math-pedagogy

[long pelican]

is that the GPA of the person for the class or the person over all their classes?

What does it mean for a calc course to aim for a 2.4-2.6 GPA average?

[strange bronze]

for the class

so a 3.6 gpa average means most students got around an A-, a little under

[long pelican]

That's pretty good

[pastel sundial]

slow paced compared to actual analysis
Meanwhile my analysis class, intended to be taken 4th-6th semester also didn't get to derivatives until the very end

Also holy shit that's low

That's like C+ to B- average

[tepid smelt]

I mean the sad truth is most kids even struggle when spoonfed the material. Icy I agree AOPS is a great resource for k-12 for motivated kids. I teach my own children with that curriculum and I am just as impressed with the elementary curriculum they have. I think mathematical literacy should be emphasized more the issue is as a teacher I have so little time just to cover what's required that I just don't have as much time necessary to develop it for my students

[long pelican]

Struggle even when spoonfed the material makes sense in the absence of math literacy. You have no recourse but to memorize the spoonfed material at that point, and memorization is hard and not fun.
In fact, I believe if you start the year and spend a month on math literacy, the rest of the curriculum can be learned exponentially faster. You'll be able to teach twice the material in half the time. I've personally witnessed many people struggle with math, learn math literacy, and find it super easy after that.

[winged urchin]

I would love to see a not insignificant amount of a math class to not be solving anything, really avoiding the application of math but just grading how well they use the language. Just like in English or other language classes

When I tutor I call it mathematicalization lol

Where you read some statement written in normal language and just want to write the same thing but in symbols and equations

We could go real kooky and be like... If we had the statement "Whales are a part of the animal kingdom" and 'mathematicalizing' that into like... "Let W represent Whales and A represent the Animal Kingdom, then we want to say W is a subset of A"

Just a while year where all they do in class is try to write and understand statements that can be said using mathematical notation

Introduce logic too!

I think even elementary kids could learn it honestly

[tepid smelt]

I don't know about that but I do try to sprinkle it in but the students really need to start seeing it at the elementary level. By the time they get to HS they are just so far behind its a constant game of catch up. The elementary level is where there needs to be a lot more focus in general. I think the whole k-12 curriculum needs to be redone I don't know why there is so much resistance to that though.

[long pelican]

I'm finding it more and more it's not worth it to talk about anything until they can think mathematically

[pastel sundial]

how would you say students should be taught to think mathematically

[long pelican]

My hot take on that is that teachers are already trying to do that, and know how to do that (in a general sense). However, the math they teach, the way they're taught to teach it, is so confusing and unlearnable (in its presentation) that it isn't conducive to students thinking mathematically about it

[pastel sundial]

so the problem is the curriculum then

[long pelican]

Yeah I suppose, the alternative is to expect the average teacher to have and teach a more connected understanding of math than the most popular textbooks, which is... a stretch to ask for

[pastel sundial]

it's an interesting thought

[long pelican]

https://math.berkeley.edu/~wu/Stony_Brook_2014.pdf here are some slides by Wu who has been writing on this topic for a long time

[pastel sundial]

ooh this looks very interesting

[pastel horizon]

Just read something quite interesting similar to this actually. They suggested something called "number sense", a conceptual understanding of how add, times, divide and subtract work rather than just a procedural understanding. Say for example, working out 21 - 6 as 20 - 5 instead of counting back 6 times

[winged urchin]

That's something that's been around for a bit I believe. It's kind of at the core of the 'new math' idea. Flexible numeracy

Of course it is useful to be able to see calculations in different ways but how it's done sometimes leaves something to be desired in school, I think

Not that I've spent a whole lot of time looking into that, just I've seen some 'number sense' kind of work for elementary students

[long pelican]

I watched a teacher try to diagnose why a student wasn't understanding where to put 1 and a fifth on the number line in front of him, and I could immediately tell the teacher wasn't asking the right questions. "Which number goes here? Which number goes here? 2 is bigger than what?"
Asking good questions is like debugging a program. Teachers should be taught debugging... for example, binary search to find the error is one of the first things you learn from debugging

I also noticed that the answer was probably that the student didn't remember an important definition, or never learned it, but the teacher kept asking him questions assuming he knew the definition of things involved

[pastel horizon]

I guess the idea is if you can spot patterns in numbers you can understand how they're generalised with algebra. And algebra helps with more topics

[winged urchin]

Yeah it's certainly useful generally. It's unfortunate that the BEDMAS stuff kinda is counter to the flexibility we want when we move to algebra

A student will see 2*(3+4) and always do the 3+4 first because BEDMAS and may not necessarily think of expanding. Albeit in this case it's probably more efficient to do it with BEDMAS

But then when they see like 2*(3+x) then sometimes they get confused because they can't really 'do' the brackets in BEDMAS way

To what you said Icy, I definitely like the diagnostic feeling of helping a student just asking question after question ahah

Sometimes I will say though, that it isn't them necessarily not knowing a concept it's just them being silly in the moment and forgetting what they're doing or not staying conscious through the problem solving

So sometimes asking really simple obvious questions does help re-align them aha

[long pelican]

Yea, like 50% of people stuck on an algebra problem don't even know what it means to solve an equation

And every tutor just goes "Do you know the quadratic formula? Do you know how to complete the square?"

Or even worse

"Just use the quadratic formula"

[winged urchin]

I don't like the quadratic formula, for sure

Any time my students would use that, or the multiplication table or long division, I try to discourage it

Those will obviously work if you do it but they're really not much better than using a calculator

And, usually, the problems are designed so you don't need to use such mechanical processes

Although asking whether they can complete the square, that's a fairer question imo, just personally

[long pelican]

My point isn't about quadratic formula or completing the square!!

[winged urchin]

It's good to know whether they've heard of it or remember it, might trigger them into remembering

[long pelican]

It's the fact that the tutor isn't doing binary search

He's going down the ladder instead of guessing that the student has extremely deep misconceptions off the bat

[winged urchin]

I mean, I don't know how much I agree with guessing the student has extremely deep misconceptions off the bat

But I will also add

As much as I can talk about education and that sort here, in practice I am a very... in the moment, or 'organic' tutor

I don't necessarily have a plan in my head to help them, or something like that

I just act on my best instincts, and more often than not my students find it helpful and I see them able to do problems they weren't before

And there's always been a disconnect between my like... stated ideology and what I actually do in practice, but that's just me

[long pelican]

You can see it in #prealg-and-algebra, just ask "Do you know what it means to solve an equation?" and 50% of the people asking for help will have no idea what you're talking about

For those 50%, helping them by reminding them what to do and the correct steps, will not stick

[pastel horizon]

Oh the tutors asking in this server? Yeah sadly I guess it's an implicit assumption that they should know it

If it was their teacher then it's a bit more complicated maybe they already know the level of their student so they can skip some diagnostic questions

Like with lower ability, you might expect them to have trouble with negative numbers or decimals. You'd just assume someone in a higher set would know that fluently

[winged urchin]

Could you elaborate on what you mean by using a binary search to help a student, @long pelican ?

Like if someone was like... "Here's my problem, x^2 - 5x + 4 = 0, idk what to do I'm lost help plz!"

[long pelican]

It would mean to locate what level is the "last" level the student has high command of, or equivalently which level is the first level the student missed

What happens with linear questioning is that after several failed questions, the student starts to feel bad and try to say what he thinks is the right answer, making the tutor/teacher believe he gets that last question when he in fact is nowhere near that

[long pelican]

If they're completely lost and don't even know where to begin, there's in my experience about 50% chance they have seriously missing fundamentals such as what is a variable, what is an equation, what does equals sign mean, what is logic

[native hemlock]

I think the concept of a variable is genuinely confusing. Things like changes of variable in a summation or limit have never made any sense to me.

But I am also dumb.

Also you never really learn a formal definition of variables in a discrete math/proof writing class, so it’s hard to know precisely how it behaves

[long pelican]

Ye, variables and functions are like the two things everything else rests on, and the average teaching of variables and functions is so bad

Maybe fixing the way those 2 are taught will have the greatest impact

The concept of variable isn't a mathematical concept either btw; in sentences like "x+3=0 implies x=-3" the x is like a name given to a number that has a quantifier. More explicitly the sentence is "for all real numbers x, x+3 = 0 implies x = -3"

[native hemlock]

I don’t struggle as much with functions because they have a precise definition: a relation F from a set A to a set B is a function if for all a in A, there’s exactly one b in B s.t. (a, b) is in F. But I haven’t come across anything like that for variables.

[long pelican]

Heh, teachers often don't teach the precise definition and don't even realize it's needed

[native hemlock]

I never understood them until I learned the precise definition

[long pelican]

Bingo! Another statistic in favor of precise definitions for everyone (not just math majors)

[native hemlock]

Lol

https://cdn.discordapp.com/attachments/295394724443848704/898703316278706216/on-developing-a-rich-conception-of-variable.pdf

Not sure if the link works, but I thought this was a cool article

[long pelican]

Yeah I can see it

I definitely agree with the first paragraph so far

[native hemlock]

Although I do think that precise definitions can be a bit of a double edged sword. I’m currently taking calc 1, and maybe 2 weeks into the course, the professor spent a couple lectures going through the formal definition of the limit and everyone was clearly completely lost. I think it’s a bit of a balancing act trying to be precise, but also accessible

[long pelican]

The thing with the definition of a limit is that it takes 150 times to read to get it for first-timers

Having an imprecise definition means you can't prove anything (or understand the idea that they have proofs) and have to take a lot of things on faith

[native hemlock]

I also know literally nothing about pedagogy, so nothing I say has any empirical basis

[long pelican]

I told calc II students today that having to read everything in math 15 times is normal

And the textbook says the same thing more or less

The definition of limit taking 150 times also sounds normal

since it's the first instance of doubly (or even triply) nested quantifiers

[native hemlock]

Yeah

It took me well over a week to get the hang of it, and I was lucky enough to have read a book on discrete math/proofs before the class started

[long pelican]

If I think back to all the nonsensical student responses on exams (e.g. including the screenshots I shared here) I could probably say that the confusions are all traceable back to not being taught functions and variables correctly

I can't really think of any other aspect of what they write that jumps out to that extent

[long pelican]

Ok after reading some more of that article, I have a hot take which is that we should eliminate the idea of variable altogether (!!!)

