mind, "between" their high school and uni years)

[wispy slate]

Yh gonna probably make a club thingy for the smarter and more interested ones

[pastel horizon]

No

[plucky elbow]

Afaik in France, some basics of group theory are taught

I might be wrong though, but I think some of my french peers had some basics starting uni

[kindred stag]

I feel like you could probably do some (very) introductory group theory in a geometry class. Especially considering geometry is typically supposed to be a student's first look into more formal proofs.

[surreal warren]

Hi, I really like to advance math content, like formal logic and foundations, but it takes a long time (about 10 months to read an introductory book to basic logic). I believe I have to exercise my critical and mathematical reasoning first to be able to do this. Does anyone have any tips that can help me? I'm 16 years old and I haven't studied much more than elementary functions. My friends suggested that I stop studying logic while so I could study the most basic things and improve, is it a good way?

[kindred stag]

Realistically, I feel like beyond elementary set theory, you probably want some abstract algebra under your belt before tackling the rest of foundations. And to do well with that you should make sure you can do the most basic things, so if you're struggling with the pedagogical foundations, practice those before advancing onto the logical foundations.
Formal logic itself isn't too bad though so you could hypothetically do well with that but not other math. But I'd recommend focusing on other math first though.

[surreal warren]

Ok. Thank you very much. :D

[quasi musk]

I think you can introduce some ideas to motivated students in particular settings

But in general it's a hard thing to do

[quasi musk]

There's an old joke that goes "Tell me young french boy, what is 3 + 2" to which the boy responds "2+3 by commutativity"

[mint lark]

The French undergraduate says “3+2, by the commutativity of the coproduct in a category”

[quasi musk]

It's a shame

The land that gave rise to Fourier, Poincare, etc.

[spark flare]

Not in (normal) high schools. As far as I am aware there is only one notable high school in France that might be doing it in one of their class though.
Otherwise group theory is typically introduced in the first year of uni/prepa (though for prepa it depends on which specialty you got into, some don't officially have group theory)

[wise onyx]

Yes

Don't teach it like you would teach it to undergrads

[spark flare]

Bourbaki killed french mathematicians

[austere inlet]

I'd see as sort of a math outreach effort, perhaps show interested students an overview of how things like symmetries can be related to group theory

actually teaching it would be tough due to HS not being rigorous in general. (And it needn't be.) From what I recall even my peers who were into math olympiads only learnt things like algebraic manipulations and combinatorics at most -- "tricks" for their contests

[pastel horizon]

I just think that in our high schools there's not enough time to include it

There are kids still confusing square numbers with prime numbers

[grave crane]

imo primary concern should be people unable to understand percentages because that relates direclty to financial wellbeing due to taking credits they shouldnt

[austere inlet]

that's a legit concern, if HS students are graduating without understanding that then there's something going wrong with pedagogy

[muted oriole]

anyone known anything about pedagology/neuro research on how studying a number of fields at once affects learning/recall?
i.e real analysis and vector calculus at same time instead of one after the other

papers/articles would be good, if yall know of any

[pastel horizon]

Well there's actually research about the opposite, only studying one field at a time for full mastery. This is a common strategy in Asian schools particularly Shanghai and Singapore

Mastery is a big buzzword I'm sure you'll find loads of studies

[meager shore]

Hello. I'm writing a lesson plan for an activity for hypothetical Mathematics III students involving Binomial Theorem.
The activity is to calculate (and generalize) the number of ways to traverse city blocks to reach the opposite corner of a rectangular area.
I am satisfied with my explanations for deriving the solution graphically and combinatorially, but I am having trouble justifying a solution that depends on the Binomial Theorem as a starting point.
Can somebody please look at page 3 and tell me how I can make the connection here without rehashing a combinatorial argument?

[muted oriole]

Got a link/study name?

[pastel horizon]

https://www.tes.com/news/maths-mastery-everything-you-need-know this is a starting point but if you want something with more authority I'm sure you can search "maths mastery" on an academic database and get more rigorous articles

This is the latest trend in UK maths education though and it's been going on in East Asia for even longer. They learn how to do maths not just by rote learning methodology but by actually understanding the concepts

I'd probably say the only thing that should be rote learned is the times tables but even then only once they've mastered the idea of multiplying two numbers

[muted oriole]

[quasi musk]

What? I think this goes against a lot of known things about how people learn things, especially in mathematics

Particularly if you look at over-saturation of topics - so switching topics with a different point of view keeps you fresher

I think a deep dive into one subject for a month or two can be beneficial - but you can't long term try to cram one subject in without changing your point of view. You get diminishing returns pretty fast

[quasi musk]

I don't think it makes much of a difference doing it one at a time or after the other. The issue is that you're trying to compartmentalize "real analysis" and "vector calculus"

But the two go together very well - they aren't at all "separate skills"

[pastel horizon]

You don't spend extra time on a subject you just cover more breadth in the time you do have. And the order of teaching should have a logical continuation so you can build on previous ideas

So it's not like one week you teach circle theorems then the next you're on fractions

[quasi musk]

Right, but that's not what the article was advocating for

It was advocating for the class not to move on until "everyone got it"

