math-pedagogy

[sharp lodge]

So, do you want me to just write your document for you?

That's okay! We need teachers, too.

Teachers are important.

Doctors learn from teachers. I learned entrepreneurship myself, but that doesn't mean that an entrepreneurship class is something that is beneficial to students.

[rough agate]

sure but not in a math class

[sharp lodge]

There is an argument that some idiots say, "Business teachers don't know how to start a business, learn from someone who owns a business." This is false. Plenty of therapists learn from professors, and a lot of those professors are therapists as well. My business teachers have owned successful businesses.

[sharp lodge]

Yes.

The problem with mathematics instruction.

Is that, in the USA, you don't need a bachelor's degree in mathematics to be a math teacher.

You just need a bachelor's degree in any subject, and to take an exam for credentials.

It's really easy to be a math teacher, and no one really sticks to the job. So, you're guaranteed to get a job anywhere.

[rough agate]

even if you needed a bachelors to become a math teacher, a lot of math degree holders wouldnt become one because other pathways pay so much more, e.g. cs

[sharp lodge]

The benefits of being a math teacher are that

• You have the biggest impact on a person's life, besides a doctor (Since they save lives)
• You are paid enough, you only need 70K a year to really get happiness
• You have no issue finding a job, there are so many openings for math teachers.

[sharp lodge]

That was my response that I typed, to counter your argument of "more pay"

If I wasn't an entrepreneur, I would have been a math professor.

[rough agate]

im not saying that there is an issue with being a math teacher. i am stating why it wouldnt work to limit the pool of applicants to those who hold a math degree

[sharp lodge]

I see.

[rough agate]

additionally, liking maths isnt conducive to enjoying teaching maths

[sharp lodge]

I mean, you can like math. If you teach at the right places, and you have lots of energy, you can be a great teacher.

Most teachers are old, and just want their seniority pay. They don't want to retire, and they are grumpy towards their students.

[rough agate]

i also have a lot of stuff that needs to get done before tomorrow so sorry

im going to have to go

[sharp lodge]

A good teacher would just quit before they get grumpy. And yeah, there's tenure as well.

For sure, good luck, Kara.

[rough agate]

thanks and you too

[sharp lodge]

If you have any quesetions, just type it in this text channel

I apologize, I can be quite intimidating to people. I struggle with this.

[tawny slate]

i dont think it is being intimidating as much as making a lot of unsubstantiated/dubious claims that are assuming and judgy

i think perhaps you should be a little bit more skeptical and wary of the information you present to others

for example

[tawny slate]

just looking solely at the 70k a year claim

firstly, median salary of a public school teacher in the US is not great: https://www.salary.com/research/salary/benchmark/public-school-teacher-salary

secondly, what amount you need to live comfortably depends very much on your own personal goals, your location, your needs, and these vary wildly between people, not to mention that it is constantly changing as time goes on due to market fluctuations and inflation

so a lot of people may not take this kind of conversation well, as it can be interpreted as assuming, a little bit tone-deaf, even if you mean well

youre very fortunate to be financially secure and well-off, but just know that this is not the case for most everyone else on the planet and it's not due to lack of trying

being open about your own views in discussions and forums like this is obviously the entire point, but it helps to be mindful of others' situations to make yourself more agreeable

[austere delta]

Where in the world does a math teacher make 70k? That's pretty good pay

[tawny slate]

i specifically stopped teaching professionally because i was just struggling too much financially and couldnt take it anymore

[vocal phoenix]

Yeah, same

I mean, I still teach part time, because I love doing it

[tawny slate]

the housing market and rent is so bad that i even had to give up teaching part time last year

its just completely and utterly unsustainable for me, and it breaks my heart because it is the only job i really loved

[sharp lodge]

That's average. You have to consider seniority pay, whether the teacher has a master's degree (which is required in New York), wage increases every year.

[sharp lodge]

I have an adjunct professor friend, who just pre records zoom lecutres, applies for positions at CC's and universities throughout America, and has TA's do all the grading for him. He is only present in-person at one current institution, because he actually enjoys teaching. But, he makes over $1M a year, from the salaries at all these institutions.

I mean, the biggest headache he has is that he has to write tons of research papers for all the universities he is at.

So this takes up a large majority of his time.

However, once he secures tenureship, he will be living the life.

By the wway, I recommend every teacher incorporate this into their learning system.

https://www.hubermanlab.com/newsletter/teach-and-learn-better-with-a-neuroplasticity-super-protocol

These are all cliff notes, you don't have to watch a 1 hour long video.

[cloud zealot]

<@&268886789983436800>

[tawny slate]

1. that clearly says median not average
2. are you here to argue or provide assistance, and if you are providing assistance, why do you think what you're linking helps?

[tawny slate]

i was responding to this

not looking to start an argument

[quasi musk]

Chicago

[soft scarab]

california

[quasi musk]

https://www.glassdoor.com/Salary/Los-Angeles-USD-High-School-Math-Teacher-Salaries-E111599_D_KO16,40.htm

Yeah, LA is a place where you can make that much

I heard San Francisco is beginning to push 120k just for cost of living

[soft scarab]

and in san jose and other cities around there

[quasi musk]

Yeah, in the bay area

[soft scarab]

did you get a degree in math? @quasi musk

[quasi musk]

I currently hold 3 degrees in math: Associates, BS, and MS

[soft scarab]

Pure or Applied?

[quasi musk]

Associates I did in Math & Physics, BS was pure math, MS was a mixture. I did like 12 classes, and they were split down the middle on pure/applied

I'm interested in analysis/PDEs, so it tends to be split down the middle

[soft scarab]

Nice!

I’m currently working on a bachelors in Applied Math in hopes of becoming a professor someday

[quasi musk]

Have a back up plan

[soft scarab]

well not exactly since i’m kind of banking on teaching, but as part of the requirements for applied math, i chose to take econ classes. so if teaching doesn’t work out, probably somewhere in finance/econonmics.

[jovial falcon]

Is it possible to be a math prof without studying pure math?

[soft scarab]

i believe so as my current professor earned and BS and MS in Applied Math

[jovial falcon]

I’m asking because I’m also doing applied math, and I had closed the door on teaching

Just trying to get into finance

[soft scarab]

i hope to become a community college professor rather than a university professor

[jovial falcon]

Ahhhh I see

[soft scarab]

Nice!

[jovial falcon]

Make sense then

[jovial falcon]

Probably better right? Way less competitive probably

[soft scarab]

i think so at least. less stressful and i can dedicate more of my time to teaching students compared to research

[jovial falcon]

I definitely feel you on that, I love teaching math too but so scared of being asked to produce

[soft scarab]

I totally agree. i’m afraid all of my research will either be wrong or i’ll just be incapable 😭

[jovial falcon]

I do feel like there are lots of gigs in regards to teaching… like some elite private HS also in need of great math graduates

Depends on your feelings towards teenagers tho lol

[soft scarab]

that’s very true

from my experience tutoring high school freshmen…i think i’d like to avoid HS teaching at all costs 😭

[jovial falcon]

https://tenor.com/view/harram-walt-walter-albanianmeme-ron98s-gif-21423021

My peer reading my proofs be like

[soft scarab]

LOL

[jovial falcon]

Why

I would assume kids probably more curious?

[soft scarab]

I’d like to teach students who are actually interested in math

[jovial falcon]

Mos def

[soft scarab]

can I send you a friend req and we can chat through dm rather than this channel 😭

[jovial falcon]

Oh Yh sure

[short hemlock]

I was a learning assistant (sometimes called Supplemental Instruction) last semester and I found 2 things

1. Some people just aren't interested in the math and that's okay
2. Sometimes the best inspiration is to show how much you love math. Not all at once, but consistently and getting excited about math at their level

[sharp lodge]

I mean, if you click it. It was created by a reputable Stanford-educated neurobiologist. And it's completely free.

Not even God can save you, if you refuse help.

I understand that it's fun to power-trip with your Discord role on a large server. Believe me, you can do better things in life.

You're better off owning your own Discord server, to be fair. You're not being paid as a moderator.

If you really want to power-trip, be a professor and charge $200 for engineering textbooks. Then, be a loudmouth.

I'm sure you're smart enough to do that, so I hope you enjoy your career!

[sharp lodge]

If you're an undergraduate mathematics student, some advice would be to learn to type with proper capitalization. You won't get anywhere with MonkeyType lower-case practice.

[sharp lodge]

Why? You could just be a lecturer at a university, and get even more pay. Why not just get your master's degree, and then be an adjunct professor at 20 different academic institutions, and farm money with pre-recorded lectures?

You should honestly add a rule saying "No Homework Help Kids Allowed" so that you can actually have an intellectual discussion, without being swarmed with children who are struggling with their math classes.

Eh, I'll just snipe your vanity URL the moment this server ever dies, if it does.

Most websites do die, within a ten year period. This would be no different.

Acquiring and sniping domains is very powerful.

[turbid zenith]

This is a casual Discord chat.

Chill, please.

Also you’re gonna literally come to a channel about math teaching and tell us we should alienate people who come looking for math help?

I think you’re in the wrong place.

As for everyone else: best of luck to your students on finals if they’re still taking them!

[tawdry venture]

<@&268886789983436800> troll

[real jolt]

I understand you're seeking intellectual discussions, to a degree I believe you're right, but the server does a decent job at separating the homework help from the discussion channels if you'd rather not see them, you can just mute the category and not react to the role.

I've had some interesting discussions here and I haven't had to help anyone with their algebra homework.

[soft scarab]

20 different institutions simultaneously? wow!

[vagrant meadow]

tomorrow i'm covering LU factorization in my discussion but i'm wondering is there any realistic situation in which calculating an LU factorization by hand would be more efficient to solve a system? i feel like by hand it's always better to just row reduce it, and if you're using a computer then what's the point of knowing how to do it by hand? like wouldn't it be much more worth their time to cover how to solve a system with the LU factorization given (since they'd almost surely use a computer to calculate it)?

i just want to get confirmation from someone who knows more than me here that i can safely say "there isn't really any value to knowing how to compute an LU factorization by hand. but there is value in knowing how to use the LU factorization to solve a system"

[vocal phoenix]

Well, the latter is true becasue I'm almost certain that numerical methods of solving various linear algebra problems use LU extensively.

By hand, I don't have much experience but I'm inclined to agree that being able to do LU manually isn't going to be useful very often

[whole latch]

What is your favorite way to teach factoring? There are some heated discussions happening in my school right now and was curious about what other mathematicians think

[turbid zenith]

I prefer the “box method”!

This is also how I teach polynomial multiplication (since it generalizes so nicely as opposed to “FOIL”) and it makes it clear where everything comes from. You can even use the “box” for other places where the distributive property holds like multiplication of whole numbers or mixed fractions. And it can even be used for polynomial division.

For non monic quadratics I also use the “AC” method, which is where you split up the linear term into factors whose product is the same as the product of the other two coefficients.

Also relevant — I start by giving them a lot of carefully sequenced examples and ask “what do you notice, what do you wonder” rather than launching right into “step 1 do this, step 2 do that”.

[tawdry venture]

i wish FOIL wasn't a thing

it feels like such a crutch

[cosmic ibex]

Oh, don't get me started.

[tawny slate]

I really love the box method because abstractly, it is the cartesian product, which is what you are basically doing

it makes distribution very intuitive (because you can now use a geometric diagram with algebra) and it works functionally

i even used it in my videos as a representation of dimensional analysis

box method is excellent

[lethal leaf]

may have to steal that ngl

[tawny slate]

also graph is not bad, it is an alternative representation

[lethal leaf]

since factoring polynomials comes up in my tutoring as a pain point

wdym graph?

how do you use the graph of the polynomial to showcase factoring?

[tawny slate]

hold on making quick sketch

each circle is a term from one factor

one factor on top (4 terms in this example) one factor on bottom (3 terms)

each line connecting them is the product of the terms

add the edges

basically a complete bipartite graph

this representation is sometimes what i think about from an algorithms perspective

but this and the box method are functionally the same

https://youtu.be/IvQ5ag2cEx8?si=gCwKs1N1-XqAffuC 3:57

[lethal leaf]

Oh that kind of graph

[near harness]

Relying on the logical structure of an acronym instead of the logical structure of the math itself 💀

[cosmic ibex]

The anagram acronym doesn't even encode the relevant logical structure.

(a+b)(c+d) = ab + ad + bc + cd
First two, Outer two, Inner two, Last two.