"solve x+3=5" --> "Find all real numbers x such that x+3=5" --> no need for the concept of variable here

"There exists a real number a such everyone has at least a pieces of candy" --> no need for the concept of variable here

the x in f(x) is a name given to the input to the function --> no need for concept of variable here

Also completely eliminate the "equivalence" between y and f(x) -- y is only related to f when passing to the graph of f

If x and y are constrained by an equation like x^2+y^2 = 1, this is simply a subset of R^2

If one quantity depends on another quantity, that idea is captured by the concept of a function

[tepid smelt]

I did a lot of this with my daughter as a game starting in 1st grade. I don't know if it helped her conceptual understanding but her mental math improved tremendously. She can easily hold longer calculations in her head compared to her teachers even in the 4th grade. I think many young kids don't realize how flexible arithmetic can be. They see it as a rigid procedure

[pastel horizon]

I mean, do they even understand the significance of that question? That a quadratic = 0 implies a curve intersecting the x axis twice, once or not at all

[pastel horizon]

Yeah and that has a knock on effect when they see the rest of mathematics as rigid procedures. It has some benefits for sure

Like even when you talk about proofs there's not necessarily one correct way to prove something

(there are certainly wrong ways though!)

[pastel horizon]

The most common misconception would you say?

[long pelican]

Where's the misconception, the question or the explanation?

My take for the most common misconception is any typical student's idea of variable or function. The whole of their understanding is one giant misconception

[pastel horizon]

The question I meant, not seeing that actually they are the same number

I feel like on most q and a forums for maths the most common question is related to 0.9999... = 1?

[long pelican]

It's actually a really subtle question and the mathematical answer to it is that one defines the value of 0.9999..., which is an infinite series (!), to be its limit, or more simply, the least upper bound of the increasing sequence 0.9, 0.99, 0.999, ... As you can see, defining the infinite decimal as the least upper bound is the only true way you have of answering the question of the type "But 0.999... is a process that never ends!"

[pastel horizon]

Yeah the series is probably my go to answer as well

[quasi musk]

For a lot of people that's a little too high powered

For lower level I just do the basic

1 = 3/3 = 3(1/3) = 3(0.3333...) = 0.9999....

It's not entirely "correct" but it's a good reason why one should suspect it to be true

[flat cargo]

I tried to explain to some of my non mathy friends and they rejected the 3(1/3) explanation, I then told them 0.999... /= 1 breaks the archimedean property and you get all sorts of contradictions with any other definition, their response was "I'm ok with contradictions"

I think I was getting trolled

[quasi musk]

How did they formulate their rejection

[pastel horizon]

They just wouldn't be convinced that they are the same thing

[quasi musk]

Ask them what amount of evidence would convince them that they are the same thing

Or are they assuming at the outset that they aren't the same thing

And this assumption cannot be changed

[long pelican]

Zeno's paradox is more or less the same thing

[pastel horizon]

Well for example here it seems like they're already assuming the two aren't equal

[long pelican]

They definitely don't look the same... assume unequal until proven otherwise?

[quasi musk]

I guess the burden of proof is on you

So you provide a proof

Then the burden of proof is on them to show why that isn't a "proof"

[long pelican]

Can't be proven or disproven until the definitions are clear

[long pelican]

Thoughts on this hot take?

(taken from https://faculty.utrgv.edu/eleftherios.gkioulekas/OGS/Santos/santos-precalculus.pdf)

[pastel sundial]

Yeah seems reasonable

[winged urchin]

Thanks for explaining your position before Icy, got busy but it helps to see where you're coming from

Now on this...

Hmm

[pastel sundial]

||Though I am ever so slightly bothered by the use of "he" to refer to everyone||

[long pelican]

The book was written in 2008 so he can be given some slack

[pastel sundial]

I don't even think it's unreasonable here. I'm just a bit scarred from reading a board game rulebook yesterday that used "he" exclusively over many pages.

Regarding the actual position though idk

[winged urchin]

I think I would be happy if the student could make a decent attempt at exploring the possibility of proving sqrt(2) is irrational

Like... if they just said like "I'll try proof by contradiction and assume sqrt(2) = m/n where m,n are integers"

And then just couldnt get anywhere or whatever

I'd be happy with that

The other points, though you didnt underline them, are all really just memorization

[pastel sundial]

Like a precalc student should definitely know sqrt(2) is irrational, and should have the mathematical maturity to understand a proof, but I'm not sure if they need to be able to recall/rederive the proof off the top of their head.

[winged urchin]

Whereas the sqrt(2) thing could be memorization sure, but has more play to it

[pastel sundial]

An instructor should definitely know the proof though

[long pelican]

Actually the other points are also proofs too, for example, the equation of a line being of the form y = mx + k comes from looking at similar triangles

not immediately obvious

[pastel sundial]

How is equation of a line memorization?

[long pelican]

Same with slopes of perpendicular lines (similar triangles)

[winged urchin]

I mean the equation of a line can definitely just be memorized

You can certain derive it, and that's better

[pastel sundial]

Yeah but it should be clear why it is what it is

[long pelican]

Equation of a line is one of the thing Wu slams K-12 textbooks heavily on, namely that they never prove that the graph of a linear function is geometrically a straight line nor even give a suggestion that it's a proposition that needs to be proved

[pastel sundial]

Icy do you think K-12 could cover more material if it did a better job of it

Like elementary algebra is not a hard subject once you get the concepts down

[winged urchin]

If it didnt focus so much on the basic operations, for sure

[long pelican]

Yeah, quite easily

[quasi musk]

This is true, but most beginning students don't think in terms of definitions

[long pelican]

There's a point in any math student's journey past which their math learning really accelerates

[quasi musk]

They just think in terms of the object

[long pelican]

it's something to do with understanding that things have definitions, that math expressions have meaning, that you can do logical thinking with math expressions

K-12 ought to have students reach that by 6th grade or something

I think 6th grade was my own point

[pastel sundial]

How much math should the average person learn before they graduate highschool

[winged urchin]

Instead they have to do three and four and more digit multiplication, addition, division, BLEH

[pastel sundial]

Like would it be unreasonable for students to have leaned what we now call precalc by the end of 9th grade

The important parts at least

[long pelican]

I don't think so at all

[pastel sundial]

Then like have electives in hs

[long pelican]

That would be so nice

Graph theory, combinatorics, number theory, TOPOLOGY even

[pastel sundial]

12th grade homological algebra when

[long pelican]

When Namington becomes president in 2024

[quasi musk]

I think math is a waste of time for most high school students

and I think the bar to graduate has been lowered significantly

Like, we don't teach classrooms of 14 year olds algebra because it's likely to benefit most of them

Because by the numbers it doesn't benefit most of them

[pastel sundial]

When I said intensives I moreso meant like applied mathematical thinking

[quasi musk]

It benefits a select few of them

[long pelican]

I could agree it's a waste of time in the sense that they're trying to get high school students to think critically and logically using math and never succeeding because of lack of logical reasoning instilled by teachers

[quasi musk]

I don't know if it's due to a lack of logical reasoning instilled by teachers

[pastel sundial]

Where students could choose to take computer science, statistics, calc (with a focus on physics/engineering, or pure math)

[long pelican]

I've seen the reality of how teachers teach sadly

[quasi musk]

I am the reality of how teachers teach, I just think that regardless of how well you teach, prepare, etc.

Students will be somewhat disengaged

[winged urchin]

Bring back Euclid's Elements and more geometry, ahah

[long pelican]

A good reason to be disengaged is being lost

And you're lost because you have a horrible foundation of the notion of variables and functions

and it's never being fixed

[pastel sundial]

Like I feel like mathematical reasoning is what's important to teach these days more so than math itself, especially with the growing importance of computer science.

[quasi musk]

Even if you give them 2 hours of dedicated support each week for their class where you go in and personally help them

They will still not try very much

[long pelican]

Well under 50% of my current students fall in the category of not trying I'd say

[quasi musk]

I'm at like 90%

[long pelican]

Oof

[pastel sundial]

It's honestly tragic thinking about what the average person's experience with math is

[long pelican]

I could see that more or less; my students were more or less all A students in high school

[quasi musk]

I don't think it's tragic, I think it's fine. Most people for a very long time had zero education

So this is a step up

And we'll progress, slowly but surely

Yeah, whereas I'm teaching in the ghetto with a very disenfranchised group of students that don't traditionally value education that highly

[pastel sundial]

I honestly wonder if the reason I like math so much was never having to go through the standard K-12 education system

[quasi musk]

Even the senior year math courses aren't that great

[long pelican]

omg me too

[winged urchin]

I went through the normal system but then by grade 12 I was exploring math on my own

[pastel sundial]

Homeschooled K-8, weird private schools 9-12

[long pelican]

I learned algebra from reading Zaccaro's "Challenge math" in elementary school given to me by a really great enrichment teacher

[winged urchin]

In grade 12 I was trying to figure out what the calculator did when I pressed the square root button lol

[long pelican]

That way I didn't have to learn algebra via the normal classes in middle/high school

[quasi musk]

I failed almost every math class past grade 5, except for grade 11 where I made an A

[pastel sundial]

I leaned a lot of math early on from the Murderous Maths series

[winged urchin]

And I ended up re-deriving the Babylonian square root method in a messy way

[pastel sundial]

And then had some tutors

Incidentally the two years where I had the closest to a standard math education were 9th and 10th grade and those were also the worst

I remeber we did basic matrix stuff in 9th grade for solving systems of linear equations, and it was taught just like a computation.

Like we did it with taking the determinant and stuff

And the teacher taught this really weird method of taking the determinnet of 3x3 matrices where you like wrote the first two columns again forming a 3x5 grid and then did some multiplication along the diagonals.

[winged urchin]

That's actually fairly common imo

[long pelican]

Oh yeah that's to see the diagonals instead of having to wrap around

[winged urchin]

There's that, which is special for the 3x3, and then the cofactor expansion

(I always teach cofactor expansion, heck on specialized methods)

[quasi musk]

The worst is the checkerboard

[pastel sundial]

And there was literally no explanation of why any of this was at all a logical thing to be doing or why it somehow gave the correct solution.