[pastel horizon]

Realistically that's never going to happen

[quasi musk]

I mean that's literally what the article you linked to suggests

And it is unrealistic

So your advice right now is different from the advice above

That you also gave

[quasi musk]

In particular this

[pastel horizon]

I think it works well for challenging and extending topics but it's up to you how you scale back for lower ability

[quasi musk]

I think you get diminishing returns on doing a deep-dive. As a student, I wouldn't focus on a singular topic for longer than 4 weeks because that's when I begin to feel oversaturated and I change what I'm learning

[pastel horizon]

But rote learning with lower ability doesn't work long term anyway either since they'll just forget it after passing tests

[quasi musk]

Well it depends. I think this is why interleaving works really well

[pastel horizon]

Yeah for sure

[quasi musk]

Like if you have to cover chapters 1-10 in some math book

[pastel horizon]

That's something used in our mastery scheme of work

[quasi musk]

For the academic year, roughly a chapter a month

[pastel horizon]

And retrieval practice which is another buzzword

[quasi musk]

You put parts of chapter 2 while you're learning chapter 1

You spread out the material throughout the entirety of the semester. For instance when I was learning Calc 3

[pastel horizon]

Yeah so like in your analogy with retrieval practice you could have questions from chapter 1 as a starter while you're on 6

[quasi musk]

I had a professor who used this to give us 10 weeks to learn green, gauss', and stokes

Exactly

I think this is where I get the best reactions from students

Because the final is cumulative (usually)

[pastel horizon]

It's quite effective you teach them to recall and use knowledge

In the long term memory

[quasi musk]

Yeah ~ that's why my professor did from calculus 1 all the way through calculus on manifolds

I had the same prof and he set up the curriculum that way

[pastel horizon]

That's cool actually

[quasi musk]

We learned both proofs and computation problems throughout the entire sequence

And when we did calculus on manifolds he'd often give us rephrased calc 1, 2, and 3 problems in R^n

Or make us translate back and forth

Because of that my long term memory on math is super good - but I've always had to deal with poor short term

[pastel horizon]

I wouldn't say mastery is necessarily the same topic. Like say if it was fractions, lesson 1 could be simplifying, lesson 2 could be adding fractions with the same denominator followed by different denominator. Then just extending the topic

[quasi musk]

The way I've seen schools implement mastery like Kumon

[pastel horizon]

But it's not like the whole month is spent purely on adding fractions

[quasi musk]

Is the same topic over and over until you get 100%

[pastel horizon]

Nah I don't like 100%

That's putting pressure on learning as well

And they'd just end up getting pissed if they missed one mark and have to resit something they clearly understood

[quasi musk]

exactly

[pastel horizon]

Mastery for me is more about just them understanding it conceptually rather than procedurally

[quasi musk]

So in measurements for mastery, how it works is

[pastel horizon]

Then 20 years down the road, hopefully less people will say "thanks for getting me an A on the exam, but I don't remember any of it now"

[quasi musk]

They just keep re-doing the same stuff until they get x% right

So instead, using an interleaving technique and giving students the chance to go back and do it again in more detail

Let's say you introduce derivatives in chapter 1, and you warm them up on lighter problems for product rule

For exam 1

then in exam 2 you have harder problems for chain rule involving product rule

etc.

So you add more detail and interleave it throughout

I gotta go teach rn

[pastel horizon]

No worries I'll catch up later. We don't quite do it the same way

The main thing is instead of just reteaching a topic they failed at, that would end up being the subject of targeted homework while teaching new topics.

But yeah as mentioned before there's interleaving and use of retrieval practice in starters

And the exit tickets nip misconceptions in the bud before it becomes a problem

[viscid verge]

i hate my high school math class for this reason unfortunately, but i love math

a lot more of "practice this specific type of question because it will show up on the test" and less of "lets understand where this came from in such depth that we could have made it ourselves if we had to"

[austere inlet]

frankly most of the time not even the instructor understands the material with such depth, especially so at the HS level. Not everyone is that passionate or has enough time

[quasi musk]

The people that understand HS math at a deep level usually don't teach HS

Because they go use that stuff to go do other things

[austere inlet]

exactly

[pastel horizon]

This would not happen in the UK. Teachers need to show evidence of their subject knowledge and would very quickly get schooled by OFSTED if this was the case

[viscid verge]

yes i like to explore math on my own because of that, and i asked my teacher a question about it and she said that the topic is for next years course and somewhat scolded me for asking such questions without focusing on what is being taught now

i think a problem with some math teachers is that they dont like to be wrong or not know something, but i guess that applies to everyone bc nobody likes to look "bad"

thankfully this server is much more welcoming

[austere inlet]

that's a pretty awful and demotivating attitude from a student's POV, kills their natural curiosity and enforces the (sadly common) idea that the math curriculum is a set of rigid rules to be memorized rather than a set of ideas in fluid interaction. I agree it's best to explore math in your own and stay away from that kinda people if you care about actual understanding.