[rough agate]

its nice to teach all

but also knowing when to use them

always nice to try box method first

if it doesn't factorise with box and x coefficient, with a monic quadratic is an even integer then complete the square

then quadratic

I feel like quadratic is emphasised too much as a last resort even thou it's a really solid method

sure the arithmetic isn't nice

but if the coefficients are all in the tens then I don't know why you would waste your time with box

[tawny slate]

i mean, if we want to go there

if we are specifically teaching how to factor quadratics, i would start from the very basics

like, im talking first teach the "basic algebra" involved and the traps to avoid, followed by fundamental theorem of algebra

(dont start with the full theorem, start with the simple explanation and tack onto it as they learn more, be clear that you are using a lies-to-children)

the point of it is the abstract theory and to avoid traps, two examples:

if kids dont remember that cubics always have 3 solutions, they will easily miss some when solving them

also some problems or methods rely on the fact in order to make things easier

[vocal phoenix]

You teach the Cardano method to kids?

[tawny slate]

what no, i just said its the abstract that matters, not that you can solve for every root

the second example is like when it actually matters for problems or methods

like if you are solving a quadratic and you happen to know one root thats not good enough

but if you know both roots (maybe by pure guesswork or something) then you can be sure that is the full answer

the fundamental theorem of algebra is only an abstract tool on the more advanced problems, but critical as guidance

as a tool it is everywhere in contest math for sure

but seriously there are a lot of traps when it comes to even teaching basic quadratics

[rough agate]

oh yeah thats definite

quadratics comes certainly after linear

and also freshmans dream

[tawny slate]

like one of the most common errors is solving something like x^2 = 5x

its almost muscle memory for many students to divide both sides by x

and now a solution is gone

the dividing both sides by an expression that loses solutions is easily one of the most poorly understood common mistakes

after that i would teach vieta's from first principles before completing the square or quadratic formula

because then even things in double digits, as long as the solution is nice, there will be students who learn to eyeball it using sum/product methods

builds good intuition where it is useful and needed

[rough agate]

intuition is really underrated

but also pretty difficult to teah

[tawny slate]

i think this is a really good framework to approaching the topic, but even within that framing there are a lot of ways to go about teaching it

[rough agate]

yeahh thats true

[near harness]

I agree. I think the quadratic formula is great but the problem is when people use it as a crutch without being able to derive it themselves so it just becomes this mystical tool that works for some reason.

And then you get all of these ridiculous mnemonics to memorize it

[rough agate]

yeah real

[turbid zenith]

🎵
x equals negative b
plus or minus square root
b squared minus 4 a c
all over 2 a
🎵

[shadow flower]

i hate that i read that in the correct tune

[turbid zenith]

Glad I could be of service

[soft scarab]

my ptsd has resurfaced……….

https://youtu.be/VOXYMRcWbF8?si=VFxzsioSja67K_k9

if i have suffered the wrath of this song, so shall everyone else

[rough agate]

ive never seen that one before

[tepid smelt]

Yeah this method is standard in all the new curriculum I have seen. They use it for factoring and dividing polynomials also.

I also prefer it

[soft scarab]

I’ve seen teachers teach the box method and it feels like a crutch for multiplication/factoring of polynomials. Its def a good visual representation of what happens. I think just straight factoring using guess and check and the diamond method is a good way to show the dynamic between adding and subtracting certain terms though. Guess and check I feel like is especially good for helping students to become quick with their quick multiplication skills and ability to find factors

Though i would suggest not to rely on guess and check unless you’ve already mastered factoring by grouping, box method, the diamond method, and other ways

[turbid zenith]

I still use it in practice myself if I'm multiplying anything bigger than binomial by binomial.

Like I used it when I was working out why α(a + bi + c√2 + di√2) = a - bi + c√2 - di√2 is a ring homomorphism. It helped me keep the terms organized.

So I don't see why it should be called a crutch.

[tawny slate]

the box method doesnt make it NOT guess and check

the only thing the box method does is organize what is happening

whatever rule you are applying to the box method to get to the right answer is being applied the normal way too

it is strictly a visual representation

more importantly, it is easily extendible to all distributive applications and i see no pedagogical reason it hinders anyone for any reason

so i would go further to argue that it is not a crutch

[cosmic ibex]

If I understand the answers I've gotten in #prealg-and-algebra, all those methods are typographical variation of "hope the roots are integers and check all divisors of the constant term until you get the right middle coefficient"?

Is factoring a quadratic over Z specifically really so important in practice that it makes sense to drill kids in a semi-bruteforce method for it? Most quadratics I meet in my day-to-day math musings turn out to have irrational roots anyway.

(I'm assuming there's drill involved because otherwise being particular about where on the paper which intermediate numbers are written seems to be pointless ...)

[tawny slate]

i dont think quadratics over Z are specifically important, i think vieta's formulas is important

and quadratics are simply the first nontrivial case where understanding vieta's gives you a significant advantage over factoring over Z

also i think it just abstractly gives students better intuition and insight

being aware of the more nuanced differences between Z and Q and R i think are important, because it could help dictate how they think about problem solving in general

like if they are looking for answers in Z they know that a brute force approach is exhaustive, where it is not in R and sometimes in Q

thats my take

[soft scarab]

By, R, Z, And Q, are you referring to the real numbers, integers, and rationals?

I also feel like so much time is spent on just plugging and chugging rather than being shown the derivation of say the quadratic formula or showing why factoring works/the theorems. I wouldn’t have learned how the quadratic formula came about without researching it on my own time

[rough agate]

the one issue is that understanding about how the quadratic formula is derived is difficult to assess

because even then people could just memorise the derivatioon and write it down on a test

and so we need a scaleable approximation for understanding

plugging and chugging is not it but it works well enough

[tawny slate]

i think using the quadratic formula as both a pedagogical and narrative climax to the subject of quadratics is honestly beautiful

you could derive the quadratic formula by simply completing the square, but this is tedious and not interesting, not something the kids would get excited by

they care more for the result than the process

but by employing every concept at the student's disposal to derive it, by splitting it into manageable steps, to finally reach the point where all quadratics are now solved algorithmically

students are not only tested on their overall understanding but are also rewarded at the end, and feel it

here is how i outline my teaching of quadratics (borrowed heavily from AoPS but with my own flair added):

• first teach the most basic quadratics like x^2 = 4
• now crank it up: x^2 = 5x (there are traps here, introduce that there are always two solutions and that factorization is always strongly preferred over dividing an expression on both sides to avoid the risk of losing a solution)
• extend the factoring idea: (x-2)(x+3) = 0
• expand and ask them to solve but dont tell the students what you did: x^2 + x - 6 = 0 (when they give up because its too hard or forget a solution, show that this is the previous problem expanded, so it is the same answer
• explain this is the motivation for factoring quadratics, try to teach them to reverse engineer this process by guess and check, tease the quadratic formula's existence for solving unfactorable quadratics
• introduce vieta's formulas
• factoring II: when a > 1, use the ugly middle coefficient as a test
• introduce the three forms of quadratics: standard form (ax^2+bx+c), factored form (a(x-p)(x-q)), and vertex form (a(x-h)^2+k), and that to master quadratics, the goal is to be able to convert any form to any other form
• with that motivation, realize that factored form literally hands you the solutions, vertex form is not bad at all to solve, standard form is what is hard. the goal is to convert standard form into anything else
• to convert to vertex form can be thought of as completing the square (which should be taught), but also teach an intuitive form of it by graphing the quadratic as a function, showing its symmetry, its vertex, and by introducing function transformations (this may need its own lesson, which could mean that quadratics need to be revisited)
• quadratic formula derivation is as such: using function transformations, convert standard form to vertex form, then solve. <- insanely easy in concept when framed this way, using tools the students have learned already
• the only issue now is that the quadratic formula now introduces complex numbers, which is its own lesson from here onwards

if anyone has any suggestions or improvements on this outline im all ears

[tawdry venture]

yeah honestly i like this a lot

[rough agate]

that's honestly pretty cool

its not too different from what we have here with the exception that we dont derive the formula

and also no vietas

[soft scarab]

i feel like that’s a pretty good layout. i think also teaching about the discriminant really helped with my understanding of quadratics. i didn’t hear about vietas formula until calculus, which by then i had pretty much mastered solving quadratics. my teacher didn’t even call it “vietas formula,” i just remeber it as like “how to find the sum of the roots quickly.”

[soft scarab]

i wouldn’t expect a teacher to asses a students ability to derive the quadratic formula, but i would appreciate seeing the derivation in class.

i highly doubt most Algebra 1/2 students would be able to successfully recreate the derivation for the QF on tests

[dapper flume]

Definitely not on tests, but they can get surprisingly far with a couple motivated hints

[soft scarab]

True that

[dapper flume]

It's a great inquiry activity. "How do you tell a computer to solve a quadratic" type of thing.

[soft scarab]

idk why but when i read your username i thought of it as dy/dx-ia 😭😭

[dapper flume]

Lmfao

[soft scarab]

Yeah a good way to get them to think

[tepid smelt]

That's pretty much how most highschool curriculum approach the subject with sadly the exception of no vieta or complex numbers and more applications with projectile's and area problems. There are generally no inequalities also which I always found strange.

I wonder if there is any benefit of showing the other form of the quadratic formula which is 2c/(-b-+√b^2-4ac) other then a cute problem to show they are equivalent.

[soft scarab]

is that not the standard way to write the quadratic formula?

Oh wait it’s 2c on the top

[dapper flume]

I always make sure a step in our derivation looks like this, to really highlight the geometric meaning $$x=\underbrace{ -\frac{b}{2a} }{ \text{axis of symmetry} } \pm \underbrace{ \frac{\sqrt{ b^2 - 4ac }}{2a} }{ \text{two equidistant points} }$$

[burnt vessel]

dysxleia

[dapper flume]

its 2c on the top. It's an uncommonly used form because the square root ends up in the denominator

[soft scarab]

i think splitting it up like that also helps the student not to forget to divide the -b by the 2a

[dapper flume]

yeah that's true too. They'll be able to gut-check that a bit easier

[soft scarab]

my bad, yes the top 😭

@dapper flume how long did it take you to learn latex, i’ve considered learning the syntax myself

[dapper flume]

way less time than I thought it would. Something like a month or two to be able to do 80-90% of what i need to do. I also use a plugin in obsidian which lets me write it faster with shortcuts

[soft scarab]

Dang, nice.

[tepid smelt]

https://math.stackexchange.com/questions/49229/why-can-all-quadratic-equations-be-solved-by-the-quadratic-formula/49243#49243

Reminds me of some other approaches to prove the formula

[dapper flume]

I like it a lot. Inquiry is always a great bet for monumental problem-solving achievements like this.

With this itinerary of yours, I would focus very hard on encouraging students to try and figure out these strategies. The quadratic formula is a perfect place to exemplify problem solving skills such as

• "what is preventing me from doing this easily? How do I get rid of this?" (An example might be when ax^2 + bx + c = 0, that leading coefficient is hard to deal with, so dividing out by it helps make progress)
• "what forms of the problem do I already know how to solve? Can I convert this into that form?" (An example is what you said about vertex form by completing the square).
• "What should my end result look like?" (This comes from both the geometric intuition of the solutions being equidistant from the axis of symmetry at x=-b/2a, but also from the fact that the answer should look like x = [something in terms of a,b, and c, but not x])

etc.

[tawny slate]

man there are a lot of approaches here
not all of these are directly applicable to teaching the subject imo, especially when time is limited, but i wonder if i can do something with them...

[dapper flume]

Scrolling down reminds me of this particular image

[turbid zenith]

Reminds me of IRC.

[rough agate]

i know you didn't ask me but I thought I'd give my experience on this

which is that up to and including precalc content takes less than half an hour to be able to write comfortably fluently on latex

like that part is basically natural language

but starting with integrals, syntax gets a little more complex

but either way if you actually sat down for a while and tried to learn all the stuff then it wouldn't take much time

[hardy brook]

integrals is also fine as long as you understand what is happening. The problem with our beloved LaTeX is really the fact that macros are not hygenic. Which means that doing anything where the objective has depth, i.e. a complex layour with many components is virtually impossible without spending a lot of effort.

[austere inlet]

... which is why packages and user-defined macros are a thing

admittedly it is a bit weird for many people at first and the "LaTeX way" to think about things takes years to master, I've found

it's definitely easier if you come from a background that has some programming skills, which isn't surprising because TeX was invented by a programmer (Don Knuth)

[austere inlet]

I like this way to summarize inquiry-based problem solving

[austere inlet]

incidentally, this year I'm teaching problem-solving skills to 10th graders in an afterschool club type thing that our uni organizes -- it's my first time doing this type of stuff and it doesn't help that my social abilities are a bit lacking hahah

but I try to inquire about e.g. their proposed solutions or build from correct ideas they come up with in an otherwise not-perfect solution, try to emphasize things such as working with simpler particular cases and then generalize etc.

[dapper flume]

That's a pretty good approach for a club too

I always say that a teacher should never teach as if students have already learned problem-solving skills.