[quasi musk]

Plus, minus, plus, minus

[pastel sundial]

And I legit remember asking the teacher why any of this worked and he was like "lol I don't know"

t i l t

My main other memory from that class was doing a matrix problem, being told I got the wrong answer, checking my work 3 times and being unable to find the mistake, and then it finally turning out I had done (-1)(-2)=3

[quasi musk]

I guess a good answer is that determinants are just important in linear algebra, and computing determinants of matrices is a way of understanding different aspects of the matrix

e.g. invertible if and only if det(A) is not equal to zero

[pastel sundial]

No he didn't even explain any of the linear algebra

[quasi musk]

But the cofactor expansion being what it is

[pastel sundial]

A matrix was just a magic box we put the coefficients into

[quasi musk]

Having motivation would be difficult

[long pelican]

The teacher not knowing the motivation is pretty

[quasi musk]

I think it's pretty common

It's very hard to know the motivation for every piece of math you're teaching well

[pastel sundial]

We didn't even do Gaussian elimination

[winged urchin]

Yeah especially at the elementary/highschool levels

Like you'd hope a professor of math would know the motivation

But before that, teachers can't really be specialized unless you're at a private, rich school or something

[quasi musk]

I mean, for things they're experts in or teach all the time

Fun fact: private schools don't have to hire certified teachers

So many teachers in private school are less qualified (at least by state standards) to teach

[long pelican]

I think disinterest starts early and builds on every year their fundamental confusions about the language don't get resolved

A 5th grader who doesn't get fractions for the whole year needs a lot of willpower to stay interested

[quasi musk]

I think even in an ideal world if you were able to get rid of fundamental confusions

Students would be disengaged

I think even in an ideal world if you were able to get rid of fundamental confusions

[long pelican]

How many though

[winged urchin]

How many get disengaged in other subjects?

[long pelican]

If I think about English, it wasn't the most interesting class for me, in fact, the books were very hard to read, but I mean I graduated high school being able to understand Catcher in the Rye

Surely the equivalent of knowing how to prove sqrt(2) is irrational?

[winged urchin]

I like that claim ahaha

Although those are some very different apples and oranges but English was also my thought

I could clearly understand it, but idk if I was engaged?

[pastel sundial]

How are fractions generally taught anyways

[long pelican]

I can answer

They're taught purely by analogy

No definitions whatsoever

Pizza slices, parts of a whole, number line

[winged urchin]

Like, ratios

Yeah

Real life examples

[long pelican]

Seen some videos briefly, students are expected to get the underlying connections implicitly

And when a student was having trouble

[pastel sundial]

That's like awful

[winged urchin]

Which honestly, I think is a decent starting point? You always want some physical, real world examples to ground their understanding

[long pelican]

the teacher would help the by asking procedural and memory questions

like "Do you remember what always goes here?"

[winged urchin]

What would starting with the definition look like?

[long pelican]

Wu gives a good candidate

a/b is the size of the "vector" in R^1 (obviously not that language) when you take the vector a and split it into b equal parts

Almost made a false claim there lolol

on the number line

So Wu uses the number line as a basis for everything else

[winged urchin]

Idk, that doesn't exactly make me super happy to tell a student at that age

[long pelican]

It's a bit abstract for sure but at least you can do logic with it

[winged urchin]

But with a lot of this, I would love to see a proper attempt made, of course it's quite hard to do

[pastel sundial]

Honestly thinking about this right now, fractions seem like a pretty strange concept

[winged urchin]

Shame we can't be like, "Here's all our different ideas of how we could teach math, now lets test it on thousands of different classrooms and give a whole bunch of students entirely different teaching methods and see what's the best! Yeah1'

[pastel sundial]

Like we're all familiar with them, but to someone that isn't

[winged urchin]

WHat if you taught it like history?

The history of math

[long pelican]

Equivalence classes of pairs of integers a/b with a/b ~ c/d iff ad = bc xd

[winged urchin]

Whole numbers for trading

Integers to record debts or stuff like that

I wonder if a bit of a history lesson could give grounding to it

[pastel sundial]

The continuum is strange tbh

[quasi musk]

The more you think about them

The weirder they get

[long pelican]

I wonder if historically fractions were treated as the scalar field for 1 dimensional vector spaces

[pastel sundial]

Like the number line

[long pelican]

e.g. 3/4 of this land

vector space is the possible sizes of land

and 3/4 is the multiplier

[pastel sundial]

Weirdest part of R is that it's countable

[winged urchin]

?_?

[pastel sundial]

Despite all the proofs that it isn't

[long pelican]

I can name a weirder property of R; it's well-orderable

but good luck naming a single well ordering

[winged urchin]

What do you mean Namington? =p

[pastel sundial]

Yeah it's easy to well order countable sets

[long pelican]

Wait a minute

[pastel sundial]

(I'm meming)

[long pelican]

R is countable...

[winged urchin]

(okay lol)

[long pelican]

Burn the finitist

[pastel sundial]

Jokes aside though R is freaky

[long pelican]

We could just work in the smallest field containing sqrt(n), pi, and e

[pastel sundial]

Like how strange R seems is proportional to how much you've thought about it

[long pelican]

namely Q[sqrt(n), pi, e]

but what about sqrt(pi)...

dammit

[pastel sundial]

The algebraic closure of Q[pi,e]

[winged urchin]

And tau

Cant forget tau

[long pelican]

WAU

Ok so if I were to list the gatekeepers of high school math in order:

1. functions and reading/writing stuff involving functions properly
2. variables and reading/writing stuff involving variables properly
3. fractions
4. Lack of exposure to sets? Not even set theory, just expressing things in terms of sets

[pastel sundial]

Honestly what would you lose just working in the algebraic closure of Q[pi,e]

Besides every theorem in analysis

[long pelican]

You lose every rational ellipse's circumference

I have a feeling each of those (up to some equivalence class) are algebraically independent of each other

[winged urchin]

Kinda with sets, but just logical operators and statements

You've touched on those, but if kids could work with logic like they can with equations

[long pelican]

Can't win

[winged urchin]

That'd be a big step up

[long pelican]

Oh yeah definitely, but to work with logic you need things to do logic on

Don't think school math provides such things ironically

because they butcher fractions, variables, sets, and functions

That's like everything you do logic on in algebra

I once had an exam question saying "Let f be the function which assigns to each real number x [a probability of something in terms of x]" and part a was to prove a very trivial property of f, and part b was to connect f to another function (also trivial)

60% of students couldn't even read it because f wasn't given by an equation

So that's evidence they don't even have a baseline of what a function is beyond "do procedures to equations", much less doing logic with them

40% of people did manage to prove the trivial property in part (a), which is now impressive to me

Exercises typically done with functions:

1. Is this a function? Is this not a function? Do the vertical line test
2. Find the domain of this function. (Solve for denominator =/= 0 or thing in square root >= 0)
3. Find the inverse of this function. (Swap x and y and solve for y)

oh and 4. Find f(something). (Plug in and simplify)

Everything is like, do something to the thing, rather than think about the definition and make logical steps

Fun story, I remember in 11th grade in math team there was a problem in one of the rounds that was something like

And I was the only one in my school to understand let alone solve it

But like, this is something everyone learning functions should be able to do

[winged urchin]

Sometimes I wish problems like that were written as "Find as many functions as you can that satisfy..."

Maybe then some students would attempt to find one or two at least and then maybe they'd understand it enough to get there?

I think sometimes they just dont think they know how to find 'all' functions and throw their hands up

[long pelican]

Using the definition of a function it should be logically evident

Assign 1 to something, assign -1 to something

1 can go to 0, 1, 2, and -1 can go to 0, 1, 2

That's a prime example of why a precise definition goes a long way

[winged urchin]

You know I was thinking, and there's two thoughts that come into my mind as possible uh... concerns when arguing for precise definitions

1. First and easiest to see I think, is the accuracy to which students remember them in addition to their own confidence with the recollection even if they are accurate. Obviously it's easier for us to remember definitions and be confident in that memory

2. Kind of queueing off of your explanation before about possibly causing uh... feelings of giving up in asking linear questions.. But in this case I think I recall sometimes when I'll try to emphasize the precise definition and sometimes students seem like they get put off by the correction

Kind of like when you correct someone's grammar, just feels annoying and useless

If you're like, "What do is the definition of X?" and they don't answer exactly right so you have to correct them if you want precise definitions

Or also, sometimes they answer in their own words and so are usually inexact and you have to correct them with the precise definition again

And then eventually it might just seem like rote memorization again which is somewhat vilified

[long pelican]

I guess the teacher has to demonstrate many times in class the use of a precise definition and the dangers of an imprecise definition

precise definition serves logic

Teacher has to demonstrate explicitly that they’re using the precise definitions to build everything else using logic

So say logarithm rules

The reason for them is based on both the precise definition of logarithm and the corresponding exponent rules, and should be front and center in the lesson rather than as a side remark

But yeah, the idea can definitely be a double-edged sword with teachers who don't fully understand the point

I can definitely see a botched implementation having teachers using precise definitions just to be pedantic and little else

Sort of like New Math in the 1960s

[astral laurel]

On 4 I think the curriculum does not encourage it enough. The rest are possibly failings of curricula since I do think they are intended to be taught

[lethal leaf]

Yea I think any high schooler can understand the concepts of basic set theory and injective/surjective functions

I think any high schooler can understand alot of more basic math concepts that aren't taught because the curriculum is just designed as a calculus pipeline

[wispy slate]

it's like they think calculus is the center of everything

[pastel horizon]

It is for something like engineering tbh. But maths isn't just about training engineers

For chem eng at least anyway, the core principles where that mass and energy were conserved in a closed system; and that we analysed systems at a differential scale (so looking at for example how temperature changes across a differential length dx for example). That's the main reason why in that situation calculus dominates because it's simple to study how systems change on a differential scale.

The other half is underpinned by numerical analysis which sadly is only touched over at secondary school

[pastel sundial]

Is calc even that useful?

[pastel horizon]

For engineering yes it certainly is

[pastel sundial]

Like for engineers and shit, sure. But for math students is there really much use for calc?

Like why not do discrete math/intro to proofs instead and then teach calc with analysis

[pastel horizon]

It's like 1% of the entirety of maths. Maybe even less

But you can make the same argument for another topic

Once you leave secondary and move into higher education it becomes obvious that calc isn't even the surface of the whole field of mathematics

But yeah it's probably seen as a big deal because it underpins a lot of STEM material and this is something valued more by society sadly

In the UK curriculum it's not really seen as a big deal, it's just a part of our A level course that 16-18 could choose to study.

[winged urchin]

I mean, Calculus might get a little too much attention but it's not entirely unwarranted

Calculus is pretty spectacular

The ability to think of change in an instant or to sum up an infinite number of lower dimensional quantities to actually get something and that something has meaning? Pretty rad

I don't know why they focus on addition, subtraction, multiplication, division for so many years though

They simply must know how to do multi digit multiplication! Because there will surely be a lot of problems in highschool/university where you multiply 534 by 253 without a calculator

[normal oriole]

Not really sure what a kind of "use" math students are looking for tbh

[strange bronze]

yeah thats kind of nonsense

being able to compute derivatives and integrals are just a basic operation for higher mathematics

in the same way that \sums and exponentiation are

that isnt to say you need to know all the slick calc 2 integral tricks

you dont

but you need to know the basics so you dont get "garbage in garbage out"

i'll agree that it doesn't need to be a separate course from analysis, though

[long pelican]

I'll add be able to compute derivatives, integrals, and know the definitions of these things at a basic level of being able to do logic with them

[strange bronze]

but if you bundle calc and analysis together, prepare to make your course a lot slower paced

unless youre teaching at a top 20

even keen mathematics students can get overwhelmed if you go too fast paced with that content

[long pelican]

I teach at a top 20 and I get students who think f(x) and f(y) are two different functions

[strange bronze]

hopefully not math majors?

