#math-pedagogy

1 messages · Page 15 of 1

rough agate
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i uh

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doont think imaginary numbers should be taught with mathematical trigoonometry

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but you doyou i guess 😃

rapid tusk
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how do yall deal with the old issue of people not understanding what the square root does

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like “why can’t we write $\sqrt{9}=\pm 3$”

burnt vesselBOT
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elrichardo1337

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soft scarab
# burnt vessel **elrichardo1337**

because the square root function is defined to be the positive number, if it gave us both positive and negative outputs it wouldn't be a function

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-3 is a square root of 9 but when saying "sqrt(9)" then you're by default referring to the principle root

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@rapid tusk

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t

rapid tusk
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see i try to provide that explanation but often it doesn’t click with whoever im explaining it to lmao

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i think it stems from a failure to separate “taking sqrt of both sides of an equation” and “plugging into the sqrt function proper”

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(to be clear: I don’t have any confusion myself for the matter lmfao)

tawdry venture
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"taking sqrt of both sides" is a bit of a misnomer

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emphasize that sqrt(a^2) = |a|

rapid tusk
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yea

tawdry venture
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thus when you solve the equation x^2=9 for example you first sqrt both sides genuinely (as in really apply the sqrt function to both sides)

rapid tusk
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not long ago I was trying to explain that to someone and it just 
 didn’t register with them?

tawdry venture
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so you get sqrt(x^2) = sqrt(9)

rapid tusk
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yea

tawdry venture
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then lhs becomes |x| and rhs becomes 3

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just 3, and not ±3 since sqrt() only ever returns positive values

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is how i would explain it

tawdry venture
rapid tusk
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they were getting hung up on “but does that mean sqrt returns two values” 😭

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I get the impression they’re way in over their head

elfin sleet
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I distinctly remember feeling like I didn’t have any intuitive understanding of why multiplication was commutative. This was like, ages 9-12 maybe? I was aware of the area thing, but I guess since it wasn’t how I primarily thought about multiplication, it didn’t feel like a satisfying explanation. Idk to what extent this anecdote is relevant but I just thought I’d put it out there since you seemed to have trouble imagining that someone would not intuitively get it

quasi lava
rapid tusk
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😭

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their tangent on “i like being smartest person in class” was just

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??????????

quasi lava
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LMAO

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This prealgebra shit getting serious

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About the question itself tho maybe you could point out that it’s a useful convention cause root of x isn’t a function if we don’t define it as the principal root

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So it removes ambiguity cause it only maps to one specific value

rapid tusk
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how does one proceed if even that explanation fails

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and they simply don’t have any of the foundational skills to understand the distinction

quasi lava
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Like of why it’s convenient to define it that way?

rapid tusk
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im thinking more “general number sense”

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it turned out that they were asking why sqrt(6+sqrt(6+
)) was 3 and not -2 (or smth very similar)

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which

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basic number sense should tell you why that is ?

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they were like “why is it not both” or smth

quasi lava
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I can’t think of anything tbh if we’re talking about kids who haven’t done anything with convergence

rapid tusk
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how is the sqrt of something that’s clearly positive going to come out negative 😭

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and yea convergence is another way to see it

quasi lava
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Lmfao I think it all just boils down to like

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Something that eats 2 numbers and spits out some value should have an inverse that eats that value and spits out 2 numbers

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That’s what they’re thinking ig

rapid tusk
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yea

quasi lava
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It might just be one of those things that you only get a precise handle on later like how ppl usually encounter cross products way before building up wedge products and stuff

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But it’s poorly explained initially

minor turtle
# rapid tusk they were getting hung up on “but does that mean sqrt returns two values” 😭

i find this kind of subject hard to teach at every level, since we are told (against our intuition) that things such as f^(-1) must be functions, and therefore we must restrict our domain of f for such a thing to make sense, and then later students are then told the precise opposite, simply taking f^-1 as a map between powersets, and being asked to entertain such absurdities as "f(f^-1(x)) can be empty, and f^-1(f(x)) may have multiple points in it" and they now have to unlearn everything they were taught by an authority

rapid tusk
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yea

minor turtle
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becuase such a solution could be negative

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i really could define the square root function to be a truly terrible function

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if i wanted to

rapid tusk
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ig lmao

minor turtle
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i really could!

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students do so every day

quasi lava
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Also how you doing btw @minor turtle ur the one i talked to like a month ago about tensors right

minor turtle
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maybe, i do like tensors.

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I'm doing well, I might be tutoring for a while before i find some other job

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I just recently graduated undergrad

quasi lava
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Ah nice

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You suggested to me to read ITM by Tu I’ve been doing that

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It’s really good

minor turtle
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perhaps you define it like I do, where you take f(x) to be the positive solution of y^2 = x if x is rational, and the negative solution if x is irrational

minor turtle
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oh the notation

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i thought you said "notion"

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I like the notation

tight star
quasi lava
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There was something about it i was gonna say i forgot

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Rip

tight star
daring gulch
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Oh! How about using exponents?

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We know that $3^2=9$. We can clearly see here $3$ is positive. We also know $(-3)^2=9$, but we wouldn't assume that the base in $3^2=9$ to also mean negative, now would we? Similarly, $9^\frac{1}{2}$ is assumed to always equal $+3$. If we want a negative answer, then we put a negative in front, like so: $-9^\frac{1}{2}=-3$.

burnt vesselBOT
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⚞Shidouuu⚟

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daring gulch
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Or, how about this:

cloud zealot
vagrant meadow
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personally I've been pretty lenient with late work (up to the limit my primary will allow). I want my students to succeed, and I'd rather they do the assignment (so they can learn from it) whenever that happens to be.

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once I started a policy of only grading a few questions, that really brought down the workload as well. I can see a late work accepted policy working well with that.

vagrant meadow
# daring gulch I suppose you could say square root only returns two values if you're solving fo...

you can have a well defined function on R+ where x^2 is sent to a set of two elements {x,-x} (or one element if x=0). this is generally what we use (though not explicitly) to solve equations with square roots. since the positive and negative root could be solutions.

but in general, the square root (on R) is a function to R+. we just define it to be to the principal root. but I personally hate how this is often used as a "gotcha" question for students (where any answer that isn't just the principal square root is wrong).
I don't personally have a problem with referring to -3 as a square root of 9, as it's a perfectly valid solution to x^2=9.
that's just my take on it. but tbf I'm generally very loose with notation for the service of intuition (and the fact my ADHD brain thinks faster than I can write/type)

daring gulch
# daring gulch Or, how about this:

When finding the roots of a quadratic function, we are sometimes able to factor it such that we arrive at two solutions, as in $y=x^2-1$: $0=(x+1)(x-1)$. If we divide one of the terms, say $(x+1)$, that would leave us with $0=x-1$, which has a singular answer of $x=1$. Normally, we think of $y$ as a function of $x$. But what if we were to define $x$ as a function of $y$, only $y$ always equals $0$? We know functions can only have one output for each input. $0=x-1$ would be a function. But $0=(x+1)(x-1)$ would not. The same can be said for an equation such as $4=x^2$. When finding the roots of a function whose $y$ is a function of $x$, we assume that any amount of $x$'s may equal $y$, in this case 2 $x$’s. If $x$ is a function of $y$, that cannot be the case. Hence, if $x$ were a dependent variable, we are forced to define $\sqrt{x^2}$ to have only one value, which is positive.