Chances are, their teacher's did not emphasize them. And if they have, everyone could use more reinforcement.

I love math but I constantly remind myself that those transferable skills are ultimately the reason we still teach math as a core subject.

[soft scarab]

thanks for the advice. the syntax looks a bit complicated, but i'm sure i could get it down if i dedicate some time to learn it.

i just have to find the time to learn it...

[shadow flower]

it’s very intuitive for the most part

[rough agate]

yeah the thing is if you look something up each time you want to write it, it'll be more time consuming in the long run than if you just took the time to look at it

[vagrant meadow]

i just can't help myself injecting DE into my linear algebra worksheets

i can't use the term eigenvector yet, so i've been calling them "magical vectors" with my students 😆

[tawdry venture]

what's the intended answer for 6a? "D is the derivative operator on R_2[x]$?

[vagrant meadow]

yeah. "the result is the derivative"

i like questions with one sentence answers.

especially since many of these students haven't developed mathematical maturity yet (and most aren't math majors). so the purpose is to expose them to applications of linear algebra they may end up seeing again later in their field.

[vagrant meadow]

at this point, to them "linear transformation" means multiplying by a matrix. so i'm trying to help them abstract their idea of what a linear transformation can be, and how many things they are familiar with are linear transformations

[noble hare]

Do they know yet how to find a matrix of a linear operator wrt a basis? Bc if so would be good to get them to do that for D, i remember it being eye opening the first time I did that

[vagrant meadow]

they should imo. last time I TAd the Prof went through

• vector/subspaces
• column/null space and linear transformations
• basis and the matrix for a linear transformation
  in one class. this quarter the Prof is doing each one lecture at a time. so I spent Tuesday entirely on just identifying subspaces basically.

this class is an absolute mess

[vagrant meadow]

screw it imma change it to a question that kinda goes into that

[drowsy otter]

Hello. Me again. My student is learning how to multiply fractions. Thing is, my student is doing the multiplication correctly, but is not really answering the question.

For example, if I ask 3/7 x 4/8, the student is doing 3 x 4 = 12, and then 7 x 8 = 56, however, the student has a tendency to just stop there and not put the "over" between the 12 and the 56. The answer is 12/56, not 12 and 56. The latter is not clear.

How do I get the student to answer the question fully so that their answer not only makes sense to them, but is explicitly clear? The reason I ask is because the student will need to make it clear in the exam so as not to lose marks that they are capable of getting. Thank you guys.

[tawdry venture]

have you already pointed this out to your student?

[drowsy otter]

Yes, I did. However, the student has it in their head that the multiplications are sufficient. They are NOT, as the fraction also needs to be formed correctly.

I would like to reconstruct their 'concept image' as it were.

[tawdry venture]

try asking them to do some easy ones orally, like 1/3 * 1/3

or 1/5 * 1/5 is probably better

[drowsy otter]

Good idea.

[tawdry venture]

if they try to answer "1 25" emphasize that it's "one twenty-fifth"

also try having them do fraction multiplication as part of sth more complex?

such as 1/2 * 3/4 + 5/6

[drowsy otter]

I haven't introduced adding and taking away fractions just yet. However, something complex might work. I'll give you a screenshot of a slide I have, for example.

Thank you Ann.

[cosmic ibex]

It sounds vaguely like the student isn't really aware that fractions mean something, and just considers them a way to write down two multiplication problems in one. If so, you'd need to work on getting that meaning through rather than focusing on recipes for particular problem types.

It also sounds a bit iffy to train multiplication of fractions before even speaking of adding them. Addition of fractions can get fairly good intuition with just a number line, but getting multiplication to make intuitive sense (rather than just a set of rules to follow) is more tricky. And then having introduced addition first will have a much better change of driving home the point that arithmetic on fractions can be meaningful in the first place.

[drowsy otter]

I get what you're trying to say. If you introduce addition of something before multiplication, it makes sense.

However, in this case, it's harder to add or take away fractions (especially when you have different denominators). I know that from experience, both as a student and as a tutor.

It's to the point where adding and taking away fractions could be considered a GCSE question (foundation tier), even though it is introduced at primary level (which is what I teach for this particular student).

[cosmic ibex]

But if you teach multiplication without also making sure there's a sense of what the multiplication achieves, then no wonder the student won't feel it particularly important to write the result down as a fraction.

[austere delta]

Isn't the fact that addition is harder another reason to spend more time on that rather than multiplication?

[drowsy otter]

You two might make a good point. Adding and taking away fractions might be the way to go after all.

I was going to teach that at the end of the module, but you've convinced me. Thank you very much for your help.

[austere delta]

Yeah, I think the only reason to take the simpler things first would be as a stepping stone to the more difficult topics.

But here I think it is somewhat counterproductive. Spending a lot of time drilling that
a/b x c/d = ac/bd
Without properly conceptualizing how fractions go together might lead students to think
a/b + c/d = (a+c)/(b+d)

[drowsy otter]

You're right. I wouldn't want that (though it still might happen regardless!)

[drowsy otter]

Thank you all very much.

[jade sphinx]

Hi, i have a question. How can i study in a University without a high school diploma. I'll study math btw.

[austere delta]

Think it depends a lot on which country you're in, and possibly on why you don't have a high school diploma.

[cosmic ibex]

It's also not on topic for this channel; try #advanced-lounge .

[vagrant meadow]

i do empathize with the desire to skip over the more difficult/basic cases and start with the case that is easier computationally. sometimes i do this for a bonus/fun topic that isn't strictly part of the course content, but for something essential like fraction arithmetic, i wouldn't move things around.

[next sinew]

I actually hold the spicy opinion that the multiplication of fractions should be taught first and fractions should be taught, first and foremost, as objects that play nice with multiplication

with the caveat that apriori, we don't really know what fractions are, until we start exploring their properties

and often in math, what something is is less important than how it behaves

I would teach the fraction 8/5 as the number that when multiplied by 5 gives 8

you can begin by playing some games and noting that such a number doesn't exist in the sense that there is no integer with that property

well, it turns out that such a number can be defined but that, of course, will just get sidestepped because again, what something is is less important than how it behaves

so we can just assume that a number like 8/5 exists and see what happens

well, obviously 8/5 times 5 will yield 8, by definition, that's cool

[cosmic ibex]

You'll then incur an obligation to explain to the kids why we can't just assume that a number like 8/0 exists ...

[next sinew]

well, yes, you're right, it's dangerous to assume that an object with certain properties exists, because sometimes it genuinely doesn't

but I'll continue on with my story

[cosmic ibex]

(And also, denying them a number-line intutition for Q seems like a colossal disservice).

[next sinew]

But we know that multiplication means adding 8/5 to itself 5 times, and if the student is more advanced then you can remind them of the distributivity argument that makes it work for natural numbers, given that multiplication by 1 is defined to do nothing

So we find that (8/5) + (8/5) + (8/5) + (8/5) + (8/5) = 8

So 8/5, whatever it is, has the property that five of that number is 8

and I guess this is where you could start talking about a number line if you wanted to

and how certain rational numbers are bigger than others in a very meaningful sense

[next sinew]

and if the student doesn't believe this, then all you need to remind them is we want distributivity to hold, and in an ideal world the student would of course have an appreciation for why distributivity is a beautiful and intuitive property

Because
( (8/5) + (8/5) + (8/5) + (8/5) + (8/5) ) * 5

= (8/5 * 5) + (8/5 * 5) + (8/5 * 5) + (8/5 * 5) + (8/5 * 5)

And now, by definition, those turn into eights

So 8+8+8+8+8 = 8*5

which we know to hold

hmm, convenient

[spark vale]

Do you really want to mention distributivity to 7y olds

[next sinew]

And once the student understands the definition, and understands that adding a rational number to itself some amount of times yields an integer (and thus can be, in a sense, assigned a size, and you have the number line etc.), you can start to think about multiplication of fractions

So we have our 8/5, let's multiply it by 2/3. What is that number? Well again, we don't know. But let's think about how it behaves. We know that fractions behave nice with multiplication, so let's try to do that

Let's multiply it by 5

(8/5 * 2/3) * 5

= (8/5) * 5 * (2/3)

Multiplication is commutative and associative

By the way, commutativity follows from left and right distributivity, so I really don't understand why commutativity is often taught before distributivity, even though distributivity is the more fundamental and the more intuitive property lol

[next sinew]

anyway we find that multiplying our product object by 5 gives 8 * (2/3)

that's already a simpler expression because 8 is an integer, and we know what integers are

so maybe we might be inspired to multiply the expression by (5*3)

And obviously by shuffling stuff around, and relying on the definition of a fraction, we find that multiplying (8/5 * 2/3) by (5*3) gives (8*2)

So again, we don't know what (8/5 * 2/3) is, but we know how it behaves

But wait, if multiplying (8/5 * 2/3) by (5*3) gives (8*2), then that's the same as saying (8 * 2) / (5 * 3)

right, that's saying "multiply this quantity by 5*3 and you get 8*2"

and we accidentally found out that (8/5 * 2/3) is exactly the number described by (8 * 2) / (5 * 3)

And generalizing this line of reasoning we find the general product of two fractions

If we think about what (ac)/(bd) is, then multiplying it by bd gives ac, but (a/b * c/d) also has that property, so thus they're the same thing!

Of course this raises the question, "wait, just because they behave the same, how do we know they're actually the same"

But at that point, I would just reassure the student that there is, in fact, a way to actually DEFINE these strange objects using set theory, and it's not even that bad. But for actually doing mathematics, it suffices for us ignore such details, and treat any number with that property as one and the same thing

I mean, for one, 2/3 and 4/6 are treated as exactly the same number

and yet, people usually don't bat an eye

so truly, the details will only be relevant once the student is more mature (and even then, not really lol)

[next sinew]

if you ask me, then distributivity is like the holy grail of mathematics. Even commutativity is sometimes dropped, like matrix multiplication for example doesn't always commute, but once your operation isn't distributive, you're quite lost

And there is no real world illustration of what an operation being "commutative" really means (besides counting a collection of objects arranged in a rectangular shape in two different ways and noting that they're the same, but I dislike that approach because it's like, we do this thing and it works but no one knows why)

but to me at least, distributivity just makes sense

Like if we imagine an alternate timeline where counting is actually done with squares then we could define multiplication of two collections of squares as inserting the squares of the first operand inside the squares of the second operand and counting the inner squares

It's quite clear that shoving one square inside n squares versus shoving n squares inside 1 square yields the same amount of inner squares

and if addition is defined as just merging the collections, then it's quite clear that this operation is also distributive because if you imagine the merging of two collections by doing some sort of translation of some objects in your mental landscape then obviously that won't affect the amount of any squares in the end

[austere delta]

I'm not sure that's very clear at all. Certainly much less clear than the fact that an nxm rectangle is the same size as a mxn rectangle

[next sinew]

And from distributivity, and the behavior of one square, we find that the multiplication of natural numbers commutes because multiplication by one commutes

And to me at least, each individual step in that line of reasoning makes sense

and the equivalence to whatever traditional multiplication you may define is obvious

and ultimately you show commutativity

But the n x m vs. m x n rectangle, to me, is a lot harder to accept

Like, how do I know that if the rectangle is big enough then at some point something weird doesn't happen and shit just breaks for some reason

I'm pretty sure that anyone who's convinced by the rectangle argument has only ever imagined a finite collection of finite rectangles

which doesn't feel satisfying to me

I feel like it's missing the underlying reason and the pure black magic that is the commutativity of multiplication

I mean, okay. Is the rectangle argument convincing? Yes, of course. And is there any real pedagogical value to thinking about commutativity in terms of distributivity? Maybe not.

But the point I'm trying to make is not "make all students understand this line of reasoning" but rather "why is distributivity such an overlooked property of multiplication"

Everyone and their grandma knows that multiplication is commutative, but if you ask serious, well-educated adults, then there will be a lot of people who will not be able to tell, without a calculator, that the quantities 22 * 99592 + 22 * 23 and 22 * (99592 + 23) are actually the exact, same thing

even though to ME, that's the ONLY thing that makes multiplication so special. I can come up with plenty of commutative operations that are completely uninteresting piece of trash operations, but if an operation is distributive, and the whole set is for example generated by the multiplicative identity element, then oh boy that's a very generous amount of assumptions

[next sinew]

I can only refer you to this message

[wispy slate]

That is awesome.

I wish I was taught like that for linear algebra as a science major.