[long pelican]

Nah

You want to know a funny one I saw recently

Q: Does the differential equation y' = y^2 have 1/(C-x), C in R as its set of solutions?
A: No, needs C =/= x

Me: ...

[strange bronze]

that looks like a student who didnt know what it means for a function to solve a diff eq, and just made up something that sounds right

[long pelican]

At least 30% of students are probably like that

on the last exam on a question that literally just asks to check a function satisfies a differential equation and an initial value condition y(pi) = 1, they plugged in pi to check the differential equation too

[strange bronze]

i wonder if explicitly writing out y'(x) = (y(x))² would help with this

[long pelican]

You know what

That was actually what the question stated

[strange bronze]

...oh.

well.

[strange bronze]

i guess that answers my question

[strange bronze]

oh god

well, that would at least make it a lot easier to solve most diff eqs

make everything a constant function

[long pelican]

My official diagnosis for like 80 of the 87 people in this class is doesn't understand functions and/or variables

They got A's in their classes by memorizing examples

They study for this class with the same method

Here's the weird part

In office hours today when I explained, using chapter 1 of the textbook, what a function is, what a graph of a function is, the student understood it and said wow it makes so much sense now

So I'm like.... ok so these students are obviously smart so the reason they don't understand functions or variables must be their teachers'/curriculua's fault

(again this is a top 20 school)

[earnest trail]

damn

that's the issue with school

if teachers taught at a comfortable, in-depth pace that allowed students to gain the proper understanding, everything would be fine

but the curriculum doesn't allow teachers to have that freedom

and also it would take too long

[long pelican]

Is anyone familiar with the "equivalent vs. equal" movement in schools

Like 2/4 and 1/2 are now not taught as equal, only "equivalent"

This seems very disastrous

It's taking students further away from the idea that rational numbers are numbers

[vagrant meadow]

Update:
Well the exams were graded and she got a 23% 🤦‍♂️
Technically an improvement from a 15% on the first exam -_- 🤦‍♂️ 🤦‍♂️

[long pelican]

23%

Did you hear from her after she got it back?

[vagrant meadow]

Yep. She wants more sessions before the next exam

[long pelican]

Is she admitting she has no clue about anything yet?

[vagrant meadow]

Nah :/

[long pelican]

That is one hell of cognitive dissonance

[vagrant meadow]

I'm tempted to ask her if it's time for a change of approach, and if she'd like to try things my way

Seeing as i got 100% on the same exam (not a huge achievement since I've taken the class before but still)

[long pelican]

Oh can I ask what the exam questions were like?

[vagrant meadow]

Not bad. Pretty straightforward and standard diff eq questions. Pretty good honestly. When i get back i can probably post the questions in a thread or just DM you (we had to turn it in digitally one page at a time so i have pictures of my work)

Even though it was an in person exam

[long pelican]

Ah so not even theoretical questions

Do you remember if there were any non-computational questions where you had to do logical reasoning on there?

[native hemlock]

smh rational numbers aren’t numbers

[long pelican]

Imagine this being the explanation: some math teacher took a university math class, learned about equivalence relations and took the wrong idea from them and decided that $\bQ$ should be what you get just before quotienting by the equivalence relation, and decided to share this false epiphany with elementary students

[burnt vessel]

Icy001

[long pelican]

which then became popular and spread to other math educators for who knows what reason

anyway that's my take on "equivalent vs. equal" for fractions

[real mauve]

you reminded me of something i saw way back when i was taking diff eq. there was something like poly(x) = c, for some poly and a constant c, and one had to find the roots. the lecturer had asked a student to solve this on the board. the poly was a quadratic, so the person factored it and set each linear factor equal to c

mind you, diff eq in engineering is a second year course requiring a bunch of other stuff (at least where i did it)

[long pelican]

Wow

Was the professor taken aback?

Surely she did not go "Yep I expected this"

[real mauve]

yes, she couldnt believe her eyes

struggled to ask what they were doing lol

surely if it works for 0 it works for other c 😌

[long pelican]

Classic case of the failures of memorization-based learning: master a technique, get 100%, fast forward some time, get a little rusty in it, and completely butcher it

I am glad I will not completely butcher something I am rusty in

[real mauve]

🍗 🔪 🥩

[topaz scarab]

My most generous explanation is brain farts
especially if some students are shy, and when they walk to the front, the brain just stopped working and the only thing that kicks in in muscle memory.

[topaz scarab]

I blame the exams. Can't really blame the players if the game is broken.
If you make exams that can be done by "memorizing examples", then that is all your students going to learn.

[grand wharf]

how can discrete mathematics be better integrated into the curriculum?

[tawny slate]

I agree that the equivalent vs equals thing is dangerous

To me, equivalent is a generalization of equals

It's okay to say 2/4 is equivalent to 1/2, but it is not okay to say that 2/4 is not equal to 1/2 or to discourage that labeling

Because that is literally how we use the equals relation

[pastel horizon]

Well they're normally taught as "equivalent" fractions in secondary school but I get that's just an example

Tbf as well equivalent fractions is just a name I don't think there's much discussion about what equivalent means other than we are multiplying by 1

[strange bronze]

the terminology i heard growing up was "equivalent fractions" yeah, but i never heard them say that "equal" is incorrect

if theyre actually saying that, then uh

thats clearly a failure of education somewhere

[long pelican]

https://www.quora.com/Are-1-2-and-2-4-the-same-number

Just read some of these answers

[strange bronze]

(probably in educating the teachers)

[long pelican]

Apparently this misconception spread to Chinese education???

Very surprising

[pastel horizon]

The white rose maths way is normally pictorial representation so say 3 out of 6 boxes are shaded, write that as a fraction. Would 1/2 be a valid fraction?

[long pelican]

"Is 1/2 a valid fraction?" Isn't even a mathematical question jeez

[pastel horizon]

Yes, you can say for every 1 shaded box, 2 are unshaded. You could deduce 2 for every 4

[long pelican]

what is 1/2 or 3/6 the answer to: it's how many full boxes are shaded

[pastel horizon]

Yeah I kind of paraphrased it

[long pelican]

Treating the 6 boxes as 1 full box

[pastel horizon]

The idea is the students should realise they're not just counting boxes

[long pelican]

A simple definition exercise could clear it up: a box is one of the 6 boxes, a full box is the collection of 6 boxes

[pastel horizon]

Then there's more questions where the boxes aren't the same size

[long pelican]

Like, a soda box for example

6 sodas in each box

3 sodas is how many soda boxes?

oh boy

[pastel horizon]

Make them work it out first then ask them to come to the front

You shouldn't ask them to solve it in front of everyone, more like a demonstration of their thoughts after they've had a go. My opinion anyway

[long pelican]

https://www.nytimes.com/1993/09/03/us/study-finds-most-students-lack-reasoning-skills-in-mathematics.html
Improving exams means results like this... can the education establishment handle it?

[pastel horizon]

If it's not examined then how can you expect strong reasoning skills

It's a self fulfilling prophecy

[long pelican]

There's gonna be at least a time delay

Improve examinations, force teachers to improve their reasoning (1+ year), and finally force teachers to teach the improved reasoning (1+ year)

Can the establishment handle it?

[pastel horizon]

That's pretty much how it went down in the UK. Hard to say actually since there was COVID disruption

The last two years were just teacher assessed grades

I think it's gonna trickle through shortly soon though, I think I was 2 years before the infamous Hannah's sweets question and coming into the profession just now

[tawny slate]

I mean ok sure
Saying 1/2 is not the same number as 2/4 is like saying that 1+1 and 2 are not the same number or that sqrt(2)^2 and 2 are not the same number or that 0.5+0.5 and 1 are not the same number

Sure, I suppose?

Not quite sure what purpose that serves pedagogically aside from like questions involving domains

[pastel horizon]

I'd say the mean issue pedagogically speaking is more understanding what an equation actually represents vs an expression

[pastel horizon]

Seems like there's issues in all aspects of the curriculum tbh. Could you really put something this basic down to COVID though? The problem with variable and function meaning is a lot more technical imo

[topaz scarab]

That's great and smart, completely agree.

[pastel horizon]

I think as well my other tip would be if you know they're someone who might struggle to articulate themselves in front of a large audience, encourage them to bring their notes, book, whiteboard whatever with them to the front as a prompt

[rugged bobcat]

Man I did that a few times to try and up student engagement

But i did not put as much thought into it as i should have

Ended up accidentally humiliating a student and I felt real shitty about it

[pastel horizon]

How are you structuring it?

[topaz scarab]

Ah, reminds me of my girlfriend, she's an English teacher. Middle and high school.

I think the pendulum swung from one extreme to another. At first it was grammar, spelling, and punctuation only! And then it was: creativity and expression only. I think there should be a balance in between. Maybe grade 7 is actually a good transmission in English, let the kids understand the joy or stories, poems, musics, etc 2. And then learn the proper rules.

I can also imagine math swinging the other way, students studying essay about category theory and equivalence relations, but don't know how to add fractions.

[pastel horizon]

Something like remembering a capital letter is easier than fractions. It would be more like if a student said "I don't know what 2 X 3 is"

It should be a reflex, something you don't need to think about

[topaz scarab]

I only do this to my advanced class, and I only take volunteers.

But I can totally imagine me saying: "Alice, your doing question 2 on the board in 5 minutes, Bob, you're next after Alice" and I wouldn't complain so much if they double check with their friends.

[pastel horizon]

I think you need to pick volunteers for sure. Calling random students is a risk

[rugged bobcat]

This was years ago, but the circumstances were pretty unique. It was a class of youths and teens with varying degrees of autism spectrum disorder, and very large differences in math level. So you had some students learning trig while others were learning arithmetic.

I started having students come up and write answers that I knew they were likely to get correct, so as to give them a confidence boost and get them to socialize a bit more with the class.

But I'm not always able to check answers, and this one student who looked up to me went up to the board to solve an equation, but completely misapplied order of ops. Normally if there's a mistake or something I can at least give a quick correction or offer support for getting close. But with this there wasn't really anywhere to start since it was clear the student wasnt at the level for that problem.