[austere inlet]

my experience with actual mathematicians is the exact opposite, they're (and we're) mostly open to discussing ideas, learning new things and seeing them from a different perspective

Dunning-Kruger effect

[pastel horizon]

Teaching maths is just as much about learning as well

[viscid verge]

ah thats good, i just said from my experience

im sure there are great teachers out there

[austere inlet]

I didn't mean HS math teachers, but research mathematicians who do undergraduate teaching as well

honestly most HS math teachers I've known or heard of are either nothing special or seem close to your experience

and there's like the one oddball who's actually motivated to teach

[viscid verge]

oh yes i assume that someone who decided to take on math as something theyd like to pursue in the future are also passionate enough about it to learn more

my only goal is to learn more about math and its core concepts

often i think the big ideas are ignored with more emphasis on practice with a specific subconcept

i am tutoring people of different levels (algebra to differential calculus) at my school, specifically because i need some extra credit in my math class. to be fair to teachers, its already hard for me to explain concepts to students one-on-one, and teaching it in a large class must be even harder

so resorting to allowing students to memorize and do repeated practice with a specific question type without explaining the "whys" or "hows" is the easiest solution that gives temporary results

[pastel horizon]

On the flip side, I've seen some researchers who I'm sure are really bright academically but were terrible when it came to teaching in my undergraduate course

The shortcuts don't really work short-medium term either since it only means you have to reteach again for GCSE

[wispy slate]

So I'm starting up the currently inactive mathematics club at my community college
There's little interest but we have 3 club officers including myself and around 13 people in the discord who have all shown interest in attending meetings but only 2 people besides myself have ever shown up to the first two meetings
We just got interest from prospective advisors so that's exciting!
But an issue we've ran into lately is people not wanting to come because they are misinformed about what we're doing-- we advertise ourselves as hosting short talks, activities, and events but it isn't clear to most people that it's casual, recreational, and focuses on the beautiful and playful side of mathematics rather than trying to get people to flex and work their muscles of rigor for some specific purpose. It's a club, not a class.

Because of this common misinterpretation, I want to rename the club to something a little more marketable

The first couple ideas are a little cheesy but I kinda like them. Any more are appreciated.
-(The?) Math explorers (club)
-Math is awesome club

[pastel horizon]

I don't think they will work with teenagers. You just want to be honest about what you're doing, which topics are you covering specifically?

I would probably pick a famous mathematician and name it after them. They'll know it's related to maths but it also doesn't seem like an intense maths session. Then you can maybe create posters for the topic of the week in the maths department (e.g. Mandelbrot set)

[wispy slate]

You wouldn't think so but the math club at the University I'm matriculated in (RIT) is called PiRIT (like Pirate) lol and they get very good turnout

[pastel horizon]

PiRIT is a cool name

[quasi musk]

What we did was we hosted homework night

And had free food, drinks, etc. and just worked on cool problems

Then we presented talks or interesting problems only once a semester

[wispy slate]

Thanks for the suggestions 😃 I like the free food and drinks idea

and the homework night thing could work too if people have enough interest

[quasi musk]

Go around to classrooms and advertise it. 100% works

[winged urchin]

Oh the homework thing is a nice idea actually

That's a way to get new interest

Student is like... Uhhh, I don't wanna do math club stuff grosss. But hey they'll help me understand my homework this night?

I guess I'll come just for that

Then they could go and potentially get interested in other things in the club

[quasi musk]

Exactly. We did that every night on thursdays

[viscid verge]

#math-pedagogy message regarding this, does this change in college?

i was thinking about minoring in math in college

but i want a heavy focus on things that truly make me get that "aha moment" rather than rote memorization as ive seen in high school classes

[strange bronze]

once you start taking proof based courses, yeah.

depending on school, that may happen right away or may take till 3rd year, or anything in between

[fleet isle]

Couldn't agree more

@strange bronze What is C* what's the thing with this? Can someone explain what this is? I'm such a rookie x)

[austere inlet]

yeah, so it likely won't happen if you're an engineering/bio/econ/etc. major. Sometimes the profs do bother explaining the "how"s and "why"s (since in most cases they are or have been math grads themselves) but I find that students in these areas tend to resort to memorization anyway

[austere inlet]

sounds like you'd enjoy a math minor then, it should have at least one proof-based course like real analysis, linear algebra or at least an "intro to proofs" kinda class

[pearl onyx]

I’m starting up the math club at my high school, and I cant decide what to do yet with the club. I’m wondering if you guys have some suggestions.

In the past, the club was purely competitions (you show up, compete, and leave). I really hated that because we never learned any math and everyone sucked and people started leaving. So no more of that. There are two options (or perhaps combining both):

1. Compete one week, then review and learn the next week. Everything would be geared toward competitions. This way we wouldn’t suck as a team, and I could teach a little math.

2. Make it like a math circle. No competitions, just me leading interactive activities and doing interesting math in a casual and fun environment. I personally would like this more because it’s more interesting.

My issue here is that I don’t know whether to let the kids in the club decide which option they want, since they’ve never done either of those things before. Also, if I do the second option, I’m afraid most students aren’t intellectually curious enough to be motivated.