[next sinew]

To illustrate this point, all the commutative binary operations over the natural numbers are characterized by assigning arbitrary values to each pair of elements

All the left-distributive binary operations over the natural numbers can be characterized by what multiplication on the right by 0 and 1 does. So you just need two numbers associated with each element. And if you also include right-distributivity, then you get that multiplication by zero is always zero, and if multiplication by 1 is commutative then the whole operation is commutative, and you only need to associate one number with each nonzero natural number

[elfin sleet]

In light of this message, how much of this are you saying should actually be taught to actual kids? Because you started by saying that multiplication of fractions should be taught before addition but your reasoning seems to involve teaching all this stuff that you agree maybe isn’t appropriate

[next sinew]

The only part I think might not be relevant to kids is how exactly commutativity follows from certain other (in my opinion) simpler assumptions like distributivity. Everything else I do think is highly relevant to anyone of any age

I mean, teaching fraction addition and fraction multiplication kinda goes hand in hand

I guess I changed my mind a little bit. I think fraction addition should probably be taught "first". But the whole story is a little complicated

I would teach it like

-> multiplying a fraction by the denominator (my definition)
-> adding a fraction to itself the amount of times specified by the denominator
-> understanding how fractions can be interpreted as quantities, being "big" or "small" and being able to be compared (although I would not define precisely what a<b means for two fractions)
-> understanding what two fractions being different means on a conceptual level and how fractions can be reduced, yielding the same underlying quantity
-> learning how to sum together two fractions with the same denominator
-> learning a method for computing the sum of two arbitrary fractions by forming a common denominator
-> fully understanding fraction multiplication
-> revisiting fraction addition to derive a formula that doesn't require computing a common denominator
-> revisiting comparing two fractions together (and defining a<b) and revising all that has been learned

I think it's pretty cool that after covering fractions properly, the addition formula actually isn't too bad either. Obviously multiplying a/b + c/d by bd yields ad + bc, which means the sum is (ad + bc) / (bd)

For understanding the multiplication of fractions, this line of thinking is necessary

For understanding the addition of fractions, this line of thinking is optional (because one can also just form a common denominator which reduces to a simpler problem. But of course I would argue that finding the common denominator is only simple because it's been drilled into our heads, there's nothing really creative about the process)

And I wouldn't do this because the current way of doing things is actually adequate and I'm just a sadistic mathematician, the current way of doing things is DEMONSTRABLY INADEQUATE when you observe how much students struggle with logarithms even though it's the exact same thing with different operations at play

The only reason this seems difficult to us is because the current way of teaching fractions is so bad that the proper way of doing things looks way different to what we're used to

But for me at least it's not difficult to imagine a generation that had access to the right insights at the right time in their development

[tawny slate]

so i disagree very strongly because while there are bits in your wall that are good

this is, in my opinion, not helpful in context

the very issue this particular student is having is that they are not understanding what a fraction is supposed to be or represent, and so reducing the problem in this way to its foundations is only going to make it more abstract and exacerbate this issue

here, the addition of fractions should be taught first, starting with something as simple as 1/3 + 1/3 = 2/3 with like denominators

then slightly harder with something like 1/2 + 1/4 = 3/4

piggybacking off of existing intuition with quantities

this foundational stuff is useful later, but not in this context, when the student needs a motivation and direct application

[midnight scarab]

To quote Poincaré, there are two ways to teach fractions: with apples and with pies [and now with pizzas]

[tawdry venture]

i think this borders on teaching kids to say 2+3 is "3+2 bc addition is commutative" before you ever tell them it's 5

[soft scarab]

I once read this quote and it went, “the reason why many students struggle learning math is because it is taught by mathematicians and not teachers.” I somewhat agree with it. I feel like sometimes it seems professors are more interested in their research than proving quality lectures to their students. Additionally, although one may be amazing at research, that doesn’t necessarily mean they can teach the subject well. As a community college student, my professor doesn’t do research and thus he can focus on creating quality lectures and spending more time helping his students. He’s by no means the best mathematician in the world but he is amazing at teaching. I think I used to assume that if you’re outstanding in the subject, that means you can teach it well, but now that I look back on it, teaching is a completely different skill that one must learn. What are your guys’ opinions?

[tight star]

Yeah I think if you’re good enough to be a professor you might just forget or not know what parts of the subject are hard to people who learn it for the first time

[austere delta]

Quote doesn't quite check out, given how many people struggle with math long before university. And people teaching K12 surely counts as teachers.

[midnight scarab]

From my personal experience many students struggle with math at school because their teachers are just bad at math...

[tawny slate]

you obviously need to be good at math to teach it

but the skill of teaching is distinct, i would say

and there are a lot of reasons for this

• you have to know what the student doesn't know to help them
• you may have forgotten what it feels like to not know
• you might not realize that when explaining a concept, you're applying ideas that you already understand but the student doesn't
• you might have difficulty putting into words whatever you understand
• from a neuroscience perspective, you can only learn through active attention and engagement, so being able to inspire or motivate students helps a lot more than many realize
• for all of the reasons listed above, teaching requires at least some minimal preparation, which is made more efficient through experience and proficiency

when i teach a subject, i try to think about a few layers:

1. what is my goal?
2. what is expected of my students before I even begin?
3. can i design a lesson plan that tells a good "story" while delivering on the goal?
4. how can i make sure that i can execute this lesson plan, ready for unexpected outcomes and sticking to the plan, according to the students' needs?

3 and 4 often times iterate dynamically on each other as you pick up experience and adapt. the "story" is, in my opinion, a good metaphor for what you are doing, because a good story gets students emotionally invested and makes your lessons easier to follow. your real goal is more important than just teaching a skill, it is about higher order attributes like confidence, creativity, passion, etc

human beings are not calculators, and often times those who passionately care about math dont care enough about the fuzzy and social and emotional that make us who we are (past me)

some dude, maybe feynman, was like "if you cant explain it to a 5 year old, then you don't understand it" or something

a little hyperbolic but i agree with the gist

[tawny slate]

oh and i should also mention, there is far more innovation in this space than most people can imagine i think

there are lots of new ways to teach literally the most basic of math concepts, like counting or addition, that even smart people wont ever see without active consideration and planning

you dont need to be a super math expert phd to contribute to math, let alone math pedagogy

[tight star]

What do you mean by this?

[tawny slate]

for instance, pedagogically we dont really teach rigor in early education, we generally care more about students getting the general concepts before digging into foundations

when they start doing algebra, we at least teach them how to substitute values to check their answer, and while it is not fully rigorous, it is still an excellent place to start

we certainly dont hit them with things like domain/range and logical equivalence

but we do cover it later when they learn about functions and proofs, in order to fill those gaps in understanding and develop mathematical maturity

but at least here in the US, in my experience, we neglect combinatorics

almost no students understand how we rigorously check answers to a counting problem

combinatorial problems are infamous for students getting wildly wrong answers and having zero idea how to justify or tell

perhaps emphasizing one-to-one correspondence and demonstrating more examples of constructive counting, students can develop stronger and more rigorous intuition

is there a way to incorporate that into earlier education? ive been able to do this with middle school students, but perhaps someone else might eventually be able to improve on it and allow even younger students to better understand counting

maybe cover number bases earlier than later?

perhaps teach students two different counting conventions, indexed at 0 and indexed at 1?

not saying these are good or bad ideas, just that there is a vast sea of approaches to take and experiment with, and few people are doing it

[next sinew]

I wasn't talking about how you should tutor a specific student, because in that case you have a very finite amount of time and resources and great pressure to succeed based on very simplistic metrics like "can the student carry out a fraction computation on pen and paper"

Rather I was talking about if, or when, the whole curriculum gets redone, that's how I would do it

And when I say "that's how I would do it", I don't mean copypasting my discord messages into a latex document, I mean taking the central ideas and then making them accessible to students by turning things into fun games and having bright colored textbooks and whatever

Because right now I feel like there are very fundamental ideas about fractions that could be conveyed to students but aren't

[next sinew]

This is a weird message to me because if it implies that this "foundational stuff" is useful to be taught later, then actually, I don't think it is. Eventually the bright students will have figured out how subtraction, division, logarithms and square roots work the hard way and by then any "foundational stuff" is just gonna feel overly complicated and "we know this stuff already"

If you mean that the "foundational stuff" is taught later, then uhh, feel free to ask your local math grad student how the integers are defined from the natural numbers and you'll probably hear the usual "well for each natural number, we define its additive inverse to be an element with this property", as if the existence of such elements is so obvious that it needs no further specification

And to be fair, the constructive aspects of this "foundational stuff" are indeed not relevant for anyone but the most devoted mathematicians, but the motivation behind constructing such objects is hardly foundational at all and is in fact the reason why math isn't just a concise way to write down what fraction of the pie you ate, but it's in fact a seriously generalizable set of problem solving tools

[midnight scarab]

Usually we are taught how to define fractions over an integral ring. The same idea can be used to define Z as a quotient on pairs

[midnight scarab]

If you want to explain the power of math from the generalisability, you first have to naturally introduce the more general objects for which this abstract perspective would be helpful

If you're teaching a 5 or 7 yr old fractions, then yes he'll be mainly interested in counting stuff

Though you can show how "counting" can be quite general

And I mean the fraction of an object vision is pretty much the same as "the quantity that multiplied by b gives a"

Also, even axiomatically speaking, integer multiplication is defined as iterative addition

So I think it's backwards to say that this iterated addition vision is a consequence of artificially introducing multiplications by 1 and then collecting them together using distributivity

[next sinew]

By integral ring you probably mean integral domain?

What pairs are you considering in the definition of Z, since the natural numbers don't form a commutative ring (because additive inverses don't always exist)

As far as I can see the field of fractions over a commutative ring is how you could define Q from the commutative ring Z

[midnight scarab]

Yeah, integral domain.

I just said it's the same idea

Namely, (m,n) represents m-n

[next sinew]

Going from N to Z is certainly a similar process but it's not precisely the same thing

[midnight scarab]

I.e. (a,b)~(c,d) iff a+d=b+c

[midnight scarab]

The same idea can be used to define Z as a quotient on pairs

[next sinew]

Alright fair enough

[midnight scarab]

Which is verbatim fractions but with addition. Of course you still have to verify by hand that you get a group but that's pretty easy

[next sinew]

How much of this is you being taught all this vs. you learning about the field of fractions over an integral domain and then connecting the dots that aha we can similarly define Z as N x N?

[next sinew]

You probably mean natural number multiplication because integer multiplication is just natural number multiplication with extra steps?

[midnight scarab]

Iirc I didn't come up with it myself. I think it was mentioned in class but I was already familiar with the idea by then

[midnight scarab]

Yes

[next sinew]

I'm not trying to make any claims as to how the multiplication of natural numbers should be taught, but whatever definition you choose I think multiplication by 1 will quite trivially commute with any other number, and at the very least that's a simpler observation than noticing that the multiplication of any two numbers commutes

[next sinew]

Of course my experiences are shaped by my own university whose teaching goals don't always align with my learning goals or the learning goals of the larger mathematical community so it could be that all of this is actually taught elsewhere and I'm just living in a third world country

But even where it is taught, it's taught in the context of abstract algebra, like you clearly used abstract algebra terminology for example, and I don't think that's necessarily a good thing

Because algebraic structures are abstract while the set of real numbers is so ubiquitous in mathematics that I think it would be helpful to explicitly define what numbers actually are before graduate level abstract algebra

and again, if that is actually the norm, then sorry for living in a third world country I guess lol

[tawny slate]

oh i see, sure

wasnt aware the topic had shifted but ok

[next sinew]

Whenever I talk about math pedagogy I always assume that the teacher is arbitrarily competent and has sufficient time to spend with the student because it's more fun that way

but I should learn to explicitly give such disclaimers

[tawny slate]

i think those general concepts outlined like that is fine, i just didnt realize the topic of conversation shifted there, that's all

[next sinew]

I mean, I'm a mathematician, I don't give advice, I give ideas

I leave the practicalities for the engineers

[next sinew]

Yes, I agree that apples and pies are genuinely helpful ways to view fractions, but in an ideal world I don't think they should be the foundation upon which you build your fraction understanding, because apples and pies will never help you understand how to multiply two fractions together

Fraction addition is as far as you get with apples and pies

and even that most students really struggle with

I remember as a kid I would be so confused as to why multiplying by (1/2) and dividing by 2 is the same thing, and I would wonder why division is the same as multiplying by the reciprocal, and now as an adult I know that A/2 and (1/2) * A both have the propery that when you multiply them by 2 you get A, and thus they're equal

It really was such a simple thing

eventually I would just accept that it just works that way and move on

[long pelican]

Hey that's a mini-example of unconfusing yourself by using a universal property!

[next sinew]

But I had so much confusion with fractions growing up because no one ever told me what they really were

all I got was apples and pies

[long pelican]

I like the idea of putting the universal property of fractions front and center: a/b is the universal object which gives a when multiplied by b

[next sinew]

I don't know category theory enough to understand the nuance of that statement but my gut reaction is an empathetic yes!