burnt vesselBOT
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⚞Shidouuu⚟

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daring gulch
daring gulch
tight star
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Yeah properly understanding functions is a pretty key step in maths education

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They’re everywhere after all

daring gulch
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[Insert joke about those damn functions being everywhere]

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Idk too tired to think of something that makes sense

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But yeah I never really bothered to "learn" the difference between the square root function and square rooting "both sides." I just grew my intuition by doing problems where each makes sense in context. Context is one thing earlier math education definitely lacks, but hey, whatreyagonnado?

cosmic ibex
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Square rooting both sides is just applying the function too. The trouble comes if one of the sides was already a square and you now have something like sqrt(42) = sqrt(f(x)^2). Reducing the right-hand side of that to ±f(x) is not because the square root is suddenly multi-valued -- it always produces the single result |f(x)| -- but is just a way to encode that we don't yet know whether or not f(x) needs to be negated to give that |f(x)|.

daring gulch
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I think I see what you mean. So students may be mistaking ±f(x) to mean two separate answers, rather than meaning f(x) may be positive or negative but not both?

cosmic ibex
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Yeah -- and it ties into students' tendency to think that -f(x) necessarily produces a negative number.

daring gulch
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I've most certainly abused ± to hell in that case...

tawny slate
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genuinely, i teach trig by indirectly teaching euler's formula, its not as controversial as you might think

basically, you teach one single definition of trig functions: that starting from the origin, draw a ray going right. rotate ccw the degree amount. the point of intersection with the unit circle is (cos, sin)

(to be clear, you dont start immediately with this definition, you should imo still start with the basic right angle triangles, you build up to this definition by domain extension)

this singular definition prevents all need to memorize anything in trig, and it transitions very naturally and beautifully into complex numbers by introducing polar form, which then effortlessly derives even the trig sum identities that are so nasty

cosmic ibex
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It would probably sound less controversial if you call it "teach the unit-circle definition" instead of "Euler's formula, indirectly".

tawny slate
# rapid tusk they were getting hung up on “but does that mean sqrt returns two values” 😭

I have a real good trick for this:

i teach that by definition, the sqrt always returns the nonnegative value, that this is convention, not mathematical truth. "it is because it is"

now the students will have questions, which you are prepared to explain:

  • why do we choose that convention?
    suppose you accepted both the positive and negative values. what is sqrt4+sqrt1? is it 2+1? -2-1? 2-1? -2+1? there are 2^n different possible values for this expression, so thats not useful at all

  • so how come x^2 = 9 -> x = +-3?
    well, we are saying that both values make the equation true. remind them that solving an equation means finding all values that work, and excluding any of them gives a wrong answer. have them try by plugging in. emphasize that the +- symbol is really doing this: x = +-3 ~ (x=3 or x=-3)
    meanwhile, if this gives a good precursor to quadratics too. notice that quadratics always have two solutions. if you dont add the +-, you dont have two solutions. notice also that when you take an equation like x=3, square both sides, you gain a solution you shouldn't have. basically, rooting both sides you must include +- because you are lowering the degree, losing solutions. anytime you square both sides, you increase the degree, potentially getting extraneous solutions

  • what if we want the negative root?
    write -sqrt x

  • why don't we take the negative value then?
    because we use the positive values more and will make it more convenient for more advanced reasons later, such as the fact that negatives dont have a square root in the reals, so if you put a square root inside a square root you have to add a minus sign to cancel the assumed minus sign, weird. again, convention

tawny slate
# cosmic ibex It would probably sound less controversial if you call it "teach the unit-circle...

absolutely, but the point is the person i was responding to had some skepticism about this, but from the point of view of the educator, i think there is actual value in describing it this way

i have quite an aggressive approach towards pedagogy, part of why i like this channel so much. i want to reimagine math education by thinking about what is it that we want to teach and working backwards, completely independent of however we were originally taught in school or what a textbook might say. i have a lot of confidence that even basic math concepts still have a lot of room for improvement in terms of how we teach them

by associating the "unit circle definition" with "euler's identity, indirectly", i am hoping to get educators to warm up to this idea and challenge themselves: what is the end goal of my teaching, what am i preparing students for? are "advanced" concepts really that advanced?

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@rapid tusk sorry for ping, added one more bit to the square root thing on the bottom

rapid tusk
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aight aight

cosmic ibex
tawny slate
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yeah, i very much agree. to be clear, I'm not saying "system bad" necessarily, i do sometimes converge on a lot of the same ideas and approaches as the system does, i am merely expressing that on a personal level i want to promote a more critical element to pedagogy because i believe it fosters a stronger motivation and creativity to rethink how math is taught

elfin sleet
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I have a very strong mental image of the real line (it’s not a straight line in my head though) which I rely on heavily for all arithmetic

tawdry venture
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what kind of line is it, out of curiosity

tight star
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I think I have a categorical bias here in that

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I much prefer thinking of addition and multiplication as decategorifications of operations on sets

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Namely (disjoint) union and Cartesian product

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And I guess you can derive repeated addition from this viewpoint but

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Idk, just
 I wish this is how it was taught

modern trench
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Induced morphisms on the skeleton of Set

tight star
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FinSet really

modern trench
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Though how do you do that for non-integers

tight star
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Oh I mean specifically counting numbers

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I guess otherwise area

vocal phoenix
tight star
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I kinda draw a distinction between counting numbers and naturals

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Again this is probably my visual bias but

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To me counting is about finset

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And naturals are about induction

modern trench
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Is this cardinals vs ordinals catthink

vocal phoenix
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I teach at university level so I've never had to teach people the basics of numbers, but in general my preferred interpretation of numbers is geometric.

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As points or even vectors on the number line

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With addition as composition of vectors, and multiplication as scaling with possible direction change.

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That's mostly what I start with when I introduce complex numbers

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I motivate it it as moving from a line onto a plane

tight star
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yeah the number line is nice

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though it doesn’t make things like commutativity of multiplication obvious

vocal phoenix
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True

tight star
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This is where I wish “naturality” was taught earlier

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I feel like it’s not any one perspective that is best

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But the ability to translate between them, and know which one is more useful for your situation

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Which is the philosophy of category theory

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And what naturality is all about

modern trench
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How does naturality come into this

tight star
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Have you met quivers?

modern trench
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I've read the definition

tight star
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Right

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My definition of naturality is essentially a quiver homomorphism

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The idea is that you have a function F which relates inputs to outputs, so X becomes F(X)

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But maybe your inputs aren’t just some set, but are related to each other in some way - they form the vertices of a graph, with edges serving as connections

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I.e. a quiver

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Then, F is “natural” if, whenever there’s a way to relate inputs, there’s a corresponding way to relate outputs

modern trench
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So essentially functoriality

tight star
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So you upgrade F from a function between sets to a function between graphs

tight star
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Functoriality is like an upgraded form of naturality

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It means that the relations on outputs compose nicely

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But to show that, you first have to show they exist

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That’s what naturality is about - that there exists a way to relate the outputs (given a way to relate inputs)

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At least, in the way I use it

modern trench
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Oh hm

tight star
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So if you have a relation f : X -> Y

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Naturality is about the existence of F(f)

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Extending F to act on edges

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Functoriality is a stronger condition - it means F(f o g) = F(f) o F(g) and also F preserves identities

modern trench
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How does this form of naturality come into the above discussion?

tight star
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So for example

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Suppose you have two collections of things

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Like (2 cherries, 3 bananas)