[pastel horizon]

Personally I'm not sure if I like competition maths in school. As you said they aren't really learning anything from it but I also feel like it can create more anxiety in the subject

Also, if you do the second option the students who turn up will be curious. They're turning up out of free will

I think the worst case scenario is you can watch a pop maths YouTuber and have a quick discussion about the content. It's an easy win in my opinion but doesn't take much planning

[boreal yacht]

As a student, it seems like there is less motivation to do competition problems. Competition problems aren’t inherently interesting in the sense that they don’t really develop/motivate any new mathematics. They just appear as hard problems for the sake of hard problems

Idk it’s also possible that I’m the only one who sees things that way

[pearl onyx]

Yeah

Competition stuff is great for developing problem-solving skills, but that’s about it (at least with the type of stuff we’re doing)

I really want to show people a bit of higher level math, because their current perspective is so narrow and sad

They have never done or seen real proofs (beyond the stupid two-column geometry ones) and think math is a tool to apply to the real world. They’ve never even heard of a set. It’s just insane to think about all the math they have no idea about

[boreal yacht]

i mean that was basically me until i started doing my own thing. School is a terrible place to develop any good idea of what actual math is like. I understand it because teachers are under contract to teach exactly the math that everyone needs to know, but you would have no such obligation within a club.

Ultimately, a math club focused on competition or a math club focused on topics would both give people a better perspective, but I just think competition math is limiting the scope of new-mathematics-learned to a few facts in NT and like 4 standard inequalities, which is also kinda "narrow and sad." Besides, its not like there is a shortage of hard problems for practice outside of competition math.

[pearl onyx]

Yeah

Wow that’s hilarious, it lowkey is just some NT and inequalities lol

[pearl onyx]

I wonder if I could do something like I throw 4 problems on the board and anyone can choose anything, and they’re like long-term hard problems you do over time. That way I can teach at the same time and provide some interesting context to the problems, new types of math developed out of those problems, etc

For example

“What if we omit the parallel postulate?”

“Can you find a way to count the rational numbers? How about the real numbers?”

“The quadratic formula is a way to immediately find the roots of any polynomial with degree two. Is there a “quadratic formula” for degree three? What about four? n?”

…

You’re right, none of this fascinating stuff would come through doing competitions problems

[strange bronze]

those are all googlable

[pearl onyx]

True

Would you suggest a problem that’s not literally those questions, but they motivate them

I want a natural way to introduce those things instead of “okay now we will learn about Dedekind cuts, they are defined as …” with no context

[boreal yacht]

i think this is a really fun idea. They don't even have to be framed as problems to be answered. Just concepts to be explored and eventually understood.

in that sense, the googlability is not really an issue. its not like a kid is going to google

Can you find a way to count the rational numbers? How about the real numbers?
see the answer
yes, no
and be instantly satisfied.

[pearl onyx]

Lol yeah I’m not giving them a grade. Besides, they would know that ruins the whole experience

[boreal yacht]

i don't even think it ruins the experience. Not until they know why the answers are what they are

[pearl onyx]

No I mean if they looked up the explanation

It dulls the surprise

It’s more exciting to discover there are different sizes of infinity yourself than read that some dude figured it out 100 years ago

[boreal yacht]

ye, sure.

[pastel horizon]

A kid won't know what to Google and even if they did they won't be able to understand the technical language of the results that appear

[pastel horizon]

Also what's the problem in seeing maths as a tool to use in the real world? I get you're trying to make them appreciate the pure side of it but there's also merit in bringing in the STEM side

Maybe I'm just biased I'm a maths teacher with an engineering background

[pastel horizon]

Did someone ping me?

[fluid estuary]

Does anyone have good graduate-level instructional material on spectral theory and generalized Fourier analysis? I’m not totally sure what I’m looking for, but I think that I want to use something along the lines of this paper in my research (though i am not specifically working with fractals): https://arxiv.org/abs/math/0606349

I’m familiar with the content in the first half-ish of Rudin’s Analysis on Groups, but it’s been over five years since I’ve done much serious in that vein.

[wispy slate]

i think maths as a tool is way over represented

people always ask why is this useful

no one thinks maths is an art

[tepid smelt]

I don't see anything wrong with presenting it to students as both

[tepid smelt]

I mean at the high school level at least it's hard to find really interesting problems outside of contest math. Also a big part of these contests is the socializing of kids who just really enjoy working on hard problems. Also these problems can spark interest in other areas of math. A lot of these kids at my math club also just enjoy the competitive aspect.