[long pelican]

I thought of it because of this very recent paper https://arxiv.org/abs/2405.10387 -- very good reading if you are acquainted with any of Grothendieck's work on algebraic geometry, or Langlands

(no prerequisites to understand the paper, though)

[next sinew]

I'll give it a read :D

it's a good way to procrastinate some more on my thesis

[cloud zealot]

#foundations message

this article was recently discussed in #foundations

[long pelican]

Yeah

[faint cradle]

How about a=0, and b=1 versus b=2. Is there some unique factorization through some universal object here? And what would b be for a=0 ?

[long pelican]

No issue here. For a=0 and b=1, a/b is the "universal number which when multiplied by 1 gives 0". That number must be 0. Same for a=0 and b=2

[tight star]

what difficulties do people usually have when learning set theory?

[vocal phoenix]

I'm not sure how much this qualifies as "set theory", but people often struggle with distinguishing between the concepts of "belonging to a set" and "being a subset of a set", i.e. if X = {{a},{b}}, then they'll have problems answering whether a is an element of X.

And especially they struggle when you've got sets of sets, like when sigma-algebras become a thing

[tight star]

Yes I’ve seen this too

I think it’s because irl sets are always transitive

In the sense that

If you have a ball in a bag, and a bag in a box, then the ball is in the box

[midnight scarab]

They're all collections of atoms/points in space so they're all sets. So technically it's all about inclusion which is transitive

[tardy ember]

yeah i think it depends where you mean on the scale between "first encountering the concept of 'set' at all" (possibly with just like, sets of real numbers) and like, forcing

[tight star]

Probably the former

[tardy ember]

on the #proofs-and-logic end of things, yeah definitely the distinction between element and subset is a thing

i think in particular i've seen it before with the empty set

[tight star]

Yeahhhh

How do you usually fix that?

[tardy ember]

possibly we should be counting $\varnothing \subseteq X$ as its own thing here

[burnt vessel]

bee [it/its]

[tardy ember]

but then once you've explained why that's true you also end up explaining that this does not mean $\varnothing \in X$ is always true

[burnt vessel]

bee [it/its]

[midnight scarab]

I've always wanted to complain about things like "a line belongs to a plane", especially when written $\Delta \in \pi$ which leads to set-theoretical nonsense like $P \in \Delta \in \pi$

[tight star]

Mhm, when it’s really that a line is a subset of the plane

[burnt vessel]

afqt

[tardy ember]

although that's probably more of an issue with vacuous truth than with sets in particular

[bronze stirrup]

im writing notes on calc 3 for incoming freshmen to use, how is this introduction

(for context, our calc 3 does linear algebra for 1/2 of the semester and calc 3 for the other half)

[vagrant meadow]

any suggestions for explaining span? I try to link the linear combination definition with a geometric picture, the idea of generation, and the subspace idea but no matter how I do it, it often feels like it just doesn't make sense to them.
it's a little concerning when a student asks about it near the end of the class, when I feel like we've been using the concept the entire course.

[bronze stirrup]

i feel like the geometric idea is objectively correct

[vagrant meadow]

I like how you mention a lot of good "reasons to learn" the subject, and a concrete application with sqrt(pi). wonderfully mysterious and tantalizing (why does pi show up there?!)
I see you mention statistics, but maybe throw in the word "probability" as well.
I also like how you give concrete applications of linear algebra without going too in depth.
i think it's a great length (not too long students will definitely not read it, and with enough content to be worth reading).
I can say that reading this would definitely motivate me to want to really understand the course content, at least.

[tight star]

So what do you mean by the geometric picture/generation/subspace?

And what difficulties do they usually have with it?

[tight star]

Also, with set theory earlier - i think one common difficulty I’ve heard is that there’s a lot of definitions to remember?

[cosmic ibex]

Hmm, perhaps it's just the typography (the pink box attracts a lot of visual attention), but I think that can easily read like the main motivation for multivariable calculus is to be this One Weird Trick that lets us evaluate Gaussian integrals.
It is fine to mention it in an introduction, but I'd put at least a paragraph break after "3D motion, solid objects, and more", and consider losing the pink background. It might even be good to put it as an explicuit teacher after the roadmap.

[tall bolt]

I think that boxes like that can be good if they’re used appropriately and sparingly to really highlight the point, but I’m not sure how I feel about the pink, just feels like an odd colour choice. It’s not quite bold enough to be a point you’re really highlighting imo

[bronze stirrup]

its the latex template

[tall bolt]

I stand by my point though

But I also agree with tropos

[bronze stirrup]

yes, i agree though

i'll remove the box

also not sure if this is gonna be good pedagogy

the course im making this note for only works in R^3

[cosmic ibex]

Hmm, it looks weird just to define a scalar as a synonym for real number. That will make it look like it's just new words for the sake of new words. Could you manage without ever needing the word "scalar"? The only urgent need for it I immediately see is to motivate terms like "scalar product", but you can just call that dot product instead. :-)
If you do need the word, I think you'll have to at least reveal that the scalars can be other kinds of things in more advanced settings. If they haven't yet heard of complex numbers here, it might be difficult to use them as a motivation, but you could explain that, for example, in higher algebra (field theory) it is useful to consider generalizations where the scalars must always be rational numbers ... but in this course they're always just reals, e.g. such that we can be sure the square root of a scalar (if it exists) is always a scalar.

[tall bolt]

Yeah I’m not sure how I feel about not just defining them properly to start with. This notion of a vector should already be familiar from Highschool anyway so I think the only reason to mention it would be to say it’s actually more general than this

And if you said the first half of the course is just LA then I’d definitely just define vectors, and it lets you give a better definition of a scalar, because just saying they’re real numbers isn’t great imo for basically why tropos said

[bronze stirrup]

i don't think i can abandon bringing up scalars

[tall bolt]

I do see the rational though, if you only deal with R^3 I’d probably give the defintion and say something like, for the pourposes of this course all vectors will be in R^3 so have this form/can think of them like/ etc

Something to that effect anyway

[midnight scarab]

I think that if you're going to be stating the actual def of a vector space, you should also give examples that explain the usefulness of the generality. In particular spaces of functions and polynomials

[elfin sleet]

I think that if you haven’t mentioned i,j,k yet, the part about expressing in component form will be confusing

[vagrant meadow]

the idea that the span of some vectors is a subspace (like a line or plane or higher dimensional object) that the vectors generate with linear combinations. so two vectors in R3 can generate a plane. and i usually accompany this with some weird hand gestures.

[tight star]

Sure, that makes sense

[vagrant meadow]

usually it's just "what is span" and that's all they ask

[tight star]

huh…

hmm

are they given enough examples?

[vagrant meadow]

and i feel like i've put in a lot of time trying to explain what it is intuitively. like i try to hammer home what it is and how we've used it over and over all quarter.

and how central a concept it is

[tight star]

like if u give them two vectors

[vagrant meadow]

and then they ask "what is it" and idek

[tight star]

can they find the span?

[tawdry venture]

can they tell you what the span of, say, (1,0,0) and (0,1,0) is

[tight star]

yeah

[tawdry venture]

or some other easy one that doesnt take any computations

asking eigentaylor here

[vagrant meadow]

i would hope so, but tbh probably not

[tight star]

hmm..

have you tried?

[vagrant meadow]

i don't really like the structure of this course

if it were up to me, i would have introduced the idea of a basis NOT in week 8 out of 10

[tawdry venture]

week 8 out of 10??

[vagrant meadow]

so only now are we talking about how to describe a subspace as the span of some vectors

[tawdry venture]

tf do you spend the first 7 weeks on ???

???

[vagrant meadow]

I KNOW

[tawdry venture]

are you at liberty to show what the rest of the course is like

doxxables blurred obv

[vagrant meadow]

this is the "Syllabus"

no determinants even

eigenvectors in the very last week aslhdosajhdfoufsafouad

[tawdry venture]

aaaagh

[tight star]

hmm….

[tawdry venture]

how tf do you do sol sets of linear systems w/o the concept of a basis

what the actual fuck

how do you do the matrix of a linear map w/o bases

this makes no fucking sense to me

[vagrant meadow]

no exams or quizzes or a final. originally it was 80% online homework and 20% whatever i give them, but the prof decided halfway through week 7 that (because his lectures are trash and his lecture slides are just TEXTBOOK SCREENSHOTS so people stopped attending) attendence would be mandatory and 10% of the grade. it's just a complete cluster fuck

first week he was like "yeah no exams or anything. but if you (the TAs) want to give them an exam you would have to write it"

[tight star]

yeah i don’t understand how this would work

[tawdry venture]

is it just "stuff the coeffs in a matrix this just works i prommy"

[tight star]

maybe if you only use the standard basis

then you could do it

[vagrant meadow]

i did my best. i explained that the definition of matrix multiplication (linear combinations of the columns) necessarily make it so that the ith col is precisely the image of e_i.

so yeah standard basis. but no intuition for like... what it is or the idea of a basis in general

[tight star]

i see

[midnight scarab]

Seriously if I had to TA for such a class I would just despair

[tawdry venture]

this is such a shit class

[vagrant meadow]

it's just like "oh i can write (x,y,z) as x(1,0,0)+y(0,1,0)+z(0,0,1). that's a pretty stupid thing to do"

[vagrant meadow]

with the freedom i took the opportunity to sneak in interesting applications and more advanced concepts in a digestible way. so i've been kind of building up eigenvectors since week 2, even though they'll only learn it formally in week 10

[tawdry venture]

i think this works only for spans of a single vector

[vagrant meadow]

i did give them this question this week actually

it's also hard because this is a lower div linear algebra class. so no proofs even. it's just pure computation

*even though most lower div linear classes have proofs in my experience

this is just a fucked up class in general

[tight star]

hm

i wonder if this is just them forgetting definitions then

[tawdry venture]

how'd they do

[tawdry venture]

if it's lower div, could it be that they don't realize the importance of definitions itfp?

[tight star]

maybe?

im not very familiar with this education system

idk how you convince someone that definitions are important

[vagrant meadow]

okay ig. it'll be a while before I collect it

it was hard bc I had to do that day over zoom for health reasons and nobody had their cameras on lol

[real jolt]

Zoom classes were the worst.

I'm glad the COVID stuff is over.

[surreal grove]

can anyone help me find the proof for parts d and e?

[cosmic ibex]

Not in this channel. Try #probability-statistics.

[faint cradle]

So for a=0, what is the universal object a/b which gives a when multiplied by b? And what's the proof that the 0/b object you've chosen is universal at all?

[long pelican]

I don't understand your confusion, what I said seems pretty straightforward

[bronze stirrup]

yeah this is the problem i am facing too

i can't really do anything meaty when the course is 7 weeks and isn't made to be rigorous

[quasi musk]

I gave a question to my high school calculus students today (teach at an after school program)

Was wondering how y'all would explain this fact

$$\lim_{x \to \infty} \frac{\sqrt{x^2-5}}{3x+7} = \frac{1}{3} $$

[burnt vessel]

MoonBears-C-

[bronze stirrup]

as x becomes big, we have that -5 and +7 becomes really small and insignificant

[quasi musk]

One of my high schoolers gave something like, for large values of x, $\sqrt{x^2-5} \approx \sqrt{x^2} = |x|$

[burnt vessel]

MoonBears-C-

[quasi musk]

or rather, x

[bronze stirrup]

when x is a small number like 1 or 2, we have that the -5 has a large impact on x^2-5 and +7 has a large impact on 3x+7

[quasi musk]

Yeah, I wonder what the optimal 'precise' way of doing that is

[bronze stirrup]

uhhh

imagine a glass pitcher with a small amount of kool aid

[quasi musk]

I showed them that for large values of x you can write $$ \frac{\sqrt{(x-1)^2}}{3x+7} = \frac{\sqrt{x^2 - 2x + 1}}{3x+7} \leq \frac{\sqrt{x^2-5}}{3x+7} \leq \frac{\sqrt{x^2- 5 + 5}}{3x+7} $$

[burnt vessel]

MoonBears-C-

[bronze stirrup]

if you pour a small amount of water into the kool aid, it'll still taste like kool aid but if you pour a large amount of water it gets diluted and eventually you can't tell the difference between the taste of this mix and regular water

[quasi musk]

Since the square root function is increasing

Emphasizing this is true for large values of x. Now the lower bound goes to 1/3, and the upper bound goes to 1/3

(This might be my inner analyst coming out)

[bronze stirrup]

i think considering high schoolers it may be just better to show that the constant terms become insignificant

[quasi musk]

Yeah, that's certainly the idea. But there is something nice about doing things precisely

[bronze stirrup]

is it worth it though?