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You can union them together to get (5 fruits), and then take the size to get the number 5

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Or, you can take the sizes individually, to get (2, 3). And then add the numbers

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The idea here is that “size” is natural

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The inputs are (2 cherries, 3 bananas), and (5 fruits), and there’s a way to relate them - the second is the union of the first

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This gives a way to relate the outputs - 5 = 2+3

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With the number line vs sets - you can imagine “size” taking a set to the corresponding position on the number line

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You can add the sets together by union

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Or you can add points on the number line together by shifts

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Again size is natural - both ways give the same result

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So you can choose which one is more convenient

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This is the essence of naturality, as I understand it

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Size is like a “homomorphism” between two different representations of numbers

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As collections of things, or as points on a number line

modern trench
tight star
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So have you gone through the def of a quiver hom

modern trench
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Nop

tight star
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I would recommend doing that

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Note that every quiver hom is a natural transformation

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It’s essentially a “graph homomorphism”

modern trench
tight star
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Well a quiver is a functor right

tardy ember
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wait, what?

tight star
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Is this not how they’re usually defined?

modern trench
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Oh the nLab sense

tight star
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Yeah

modern trench
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Ok I see it

tardy ember
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a quiver is a set of vertices, a set of edges, and for each edge a source and target, right?

modern trench
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Yea

tight star
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Yes

tardy ember
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how would that be a functor, what are the categories here

tight star
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You can describe this as a functor

modern trench
tight star
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It has two objects, E and V

tardy ember
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and two morphisms from E to V

tight star
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And two nontrivial morphisms

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Source and target

tardy ember
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right i get it

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and then the codomain is Set

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(or more generally C for a quiver in C)

tight star
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Yes

tardy ember
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yeah ok and then natural transformations are just the obvious notion of homomorphism

tight star
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Yes

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And this is my candidate for what naturality should be thought of

modern trench
tight star
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Oh I guess

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Vertices are either a pair of finite sets (X, Y), or a single finite set Z

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The edges are from (X, Y) to Z if Z is the (disjoint?) union of X and Y

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For the output - the vertices are either pairs of numbers (m, n) or a single number k

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And an edge from (m, n) to k if k = m + n

modern trench
tight star
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I mean that works as a quiver

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It’s just edges and vertices

elfin sleet
# tawdry venture what kind of line is it, out of curiosity

Oooh yes I would love to explain it if you are curious. It’s a path that mostly travels in straightish lines with turns at round numbers. For example, [0,100] looks something like the drawing I’ve attached. The reason it doesn't look to scale is that it's a path in 3d space and I've drawn it from the perspective of 0 so the bigger numbers look farther away. There is repetition in the shape of the line; for example, what I drew is also the shape of [100,200] from the perspective of 100, and the shape of [0, 100 000 000] if you zoom out enough that 1 million becomes your unit.
In addition to the two dimensions I drew, there is also height. 0 is at height 0 and the path of positive numbers starts out by sloping upwards at an angle of about 30 degrees. This slope decreases with a limit of 0 which is never reached, so that for very large numbers you are almost travelling purely horizontally. The negative numbers are a reflection of the positive numbers, with an initial slope of about 30 degrees downwards and also tending towards slope 0, and the same shape in the other two dimensions. So for numbers of very large magnitude, the positive path and negative path are almost parallel.

IMG_4370.png
tawdry venture
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so like a fractal repeating across orders of magnitude?

modern trench
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Symathesia

modern trench
tight star
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Also while every “natural function” is a natural transformation

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Cause they’re quiver homs

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The reverse is also true - any natural transformation can be viewed as a quiver hom

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This is essentially because any functor can be viewed as a quiver hom, since there’s a forgetful functor from categories to quivers

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Well, small categories at least

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And every natural transformation can be viewed as a functor

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This is the sense in which I mean they’re equivalent

elfin sleet
# tawdry venture so like a fractal repeating across orders of magnitude?

I guess yeah that’s kinda how it has ended up but like any mental image it’s got a lot of imperfections and stuff - I’ve smoothed over some inconsistencies in this description. And it’s not like it was deliberately created all at once, it’s just what I pictured when I was a tiny kid learning to count and it gradually got added to as I learned more numbers

modern trench
tight star
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Oh hmm

modern trench
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Or is this unlike a category and the actual objects are important

tight star
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I mean both are

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You start with a function

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This maps inputs to outputs, so vertices to vertices

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Naturality is about upgrading this to map edges to edges

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And it turns out lots and lots and lots of mathematical constructions are “natural” in this sense

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In a sense, this is the simplest possible kind of “homomorphism”

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And in my experience, people understand the concept quite quickly even without any prior exposure to category theory

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Cause people can usually pick up basic graph theory quite quickly

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This is how I aim to present naturality in my talk tomorrow as well

tight star
rough agate
tawny slate
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if it were me, id just indirectly or directly teach complex numbers anyways

rough agate
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that works when the students want to go above and beyond

silent acorn
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What is crt

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Nrt

rapid tusk
silent acorn
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Yes i googled about it@rapid tusk

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I am cofused becauae there are two different things

rapid tusk
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what are the different things?

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also i'm not convinced that it's on topic for this channel

silent acorn
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Is it not math pedagogy thing?

vocal phoenix
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Yeah, it seems to be some kind of quiz about ways to teach and evaluate students, so I think it makes sense in this channel.

tawny slate
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would you face some kind of penalty for including it in your curriculum?

rough agate
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yes if the students dont want to learn about it

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which is the case

tawny slate
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what

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why does it matter if the students dont want to learn about it? why do they get to dictate what you teach? i dont get it

shadow flower
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presumably they aren’t going to know if you’re teaching something outside of the curriculum as well

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so they may be more motivated than you might think

tawny slate
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yeah dont tell them they are learning something extra

vagrant meadow
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I hope you don't mind if I steal that haha

tawny slate
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i do want to make youtube vids for all of these ideas eventually but

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it will be a few years before theyre all done

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sharing in the meantime

vagrant meadow
tawny slate
long pelican
cosmic ibex
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Well, if those teachers are being judged exclusively on how many of their students master suh-and-such mechanical skills at the end of the year, it's natural that they would prioritize getting as many as possible over that particular bar, and not worry if they do it in a way that will hurt the mid-range students later.

tawny slate
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the way i motivate my students is by telling them math's everyday functional use is making it less likely youll get scammed or taken advantage of

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then i use all kinds of examples ranging from gambling/expected value, counterintuitive probability problems, birthday paradox, bayes rule, common logic errors (if p then q != if q then p), etc, show how you can exploited in economics, politics, like misleading policy, invalid biases and judgments, money scams, even something as simple as splitting a cake or playing a board game

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i have a mental and physical list of like 50 examples

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so the students really understand that even outside mechanical skills for their career, its useful and important

tawdry venture
tawny slate
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text file, but its very messy

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i can clean it up and post it here

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it also has a ton of other garbage notes related to math haha

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gonna need some work

tawdry venture
tawny slate
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lemme spend some time cleaning it

tawdry venture
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ping me when you do

tawny slate
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oh god this is such a mess, organizing this is so much harder than i expected

long pelican
cosmic ibex
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I'm not sure it's a rebuttal as such. I'm just pointing out there's something causing the incorrect goals.