[spark flare]

Tbh I think presenting to students why it's useful is how you make them like maths

[tawny slate]

I just tell my students that it's hard to explain why a concept is useful if you don't understand it

Once you understand a concept, you begin to see lots of places where it appears or is relevant

No one would look at someone's full head of hair and go "o we can learn something about calculus on a surface, and yet the hairy ball theorem is named as such

The more abstract a math concept is, the more it is useful as a philosophy rather than a practical skill

And a lot of times that has a much more important deeper implication than any applied math in a job

[pearl onyx]

There’s nothing wrong with it. I’m saying there’s a problem with thinking that’s the only reason people do math. I personally really like cryptography, for example, so I might introduce that along with basic group theory

I’m trying to eliminate the notion of “why would we learn this if you can’t use it in the real world”

It’s okay to study something simply because you find it interesting

[wispy slate]

i respect this so much

people flock like hers of cattle to see the mona lisa

no one ever asks

how can use the mona lisa

you try tell someone that

[quasi musk]

Use art sales to smuggle drug money, or other illegal activities. Also useful for avoiding taxes

[wispy slate]

there is a group if 7 dimensional spheres that are homeomirphic but not diffemorphic

[quasi musk]

Like if you try to move $500 million dollars worth of good across international borders, you might be in for a bad time

But now you're moving art

This is sacred and cultural and historic

[wispy slate]

and that taking two such spheres, cutting out two discs, and glueing them together

is isomorphic to

rotations of a 28-sided polygon

and all of a sudden

“but why is this useful 🤔🤔🤔”

[quasi musk]

For the vast majority of ppl learning math up to calculus and beyond is mostly a waste of time

I have a lot of sympathy for people that don't want to learn math for a degree

[wispy slate]

but

[quasi musk]

If I had to take 4 semesters of painting, and my paintings had to match certain styles

I'd drop out

[wispy slate]

is not only acceptable but cool to not like maths

[quasi musk]

I hate making Art

[wispy slate]

it really angers me so much

[quasi musk]

Why?

[wispy slate]

so many teachers i know

say that they “don’t understand fractions”

[quasi musk]

Fractions are hard

I still struggle with fraction

[wispy slate]

and is fine to not understand something

[quasi musk]

Albeit in a different way

[wispy slate]

but the attitude is

i don’t “get it” because i’m not a maths person

i’m not going to try learn

because it’s not something you can learn

you jus either get it or you don’t

and these peopl teach kids

[quasi musk]

Why does that make you angry?

Like someone tells me "I'm not a math person"

and I'm like "Cool story, you and everyone else"

and then I move on with me life

The only time I pretend to care is with my students

[wispy slate]

it makes me angry that they are the people that get to teach kids

i lament how bad a school i went to

[quasi musk]

Eh it happens

[wispy slate]

yeah it does

but i still don’t like it

and there will never be anything i can do about it but complain

[pearl onyx]

You’re right

You want other people to enjoy this subject as much as you do, so it’s pretty frustrating when people paint a totally false image of it.

The whole “I’m not a math person” thing is problematic because it not only provides people an easy excuse for not trying, but it actively harms the subject by portraying it as some elite intellectual club you can’t join unless you’re a certain type of person. This pushes away people who otherwise might really enjoy math. How sad is it to consider we might have millions of adults walking around who would absolutely love this subject for what it truly is, but instead they’ve never been exposed to it because of some silly notion perpetuated by teachers and students?

[pastel horizon]

Craig Barton says when students ask something like that it usually just means they don't understand it. It's like a defence mechanism. You'd rarely expect your strongest students to ask when they're ever using something in real life unless they genuinely want to know

I wouldn't go out of my way to specifically find an example for them either, maybe just dance around a bit then break it down for them

[wispy slate]

hearing Emily Riehl day her

say* her

bit about

i’m not even going to try to the dane around this might have applications

this is good purely because it makes the world a richer more interesting place to live and that’s good enough

lBefore hearing that

if someone asked me why i like maths

i would have said

because it’s so fundamental

and is what is universally true

but really

i just like it because it’s cool

there so many people i know who

liked maths in high school

should have studied maths at uni

but did engineering because

they were told

if you do engineering you get to do hard maths and get a job

and this is just from teachers not knowing what maths is

and what these students are enjoying

(obviously some people do enjoy the application)

but the people i know

[austere inlet]

hard maths = calc 3

[tepid smelt]

Yeah its almost always a knee jerk response when a student is frustrated. A big thing is later in life certain concepts could have a use because they can see a situation that could use a fact they learned where if they never learned it they wouldn't see the connection. I often go with a lazy redponse in that because your frustrated your brain is getting a lot of growth because its working hard. Its like what is the use of running around in a circle on a track you likely will never do that in real life but its good training for your cardio. The big issue with math education is many kids are yars behind they can't properly engage with the material so they get angry. The students who are not don't often ask about the use because they are engaged with material

I don't know how to fix the issue of so many kids being pushed along in math until high school where you can actually fail. I think one would to appropriately place kids and not force them all to do years of math.

Kids like to do well in large part so when they are understanding something they are just happy to be right it just sucks so many can't do well due to the failure of our education system

Like when I give low floor high ceiling puzzles kids don't ask what is the point because they are just happy to be doing something correct and can actually engage with it

[tepid smelt]

I don't see this problem a lot at the high school level but it is a huge problem at the elementary level where we really need math teachers with a math background teaching kids. The issue is funding and districts won't hire dedicated math teachers at this level but will for band or even science at times.