[quasi musk]

I was wondering if I overdid it by showing that

Or if there's a cleaner way to get the result I want, without waving my hand

[bronze stirrup]

multiply both the numerator and denominator by 1/x and then bring that 1/x into the square root

[quasi musk]

Yeah that'll do it

[bronze stirrup]

yes, i think that's better

you can probably make an analogy of "small terms become insignificant" with this too

[quasi musk]

Much cleaner.

Thank you

[bronze stirrup]

[quasi musk]

And that goes directly to the idea that 5/x^2 goes to zero, 7/x goes to zero

[bronze stirrup]

yes

maybe you can also use ratios too

"as x gets large, (x^2-5)/x^2 approaches 1"

the difference that the -5 makes in the ratio becomes so tiny that we can just drop it

[quasi musk]

Yeah, I did that earlier in the year when we focused a lot on limits

But I've been teaching integrals, gave them their final test for the year. Then went over the problems, and the only thing I could think of was "make square root go away"

[bronze stirrup]

im bringing up the space of all continuous functions in R as an example for showing that vector spaces can get funky and i'd like some good examples to demonstrate why we care

[rain hawk]

Why we care about the space of continuous functions or why we care that it gets funky?

[bronze stirrup]

why we care about the space of continuous functions

to someone who is just beginning calc 3

i've got an example of why we care about scalar fields besides R from quantum mechanics

[quasi musk]

It just tells us what math operations we can do on continuous functions. If we add two continuous functions, we get another continuous function. Etc.

[bronze stirrup]

actually i think it may be just best not to bring it up?

[quasi musk]

For calc 3 you might want to mention "there are more abstract vector spaces, but for now we'll just think about R^3"

[rain hawk]

tbh the vector space of continuous functions is pretty stinky, but it's very useful to know that continuous functions have a vector space structure when you're talking about continuous functions

[bronze stirrup]

i can't think of other funky vector spaces that would make sense to bring up

[rain hawk]

the vector space of smooth functions comes up a ton in AG

[bronze stirrup]

ya but i can't really bring up AG in calc3

[rain hawk]

true

[quasi musk]

You kinda can

Zero sets of polynomials?

x^2 + y^2 + z^2 = 9

[rain hawk]

all the applications of the vector space of smooth functions come up after calc3, like the other one I'm thinking of is the solutions to PDEs

[bronze stirrup]

yeah they all come up after calc 3

[rain hawk]

the thing is no one really wants to work with infinite dimensional vector spaces raw usually, you want to season them with a topology or a norm or sm

[shadow flower]

i think at one point in this channel there was talk about vector spaces having structure where addition was replaced by multiplication and multiplication was replaced by exponentiation

[bronze stirrup]

i can bring up a different function space for fourier series, but i think i'll probably just say "there are more abstract spaces that are of utmost importance in many fields than what we will get into here"

[shadow flower]

like as long as you have the operations be the “next” one in line, the vector space maintains structure

it’s pretty funky

[rain hawk]

exp(ab) \neq exp(a)exp(b)?

[shadow flower]

i’d have to go find it again but it was something of this flavor

[bronze stirrup]

next thing i'm doing is showing details of vectors in R^3

and the norm

[shadow flower]

this is it actually

[rain hawk]

ah neat

[shadow flower]

i think it might work with addition being exponentiation and multiplication being tetration but i’d have to sit down and go through it

[bronze stirrup]

it's hard going from the fine line between underdeveloping ideas and having people go "lolwhat" and going too technical

[bronze stirrup]

ok, i think i got it for introducing vectors for calc 3

[noble hare]

Looks nice! I would probably not call R³ a "plane" though

[bronze stirrup]

i think i wrote plane on accident

[noble hare]

Ah! There's two instances if you missed one of em, in the para above the fact and in the para below defn 1.2.3

[wispy slate]

I think for linear transformations, we can still conceptualize the map through its act on vectors and its properties.

But on the other hand, to represent a linear map as a matrix…

Yeah…

[wispy slate]

,tex You could draw out the linear map T: V \to W where V and W are vector spaces over the same field. The linear map T satisfies linearity: T(av + bw) = aT(v) + bT(w) for all v, w \in V and a, b \in \mathbb{F}.

[burnt vessel]

akarii44
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[wispy slate]

This is so fun.

[vagrant meadow]

can anyone give their thoughts on discussion posts as assignments?
due to one worksheet bleeding into the next week I dont really feel like giving another worksheet for a half a week (on material I've already covered with them... oops).
was considering doing something fun like a discussion post asking them to research an application of linear algebra related to their major/intended career path.
in y'all's experience, how successful are things like

• word minimums (like 100 words)
• requiring comments on other people's posts
  not sure if it's bettef to relax the requirements and let them put in effort on their own accord or to force them to put in some effort. open to any other input/suggestions as well

[cloud zealot]

Having been assigned discussion post assignments, not necessarily for math classes, multiple times, I have not seen anyone take them seriously, and I don't take them seriously myself. One might imagine a grand public forum where students engage in lengthy dialogue, but there's no connection to be had in the first place. It's not organic the way a spontaneous classroom discussion can be and it's not a subreddit.

By nature, any conversation is going to be temporary as well, basically gone after the term ends. Also, no one bothers to continue a thread after the assignment is past due even if they find a post interesting as they need to work on another assignment, possibly another canned discussion post.

Basically, they're busy work and despite being called discussion posts, genuine scholarly discussion does not exist in them.

Discussion posts are an ugly symptom of a trend of colleges assigning more work (or quantifiable things they can add to the gradebook) but deemphasizing learning.

https://old.reddit.com/r/AskProfessors/comments/17a4owc/was_college_harder_back_in_the_day/k5ajfxz/

https://old.reddit.com/r/AskProfessors/comments/17a4owc/was_college_harder_back_in_the_day/k5bhnnz/

[bronze stirrup]

part b is a pretty decent curveball to make sure someone really understands the concept of the basis

[wispy slate]

I wholeheartedly agree with this too.

[vagrant meadow]

are you saying there's absolutely no value to be gained in them?
idk i always took discussion posts at least somewhat seriously. i didn't love them, necessarily, but i did find them a nice change of pace (because i'm a babbler who likes to talk and talk and talk).

[vagrant meadow]

unfortunately, though, this is kind of the case. i need an assignment for each week. based on how the primary prof has set things up.

but i find it hard to believe a discussion assignment can't be done well or encourage learning

[cloud zealot]

A PhD student is already a hard worker. You are more the exception than the rule. Discussion happens when students are genuinely interested in learning the material and want others to bounce ideas off of. I understand if you need to get something in the grade book each week. I only want to temper your expectations.

[vagrant meadow]

really, I'm not expecting amazing discourse, I just want them to spend 30 minutes looking into how linear algebra might be useful to their intended career path. and that might give me suggestions for problems to put on their final worksheet.

[agile elm]

Hi! Something I've been wondering for quite some time is if there is research on cognition and math, in terms of what can you teach a child and what can they understand at age x?

If said child is developing within a standard deviation or two of the average

Also how is the development of logical understanding of children? A common challenge for kids is to understand/accept causality in part because they are so egocentric

À follow-up is How much of any particular countries' math curriculums are based on what research has shown children can understand at a certain age in terms of math topics.

As an example for instance. In my country kids aren't taught about the existence of negative numbers in grades 0 or AFAIK until grades 7-8. If you use the centigrade temperature scale, in many countries of the world, the temperature will go below 0, and so you have this physical phenomena to use as an example

[strong hearth]

have a look into Piaget's theory of cognitive development

especially the concrete operational stage (approx 7 ~ 11 yo) onwards

[agile elm]

Interesting read, but quite biased as the assumption seems to only ask US professors

But perhaps I'm reading into it

[agile elm]

Hah I know of it!

Nothing I have studied though

Thanks for the suggestion, I will look into it further!

I have watched some YouTube video experiments on child development, incredibly enlightening!

[strong hearth]

I'm not done yet, haha!

Here's an example of mapping the stages to mathematics education, context being US https://files.eric.ed.gov/fulltext/EJ841568.pdf

[agile elm]

Perhaps I should have started by saying I'm not a a teacher, but I have worked as one during a study break. As a substitute but I had mainly longer assignments and just lots of lesson planning, making exams, grading. Also was a class mentor (?) for a semester. My background is in Earth science

I'm really interested in teaching though, especially math and ofc earth science

[strong hearth]

I teach but I'm not a teacher, I'm an education researcherbut not a researcher in paedagogy

[agile elm]

(hydrology /hydrogeology)

I "had" (wasn't forced to) to read up a lot on my counties grading systems for many subjects and other stuff

Would be interesting to see if there is any research on when countries introduce concepts X, Y, Z etc

[strong hearth]

I'd be thankful if countries base their curricula/assessment on any research/evidence at all 😄

[agile elm]

The Finnish school system is considered incredibly good for instance but that's not exactly what I'm asking about

[strong hearth]

self-regulated learning is pretty effective and that's what the finnish system uses

[agile elm]

Oh indeed! In my country, Sweden, it's very politicised. Everyone and a very party has opinions on education but I doubt any know of actual research on it!

[strong hearth]

Trust me i've been a teacher for 30 years I "know" what works

the reality we face

[agile elm]

On the one hand what you are saying is meant to be dragicomedy (?)

[strong hearth]

I shall disambiguate

[agile elm]

But on the other hand, perhaps that's better than governments or a politician appointed to be in charge of education to just come up with stuff and build monuments

Sry it's still early and so my writing is bad

[strong hearth]

all good, i need a nap too

[agile elm]

The following is more of a speculative nature
In sci-fi kids seem to be portrayed to be able to learn much more and more quickly than today, in the Expanse for instance

I've been thinking about this topic for a long time

[wispy slate]

what are some research areas that combine geometry (i am really interested in riemannian manifolds) and optimization, that are rigorous but at the same time practical

[real jolt]

I think that's a question for #geometry-and-manifolds

[hollow musk]

that channel is now closed; try #diff-geo-diff-top instead

maybe #optimization also

[worn tiger]

Some books define:

but then also we have that a function is a cartesian product, because it is a relation with certain properties, can both of these definitions work at the same time, because then it would be circular, no?

[tight star]

So a function from A to B is a subset of A x B satisfying some properties

[worn tiger]

yes and A x B is defined as above

[cosmic ibex]

If they go to the length of writing that out symbolically, they really also ought to show the quantificatoon of i explicitly...

[tight star]

In this case you want a function from I to $\bigcup_{i \in I} A_i$

[burnt vessel]

Pseudonium

[tight star]

This is just a subset of $I \times \bigcup_{i \in I} A_i$

[burnt vessel]

Pseudonium

[tight star]

Satisfying some properties

So i think - probably that def only makes sense for quite large Cartesian products

I don’t think you could use it to define A x B

[worn tiger]

this doesn't make sense because we are defining x

[tight star]

Cause as you say, defining what a function is requires a notion of Cartesian product

[worn tiger]

we don't have access to it in our definition

[tight star]

So it’s like

You can say what a binary Cartesian product is

Of A x B

[worn tiger]

sure and what is that

[tight star]

And then that allows you to do Cartesian products of any size

[tight star]

Set of ordered pairs?

[cosmic ibex]

But right: you cannot both define all tuples in terms of functions and at the same time define functions as sets of 2-tuples.

[worn tiger]

and ordered pairs would be sets e.g) (a, b) = { {a}, {a, b} } ?

[cosmic ibex]

Yeah, usually.

[worn tiger]

else if we used functions we have the same problem again I believe

[tight star]

Yeah

[cosmic ibex]

This means you have two different definitions for the notation (a,b) -- either just as an ordered pair, with the specific Kuratowski definition, or as a general 2-tuple which is a function from {0,1}.
The ambiguity is more or less unavoidable if you want to reduce everything to set theory. Fortunately it's easy to show that the two definitions behave identically as far as the standard properties of tuples are concerned.

[worn tiger]

Yes, so assuming we use the { {a}, {a, b} }, definition as tuples, and then use that in A x B, and then define functions on top of that, then we can show that there is a function that return's the i'th element of a tuple as a corollary right?

[cosmic ibex]

By the way, if you're learning this instead of asking how you should teach it, this discussion belongs in #proofs-and-logic, or possibly #foundations if you want to get super philosophical about it.

[raw sedge]

Hey! I’m looking for a reference for this type of diagram of a transformation about how we can change the hard problem we are trying to solve into a more tractable problem using a convenient transformation: eg polar coordinates, u-substitutions, laplace transforms, etc. I forget the name/search term I’m looking for but I know I’ve seen almost this exact diagram before

And preferably not something super rigorous like a functional analysis type thing- just an intuition level explanation would be appreciated thanks!

[tight star]

Reminds me of that diagram from the diagonal argument vid

[cosmic ibex]

"Reduction"?