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There are probably teachers out there who simply don't understand the knowledge they're supposed to teach well enough, but I don't feel qualified to make statements about how large a part of the problem can be attributed to that.

long pelican
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I agree, well somewhat. The people designing the NCLB stuff and tests had a choice of what skills to test, and they decided to test mechanical skills, and later on when they were asked to design test questions for higher-order thinking, they made a mockery of higher order thinking because they frankly did not have the skill to design such questions. So you could argue the cultural belief that math is mechanical skills is the cause of the tests being focused on the wrong skills, leading to teachers being pressured to have incorrect goals

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You need a baseline level of creativity to design questions with enough variety to beat the Goodhart's law effect

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(https://en.wikipedia.org/wiki/Goodhart's_law)

Goodhart's law

Goodhart's law is an adage often stated as, "When a measure becomes a target, it ceases to be a good measure". It is named after British economist Charles Goodhart, who is credited with expressing the core idea of the adage in a 1975 article on monetary policy in the United Kingdom:

Any observed statistical regularity will tend to collapse once...

cosmic ibex
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Yes, that's essentially the argument I'm making vaguely hinting at.

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That, and the fact that beating Goodhart's law requires the tests to have enough variety from year to year that students (and their parents) are going to complain that they are unfair, possibly related to a cultural belief that hard work should be sufficient to excel at tests.

long pelican
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I think it's possible to write down a more accurate set of skills that we want people to have, in such a way that it's clear to parents why we need diverse questions

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Some ideas of how this set of "standards" might go to get started:

  • Considering small cases of a problem to understand its structure
  • Applying definitions accurately
  • Coming up with logically equivalent statements of theorems
  • Finding gaps in arguments
  • Finding patterns
  • Debugging mistakes (e.g. finding where your arithmetic error was)
tawny slate
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the list you came up with is interesting, its like a concrete set of abstract skills

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i would add "identify biases and fallacies"

tardy ember
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...i'm not sure that's a mathematical skill? it sounds more like just general rationality advice

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(so still a good thing to teach obviously, but it would be a weird thing to teach while teaching mathematics)

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i guess there's the common wrong arguments like messing up with quantifiers or "p -> q, therefore q -> p"

tepid smelt
# long pelican Some ideas of how this set of "standards" might go to get started: * Considering...

However, high-stakes tests like the SAT, ACT, and AP exams often emphasize mechanical skills over these conceptual understandings. These tests are typically structured to evaluate quick, procedural responses rather than deep, analytical thinking.

I agree it's a problem but we are heavily pressured to push mechanical skills. Often even with large gaps in knowledge I can get kids to learn various standard algorithms to have success on tests.

tawny slate
long pelican
#

Goes to show how philosophy is also hugely missing in the school curriculum as well

#

(Since this is the first time I've said the word philosophy, I want to say once and for all that every mention of the word "philosophy" means philosophy skill, not philosophy knowledge as in being well-read in Hume, etc)

midnight scarab
#

Fwiw the French have philosophy but it's just about memorising a bunch of quotes and trying to sound clever

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Also my prof was an imbecile

tepid smelt
#

As a teacher in an inner-city school with primarily poor Black and Brown students, I'm often torn. Many of my students have knowledge gaps but are highly motivated to succeed. I know that focusing on standard algorithms and efficiency in solving key problems can help them perform well on high-stakes tests like the SAT, ACT, and AP exams, which is crucial for their future opportunities in applied science or engineering fields. However, I also believe in the importance of developing deeper conceptual understanding and critical thinking skills, which are vital for long-term success and adaptability in any field.

long pelican
#

Funnily enough that reminded me of a Calc 2 student I once had from France; she told me they did proofs [and my eyes lit up] but that were mostly to memorize [and my eyes lit back down]

cosmic ibex
#

However, high-stakes tests like the SAT, ACT, and AP exams often emphasize mechanical skills over these conceptual understandings.
That's part of the problem too. But there's a bit of a chicken-and-egg problem in getting that to change. The creators of tests like the SAT worry a lot about avoiding (preceived or real) socioeconomic biases in their tests. If they were to suddenly start testing conceptual understandings where one can make even a halfway convincing argument that schools in disadvantaged communities are statistically less likely to even attempt to teach that, that would not be politically sustainable.

long pelican
#

If you simply make it so that rich kids with parent-hired tutors teaching test-taking strategies would struggle to answer the questions, it'll be hard to argue that there's socioeconomic bias here

#

Or rather it'll be easy to disprove it

#

Although those same rich parents (especially the ones who are lawyers) might make your life hard by arguing some other kind of bias

cosmic ibex
#

The rich-kid tutors could quickly switch to teaching concepts instead.

long pelican
#

Sounds like a win win!

#

and then the socioeconomic bias that results won't be worse than we have now

cosmic ibex
#

Only if you make sure poor kids reliably get taught conceptual understanding too. At the time you make the change there'll be a whole cohort of them already halfway through a school system that doesn't. The ones who can afford to hire tutors will have a much better chance of reorienting themself.

long pelican
#

Notice how this presents friction for any type of change toward conceptual thinking, so a problem worth solving regardless of what happens to this particular idea

austere delta
#

Kind of a long wait though...

cosmic ibex
#

Yeah, I'm not saying one should just despair of ever making anything better. Just pointing out that even wide conceptual agreement about what a better steady state would be, doesn't automatically entail that things will happen.

#

And that the lack of things happening is not necessarily because any particular persons in the entire system want the substandard status quo.

tawny slate
#

i mean, we sorta tried that with common core, and look how thats turning out

#

the principles of common core are great in theory, but in practice parents are complaining now because they dont know how to help the students with their homework, being victims of the old crappy system

#

or the work given to students starts making up random terminology that is nonstandard or non-conventional which makes it even harder

#

turns out applying these principles requires a lot more planning than we thought, and more critically, better teachers

rough agate
#

one question uh

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how do you all explain complex numbers ?

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i did it by talking about real numbers as a one dimensional number line, and complex numbers being a two dimensional extension

#

then going into rotation and that sort of thing

#

but im told that its difficult to understand that way

tawdry venture
#

thats a good visual but as far as symbolic manipulations go you might wanna stress that i can be treated as just another variable name

#

only that it obeys this defining rule of i^2 = -1

tawny slate
#

my approach (adapted once again mostly from aops, but with some of my own added flair to connect it to higher level math):

  • first define i, say that we used to say this value doesn't exist, but we will simply make it exist by sheer will. make it practical and meaningful by saying that i^2 is -1

  • at this point students will probably question what is the point of making up useless numbers, maybe even questioning whether such a thing even constitues a number. challenge this in the abstract by pointing out that even though we cant have i apples, that doesnt make i an invalid number. you cant have 1.2 people either, but 1.2 is a valid number. numbers are valid when we find some useful application of them. mention that for now, students should focus on just the basic properties. because if you just rattled off a bunch of uses, they wouldn't get it or care. its like not knowing how useful a tool is until you know what it does. hopefully this temporarily assuages their concerns and perks their curiosity, now you have a promise to fill

  • next, mention that you can't just define anything arbitrarily, because it might lead to contradictions (feel free to use your favorite example, k/0 or whatever), so we need to first check to see that everything we expect to be able to do with the real numbers we know, we can also do with this new number

  • start by clearly defining the terms "real", "imaginary", and "complex" numbers. the first gotcha is the number 0, which is the only number that is both real and imaginary (technical definitions help in the long term imo). the second gotcha is that all of the numbers you provide should be complex in addition to real/imaginary, just as how integers are also real