[wispy slate]

In my country

In elementary school

You just have one teacher for everything

Say there are years 1-8 then the school just has 8 teachers one for each year

Often they actually end up only having 5 or 6 teachers

So classes get shared

[pastel horizon]

In my country years 1-6 just have one teacher then 7-11 they are setted according to ability. 12-13 will always be high achievers anyway since we consider that further education

[wispy slate]

we don’t get separated by ability until we are like 14?

and even then

not really

[pearl onyx]

I don’t see why we need to force everyone to take math until they graduate high school

I think making it mandatory has caused the quality of education to become much much worse, so nobody’s getting anything out of it

If it were optional, we would have math teachers that enjoy what they’re teaching. Similar to what music classes are now

[winged urchin]

When do you think math should stop being mandatory? And is there a subject you think should be mandatory all through highschool? Just curious

I also don't know about the last claim. Maybe high school teachers where you can specialize as just math. But elementary teachers, at least where I'm from, have to teach everything so there's going to be some subjects the teacher is less familiar with and therefore be less comfortable in teaching

[pastel horizon]

When do you graduate US highschool? 18?

It's mandatory in the UK until 16

[winged urchin]

ya graduate like 17 or 18 depending

[pastel horizon]

Yeah see I definitely disagree with making it mandatory beyond 16 that's when the more advanced content like calc is taught

[winged urchin]

where i am you need to either take math 11 or math 12 which is either math in the second last year or last year. Which would include either pre-calc or calc ideas

though if a student wants to go into pretty much any STEM field and go to college/university I think they want to take that calc or pre-calc course in highschool

[pastel horizon]

Yeah of course

For most they won't though and it's kind of unfair I think

I guess this goes against the earlier discussion about learning maths to appreciate it but I definitely think you draw the line there

[winged urchin]

Though that probably doesn't help just general scientific literacy

I do like the idea of students picking their courses more. I wonder what most would agree 'should' be mandatory near end of graduation

[pastel horizon]

We cut off at 16 so if you want you could look at the UK national curriculum up to KS4

[winged urchin]

Like... a language course comes to mind? But honestly I've used calculus more in real life than shakespeare

[pastel horizon]

That's everything I think should be mandatory as well

16-18 is basically your top 3/4 subjects here

But usually you either split into maths/science or Humanities/English/modern foreign language

There are people who do STEM and arts but it's kind of rare since you're usually focusing on qualifying for a specific degree

[winged urchin]

With these discussions of what people 'should' learn about though it's hard to nail it all down

Cause of course people can say like... "When will you need to factor a quadratic?" or "When will you need to remember Shakespeare or whatever historical fact?"

But then someone can counter with the more general skills.. the problem solving! The critical thinking! Etc etc

[pastel horizon]

Fluency, problem solving and reasoning are the main aims

[tepid smelt]

There should be separate tracks that can all lead to graduation. Like some kids should be able to more trade skills in high school if that is there goal. If someone wants to go into healthcare they should take more biology courses. Where someone looking to study engineering/math/physics can take more math classes to better prepare them. The idea that everyone take the same classes doesn't work and you get the mess we have now.

[pearl onyx]

Except when we make it mandatory then schools are pressured into teaching in a way where “no student is left behind”, eventually dumbing down the material to the point where there’s no problem solving anymore and everything is just a formula or procedure

My view is this: if it’s neither interesting nor useful, why should we teach it?

Current, the math education system (at least in the US) is extremely boring, confusing, and totally impractical

[tepid smelt]

I am curious what is a model another country has that is not awful? I enjoy teaching useful/interesting mathematics to kids who care but the last few years teaching in the US seeing the reality teachers face has me feeling hopeless for the future of math education. I am looking to get out now but feel bad as I do care about the future of math education in this country.

Mentoring kids is great though and I have helped a few to see great math can be and helped push a few first generation kids into stem majors. It just sucks how so many have no hope due to the failure of the system

[pastel horizon]

South East Asia for one

[severe kelp]

memorization and studying for just one exam is not a good model...

[pastel horizon]

I think the only downside with their system is burnout of students because of the high expectations and work

I also heard the Chinese really normalise algebra and just introduce generic formulas for every single concept they learn even from a young age

By the time they hit secondary level they don't need to teach algebra because it's already been normalised that much

But yeah the weakness is the cram school they force them to go to until 7pm

I also the US system sounds too negative. If you have high expectations of all students that they will understand the maths conceptually then in the long run attainment will improve but for now, dumbing it down is kind of just a self fulfilling prophecy

[tepid smelt]

I can have high expectations for my students but don't know how to help a kid with algebra when they don't even know how to add or subtract. It is really depressing how many kids are pushed through the system

[wispy slate]

I think this answer is a bad answer for any of those questions

[pearl onyx]

“Ok you’re having trouble with synthetic division? Let’s see… do you know what synthetic division is?”

“No”

“Ok, synthetic division is a way to quickly divide polynomials. Why would you want to do that? Well, if you want to find the roots of something like a cubic or quartic polynomial, synthetic division allows you to factor it into a product of a quadratic/cubic and a linear polynomial, and then finding the roots is really easy.”

*confused look*

“Oh you’re probably confused about what a cubic or quintic means. That’s just a polynomial of degree 3 or 4, respectively”

*still confused*

“Wait, do you know what a polynomial is? Or a root?”