Idk how much it actually hurt them in the long run and I probably think about it more than they do now, but it was not a good call at the time.

It was a very fun class and I had a lot of freedom in terms of what I could talk about (full disclosure: i was an instructional aid who covered the math protion of class, not the overall teacher), but almost every curriculum was individualized

But yeah, ever since then I kinda stopped calling students to the white board to show their work lol

Sorry, dont mean to derail the convo lol

[long pelican]

How difficult do you think the task of deriving an equation of the unit sphere in 3D space is for a 18 year old calc 2 student, on an exam?

[pastel horizon]

Definitely not in spherical coords right? ;)

Could they do a circle on a Cartesian plane? I don't see too much jump in difficulty from there

[long pelican]

Nah, cartesian coordinates

The other professor for this class anecdotally said her students didn't know the equation of a circle when she asked, although a good number of my students not only knew equations of circles but also ellipses and hyperbolas

[pastel horizon]

Then probably gonna be a no then

[long pelican]

But even someone who forgot, could still derive it though?

[pastel horizon]

Hang on

How do you know they know? What kind of assessment did you use

[long pelican]

Just an in class question

[pastel horizon]

Hmmm, idk I'm not quite convinced they might know how

[long pelican]

Let's say they know the definition of a sphere and the distance formula

So the only work is pure logic and understanding what an equation means

[pastel horizon]

It's just another thing putting those two together in an exam without prompt

[long pelican]

In terms of solution method, it's a pretty linear task

No creative insights or logic leaps needed

[pastel horizon]

For sure

[long pelican]

Failure to do it could indicate misunderstandings of what an equation means (most likely candidate) and misunderstandings of definitions

[pastel horizon]

I think you've already decided tbh. Idk I think you're in a better position to make that judgement

[long pelican]

Well I just want other perspectives on what expectations for college freshmen's reasoning ability should be

or what to expect of them

in math

Like, a typical calc 2 final exam on the internet

is like 80% calculate this integral

I don't do that

[pastel horizon]

That's why I think calc should not be taught separately and should be taught alongside other topics

[long pelican]

what other topics do you have in mind?

[pastel horizon]

Well in the UK, what you would call calc 1 we teach along with arithmetic/geo series, polynomials (long division, completing the square, etc), some basic trig identities, exponential and log functions

So probably more like calc 1 and precalc combined?

[long pelican]

Oo so fundamentals along with the calculus

[pastel horizon]

Pretty much

[long pelican]

Yeah if I ever do this again I'm making sure they know how to read math and think precisely up front

[pastel horizon]

Oh yeah binomial expansion that's a Core 1 unit

Core 1 would be like pre calc/calc 1 combined. Non calculator as well

So basic integrals of polynomial functions

[pastel horizon]

That's fair. That's pretty much first year maths on a UK degree course

In defence of your system though, if you study maths here, that's the only thing you are studying at uni. In the US it's a bit different you pick a major and minor. So some people would probably turn up to maths classes and not really care about the technicalities

Whereas here if you have that attitude you're off the course basically

[long pelican]

Technicalities like differentiability and continuity and epsilon deltas are things I don't even touch

[pastel horizon]

Oh nope. They would start on proofs

https://www.manchester.ac.uk/study/undergraduate/courses/2022/00590/bsc-mathematics/course-details/MATH10101#course-unit-details

Hopefully that's music to your ears

[long pelican]

ahhh that's for math majors only

[pastel horizon]

I thought so

Here you either study maths or you don't

[long pelican]

My course is somewhere in a grey area. Can't assume they know proofs or how to prove things, yet I cringe when they say something nonsensical about a function

So I'm trying to think of a set of fundamental requirements that isn't full-blown ability to prove things

[pastel horizon]

The engineering courses have maths lecturers but they are more geared towards engineering students rather than a pure background

You wouldn't ever see a maths major take the same class as an engineering major here

[long pelican]

I wonder what the engineering lecturers expect in terms of mathematical fluency

[pastel horizon]

It varied tbh. I had one who was picky with definitions, another who wasn't as concerned and a really poor teacher who seemed to kind of lack confidence and presence so who knows what he expected, but everyone got 90-100% on his exam when the average should have been 55-655% so clearly not rigorous enough

[long pelican]

Oooo!

Averages are targeted to 55-65%?

[pastel horizon]

Yeah, that's the upper second level

[long pelican]

Hm this is interesting

Do your exams also target those averages?

[pastel horizon]

An upper second is like 3.3-3.7 GPA

Just to help a bit

I think my final grade was 62% overall so probably graduated with 3.4 GPA?

[long pelican]

Whoa

[pastel horizon]

First class would be a 4.0

[long pelican]

Here, if you get 62% you're considered "WTF are you doing, are you even trying?"

with a D- grade

So how does that work in a math class specifically

Do professors make exam questions intentionally that they don't expect many people can solve?

[pastel horizon]

D- here is just a fail

Going off the website, a 30% here is equivalent to a C which is tbh barely a pass, some institutions wouldn't even count that as a pass

[long pelican]

Yeah, 62% is a D here

[pastel horizon]

For some you'd be expected to get 40% as a pass

[long pelican]

Getting 62% on an exam is usually considered very bad for most courses where the average is designed to be like 80%

[pastel horizon]

But would you curve that

[long pelican]

if average is 80% then no

[pastel horizon]

What are the raw marks

[long pelican]

That's the raw mark

[pastel horizon]

Huh. Seems like there's not much fiddling with numbers then

[long pelican]

What is most interesting to me is

If a student solves less than 70% of the problems here, they feel very bad about themselves

But in the UK, that's a good achievement?

[pastel horizon]

Well that 90% average exam was viewed by the head of department as Mickey mouse

[long pelican]

Lol

In what aspect are UK exams just harder?

or are they even harder at all

[pastel horizon]

More emphasis on reasoning and problem solving I imagine

[long pelican]

I see

I should go look at one

[pastel horizon]

I don't know if you could see any uni exam papers they might only be viewable by students

[long pelican]

https://www.maths.cam.ac.uk/undergrad/pastpapers/past-ia-ib-and-ii-examination-papers

Do you know what these are?

[pastel horizon]

A level is the closest

Wow that's pretty cool, yeah think those are undergrad Cambridge exams, should be a very high standard

[long pelican]

Totally stealing this for first midterm in honors linear algebra next semester

Dam these questions are very high standard

[pastel horizon]

I'm glad you think they are, would've been quite embarrassed if a Cambridge exam wasn't

[long pelican]

haha

[pastel horizon]

From my side, there was maths questions which most students could do but the marks get lost when you're applying knowledge. Things like "state your assumptions, what could you do to improve this design? What are the advantages/disadvantages of this design?" Etc

Unless you really know a lot of the theory and spend every day on extra reading you won't get full marks on those types of questions on an eng course

Believe it or not as well assumptions are rigorous you can't just put garbage like π=3

[long pelican]

pi=3 is not anything my students will produce

but they will produce things show they don't read what they read or write 😔

Maybe the grading scheme itself lends to lower scores in UK

Here other graders will go like correct but conceptual misconception? -3 out of 15
UK might do +3 out of 15

[pastel horizon]

Ohhh yeah that's exactly right. Each question you start on 0 and you need to earn each mark

Also, getting the answer correct (without working) is still a 0 at university level

Unless obviously it's something Mickey Mouse like "1 + 1"

[long pelican]

Mickey Mouse 🐭

[pastel horizon]

Yup lmao

I kid you not as well, it's a term professionals use informally if something is too easy

[long pelican]

Wow they really do care about precise communication

[pastel horizon]

So yeah if I go back to the 90% example, it was content that was covered at A Level (16-18) in the second year of university! So should be no surprise that it wasn't challenging enough for that level

[pastel horizon]

Yeah stuff like that would be how they distinguish a first class student from an upper second student. Your upper second student would understand it, just making a few sloppy mistakes explaining their argument

The other part of it is preserving their integrity and credibility. They need to be picky so that their status and qualifications continue to be highly prestigious

That's why I expected their exams to have a very high standard without even reading through.

[long pelican]

A level marks are also like that damn

Gotta earn every mark, any evidence of not knowing what you are doing disqualifies you from the mark

If my exams had that grading scheme for all questions the average would be like 35% lol

[pastel horizon]

Yeah, exams have strict regulations from Ofqual for A Level and GCSE

Thing is though, we kind of went with the approach that grades are increasing so exams became tougher. Seems like the US is the opposite where the standard required for each grade slipped

We have had like what 11 years of some version of a conservative government as well though

[pastel horizon]

Oh yeah and as a footnote, imagine that earning A* and A grades in those exams is barely enough to even be considered for interview at Oxford or Cambridge. They really focus on bringing in exceptional students

[topaz scarab]

I did IB when I was in highschol

IB diploma

check out this one: https://www.maths.cam.ac.uk/undergrad/pastpapers/files/2021/paperii_2_2021.pdf

that's year 12 highschool

loving it

[long pelican]

Can typical high school students in your school do these?!?!?!

[topaz scarab]

Okay, so this is how it works
there are many streams in math right? This is the paper of the HARDEST stream. so definitely not typical students by any means.
I don't think any university require math HL, all science and engineering faculties are happy with one stream lower than this.
Except for masochist (like me), or someone who is actual prodigy, there is zero reason to take math HL.

even then, I think that whole paper is like elective topics. I think you have to choose 1 or 2 of the elective topics.
So someone might be able to do graph and group theory, but won't know anything about all the other stuff.

[long pelican]

Oh I see

Imagine a high schooler choosing algebraic geometry as one of their topics

These questions just seem too hard for even the #1 high school student in math at a typical high school to solve any of

too advanced and too hard

[topaz scarab]

Imagine hiring a teacher who are competent to teach all of the topics here

[long pelican]

😱

[topaz scarab]

in my experience, studnts have no choice over the topic. The teacher will pick the topic according to their own expertise.

[long pelican]

Ohh I see

In the high school I went to there were easily no teachers competent to teach any of the topics lol

[topaz scarab]

In 2007, my late math teacher picked group theory for us. He is a true chad, never realize what a big dick energy move he did until now.

[topaz scarab]

Really? In my experience, lots of math teachers had physics degree, and would be comfortable doing the advanced calculus and the physics stuff
(then again, I'm in Australia where teachers are paid quite well in international standards)

[long pelican]

Yeah nope not here

pick 100 random high schools in this country, about 0.1 of them will have a high school teacher with the ability to teach this stuff (without doing silly things like memorization)

[topaz scarab]

I mean, you could simply search all the highschool in your country that offers IB diploma, and which out of those offers math HL. That's a pretty good estimate haha

[long pelican]

Are you counting IB calculus?