  • at this point, you can bring in the complex plane. mention that it is the real number like, but just made 2d by x+yi <=> (x,y). it is now easy to see that the real number line is embedded in it, the imaginary line is the y-axis, and since they only intersect at one point, 0, thats why it is the only number both real and im

  • now put aside the complex plane for the time being and focus on arithmetic properties. make sure they know how to add, subtract, multiply, find conjugate (gotchas are like conjugate of 5 is 5), divide. point out that you cant compare two complex numbers, like you cant say w>z if one is nonreal, without breaking inequality rules we are familiar with, so this is actually the one major property we lose

#

  • from here, you can bring the complex plane back in, and show the students that there is a geometric interpretation of adding complex numbers. there is also a geometric interpretation for multiplying complex numbers that i would recommend preparing in advance with animations. now it is very clear that adding is translation, and multiplication is rotation/dilation

  • now that your students know how to interpret complex numbers visually, introduce polar form, but dont use any of the standard notation, like euler's formula or cis, just use $r \angle \theta$, show how easy multiplication is:

burnt vesselBOT
#

CosmoVibe

148282132781858816.png
tawny slate
#

$(r_1 \angle \theta_1) (r_2 \angle \theta_2) = r_1 r_2 \angle (\theta_1 + \theta_2)$

burnt vesselBOT
#

CosmoVibe

148282132781858816.png
tawny slate
#

you do not need to introduce trig at this point, just use an approximation or simple examples to illustrate the point. if you want, maybe just briefly mention that trig will help find exact values easily, if it wont scare them

vocal phoenix
#

Yeah, there's a very nice geometric proof that multiplication of complex numbers causes the arguments to be added.

#

All you need is the fact that i^2 = -1 and some basic geometry

tawny slate
#
  • now its time to explain why we care about complex numbers. first point out that the translation/rotation/dilation all being embedded so trivially in the 4 basic operations means you can now do all kinds of fancy 2d graphical operations with ease. imagine you took a picture and wanted to rotate it 90 deg, just multiply each point by i, done
#

so complex numbers are great for 2d graphics

#

another great use is to use a single complex number to describe a wave. notice that polar form, like rectangular form, has two properties, mag and arg (i mention argument is the technical name but for simplicity i just say "angle") instead of x and y, but unlike x and y, arg is restricted and loops

#

show how this makes it great for describing waves (mag is amplitude, arg is phase), and that almost all modern wireless technology and much of electronics in general involves doing math with waves

#
  • you can even introduce the abstract math properties that make complex numbers special. if they have some exposure to polynomials or at least even quadratics, explain how beautiful the fundamental theorem of algebra is. mention that while real numbers have closure under almost all of the major arithmetic operations, it does NOT have closure under exponentiation ( -1 to the 1/2 power ), but complex numbers do!
midnight scarab
#

For the last point, there's still the principal root caveat which makes rational powers of non-positive numbers somewhat delicate

midnight scarab
vocal phoenix
#

This one

complex-multiplication-1.png
#

I'd have used different colors because the white triangle is similar to the green one on the left

#

And the upper green triangle is similar to the blue one, so I'd probably have used another shade of blue for it

tawny slate
#
  • if you want, if the students already understand functions and domain/range, you can include an eli5 example of analyticity:

consider the following:
f(x) = sqrt(x-4)
g(x) = sqrt(x)
notice the domain of f is the set >=4, domain of g >=0
so domain of f(x) * g(x) should be the set >=4, right?

however, if you do multiply f and g to get sqrt(x^2-4x), you will find that the domain is the union of the set <=0 and the set >=4. where did the negative domain part magically come from??????? we know that you can arbitrarily define and extend a function, and yet this just seemingly appeared by itself out of thin air

the key to understanding this mystery comes from understanding complex numbers

tawny slate
# vocal phoenix Yeah, there's a very nice geometric proof that multiplication of complex numbers...

the example i like most is: https://youtu.be/-dhHrg-KbJ0?si=1SCmbsbn2nJSn-Ua around 10:30 ish

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e to the pi i for dummies

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▶ Play video
#

because you can apply this idea to first understand multiplication on the real number line and then it extends very naturally

#

and then theres always like the shock factor you can give your students by just dropping things like e^i pi = -1 or that i^i is real and has an infinite number of values with no explanation

#

that should be enough for now i think

long pelican
#

In fact I think even common core was built on top of several wrong preconceptions at the start

tawny slate
long pelican
#

Two misconceptions that I can communicate right now

  • Math is important because it prepares students with mechanical skills needed for their future careers
  • Nothing can go wrong if you divide math into skills and measure math proficiency by measuring those skills
tawny slate
#

hmmm last time i read common core references i didnt feel like this was communicated

#

is this just something the policy makers personally believe or is it like explicitly in the principles written somewhere

long pelican
#

Well you do see a lot of language about higher-order thinking in the common core documents, which is evidence a lot of people involved believe in it, but this like all things designed by committee ended up being a document only consisting of what all of them could agree on

#

Also there was a lot of emphasis on research- and evidence-based assessment which really limited what they could test on

tawny slate
#

i see

#

thanks for your input

cosmic ibex
rough agate
#

I kind of just got around the "isn't it just mathematicians inventing arbitrary things" by saying that imaginary numbers have a lot of real world applications such as electrical engineering and quantum mechanics

rough agate
#

that's much clearer now

#

thank you very much

tawny slate
#

oh @rough agate sorry to ping and bother again, just remembered something

simplifying 1/i or solving i^k for any integer k is a really great instructive problem for several reasons:

  • it gets students into getting used to the difference between i and -i
  • 1/i has many different methods of approach (multiply top and bottom by some power of i, sub 1=i^4, etc)
  • it provides an opportunity to show what happens in the complex plane as you multiply by i, and can be explained as rotations
  • this simple example of rotation can be a good lead into the more general rotations when introducing polar form later on

wanted to point out this problem in particular because i think it is really important early on, in fact i may even introduce this before complex number addition

native sail
#

What's your opinion on tough and terse books where you have to fill the gaps by yourself vs books that are well motivated and explain everything thoroughly?

tawny slate
#

i dont mind filling the gaps if it is reasonable for me to know the prerequisite, but im less happy about needing to prove key theorems on my own

#

i think what is more frustrating is when they give you exercises with no answer key or solutions to compare to, that drives me up a wall too

#

well motivated and fully explained is always the gold standard imo

native sail
native sail
#

but maybe they take some of "hard work" away?

rough agate
#

i mean if you bring in the grid early enough then yuo dont really need to explain the difference between i and -i since its already built into that layer of intuition

cloud zealot
tawny slate
# native sail but maybe they take some of "hard work" away?

my feeling is that it's about the purpose
like if you have an instructive problem or a proof to a theorem that will be used later, i think these should be fully written out clear proofs
the things that you can avoid writing out and have the students do "the hard way" are like exercises that supplement the material and improve intuition and familiarity. even then, full solutions as an answer key are better than not having them

obviously there is no hard line and it's subjective at times, but that should be the motivation for whether or not something should be written out imo

minor turtle
#

and maybe another one at the very heart of motivated and flowery and lots of prose (aluffi đŸ©·)

#

if you try to be in the middle you get something like fitzpatrick advanced calculus or axler LADR

#

if you get close enough to the middle you get the average calc 1 textbook

native sail
#

Specially if you've seen the essence of calculus

minor turtle
native sail
#

wdym?