“No”

…

This is what happens like every time

[pastel horizon]

What is that, precalc? Seems like something we would save for a level

[frosty flame]

And eventually: do you know what 3 and 4 are?

[winged urchin]

Do you mean to say that I didn't really argue well for that? I didn't try to but ya.

I mean, I think people would agree you get more out of school than just what the classes say they will teach. Like if math class really just taught how to factor a quadratic and there was no other benefits to those lessons then it really shouldn't be taught in general. But at least I have this sense that we get more out of classes than just what's on the tin

[wispy slate]

You shouldn't have to justify learning something by some external reason

The reason to learn something should just be

learning in general is good

a particular thing that you learn, you just learn it for the sake of its own interest

not because learning it gives you skills

[pearl onyx]

They always have some vague answer for what a root is, and usually it’s based on the procedure rather what a root actually is. “Isn’t a root like the number after (x-) in a formula?”

[wispy slate]

one thing i think is very funny

how people just lose their minds when variables are thrown in

if you had 1/3+1/4

students won't just slap them together and go 1/(3+4)

[pearl onyx]

Nah, some things should be learned for their own sake. But many things like medicine have an external motivation, which is usually much more important than the internal motivation

[wispy slate]

(okay somme will)

but then you ask what 1/3+1/x is

and all of a sudden we start slapping things together to get 1/(3+x)

[frosty flame]

My unasked for take: At some stage you do need a general education. Like you can't really decide what direction you want to go in unless you have been exposed to enough to know what you like and are good at. You can argue about where that should stop and you should start tailoring your own education, but it would kind of be just as arbitrary as ending it at the end of high school and letting you have freedom in university. Also, if you let 12 yos decide if they want to do math, a whole bunch of them are not going to want to but are going to need it later in life.

[pearl onyx]

True. We have this dilemma where if we make it optional early, then some kids won’t be properly exposed, and if we make it optional later, then the quality of education sucks. I only started liking math around my sophomore year of high school. If it weren’t mandatory until then, I may never have found out about it.

The reality is that most students have neither the self-discipline nor the curiosity to guide their own education, so for now we just have to push everyone along and hope people find their interests

[wispy slate]

Whenever I have these conversations with my friend

He always says

“You have to remember, you really love maths; most people don’t like any one thing that much, most people are something like most of their enjoyment time is they talk to friends, go for a meal, go to a party, maybe they play soccer at the weekends or cycle or play Xbox, and if they are lucky have some gauge interest in what they do for work”

[pearl onyx]

Yup. Lol when you put it that way it’s kind of sad

[wispy slate]

Those people are probably also happier also lmao

[pearl onyx]

lol well to us it feels like they’re massively missing out, and you want to share your excitement

[wispy slate]

yeah

like

there is an equivalence of categories between

the homotopy category of topological spaces

and the homotopy category of kan complexes

[pastel horizon]

I'm not so sure I agree. Most people have something they're passionate about but that something isn't always maths I'd what I would take

[pastel horizon]

I feel like even if they don't share your excitement they still appreciate someone who is passionate vs someone who isn't

That's the best you can do really

[pastel horizon]

And yeah, it's not different to how people are forced into doing art or PE from 12 the whole point is to give them a rounded education before they're mature enough to make that decision

[wispy slate]

i really think

this whole

thing you are passionate about/meant to do

is just a movie trope

you could talk to a suprising amount of people

who don't have hobbies

(and thats fine, you don't have to have hobbies)

and i'd bet a large amount of money that if one could get some actually numbers on this

[pastel horizon]

The thing is before I joined the PGCE course I felt like I didn't really have anything I was passionate about and as a result my mental health started taking a hit

I think it's almost a necessity to have something

[wispy slate]

most people would fall into the camp of not having something they are passionate about

[pastel horizon]

And if you can't make a career out of it, plan B is to find a job that gives enough money and time to let you do it

Unless it's something really crazy like flying to the ISS obviously

[wispy slate]

i think we are just talking about different people

most people

are just struggling to find a job and pay the bills

[wispy slate]

and even this

i think this advice is biased from "survivor bias"

if you are interested in painting

its probably a good idea to not make that plan A from the start

none of this is to say

that if you feel you have a passion

that that is invalid

not at all

just that most people

are not like that at all

[pastel horizon]

Yeah it's a risky career

[pearl onyx]

Yeah most people I can think of have hobbies but not like

Something they actively pursue outside of the environment they’re asked/forced to do it in

[pastel horizon]

But what I'm saying is if you literally have no interests in anything that sends you down a negative spiral

[pearl onyx]

Okay maybe we should mutually clarify what we mean by interests

[wispy slate]

not trying to be mean

but i do think that is very different from what you said at the start

there is a chasm between a passion and literally no interests

[pearl onyx]

My friend has an interest in basketball, but doesn’t practice outside of his club

[wispy slate]

and i even think

you can have no "hobbies"

and life a fine life

most of people who have no "hobbies" still meet freinds, consume social media

watch tv

[pearl onyx]

Most of my friends don’t have something they’re passionate about and they’re doing fine