IB calculus seems way less advanced than the questions on that test

seems equivalent to AP calculus

https://ibpublishing.ibo.org/live-exist/rest/app/tsm.xql?doc=d_5_matsl_tsm_1205_1_e&part=2&chapter=2 Here are some examples of student writings who are taking IB calculus

Most if not all of them are quite unmathematical

[topaz scarab]

I'm not sure what you mean by IB calculus,
but what I meant by calculus earlier, was calculus in the options, like question 14D, differential equation

[long pelican]

Hmmm

It almost feels like we're talking about two different IB's

[topaz scarab]

Oh, this is IA (like a short math paper), I don't think this is for HL IA.
I was talking about IB HL options.

So IB has many math streams,
all streams needs to do exam and papers (called IA: internal assessment)

So yes, I was talking about IB math HL options (which would be the hardest topics)
and this is like math projects that would be done by a typical IB diploma students.

(note that typical high school students would struggle a lot doing IB diploma, the workload is higher than 1st year uni)

[long pelican]

I did a google search and I don't think algebraic geometry would be an IB HL stream

so that exam paper you linked sounds way beyond IB HL

[topaz scarab]

Huh, maybe I'm wrong, let me double check

[topaz scarab]

Okay, I think I'm wrong
huh, maybe I was speed reading through the chat, somehow got the context that it was IB, clicked the link and didn't double check
my bad, lots of drama over "can't even read"

[long pelican]

Damn, no worries

[gusty forge]

In my uni (UNSW) we have two streams for math.

B Sci(Mathematics) and B Sci (Advanced Mathematics)

we have something called a 'weighted average mark' (WAM) that's out of 100

and if you get <70 WAM for any one term you get demoted from advanced math to regular math

it's kind of scary

For reference, 85+ is considered "high Distinction. Target percentile = 90",

75+ is considered "Distinction. Target percentile = 70"

65+ is credit. Target percentile = 35

[tepid smelt]

Uh this is for the masters level at Cambridge(arguably one of the best masters programs in the world). I could only see Olympiad level students being able to do this level. This is way beyond HS level.

I teach at a large public school but in a nice area. Over half the math teacher's don't have math degrees. That is typical in the states also and is a major problem. A bigger issue is the math level of elementary school teachers is really low. Part of the reason is teaching math in the k-12 system is really difficult and few with math backgrounds choose to stick with it when easier higher paying jobs arr available.

[long pelican]

Oooooh

[pastel horizon]

Yeah I thought it said IB at first, it's not. It's actually short for paper 1 B

IB would be international baccalaureate, completely different things

So these are the types of questions Oxbridge would ask at an interview for Y13s who are looking to apply

Ahh now the first one, for some context I think only students taking further maths would know how to set up the Newton-Raphson method. So students who aren't would have to be a bit more creative

[long pelican]

My idea now about why calc is hard for many students fundamentally is because they don't understand functions and language involving functions

A lot of online resources teaching functions sort of dance around what exactly a function is. They teach that function notation is the same as y = notation (damn)

[real mauve]

one of the tell tale signs is asking for the difference between f and f(x)

how would you propose to do it differently in early stages though? i surmise part of the reason is that sets are avoided early on

[charred silo]

I can see that being a problem, but I have not encountered such students myself, with my limited exposures. Calculus is introduced here at A-Level and students that struggle these definitions typically drop Mathematics before reaching A-Level.

Hm... Not sure if knowing sets is necessary. I did not know what a set is until undergrad.

[long pelican]

A set is a collection of objects, shouldn't every student know this?

And here in the US, passing algebra has nothing to do with being able to read algebra

So naturally I get students who would not pass a rigorous algebra course

[charred silo]

I only had to deal with reals. Even complex numbers were barely touched upon.

So knowing what a set is was not too useful.

[long pelican]

I came up with an idea earlier: students actually encounter functions everywhere! A function is an assignment of an element of a set to each element of another set. On their first day of school, each of them is assigned a desk. Voila, a function!

(from the set of students to the set of desks)

[real mauve]

certainly, i just wonder whether that is too abstract to really grasp so early on

[long pelican]

What part of it would be abstract?

in the desk example

[real mauve]

i would expect at least one person to ask "how is this related to numbers?"

even though the point is that it doesn't have to be

[long pelican]

The most intuitive numerical functions they see can be be the buttons on their calculators, right?

[molten urchin]

I don't think basic set theory presents any conceptual challenge. The lingo is broad enough to accommodate tons of real-life examples. Fwiw, it was a part of my high school math curriculum and my class even covered some of the basic proofs.

[pastel horizon]

We teach a function as a mapping from domain to range at a level

Seems like in reality it's not quite precise enough, it's more like a mapping from one set to another

[charred silo]

Ah, I sort of remember that from GCSE. And it was mostly unused afterwards.

[pastel horizon]

When did you do GCSE? Think it's changed a bit from when I did it

[charred silo]

2004/5-ish.. Don't remember.

[pastel horizon]

Oh so it's the opposite. I did mine 10 years later

Well 9. 2013 GCSE and 2015 a level

They probably moved the more formal function definition to a level

And complex numbers are done in further pure maths as well. We covered de Moivres theorem, roots of unity, loci and trig identities derived from complex analysis

[charred silo]

FP-modules... I only took the easy ones. Didn't learn roots of unity till uni either!

[pastel horizon]

We didn't get a choice with what we took 😂

[charred silo]

I think A-level is also used to introduce calculus as a tool. Someone doing engineering might not care about all these things.

[pastel horizon]

FP1 was actually Mickey mouse tbh, FP2 was more challenging

I think Core 4 is harder

But then the intention is you learn FP1 at AS level so I suppose it would be harder for a Y12

[pastel horizon]

Tbh a lot of a level maths is applicable to engineering, they made us relearn it all at uni anyway

Ok maybe equation of a circle is useless but a lot is useful

[charred silo]

Yea. So students like me could catch up!

[wispy slate]

I'm trash at maths lol

[pastel horizon]

The matrix/vector topics are so important. I think it's the most underrated part of a level

Like in engineering you do work on vectors of 100 variables (on the computer of course)

Reason is once you design complicated systems you end up with loads of equations and variables in your design. The vector/matrix notation is very convenient to generalise it

[charred silo]

No matrices either. I think we did the D1 module for marks.

[pastel horizon]

That's a shame

[real mauve]

i did have some basic linalg at the end of my HS, but tbh it's useless without the accompanying discussion of (sub)spaces, rank nullity, etc

[pastel horizon]

Linear algebra is definitely not something that should be underestimated when it comes to training for STEM

Might even go as far to say as it's equally as important as calc

[real mauve]

they do largely appear together in engineering, anyway

[pastel horizon]

For sure, I just feel like pre uni focuses more on calc

I think row operations on a matrix should be taught at a level also

It's actually really neat, it underpins the basic method you would initially use to solve simultaneous equations and is more efficient than calculating an inverse traditionally

[long pelican]

https://www.maa.org/programs/faculty-and-departments/curriculum-department-guidelines-recommendations/teaching-and-learning/9-key-aspects-of-knowing-and-learning-the-concept-of-function Wow this article is so good

I like how all the relevant studies that show these conclusions were done 20 years ago and are somehow still relevant in 2021, like nothing has improved

[wispy slate]

quick question icy

what is your PhD in?

like what topic

[long pelican]

Algebraic number theory

[austere inlet]

changing or reforming things on the entire educational systems is always hard

[long pelican]

What are we reforming here? All that's needed is to (1) incorporate better function definitions and exercises in textbooks and (2) use questions like the ones in the article on standardized tests, varied enough that teachers can't teach to the test for them. Just those 2 things would accomplish something even if everything else was unchanged

[wispy slate]

random question : were you a prodigy as a child?

[austere inlet]

that's not a minor change at all, precisely due to how central the concept of "function" is in mathematics. Teaching a new definition of function implies that you're teaching a different conceptual understanding, and when proposing that sort of thing on whichever education board is in charge of the HS math curriculum you'd need to justify its effectiveness with research and evidence, and then hope that after many long discussions and meetings people will hop on board with the idea, and I won't even begin to mention that educators will actually need to learn these definitions (yes, even they as teachers might not properly understand what a function is) and why should they teach them that way. Educators, as well as parents might even oppose these changes like they opposed "new math" in the 60s, and more recently the Common Core in the US. If you want this to change in the entire educational system then you want a reform.

of course it'd be way easier to just change it locally if individual teachers or institutions are willing to do it

[long pelican]

no

[long pelican]

We are getting math reforms every 20 years. When comparing with politics reforms this is a very short time scale. There's a lot more optimism than you suggest

Besides, with all the online stuff and interconnectivity we have now, gradual improvement is a possible path

Anyway I have no idea where this thread of discussion is heading

The central point seems to be "It's hard" but that's why people are working on it

[wispy slate]

no, and I don't remember anyone that could be considered a prodigy which was unfortunate 😦

[astral laurel]

Interesting article but I still hate f circ g notation

[earnest trail]

for function composition?

[tepid smelt]

This was a really good article. One thing I have done this year with my freshman is to try out more projects with functions to make them feel more dynamic. The first project was to create a story that contained an arithmetic or geometric sequence that they had to represent using several different ways. Such as a graph, picture, at least two different forms though most just used a different starting point and also they had to create a question that there function could solve based on the story they wrote. Another on the features of functions chapter I again had them create a story that could be represented with a graph and they had to label key features(where increasing/decreasing/max/min etc) and they had to write the function for various parts and again create a question the graph could answer. It wasnt perfect but the students really loved the creative aspect and seeing how different students approached the project. It made for fun presentations also especially when they had to solve the problems the students created. I think more opportunities to be creative with math is missing at the k-12 level and should be emphasized more to give life to the ideas they are studying

I think along with mathematical literacy students need more opportunities to be creative with math. I think this play aspect is missing and can really help improve engagement. It is tough though coming up with good ways students can get creative with the math. One of my favorite projects is next semester when we start doing geometric constructions with compass and straight edge. I have them create a unique art piece combining the construction techniques they learned. They must then write down the steps to recreate it using a compass and straightedge only. I am always blown away by how creative a group of 14 year olds can be. Math is inherently a creative discipline but the curriculum does not emphasis it enough and that takes a lot of joy out of the subject

Its especially interesting when you challenge a student to create something someone else made. They really have to try multiple things out and make mistakes which is really important to highlight

[pearl onyx]

I have to teach the epsilon-delta definition of a limit, as well as how the proof works, to a bunch of high schoolers who are not familiar with mathematical formalism or jargon

I want to show them how genius the definition is, and what idea it captures, in a really interesting way

But it’s very difficult since they don’t really understand what a real number is, what “for all” means, etc etc

[long pelican]

I think two steps: teach them how to read the delta part (by eliminating the epsilon part by setting it to 0.5)

Then bring the for all epsilon in plus optionally a picture or animation

But emphasize that it does take the first timer anywhere from 1 to 3 hours to really get

And like, assign them to go ahead and spend those 1-3 hours staring at it and some examples

Maybe with your help the amount of time can be reduced to 30 minutes

I had to do it mostly without help

[pearl onyx]

Oh man, I’m given a slot of around 30 minutes, it’s starting to look a little hopeless lol

Oh ok

[long pelican]

More specifically, 30 minutes of reading it over and over again lol

[pearl onyx]

Yeah it’s not great that this is the first time they’re seeing a formal definition, especially since (even to someone fluent) it’s not an easy concept to grasp

Haha yeah, you can only really get it if you try to understand it on your own

[long pelican]

Are you teaching at a public high school or a private one?