minor turtle
# native sail Specially if you've seen the essence of calculus

they’re at least written in a way that’s not motivating for my calculus 1 students, i don’t have a specific example of an average calculus 1 textbook in mind, but i think the one that i used was a mix of half-hearted attempts at motivating equations that they just threw at you. also, the essence of calculus video series is pure exposition, and is notably not something that you’ll actually learn calculus from

native sail
#

At least in the US

#

I read thomas and thought that it explained things nicely

#

I rarely saw something unintuitive

#

but that's my experience

#

also maybe people have more difficulty with calc1 not because of the books, but because of a lack of background in basic mathematics?

minor turtle
# native sail also maybe people have more difficulty with calc1 not because of the books, but ...

perhaps that’s part of it, though i cant help but think that the perfect calculus book has yet to be written in any case. i think if someone were to write a calculus 1 textbook like grant sanderson wrote the essence of calculus, with a clear narrative etc. then i would actually recommend it to students, but i don’t think such a book has been written (maybe stewart is the closest i’ve seen to something like this? but i think stewart’s book feels more like a reference book than anything i would actually want to engage with and learn from)

hollow musk
#

I've found Spivak's calculus book to be quite well written and motivated

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but I suppose that book borders on the edge of what can be called a calculus textbook

minor turtle
# minor turtle perhaps that’s part of it, though i cant help but think that the perfect calculu...

here is a random page from stewart, where it defines the integral, and i think it sort of exemplifies the “half-terse” idea and why it’s bad. not one student is happy here, if they don’t already know what this limit means intuitively, the “precise meaning” statement is useless, and the student will just glaze over the remark because it’s totally unmotivated, and doesn’t have any follow-up. then there is a section about the history of the integral (what happened to this precise definition section?? are you just gonna leave it there?)

IMG_0924.png
#

it really felt like this was just a checkbox for a curriculum that they wanted freshman engineers to go through, and so stewart said “let there be integrals” and so there was

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honestly i feel like i’m not doing a good job in getting across why i don’t like the writing as well as i want

long pelican
#

The result of trying to make (naive) mathematicians and curriculum educators happy at the same time

#

I could probably sum up the poorness of the writing like this: it's laughably out of touch with the audience

minor turtle
#

yeah

rapid tusk
#

it’s even worse in the multivariable chapters where practically EVERYTHING gets handwaved away

#

there are very incomplete proofs of trivial cases of the really meaty results

#

all extremely unmotivated

tawny slate
#

i dont think a "perfect textbook" for anything exists

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there will always be people for whom some changes to that textbook would make it better for them

cloud zealot
drowsy otter
#

I have a question. I was hoping to ask these RUCSAC questions to my student (who is in Year 7 but working at a Year 6 level). How do you reckon I should tweak these?

I want everything to be clear so as not to invite arguments, and I also want it to be easy to understand so as to not confuse the student.

image.png
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I'd particularly like feedback on the question in red please.

noble hare
midnight scarab
#

Maybe it's clearer if you just say he took a break/leave from work

drowsy otter
#

Keep in mind it's a primary level RUCSAC question. I might consider this afqt, but I'm looking for something simpler.

midnight scarab
#

I have no idea what RUCSAC is

#

I just feel like "is allowed that much time" is a bit weird

#

Wdym allowed? By who?

drowsy otter
#

I'm going to explain each point in turn:

#

RUCSAC is a step-by-step method on answering maths questions at primary level:

#

R - Read the question
U - Understand what the question is asking
C - Choose the correct calculations
S - Solve the problem
A - Answer the question
C - Check your work

drowsy otter
daring gulch
# midnight scarab Wdym allowed? By who?

Now I like to think the one answering the question is the one giving them time. Too little, and Brett gets sick and dies. Too much, and Brett becomes a lazy bum and never goes back to work.

tawdry venture
#

i think the red one with sevenths of an hour is a bit icky -- who ever heard of time being phrased like that?

#

i think it's better if you do something with money, maybe like this:

Brett got his allowance on Monday. By Tuesday, he had 5/7 of the original allowance left over, and on Wednesday he spent another 2/5 of the original sum. What fraction of his allowance is he left with?
daring gulch
#

I'm kind of curious if there's any argument for making elementary school math more... accelerated? I wouldn't say "rigorous," since I take that to mean thoroughly explaining everything. I remember still doing multiplication drills up to the 5th grade (Florida, US). While a firm grasp of arithmetic is certainly very important, I can't help but feel the more abstract elements of math could have been introduced, or even some of the conventions used in algebra. I've always been above average in the learning department, but I had little challenge doing elementary math when I first learned it. The shift to algebra I felt was quite abrupt, with little explanation of how it connected to all the previous math I'd done. It was only in hindsight that I now know how useful all the drilling, fraction manipulation, etc. was, but I feel that should be the case before learning a topic, not after. Though, I will acknowledge the previous foundations can only be fully realized whilst doing the math they were preparing one for What are your thoughts?.

rapid tusk
#

accelerating can only do so much when students already tend to struggle with the current pace

drowsy otter
tawdry venture
#

have they shown themselves not to be good with days of the week and shit

drowsy otter
#

I think they know the days of the week. What I meant was, they might think that it's a 3 parter, rather than a 2 parter.

#

The question is basically "Compute 5/7 - 2/5", but using RUCSAC.

tawdry venture
#

what's the difference between a "3 parter" and a "2 parter"

drowsy otter
#

A 3 parter would be:

#

5/7 - 2/5 - 1/3 (for example)

#

I would never do 3-parter's, because the student is not ready for that yet.

drowsy otter
#

Whereas a 2 parter would be something like:

#

4/5 + 1/8

#

(for example)

tawdry venture
#

if you're ok with having another addition problem in lieu of a subtraction one:

Brett spent 3/7 of his allowance on a new book and 2/5 of his allowance on a Lego set. What fraction of his allowance did he spend in total?
drowsy otter
#

I think I'll chance it with the subtraction one. I'll explain that it's a 2-parter, and give the student the chance to choose the correct calculations.

#

Chances are, they'll guess add or take-away of those two fractions.

daring gulch
# rapid tusk accelerating can only do so much when students already tend to struggle with the...

That's true, and not much more can be added to what was already said about matching the difficulty of the material to the abilities of each student. But I think some things can be said about the disconnect between elementary school maths and middle school maths (up to prealgebra and maybe algebra). This is an assumption, but I imagine most elementary math teachers haven't done vigorous algebra problems (problems that require a moderate effort) in several, perhaps even dozens of years, much less have any amount of a rigorous background in mathematics. How can they connect the maths elementary students are doing now to the algebra they will be doing several years into the future? It's possible they don't know how to. Maybe the disconnect can explain why so many students do poorly when taking Algebra 1 in 8th grade.

rough agate
# daring gulch I'm kind of curious if there's any argument for making elementary school math mo...

one thing i find myself saying at least once per month is that "I could teach calculus to a year six class". While it's mostly tongue in cheek, i feel that there may just be a fragment of truth to it.

Certainly, for the mathematically inclined, elementary school math is way too slow as is, and there would be no resistance to making it faster for them, however the one issue is that you have everyone doing math at that level, so you can't just power through for the sake of a few people and then leave the rest behind.

The obvious solution I see would be to have an extension class for those who are capable of going ahead, as it makes no sense to hold them back for the sake of ease of teaching, but (at least where I live) there is a bigger issue. Elementary school math teachers do not need to be certified in mathematics to teach it. Anyone with an education degree can land a job teaching elementary school maths. While anyone is able to do elementary school maths, explaining maths in a way that catches the student's interest and helps them develop a depth of understanding isn't something that everyone is able to do.