[pallid night]

I've heard Generalised Coordinates being used as a term before but I don't think that's super common

[vagrant meadow]

so next week is the last week of the linear algebra class i'm a TA for.
they are covering eigenvalues and eigenvectors for the first time on monday, skipping the characteristic equation (since they never covered determinants at all. idk how tf they're supposed to actually find eigenvalue), and then the last class on wednesday is on diagonalization (even though they've never covered change of basis formally beyond basic coordinate vectors).
i'm doing my best to deal with it, and teach them what they need to know, but what would you all do?

[midnight scarab]

For the finding eigenvalues and eigenvectors part, I guess you can just restrict yourself to the methods for solving systems of equations that they have learned, and just show that in like row reduction you must get something of the form 0=0 (cause otherwise you have n constraint for n unknowns and a trivial solution) and that you then get an entire line (or higher dim vect space) of eigenvectors

So like focus on solving for eigenvectors, and ignore the fact that you can find the eigenvaluesa priori jusy by looking at the characteristic polynomial

For the diagonalisation part, perhaps actually focus on the whole writing a linear operator in a given basis. If you can get them to understand that 1) the columns of a matrix rep are the components of the image of the basis by the lin op in the given basis, and 2) how to change bases by conjugation with the change of basis matrix, then diagonalisation should be straightforward

(And to find the change of basis matrix, again use the columns are components idea)

[frigid briar]

Does anyone know how to show that weak solutions exist for the non-stationary Navier-Stokes equations?

[austere delta]

Maybe try #advanced-pdes

[tawny slate]

i have an interesting question i think

excluding definitions, very few math rules/formulas need to be memorized pre-univ

they can almost all be derived/proven from first principles, from the definitions

the question is: what are all of the exceptions?

i think this would be a good community effort, because either we find something we think is too difficult to prove and someone finds a simple proof, or we discover that there is something we take for granted that we can be aware of to target when teaching (and have those rare examples where pure math contributed something very direct to math education)

for starters i can think of the following:

• pi is approx 3.1415... (this is not hard to compute using basic methods, but none that i am aware of that you can just form in you mind quickly without paper/pencil
• pi is irrational (only elementary proof im aware of involves taylor expansions, pretty tedious)
• fundamental theorem of algebra

i guess i am asking from a pedagogical perspective, so it would probably make sense to separate elementary proofs that are long and tedious vs something that is just immensely difficult to prove with elementary methods at all

[quasi musk]

For Engineering/Physics/Math oriented people, the trig formulas are important to know. Especially the sine and cosine addition formulas

[tawny slate]

i have an elementary derivation of this:

use a geometric interpretation of multiplying complex numbers in polar form, then translate that statement to rectangular form

sin and cos addition derived together at the same time

requires only basic geometry, basic trig, basic algebra, rudimentary understanding of complex numbers

[rain hawk]

fundamental theorem of arithmetic can be derived, but it's usually taught well before then

[dull fossil]

but deriving something is not the same as knowing the right statement to derive. the reason to memorize internalize (i think the latter is a much better word for this kind of thing) the trig identities is so that you can recognize when it would be helpful to apply them

[tawny slate]

you are correct, so i guess i should be a bit more specific in what im looking for

some concepts, if you know them, allow you to easily derive the formulas without a reference
some theorems can be proven and easily understood by working out the logic in your mind

im looking for anything that doesn't fit these criteria, something that students would need to take at face value and memorize or accept in order to proceed

[tawny slate]

i would say this is a good, but not particularly strong example, because i do think that if you eli5 and remove some of the rigor, i think a simple division and induction explanation would suffice

still a good example though, approved 👍

[dull fossil]

what do you mean by "to proceed"?

like I definitely think students who are going to be doing related stuff should know the most important trig identities, not just be capable of driving them. similar for many laws of algebra. e.g., a^2-b^2 = (a-b)(a+b) can be easily derived if you realize the right thing to simplify, but it's very hard to realize that quickly in dense equations if you don't have the muscle memory with it

maybe you already agree with this and I am not understanding what you want

[rain hawk]

Existence is easy, uniqueness takes a bit more cleverness. Like I said, still a fairly elementary proof tho

[tawny slate]

yeah i do agree, for instance, i have totally forgotten the trig sum identities, but i can derive them in a few seconds in my head, and while i might not recognize the form in a problem immediately by memory, i do know that its like sin cos + cos sin or cos cos + sin sin or something, and thats good enough to trigger intuition to try it

basically two things need to happen: the student has enough to build intuition to tackle the problem, and any reasonable doubts about the truth of the math is settled

you can internalize the trig sum identities by memorization, but then the student might be like "but why is it true tho"

so memorization alone does not suffice here

knowing the derivation is good enough if it ignores the need to memorize and can be done within one's head reasonably

what derivations or statements arent "good enough" is my question

[tight star]

i think it’s useful to both know the derivation and have the identity memorized

like, knowing the derivation helps understanding

but if they need to go through it every time they want to use the identity, it’ll just be too slow

[rapid tusk]

thing is how do you deprogram students from the mindless plug and chug trap that so many fall into?

or “memorize the proof word for word without actually internalizing what any of it means”

[minor turtle]

i really like this question. for people going into math i think that knowing the sine, cosine, and tangents for the “nice” angles are important (even if you NEVER have to evaluate them for an assignment after hs) so that you have a sanity check against your reasoning about trig functions later. Also, when trig functions are positive, and what the convention is for the domain of inverse trig functions are two that i saw my students get hit with and blindsided a LOT, because it’ll take them longer to, say, recall what the value of cos(pi/6) is, and then forget what they were doing in the first place

in general, having things come in your mind quickly helps you think “strategy” way more i feel

also, it’s not really a given that people know how to add fractions (with whatever expressions or even just numbers sometimes) coming into calculus, and i think it’s a great disservice not to make sure that it’s literally second nature to add fractions together. i used to dislike my hs precalc teacher for doing this to us once, she gave us a whole day just to add fractions and i felt really insulted, but now i realize that i wasn’t the target audience for that, and there were people who REALLY DID need it

[tight star]

I might be the wrong person to ask for this cause like

I use Anki to memorise proofs

[rapid tusk]

this so much - even though I was never particularly good at higher level contests in high school, all the drilling it gave me on the basics has served me well in undergrad when I need instant recall, letting me focus on actual “big picture” techniques

[minor turtle]

in fact i see this with a lot of tutors, there’s such a big misunderstanding that if you tutor a “high level” subject like calculus or even like UG algebra, that you’ll necessarily get “high level” students who just need a push to do better or need your little clever insights. in reality those students are not typical. the real people who need help are usually students who were left behind because they were thought to need to socially move on with their peers, and therefore were forced into things they were not ready for mathematically, and now are struggling with WAY more fundamental issues than people expect (because socially they read as someone who knows all the foundations. this is not their fault, it’s just a consequence of the fact that we, societally take for granted that “socially in X grade/year in college” and “X level of proficiency” as the same, even though that’s not at all how it works), and tutors are not always able to see this problem

or they do and are not prepared for it because they’re teaching a “high-level” subject

and that’s a huge problem

[vagrant meadow]

how important is jordan decomp to non-math majors? not sure if i should sneak this question into my final worksheet for lower div linear algebra

[vagrant meadow]

i will say that i prefer teaching/tutoring higher level classes simply because those kind of questions are much less common. my ODE students still couldn't do integration by parts, the product rule (or a basic derivative for that matter), nor could they do the quadratic formula, but at least i didn't feel compelled to spend any precious minutes on those things.

[wispy slate]

I am surprised that I did not see this.

It is used rather extensively in biology.

We use a lot of eigenvalues and eigenvectors to study the stability and change of allele frequencies over time.

Lol.

[tawdry venture]

it lets you reason at a higher level

trying to solve a complicated trig equation w/o some basic trig values under your belt is like trying to extract literary themes from a text in a language you barely know the alphabet of

[tight star]

yeah that’s a good analogy

there’s a finite amount of conscious effort you can exercise at any one moment

so it helps if you can do more things without thinking, or reduce cognitive load

[vocal phoenix]

Yeah, definitely

[vagrant meadow]

one chem student said it wasn't really relevant to them and I commented that at least one application is being used in a lot of data analysis. but I wish I was more familiar with how LA is used in every different stem field

[wispy slate]

That depends, I have seen it been used in spectroscopy and molecular orbital theory.

I have even seen the Hamiltonian operator been used in quantum chemistry.

I never understood how science majors could dismiss mathematics like that.

Perhaps besides physics majors, lol.

[cosmic ibex]

Anything even remotely quantum will be full of LA.

[cloud zealot]

I'd wager not very. It is not a very useful decomposition for numerical analysis, and the sciences increasingly rely on numerical methods.

[pastel ledge]

Is it not considered important for e.g. theoretical work in the sciences or engineering? It was emphasized in my diffeq class back in the day, as means for practically understanding solutions to linear systems of ODEs

[vagrant meadow]

In my modeling classes, we talked about how real world data rarely produces repeated eigenvalues. and if a first order approximation does, then the real behavior is probably still a generic sink or source, and the slightly curved nature of a defective node is too much detail for an approximation anyway (however, if the approximation has a repeated eigenvalue of 0, or is singular in general, it's especially a pain in the neck to figure out the real behavior).
moreso with purely imaginary eigenvalues (i.e. real part 0). real world data almost always produces a spiral and not a closed loop.

I'm oversimplifying, but the tl;dr is that defective eigenvalues and stable centers are mostly anomalies and not usually indicative of how the system behaves. in general, the eigenvalues of the approximation are really the only useful parts (eigenvectors can give a loose idea of where the trajectories come in and out but not that well).
I'm a bit rusty on this topic so I may be leaving out some subtleties, but given you usually solve these problems with a calculator or numerical tool, the nitty gritty approximation stuff is much less valuable anyway.

many ODE classes skip Jordan stuff altogether

[vocal phoenix]

<@&268886789983436800> here as well

[toxic talon]

discord is supposed to delete these on ban -_-

[halcyon glade]

Discord auto-delete seems to have broken, yeah

[vocal phoenix]

It is Friday

[vagrant meadow]

okay so idk how appropriate a topic this is, but i just find it so weird. so i was showing my students using the trace and det to find eigenvalues and i'm wondering if anyone knows why students are so good at questions like this?

• find two numbers that add up to 2 and multiply to -3
• find two numbers that add up to 4 and multiply to 4
• find two numbers that add up to 4 and multiply to 3
• find three numbers that add up to 6 and multiply to 6
  i know these are pretty easy since they have integer solutions, but they can generally figure it out in like half a second, seemingly without thinking. whereas i ask them to do basic arithmetic of calculating A-I (or even just calculate the trace) for a 2x2 and it takes like 30 seconds for anyone to say anything

when i gave them "adds up to 2 and multiplies to 5" they also seemed to know right away there was no (real) solution. like... how??

[pure light]

pretty much the same skill is used to find integer factors for quadratics by inspection, so they probably learned it in high school algebra

[rain hawk]

this one is easy because 5 is prime

[cosmic ibex]

That makes it easy to exclude integer solutions, not so much real ones.

[vagrant meadow]

i just find it weird because they're much slower to actually solve quadratics. and just do arithmetic in general lol

[rapid tusk]

it’s just glomming onto shallow pattern recognition is it not?

bc from what I’ve seen students will just default to quadratic formula if they dont see the factorization immediately

[cosmic ibex]

That sounds like a reasonable strategy too, doesn't it?

[rapid tusk]

ig

but also all the factorization examples I remember seeing in class in middle and early high school

were either monic or trivial non monic ones

[dapper flume]

I see this at all ages K-12 tbh. It's practically a meme that 4th grade students can solve some 3x3 systems of equations if the variables are fruits instead of letters. Something about the way a problem is represented seems to have a substantial effect on how people end up approaching it. I'm unsure whether it's a matter of using tier 3 vocab words like "quadratic" getting in the way, or whether people have been discouraged to think flexibly in patterns in grade school, or what.

I plan to research this more in the future.

[rapid tusk]

it might have smth to do with how we deal with abstractions?

[dapper flume]

Elaborate because I'm interested

[rapid tusk]

say your example of systems where the variables are “fruits”

the concept of that is slightly more “tangible” than just variable names idk

(i am talking out of my ass here 😭)

and it’s weird bc people still tend to struggle a lot with “translating English to algebra” in word problems for whatever bewildering reason

[dapper flume]

Yeah I spend a WHILE teaching translation. Arguably I find that skill more important than the arithmetic solutions to problems

[strong hearth]

there is a need for more fruit maths in the primary/secondary education

[dapper flume]

I'm not sure how to think of tangibility here though. Obviously a fruit is not a number, but for some reason, it's got less barrier

[strong hearth]

gone are the days of x and y, it's apples and pears my man

[dapper flume]

I have a hunch (and only a hunch) that the reason for this pattern of "non-standard symbolism is easier for some reason" has to do with the ideas in cognitive load theory

[cosmic ibex]

It must be because using letters in particular triggers some "there are steps and rules here that one has to follow and I don't remember what they are!" paralysis.