[astral laurel]

So essentially weakening the forall to stricter conditions like Lipschitzness?

Discussing the continuity spectrum might be useful. In such cases you can just say which cases of continuity are in the syllabus

[long pelican]

No, not at all. Just taking the inner part of the definition and working with that first as a mathematical statement

[astral laurel]

Well I kinda meant that, you can directly refer to the stronger conditions temporarily

Anyway as long as you provide counterexamples of inclusion under stronger conditions but not weaker, I think it will be good enough.

I feel like introducing more material will contextualise what they need to see for continuity

And it's not like it will become a topology class

[long pelican]

Oh, Lipschitz continuity has to do with |f(x)-f(y)| being less than a constant multiple of |x-y|

But this variant is just, does there exist a delta such that |f(x)-f(y)| is less than a fixed value for all y within delta of x

... wow there's still doubly nested quantifiers here

So the original limit definition is a definition with triply nested quantifiers

Ok so we need to insert step 0

Pick x. Pick a function f. Is it true that |f(x)-f(y)| is always less than 0.5 when y is within [something, say 1] of x?

obligatory time to mention that at least one student will still be lost because they are taught that f(x) is another name for y

This kind of reminds me of Inception

Single quantifier takes on the order of seconds to intuitively understand
Doubly nested quantifier takes on the order of minutes to intuitively understand
Triply nested quantifier takes on the order of hours to intuitively understand
....
Quadruply nested quantifiers... 🌌 🧠 ?

[astral laurel]

You can use f(x_1) and f(x_2) if that bothers your students

[long pelican]

Nah I think it's definitely better to have every student not continue to believe f(x) is another name for y

I think

[real mauve]

there are x_1(f) and x_1(y) s.t. |x_1(f) - x_1(y)| < L |f - y|

[long pelican]

Every competent teacher/lecturer upon realizing that their course will be the first one their students will likely have taken taught by a competent teacher/lecturer with a competent textbook, should start their class with a crash course on un-learning function and symbol-reading misconceptions taught by high schools

[astral laurel]

Reminds me of this
https://en.wikipedia.org/wiki/Uniform_continuity#Local_continuity_versus_global_uniform_continuity

[long pelican]

This is very galaxy brain

[astral laurel]

order of logic is kinda important but learning it via continuity is

[long pelican]

https://gowers.wordpress.com/2012/11/20/what-maths-a-level-doesnt-necessarily-give-you/

[long pelican]

Lots of good comments in there too, including one by Terence Tao himself

[pastel horizon]

A level is an intense course to get through especially with further maths. It's understandable that teachers take shortcuts

When I did mine he just said "It's not further maths it's faster maths. You're doing the full A level in one year"

[long pelican]

Lol

[pastel horizon]

Just imagine that though. A two year course squashed in 1 year. Of course they ended up taking a lot of shortcuts to speed through material

[long pelican]

Honestly I feel like I can fit all the material in a small space in my head, because it's well organized. If the concepts are well organized and taught logically, they might realize there is not really much material to remember in this course at all

[pastel horizon]

The calc material?

It's actually not true btw. There's four core maths units and two optional units with a choice of statistics, mechanics or decision mathematics

[long pelican]

Don’t look at the number of units

What are the core principles underlying calculus?

You either spend time on cramming every leaf of the tree, or spend time building the tree and let the leaves grow by themselves. That’s my analogy

[pastel horizon]

Calc isn't the main focus of that unit tbh

https://www.savemyexams.co.uk/a-level-maths-edexcel-core-3/

I'd say this is the hardest unit at a level and 1 section out of 5 is differential calculus! The hard part is actually trigonometry. There's a lot of identities to memorise and the questions asked in an exam can be really challenging based on those identities. Plus they get tested for some integrals

[long pelican]

Had a look at a trig paper. Pretending that I'm a typical math student, I might need to memorize more than just the trig identities. Since as a typical math student I don't understand how to extract meaning from mathematical expressions involving variables and functions, I have to memorize how to recognize what to do based on the way the symbols look.

That's probably what's making math so time consuming

[pastel horizon]

Yeah that's another problem as well when it comes to a level, you know that the teachers before would have spoon fed them to do GCSE standard so you then have to reteach basics

Then university reteach a level because they know they got spoon fed 🤣 at least because it's a 3 year course there's more time to really stamp out misconceptions and teach precisely

[flat cargo]

I've been thinking of giving a talk on intro analysis for people with a basic calc1 background. any suggestions on ordering and what topics to cover? I want to cover compactness, properties of continuity (Intermediate value theorem, extreme value theorem) and some cool stuff about derivatives, like them obeying the intermediate value property but all of that in an hour talk is unrealistic (if I show proofs that is)

if I skip proofs I might be able to cover an entire real analysis course in an hour tbh minus the proofs which is obviously the most important part

[long pelican]

I wonder if compactness would be confusing, no matter how you phrase it. Especially with a calc 1 background the expectations to think abstractly cannot be high at all. Some cool things I might add are talking about a function that is everywhere continuous but nowhere differentiable, and a summary of nice applications like Fourier analysis

[flat cargo]

oh yeah, I should talk about the Weierstrass approximation theorem too since it's a good example of why mathematicians study polynomials. you're totally right compactness is confusing, it was confusing to me when I was learning it so probably bad to try and teach that

it's just so satisfying once you get it because you can say stuff like f cont on [0,1] => f uniform cont on [0,1]

Weierstrass's continuous but nowhere differentiable function is good to mention, but I'm a bit salty because the proof was so tedious. I typically have more motivation for proving theorems then constructing counterexamples even though I know the latter is important too

[pastel sundial]

Wait so what actually is the deal with this California math education reform thing

https://www.nytimes.com/2021/11/15/learning/do-you-think-we-need-to-change-the-way-math-is-taught.html

Like from this article

[tepid smelt]

The nested interval property is neat and can show how R is uncountable I believe and you can have a nice discussion different infinities which can surprise HS kids.

[tepid smelt]

I am a fan of jo boaler and do think more stats classes should be offered. I thinkna bigger issue is what Icy has brought up multiple times and its a lack of emphasis on mathematical literacy and too much focus on tons of procedures to gear students to pass state testing

I think it needs to be addressed much earlier though its so difficult by the time they get to HS to unlearn bad habits along will fill in years of missing knowledge while also trying to cover way too much material that focuses on state tests. I can assure you many secondary teachers try very hard but we are handicapped by admin/curriculum and lack of preparedness at the primary level

[tepid smelt]

I mean that example really stuck with my as one of the first times your intuition can be so wrong.

The overall curriculum does need a major overhaul. I don't know how I would sequence everything though. Most curriculum are so bad and we are forced to teach to it and required to meet certain pacing guides which is incredibly frustrating as it handicaps you so much. The calculus teachers are the only ones allowed some freedom but even they put a lot of emphasis on preparing kids to pass the AP exam and are judged on pass rates

[flat cargo]

nice, I'll mention it for sure I'm just salty because proving weisteross's function was nowhere differentiable was a pain for me. NIP and countability stuff is nice

[long pelican]

Read the article. I don't have any comments on the social justice aspect because I don't really think about that very much. (The troubles I am angry about are common to all socioeconomic classes.) I did have a comment about their suggestions to increase problem solving and collaboration though: I tried the problem solving approach at the start of this semester in Calc II and it fell flat because it got no engagement, because no one understood what I was saying.
Why did it fall flat? Because they lacked the ability to understand language with variables and functions, which was a prerequisite to understand the problems and logical steps! So emphasis on problem solving I would say can only come after solving the math literacy crisis.

Comments were also interesting. The comments were mostly written by current high school students.

1. There was pretty much perfect correlation between people liking math and having teachers that didn't teach math in a procedural way. Also perfect correlation between students disliking math and having teachers that taught it in a procedural way.
2. Several people mentioned they were classified as "gifted" and went through accelerated classes... and still didn't like math, found it confusing, and found it boring. Because the accelerated classes were taught in a procedural way. Conclusion: gifted people don't naturally understand math better. They still need good teachers/mentors/textbooks just like everyone else.
3. Common to all students who didn't like math is that math didn't make sense to them. The traditional interpretation of this is: we need more real world applications! Have the people who push for this ever considered helping math make sense to students... in and of itself? How to read things with variables and functions (and not teach things like "f(x) is the same thing as y")

[pastel sundial]

Yeah point 3 is what gets me

And my other issue with this plan is the elimination of tracking. The goals are noble, but grouping everyone in the same class and not letting more advanced students more on to more advanced material seems like a mistake.

I don't know too much about this plan, but the impression I have is that it feels like an example of math education reform proposed by people who don't actually understand math.

[long pelican]

I was actually quite back and forth on tracking as I read the article and still not sure what to think of it. For me the alternatives were "learn math properly from a really good teacher" and "learn math in the normal classes with mediocre teachers," and the better choice was obvious. But if the choice is between "learn AP calculus with a mediocre teacher" or "learn Algebra II with a mediocre teacher", both choices seem equally bad

[pastel sundial]

But in theory all the teachers should be good right?

[long pelican]

If anything, pushing a student who only has a good procedural understanding and no relational understanding really fast through their math program will just make their overall relational understanding of math even weaker

[pastel sundial]

Oh absolutely

But some students genuinely are ready for more advanced stuff sooner and holding then back helps no one.