Where I live, algebra is taught conceptually to higher elementary school year levels, in ther form of worded problems where you work backwards to figure out an initial amount, but variables only get introduced in middle school, where there is a jump in difficulty as you have y=ax+b in the same year that you formalise algebra.

In short, we could get away with accellerating elementary maths a little, and a decent amount if elementary maths (especially after basic arithmetic) was taught by trained math teachers. but if you're really looking for a quick fix, breaking up classes can give you a lot of room to work with.

quasi musk
cosmic ibex
#

(cue Tom Lehrer)

quasi musk
#

It's a good song

daring gulch
daring gulch
#

Actually I can elaborate, just a little bit. I think it would be immensely helpful to introduce things such as the algebra of equations without variables. I mean, many students learning algebra have trouble understanding even that "what is done on one side is done to the other." Why not focus on something like that before pre-algebra/algebra before slapping y=mx+b in your face? I that I think it would be a very natural progression to go from manipulating equations like 2×4=3+6/4 to 2x=3+6/4, and go a long way for automaticity.

tawny slate
#

i disagree that learning about bases other than 10 is unnecessary

#

not only is it applicable to comp sci with binary number systems

#

it also gives students a better grasp of the "essential nature" of our number systems. what i mean is like for example:

  • by learning arbitrary bases, you realize that 10 is chosen pretty arbitrarily, and so students learn to wary of anything that attaches extraneous meaning to bases, insulates them from numerology and other similar woowoo, makes them aware of cultural biases
  • it makes them aware of how to handle certain kinds of pattern matching and problem solving, such as realizing that if an argument might work for base 10, perhaps a general base or a modular arithmetic approach works
  • it also makes a lot of different ideas much much easier to understand. you don't have to memorize the divisibility rules and suddenly they extend powerfully, the geometric series derivation is hilariously trivial, etc
#

also, a better understanding of the number system we use in general is a key step to better understanding exponential growth too, which i think is one of those abstract ideas that is super critical for everyone to know

tawny slate
#

as for algebra of equations without variables, i teach in pre-algebra by using a weight balance scale as an analogy. adding to both sides of an equation is putting the same amount of weight on both sides

rapid tusk
#

weight balances are a good way to introduce equation manipulations yea

daring gulch
# tawny slate i disagree that learning about bases other than 10 is unnecessary

For elementary students? Not at all, I don't think. It's too much effort for a "side" topic (take that as you will) which doesn't offer as much foundational knowledge as to the main topics that are currently taught (arithmetic, fractions, properties of shapes, etc.). For middle school and above I think there may be some merit. Otherwise, I agree that learning how other bases work can indeed be very enlightening.

tawny slate
#

ohhhhh idk how i missed that important piece of context

#

for elementary students maybe a bit early yeah, not sure how i missed that, my bad

cosmic ibex
#

It's not a all-or-nothing question.
Do kids deserve to know that base ten is essentially an arbitrary convention and it would work just as well to use another base? Absolutely! (Though possibly with more kid-friendly words than "arbitrary convention").
Should kids be drilled in practically performing subtractions in base eight? Absolutely not!

#

Tom Lehrer's song lambasts the latter.
I hope it's an exaggeration for parody.
But it also feels like a depressingly predictable outcome if you force a new list of topics upon a teaching profession without also engaging with their existing notions of what kind of activity "teaching math" is.

long pelican
#

I have to say Tom Lehrer’s explanation of base 8 subtraction made a lot of sense to me, maybe against what his intent was

tawdry venture
#

you sure that's not bc you have a greater background

daring gulch
midnight scarab
#

Though this is unfortunately not always clear to educators, exam authors and gov officials...

#

Which I think was the point Troposphere was making at the end

long pelican
#

or rather, I don't think I fell prey to the expert's curse when making that comment

cosmic ibex
#

Ideally, I think, it could be taught to the point where students can say

Yeah, I can see that makes sense, and that if I were to put sufficient effort into memorizing plus and times tables in the new base, it would work just as well as base ten. I'm not going to make that effort because that would be silly. But I have tried the algorithm once or twice where I laboriously worked out each one-digit result by counting fingers/pebbles (or perhaps converting back and forth from base ten where I remember the results), so I know firsthand that it seems to work.
Unfortunately, if this means it's on the test, students and and teachers alike will be incentivized to spend far more time on it than just to get to that point.
And even if it isn't on the test ... well, we'd be up against a deep-seated cultural assumption that whenever the math teacher tells students to do something on their own, it's because they're supposed to remember a procedure for doing it by rote. So it's not something one can just up and start teaching -- without also mounting a far-reaching effort to break that cultural assumption.

hollow musk
#

Out of curiosity, I want to ask: Do you all think that word problems provide motivation for students learning mathematics in high school? Also, pedagogically speaking, are they useful in terms of getting students to internalize new concepts?

#

Furthermore, I've also always felt that many of the word problems given in high school settings seem a bit contrived (e.g. problems in 3D trigonometry where it's just a bunch of angle chasing shenanigans, but dressed up in terms of friends looking at each other from towers or something). Is there a way to design better word problems?

cosmic ibex
#

I find it pretty implausible that a word problem in itself would motivate anyone to learn mathematics who was not already motivated.

#

It's a relevant skill in itself to be able to start with an understanding of a real-world situation and find a useful way to calculate stuff about it by using such-and-such mathematics. Arguably, if you're not able to do that, you haven't really learned the math in question, even it you're able to solve a problem that is already stated in technical terms.
The pedagogical trouble is that this skill assumes that you already grasp the real-world situation well enough to start trying to match some math to it. And fairly hard to describe a situation where 3d geometry is useful, well enough for students to reliably grasp it, especially when -- for practical reasons -- your description has to be in terms of words and/or drawings on flat paper.
Sightlines between towers is pretty contrived, but at least it's a situation that minimizes the risk of students getting lost because they don't understand what the situation you're trying to describe even is.

hollow musk
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Fair!

tawny slate
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yeah its like a loop you somehow need to break. students dont care what X tool is used for because they dont see how it can be meaningful to them, but they cant understand how a tool can be meaningful to them until they at least know what the tool does, and thus it defeats itself

when i try to motivate students, i try to prioritize finding some kind of demonstrable application, like a science experiment.

if i cant find that, i try to find a philosophical argument to care about it (yes, this does sometimes result in a philosophical discussion with the students, which i think fills a related but different niche that is important, and I am also aware that i am very fortunate to be one of the very few teachers who can even afford to spend this kind of time on this)

the last resort is to try and gamify the concepts somehow and make it more "fun" in the kid sense of "fun", fun for fun sake

its definitely not easy, young students who dont have a developed enough view of the world and reality are often times not going to be motivated beyond their own interests

daring gulch
hollow musk
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For what it's worth, I've always liked the problems presented in high school physics classes more than the ones typically presented in the math classes!

daring gulch
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Genuinely if kids were taught how to solve "word" problems the way we solve physics problems it would make so many things easier.

rapid tusk
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the state of "word" problems in the k12 curriculum is absolutely horrendous yea 😭

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for example AP calc likes to ask "rate in rate out" questions where you have like a tank of water or a parking lot or smth

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and the rates of entry/exit are these completely nonsensically contrived funcs

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why yes, cars will definitely exit at the rate te^(5.9333*sin(t^2))!!111!