Exactly yeah

[wispy slate]

but for some reason you are not alllowed to call these things hobbies

i wonder is this because this

self improvement, grind life, self help

buzz

[pearl onyx]

Maybe, and this is just a guess, you might be projecting your own experience onto others. Some people need a passion for their mental health, and some people don’t

[wispy slate]

has "infected" what hobbies can be

[pastel horizon]

Yeah maybe. Although arguably their friends and family are their passion but that's moving goalposts

[wispy slate]

yeah that's definitely moving the goal post lmao

but your experience is still valid

maths gets me out of bed in the morning

(also keeps me up at night so not sure how healthy a reationship i have with it lol)

[pastel horizon]

Whatever keeps you happy works I guess

Except if it's against the law

Anyway I definitely think it's an important part of being an educator giving pupils a chance to discover themself

[long pelican]

Is it reasonable to expect freshmen taking calc 2 to have the ability to read math in written form (beyond just pattern matching problem type)?

Or, more precisely, to be able to learn that skill during the course? (In this case, many or most students obviously don't have the skill presently)

By "read math", I mean like understanding a simple definition (without help from the teacher), or understanding a relatively easy written proof presented in a calculus textbook

[quasi musk]

I mean

Are they actually reading and thinking about it

Or do they just look at it and say that doesn't make sense

[long pelican]

It's a bit of both but I think they have incorrect notions of what a mathematical sentence is saying, in all the typical ways

For example, = meaning calculation or "doing something" instead of a relation

[winged urchin]

In my experience, part of that perceived illiteracy is actually the student not really exploring the statement

Like with normal language, if you're literate you can read a sentence and more or less understand it well enough

You can be able to read a mathematical statement but still feel confused by it

And the student needs to unravel definitions, take examples, etc

[long pelican]

I don't know if you've ever looked at student work but when I do, it's pretty clear to me that the thing they're missing in terms of math literacy is widespread, and perhaps learning how to read math could help them study and learn math 10x more efficiently

[winged urchin]

I absolutely agree and believe math as a whole should be taught more like a language

Even at the elementary/highschool level

[long pelican]

I'm wondering if anyone has already tried having as a goal in a calc 2 or similar course to have students able to read (and maybe even write) math

Unlike a foreign language, math is mostly English and logic and there's just a few idiosyncracies you have to learn

like chained equals being not a run-on sentence, or that the notation f(3) means the value and not an operation or process

and in my experience, learning to read math was a lot more seamless than learning an actual foreign language

I just needed the exposure to well written math

[tepid smelt]

Not in the states. Most don't achieve this until after taking a intro proof class freshman or sophomore year. These students are capable of doing it but it needs to be taught. These students typically have done well with minimal effort due to the weak k-12 curriculum. They generally are quick to pick up algorithms or procedures and replicate it in other situations. They for the most part enjoy math so can learn the formalisms but you will likely be there first exposure to it and might react poorly to struggling for the first time.

[long pelican]

Yep, algorithms and procedures are their strong points for sure

[acoustic ridge]

https://twitter.com/eulersnephew/status/1448087613328531458?s=21

Chris Bolognese (@EulersNephew)

Calculus students are crushing the Chain Rule. This cup approach demonstrating outer and inner functions has really helped with a hands-on approach! Students built their own function today by nesting different cups together. Where else could such a nesting model be helpful?

Twitter · Oct 13, 2021, 12:46 AM

[twin lichen]

nice

[quasi musk]

I've never been a fan of these types of manipulatives

[long pelican]

not to mention the example in the picture is an example of a type of problem that's given just to prove you know how to use the chain rule and nothing else

[twin lichen]

I'm afraid I don't understand your point. They are being taught the chain rule. The exercises should primarily stress the chain rule

[austere inlet]

yeah you need to do that sort of exercise at some point, there's nothing inherently wrong with it

[austere inlet]

I think this can serve as a fun game thing but I don't think the cups themselves lend a good analogy for the derivative, just the composition

this could well be an activity to teach younger students composition of functions or smth

[long pelican]

The chain rule is a theorem with connections to everything else in calculus. Here they are just practicing doing some symbol pushing. The manipulatives encourage thinking of the chain rule as symbol pushing. That's my point

[twin lichen]

ok but they should learn how to do the symbol pushing lol

[long pelican]

I would actually argue they should learn how to teach themselves how to do the symbol pushing using just the statement of the theorem

learning how to read math and apply theorems for themselves

[twin lichen]

Hmm. I certainly agree that they should "teach themselves" as much as possible, as I don't think they'll absorb much of what's lectured at them if they don't get a chance to apply it. I am worried that this might be unrealistic for high schoolers but perhaps we are not holding them to high enough standards. I am not sure I can really judge that.

[long pelican]

It sort of recently occurred to me how weird it is we are holding high schoolers' hands on all the things that you can just do if you know how to read math

granted, the reason this is the case is kind of obvious: high schoolers don't know how to read math

It's sort of like learning classical music with the method that you just copy your teacher's playing and never read the notes yourself

[twin lichen]

Yeah.

[long pelican]

since you never learned how to read the notes