[long pelican]

My tongue-in-cheek answer to tracking is: Send advanced students to whichever teacher teaches for relational understanding

(if any)

If there are none, just have them go to computer lab and do AoPS Alcumus during whatever time their math classes would be

[pastel sundial]

Lmao

It's honestly so sad how poorly math is generally taught. Like I honestly wonder how much of my interest in math comes from never touching the public education system.

Reading comments like

I believe, like Williamson M. Evers, a source for Fortin's article, says, "math is math". Math in itself is a highly logical subject where the answer to problems are a single number or set of numbers. There is not room for creativity in math.
Is actually just depressing.

[long pelican]

Less depressing if you replace math in the last sentence with "TSM ©" :^)

Textbook School Mathematics

[pastel sundial]

Well the depressing part is that they think all there is to math is TSM

[long pelican]

Ah, it's only not depressing to me because I've been familiar with that as the general public perception for over 10 years

[pastel sundial]

Do people have that incorrect an impression of other subjects?

Maybe like history is just memorizing dates?

[long pelican]

Actually sorta, until I went to an intensive math summer camp involving lots of proof writing, I thought essay writing was all about sounding sophisticated with really big words like the books we had to read or the SAT model answers

After that summer camp I treated essay writing as just communicating effectively to the reader and I got a perfect score on the first essay assignment in the reputably hardest English class of my school

[pastel sundial]

Maybe but I don't think the average American has as incorrect a view of English as they do of math

(I say American because that's literally all I know)

[long pelican]

Sure, if English was as bad as math, we'd see people thinking English is about memorizing verb conjugations and making sure you put the i before e, and the advanced classes are just the Scripps national spelling bee

[pastel sundial]

I think the average person's impression of philosophy might be comparably bad, but that's also not generally taught in K-12 to my understanding.

(e.g. philosophy is when big words and incomprehensible arguments that appeal only to academics)

[long pelican]

Intro to philosophy classes are literally teaching you how to do if then logic

[pastel sundial]

Are they?

[long pelican]

Well and some other stuff

[pastel sundial]

That was not at all my experience with intro philosophy

[long pelican]

Hmmm

My first philosophy class was a survey on Greek philosophy as the content, but the focus on essays was using proper logic

[pastel sundial]

I think it depends on the class

My intro philosophy class was a survey on ancient Chinese, Greek, and Indian philosophy.

Though I think that was probably an exception since the professor specialized in Chinese philosophy.

[long pelican]

Yeah I figured philosophy classes vary then

[pastel sundial]

But in any case, philosophy is certainly not drilled into students' heads for 12 years via bad explanations that don't give an accurate impression of what phil is.

And having a good understanding of if then logic is genuinely a very important skill for philosophy, as supposed to TSM arguably being actively harmful for learning math.

[long pelican]

Yeah, so I think the public perception of philosophy is more like "What is it?" instead of "I was never good at it and I'm PROUD of it"

I'm briefly wondering how bad the politics have to be for all the mathematicians in America and the AMS to not be able to get rid of TSM

Who's peer reviewing the textbooks written by Larson and Holt and so on

[pastel sundial]

What do you mean

[long pelican]

So people have been lamenting math education for over 20 years (well, over 100 or even 200, but the current style of math education for over 20). Hung-Hsi Wu has been writing about the horrors of TSM since 1998, there were lots of mathematicians involved in the creation of the Common Core, yet the textbook situation has barely changed

[pastel sundial]

Yeah why is that

Like someone has to be pushing back here, but who and why??

[long pelican]

Hmm under the capitalist model, textbook writers are under competition to be adopted by as many schools as possible... so the people they want to impress are the administrators

And administrators could care less if a function is defined properly or in the lazy test-prep "f(x) is the same as y" way

[pastel sundial]

But why are the administrators not listening to the actual mathematicians

[long pelican]

That could be due to them seeking results in standardized tests first and foremost maybe

Not sure though

There's also the not-very-small point that very few administrators actually understand math

even if they are the math department heads

https://www.youtube.com/watch?v=52tpYl2tTqk
Watch this video if you want your depression as a function of time to be an exponential function

Like, up to the set theory part it's okay

but then it just rapidly deteriorates

It's the first youtube result for "functions" too

[pastel sundial]

I'd watch it but I'm afraid youtube would start reccomendating me basic math videos

[long pelican]

Incognito mode maybe!

[pastel sundial]

Actually idk what youtube would think

[long pelican]

I should tell the other math professors in the department to google "functions" on youtube and watch some of the videos

it'll be a huge shock to them

It should change the way they approach teaching freshman courses

[pastel sundial]

Recently watched:
Algebraic Topology lecture 15: Proof of the Excision theorem.
Algebra Basics: What Are Functions?

[long pelican]

[pastel sundial]

If only it was a different video like "what is the quadratic formula"

Then I could make an "average category theorist" joke

ok I'm watching this video

time to see how bad it is

a set is just a group

algebraists in shambles

[long pelican]

I can forgive that, relatively

[pastel sundial]

yeah I'm meming here

it seems ok so far

2 minutes in

oh no he's giving y=2x as an example of a function

if we treat x as the set of inputs

[long pelican]

[pastel sundial]

wait so if the free variables in a function are the set of inputs

does this prove the negation of the Foundation axiom

[long pelican]

👀

[pastel sundial]

"let's try to make a function table for the equation y^2 = x"

lmfao wut

holy shit this is deteriorating rapidly

"the equation gave two inputs for a single output so it is not a function"

doesn't explain why

the equation y=2x qualifies as a function

actually shoot me

on the "function" y=x+1

if you watched our last video about graphing on the coordinate plane, you may notice that each row of this function table is basically just an ordered pair. We can even rewrite the inputs and outputs in ordered pair form. And that means you can also graph each of these pairs of inputs and outputs on the coordinate plane. You can graph a function. Here are the points from our function table plotted on the coordinate plane. It forms a straight line and is an example of what is called a linear function.

this presentation is so bad

there's like no motivation for anything what

pulls the vertical line test out of his ass and then only gives a vague justification of why it works after explaining it (also vaguely).

and yeah I definitely see the point with TSM about conflating definitions and results.

[long pelican]

You will die after the function notation part

☠️

[pastel sundial]

"these are the most common names [of functions], but you can use others if you want to"

doesn't give an example where you write f(y)=x

it's like he's actively trying to create the misconception that y is always the output and x is always the input

[long pelican]

I think the entire idea of introducing the letter y when discussing function notation is a misconception in itself

[pastel sundial]

this is like a carefully crated video that's meant to appear like an innocuous explanation of functions but is really a scheme to sabotage future multi students

[long pelican]

Like people write sqrt(x) all the time in text, do you want them to say sqrt(x) is the same as y? out of nowhere?

[pastel sundial]

wait is he actaully

going on a long rant about how f(x) is the same thing as y

and that they're "interchangeable"

[long pelican]

Yeah

[pastel sundial]

but... why?

[long pelican]

Just passing on what he learned from TSM

[pastel sundial]

I wonder if the the issue with reforming math education is just that the vast majority of educators and administrators aren't able to tell the difference between good and bad math education.

[long pelican]

That's a really big issue indeed

[pastel sundial]

But also possibly have an implicit distrust of any reform since it's new and scary, especially reforms proposed by mathematicians who they view as wizards.

[long pelican]

and the failure of 1960s New Math as a history lesson

[pastel sundial]

like I wonder if there's an element of "of course [teaching math this way] makes sense to YOU, you're a mathematician. Us regular people couldn't possibly hope to handle the level of abstraction you somehow manage to wrap your head around."

[long pelican]

We should hit them back with

"This functions video we just watched makes no sense to anyone, including us"

[pastel sundial]

yes

[long pelican]

It's like Wu said, TSM was made impossible to learn by anything other than memorization

[pastel sundial]

why was it made this way in the first place?

like historically

how did TSM come to be

I see how its self sustaining, but how did it come into existence in the first place

[pastel sundial]

also say more about this?

[long pelican]

I observed something that might explain why TSM continues to exist

I noticed that the following groups of people:
(a) authors of school math videos explaining e.g. functions
(b) authors of discrete math explaining videos and mathematicians
are completely separate and clearly don't mingle with each other

actually

there

[pastel sundial]

yeah it's weird

[long pelican]

How is it that almost every single person making a guide for school math is incompetent in understanding what a function is

[pastel sundial]

my neighbor across the street is a highschool math teacher

at what I'm told is a very good private school

super nice guy

I was talking to him a while ago and he didn't even know what topology was

[long pelican]

Better not direct him to eigenchris's video

[pastel sundial]

like not just not knowing the basic definitions, he had literally never even heard one donut equals ONE COFFIS CUP

[long pelican]

whoa

That is pretty surprising

[pastel sundial]

right???

and this man has been teaching highschool math for quite some time

and I'm pretty sure he doesn't know anything beyond calc

like I don't think he could say what a linear transformation is

[long pelican]

New Math had a lot of mathematicians' input and tried to have children start with formal set theory, but it failed for 2 reasons
(a) Lack of teacher knowledge, leading to completely messing up the point of it
(b) it might have been too formal and developmentally inappropriate

[pastel sundial]

I honestly wonder if he even knows the formal definition of a derivative

[long pelican]

Hmmm I bet that's the same as the answer to the question "does the textbook have it"

[pastel sundial]

I wouldn't be surprised

though I'm not even sure if he teaches calc

it's weird though

how little math math teachers know

is that as common in other subjects?

oh also I'm taking a cool class next semester

it's called "Issues in Contemporary Education"

[winged urchin]

Y'all will like this. Just helped a student where the practice test for their big test tomorrow had the problem (essentially)

"Consider y = 1 - (x-1)^2. Which of the following would be a possible restriction to the domain so that f^{-1}(x) is a function?"

[pastel sundial]

it's about the current US education system and how it could perhaps be improved.

[winged urchin]

And thennnn the answers were something like

Or choices I mean (it was multiple choice)

[pastel sundial]

wait did the question use y and f(x) interchangeably

[winged urchin]

x <= 1

0 <= x <= 1

x >= 0

x <= -1

x <= 2

[winged urchin]

Implicitly, yes ahah

And apparently only one of those choices is 'correct' HMMMM

Just so much wrong in one question

[long pelican]

After reading the question, I conclude the answer is "This makes no sense. f isn't defined."

Also, domain of what? (lol)

I might borrow this for my 10th problem set for this calc 2 class

Except that they are expected to answer "This makes no sense" instead of one of the answer choices

[pastel sundial]

ok so I guess you were supposed to answer (a)? But b also makes it invertible

actual question

[winged urchin]

B and d both make it invertible ahaha