[dapper flume]

Yeah that's my hypothesis too

[strong hearth]

as people do with looking at a "wall of text" written in computer modern cf. comic sans

[dapper flume]

Older kids seem to be increasingly paralyzed by this

[cosmic ibex]

Whereas with fruits they don't think in terms of rules and therefore feel more permitted to just fool around.

[vagrant meadow]

https://youtube.com/shorts/7alnR_xYuhg?si=Chov30P5il2UUJ5o

[cosmic ibex]

And that's when you hit them with the elliptic curve meme.

[dapper flume]

I've tested this explicitly in my class. If I ask them HOW they solved the system, they seemed way more willing to "guess and check" or even reason through things algebraically with fruits. And they were more inclined to just do elimination or subsitition with variables, often without even knowing how to understand the meaning of the solution immediately

[vagrant meadow]

many animals display a sense of being able to compare "sizes" or "amounts" of food. I could see brains evolving to more efficiently do math on food

[dapper flume]

What a Freudian way to do math /j

I wonder how one would even test that

[strong hearth]

could to do a pre/post

let's do a study folks

i'll do the ethics

[vagrant meadow]

when food is involved animals can do a lot. the decades spent trying to teach non-human apes to "talk" ended up just teaching them elaborate ways to ask humans for food.

[cosmic ibex]

I'm not sure that rings true to me, because fruit algebra doesn't really look like "how many apples are there" -- it only makes sense to say apple × banana = pineapple if you understand intuitively that each single fruit is to be replaced by a number.

[dapper flume]

Anyone who's tried "menu math" probably knows it's strangely effective for teaching expressions and the distributive property

[dapper flume]

My thought as well. Although I suppose it's ALSO possible that the very sight of food is just more like... Stimulating? Maybe?

It's deep conjecture at that point ofc

I'm not a psychoanalyst

[vagrant meadow]

let's try to teach a chimp fruit math

[cosmic ibex]

My money is on "PTSD from bad algebra teaching".

[vagrant meadow]

also likely^

[strong hearth]

it can be multifaceted

[dapper flume]

It's definitely multifaceted

[strong hearth]

por que nao os dois?

[dapper flume]

I'd put money on this being the biggest weight

I don't usually have time to explain a problem before my students shut down at the sight of a longer-than-usual equation

Even I sat through Diff Eq trying to avoid doing long algebra at all costs. Hence my cognitive load hunch

[rapid tusk]

inb4 all algebra classes become r/mathwithfruits

[cosmic ibex]

Perhaps fruits are cognitively easier because they always stand for unknowns you're supposed to find, whereas letters can be either unknowns or constants, or arbitrary parameters in an identity, or whatever else we need to give a temporary name to?

[dapper flume]

Slightly tangential to that, this reminds me of how 8th grade (USA) especially struggles with variables, because they stop being unknowns and actually start being allowed to vary in things like proportional functions

[cosmic ibex]

And perhaps if you try to teach that (🍎+🍇)² = 🍎² + 🍇² + 2×🍎×🍇 as a general rule, that will begin to create the same kind of confusion?

[dapper flume]

Squaring an apple is crazy

[rapid tusk]

next we’re gonna be asking to square a circle

🤠

[upper solar]

why dont we just start using "?" as a variable

could then transition to ?x subscript etc, then further along move to just x and y

people would realise that the ? means "something we don't know"

[strong hearth]

i mean there's thousands of emojis

(🔞 +🃏)² ?

mm yes

[dapper flume]

🤓 + 👉

[rapid tusk]

(☝️ + 🤓)^🟦

[minor turtle]

lmfao

[minor turtle]

my elementary school got me hooked on using ? as a variable before I knew what variables were

[cloud zealot]

"variable" is bad terminology because we aren't also taught what logical quantifiers are and how the unknown might be quantified. also, "variable" suggests something changing but it's just an unknown

[dapper flume]

Just tell them the variable is in superposition (unknown) until it is observed and collapses (is solved for)

/j

[vagrant meadow]

sometimes it's fixed but sometimes it's not. but i agree with your overall point. parameter is a better word for something that's changing. there's just something about the term "unknown" that kind of irks me, but idk what it is about it.

[cloud zealot]

sometimes it is known

[dapper flume]

That's the word I use

[minor turtle]

i like the term “unknown” because the thing we’re trying to find is unknown

alternatively “little brat” because it’s hiding from us

[drowsy otter]

I have something to ask of you guys.

For context, I have earlier been advised by someone on this server to teach adding and taking away fractions before I go any further. I have taken this advice, and as I anticipated, my student does not get how to add or take away fractions.

I am using slides that explicitly tell the student what to do. However, these steps do not seem to sink in.

I must also note that the student has done very well when I taught equivalent fractions, but on average the student does not apply this concept to adding and taking away fractions (fraction arithmetic is, as someone mentioned earlier, a must-have skill).

I am considering a different teaching style, but I am so used to using presentations to do this, that I do not know where to start.

Now, I do not think it is easy to add or take away fractions, so it could be that I don't have to change anything, but my student is likely as frustrated as I am when they struggle with this concept.

I have attached two example slides so that you guys understand the context. Thank you.

[austere delta]

Pictures like this
https://www.geogebra.org/m/BTCSvEDZ

[tawdry venture]

does the student know how to do it with the same denoms

cause if they can't do that then you're trying to build on a foundation that's not there

[drowsy otter]

Good question. I have a slide for that too.

[cosmic ibex]

Hmm, it feels like a somewhat abstract approach to keep talking about "equivalent fractions" instead of "different fraction representations of the same number". Is that because you haven't introduced (or emphasized) the idea that there's a single thing that both 2/7 and 6/21 are ways to speak of?

[tawdry venture]

i'd also suggest displaying all this on a number line for extra visuals

[drowsy otter]

What do you mean by this? I'm confused by the line "there's a single thing that 2/7 and 6/21 are ways to speak of".

[cosmic ibex]

The notations 2/7 and 6/21 are two ways to name one and the same real number.

[drowsy otter]

Noted. Thank you.

[cosmic ibex]

And I think it is extremely important to make sure students get the point that it's actually those numbers we're interested in, and the fractions on the paper are just a convenient way to speak about (some of) them.

[drowsy otter]

I'll try to memorize that line.

"It's actually the number the fraction represents that we're interested in, and the fractions on the paper are just a convenient way to speak about such numbers."

[cosmic ibex]

Do the audience for those slides know decimal fractions?

[drowsy otter]

What do you mean by 'decimal fractions?' My understanding is fractions that involve decimals in the numerator and denominator, but what are you referring to?

[cosmic ibex]

A "decimal fraction" means notation such as 0.285714285714....

[drowsy otter]

The answer to that question is 'no'. Will this pose as a barrier, in your experience?

[cosmic ibex]

It's more that if the answer had been "yes", then you should definitely make sure to emphasize that 2/7 and 6/21 both mean the same number they already know and love as 0.285714285714... -- but without that pesky business about the decimals going on forever.
If they don't know decimals, then that point is moot, though.

[drowsy otter]

They know some decimals, for example, in the money module, they understand £3.50 and monetary amounts (in pounds and pence) up to £12.99. Is that significant, may I ask?

[cosmic ibex]

Depends on whether or not that is the way they'd already think about a point on the number line that's not an integer.

If the slides are for actual grade school kids, then my gut feeling would be that those slides look frightfully abstract and adult. I'd expect larger fonts, several concrete examples worked with diagrams and shapes, and annotated with the numeric fractions.
(But it's not like I have actual pedagogical credentials to support that gut feeling ...)

[tawdry venture]

... up to 12.99? so £13.00 and £13.01 are out of their grasp?

[drowsy otter]

I don't actually know... My money module only taught up to £12.99 (in GBP currency). But chances are, the student can add or take-away pounds and pence above £12.99 - possibly up to £99.99 (if they use the ThHTO column method).

[tawdry venture]

up to £12.99 (in GBP currency)
the way you're phrasing it, it sounds like the following:

• addition and subtraction of anything over 13 GBP
• handling money in any currency other than GBP
  have been deliberately hidden from the students

[cosmic ibex]

(Perhaps a holdover from pre-1971 days when money arithmetic did need to be taught separately from everything else in Britain?)

[tawdry venture]

but why 13 pounds? what's with such an arbitrary limit?

[drowsy otter]

Well, here's why:

When I first taught the student (it is 1-to-1), they were struggling with their times tables. I taught them the tables up to the 12 times table. When the student finally learned their times tables, they were so used to numbers up to 12, that it made sense to me to do the money module up to £12.99. Don't ask me why it made sense to me, it just did.

[tawdry venture]

but this is addition and subtraction

and also idk like

why put such a limit up

at all

if the student can handle column addition for 3-digit numbers, why not just tell them that the same algorithm works no matter how long the numbers are

[drowsy otter]

I mean, it makes sense to tell them that, that's for sure. Unfortunately, I worry that because this student is behind, that they can't handle numbers with large numbers of digits. But now that I think about it, I might be unintentionally holding the child back.

[tawdry venture]

yeah thats my point there

try out a few large number addition/subtraction problems to see how the child does

like four or five digits perhaps

maybe 6

[drowsy otter]

That could buy me some time to completely redesign my fractions slides. Thank you.

[drowsy otter]

But back to fractions. I am more confident in my ability to teach my student how to add and take away fractions, thanks to my diagrams and explanations. Thank you all.

[midnight scarab]

Wait, are you teaching fractions to a single student?

Why use slides then?

[drowsy otter]

I like to keep a record of what I teach, and besides, presentations are very presentable and compact.

[carmine ridge]

Teaching fractions was stupidly hard for me 😭

[drowsy otter]

Trust me. It’s pretty hard, but since it’s a must have skill, I recommend you don’t give up. You’re doing great.

[tawny slate]

the biggest thing to remember is to have patience when teaching fractions

fractions are a concept that we now take for granted, but they are one of the most challenging concepts in math at the time most students first encounter them

there are so many layers and steps to helping students build the concepts and understanding

try to break everything down into as little pieces as possible

[tacit crater]

12! = ????

[tight star]

What tends to trip people up?

[tawny slate]

the way i think about it is how fraction are approached and what is expected of them

here are some things that i can immediately think of:

• at this point some kids maybe have just learned their multiplication tables, and so while they may have memorized them, they may still not have fully conceptualized multiplication, which makes conceptualizing division just as difficult
• to learn how to even add fractions, they have to first know what a fraction represents, what are equivalent fractions and how to convert into equivalent fractions, adding fractions of the same denom, then finally knowing to convert fractions into the same denom before adding. this is a LOT of steps for a kid at this stage
• sometimes kids will get lazy and try to shortcut the understanding by symbology or abstract pattern recognition that isnt grounded, such as adding numerators straight across when adding fractions, or even when they just multiply straight across correctly, they don't understand what this is doing or what it means. many times teachers focus on correcting this by directly correcting the symbolic mistakes without focusing on how the student interprets the values and fundamentals

[tight star]

Hmm, what counts as conceptualising multiplication?

[tawny slate]

i just realized that the first and third bullet points are largely the same thing

so like, do students understand why 2x3x4 can be evaluated in any order

if they are just hard memorizing computation, they can get to the right answer, but they should have an intuition as to why multiplication is associative and communtative and what it means

thats my view anyways

[tight star]

For me i understand multiplication by Cartesian product

So grids and areas

I guess… it’s difficult for me to see what exactly there is to conceptualise?

[tawdry venture]

i mean yeah the area thing

and maybe also volume / 3-dimensional dot arrays for threefold products

[tight star]

Mhm

Associativity boils down to volume being preserved under rotation

As does commutativity

And maybe this is my visual bias showing but… that seems intuitive? Since physical objects don’t tend to change their area or volume if you rotate them

[tawdry venture]

tbh like

i think the number line is really the way to go w/ fractions for addition/subtraction especially

[rough agate]

what do you all find is the best way to evaluate exact trigonometric values without a calculator ?

the ones that I'm aware of are

• just memorise the table
• have a unit circle in your mind with the coordinates
• draw the triangle mentally for the value
• trace along the graph

I myself have used all except the first method in the past but I'm not sure which one would be the most effective to teach

[hollow musk]

I personally like methods 3 and 4

1 and 2 are too hard for me to remember/think about on the fly

[shadow flower]

2 is not bad if you know the pairs of values that appear, which coordinates cos and sine represent, and then use logic based on the size of the x and y distance from the point to the origin

[vagrant meadow]

Eulers formula can sometimes help lol