#

"This can be seen in the many topics that are taught without any credible motivation provided for the reader. For one thing, the book is packed with supposedly "applied" or "real-world" problems designed to capture the reader's interest, which are easily seen to be intellectually bankrupt upon the slightest critical examination. Like a politician telling blatant lies and counting on people being too stupid to see through them, so also Stewart tries to fool you with cheap smokescreens into thinking that his problems have real-world relevance. Most ridiculously, of course, this is seen in the gratuitous full-colour stock photos that are peppered in various places. For instance, pre-med students constitute a significant portion of calculus students, so let's appease them with some problems about blood pressure and drug concentrations and so on, with some pointless stock photos of blood vessels and whatnot taking up a good chunk of the page, as if the book was not bloated and expensive enough without them. These problems, like so many other "real world" ones, consist of nothing but plugging numbers into formulas. The generic template for such problems is: "The function so-and-so represents something-or-other. What's the something-or-other when the value of the variable is 53?" The disastrous message to students is clear: applying mathematics means plugging numbers into formulas. It clearly does not involve thinking about why the formula is that way in the first place. It also clearly does not involve any kind of qualitative reasoning or any kind of conceptual conclusions. Applying mathematics simply means: when I plug this number in, that number comes out."

daring gulch
rapid tusk
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đŸ€Ż

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i had the misfortune of taking aLgeBrA bAsED phYsiCS in high school

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completely toothless unless you actually, you know, have the calculus tools you need for it

daring gulch
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I took AP Physics 1 and it was highly motivating for me to take AP Physics C the next year.

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Unless you're talking about the high school physics classes which skim over the entirety of mechanics and E&M, which I cannot comment on.

daring gulch
# tawny slate yeah its like a loop you somehow need to break. students dont care what X tool i...

There's not really any great way to make early math (algebra 2 and before) itself "fun," imo, due to the lack of understanding of mathematics as a whole. Nor is there a particularly excellent way to demonstrate the applications, though they are both certainly better than nothing. What the greatest emphasis should be placed on in motivating students, I think, is making the learning itself fun and engaging, however that is highly abstract and can mean many different things to many different people. In general, I think it's reasonable to say that most K-12 teachers are simply mediocre or plain not good with connecting to their students on the topics they are teaching. What can be done to address this in curricula and how teachers are taught? It's difficult to name anything specific (but also because I shouldn't be writing this late at night), particularly some things that aren't just jabs at individuals ("Your personality sucks!"). Or maybe there is. Or maybe there isn't. Or maybe there is. Or maybe there n... Guess a general law for the statement and prove by induction.

quasi musk
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"When am I going to use this" "Ah yes! We can use these equations to model diffusion of a gas, or population of bacteria! Neat!"

daring gulch
tawny slate
# daring gulch There's not really any great way to make early math (algebra 2 and before) itsel...

i agree that the teachers arent good enough, but im not gonna expound too much due to the complex nature of things. i dont want to insinuate teachers arent trying hard enough or assign blame when there is so much that is flawed on such a basic level

that being said, i think there is a vast ocean of ways to make early math fun. i think it is difficult for sure, but i think it is a problem that can be solved by throwing enough resources at it

abstractly, its like with AI or math in general, this is an absolutely monstrously sized solution space that is poorly explored, and i think there is not only not enough attention put to this area, but also adults often times forget what it is that makes kids tick, they're out of touch. combined with the fact that to make early math fun you need a combinations of skills and disciplines to understand and evaluate and iterate, i think its not a surprise we haven't made much progress there, but i definitely think it is a problem solved by commitment and resources

like lets just assume that you wanted to make a board game, which is only one of many methods of approach. i think even few mathematicians have extensive experience in the modern board gaming space enough to be exposed to plenty of very minimalistic design spaces that are both entertaining as well as being an excellent source of analysis and problem solving. skull, take 5, no thanks, fugitive, the list goes on and on. the massive space of ideas here that could be applied to early math is merely the tip of the iceberg

I have independently discovered too many alternative methods of teaching some of the most basic math to believe that there is little to be done here

vagrant meadow
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alright im about to expose myself for being a complete idiot.

why do we make math texts difficult to read? more specifically, why do we make the problems difficult, or not include solutions? many of the problems in the textbooks i've been using for my grad classes (Folland and Evans mostly) have been really difficult for me (skill issue).

is there actually more value in a problem that just says "prove this" with no substantial hints or build-up? no solution or answer in the back? personally, i never felt there was. usually the hints are either useless or vitally necessary and part of the challenge of the problem is figuring out how to even use the hint. if you're self-studying, either get good, ask on math stack exchange (and prepare to be roasted because this isn't obvious to you), or find some random person's solutions online and hope they're comprehensible to you.

i'm just thinking from the perspective of how two years ago someone gave me a copy of Evans to learn PDEs during the summer before i took a class in it at my UG. and if i had actually taken the time to try and go through that textbook, the exercises would have been completely useless to me because i didn't know enough analysis (especially no multivariable analysis). so even though i knew ODEs really well and was interested in PDEs that text did nothing for me (it also was a real slog to use this year as well). and yet, it's one of people's most common recommendations.

and then, even on this server (and i feel is a general sentiment in the math community), there's a pressure to be as vague as possible when helping someone else. someone will just tell you "go back to this definition" with no elaboration. when i give problems to my students, i try to make them very straightforward and break them up into simple parts that build up to the result. why don't we do that more often? why is that seemingly discouraged? is this just a new idea? are math students just dumber than they were 20 years ago?

am i just stupid? am i just bad at math? is this just a hard skill issue? do people actually consider Evans good? why do people write textbook problems like this? sorry for the rant, i feel like i'm going crazy.

austere delta
# vagrant meadow alright im about to expose myself for being a complete idiot. why do we make m...

I don't think anyone purposefully makes textbooks hard to read, but specifically getting the right difficulty of exercises can be tricky. What one finds difficult varies from person to person, depending on their prerequisites, how much time they have/are willing to spend, and general problem solving skill.

I guess many textbooks are also designed to be used in a class room setting, in which case you don't want solutions, because you might want to use the exercises as homework. And you might expect students to receive more personalized hints at their level (from a TA or similar).

As for the value in problems of the form "prove this", if the tools you have learned about this section are the tools needed to "prove this", then certainly that would be an appropriate problem. And if the problem is difficult then adding a hint, making the main part of the problem into how to apply the hint seems totally reasonable.

If the hint is useless, then that's either a badly designed problem, or you just solved it in a completely different way than the author expected.

I'm not familiar with Evans, but I would imagine it be difficult to learn PDEs without knowing multivariate analysis. Is the book specifically aimed at people without such prerequisites?

I'm not sure the pressure in the community is to be as vauge as possible, but rather to give hints instead of solutions. If someone just gives you the answer, then they take away your learning outcome from that problem. If one hint is not enough to solve a problem you could always ask for more, but if you're given too much of a hint you can't unhear them.

#

You're not stupid, exercises often are difficult. And at times probably too difficult. I don't think there's anything particularly new or old about this problem. Many text books have solutions to selected exercises or odd numbered exercises, so that you both have exercises were you can look up solutions and ones that are not spoiled. Many have a wide selection of difficulties, and certainly many contain at least some problems that are not that well designed.

tight star
midnight scarab
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At my uni, there can be similar problems on exams, exercises (which are done in-session: we get some time to think and then usually one of the students presents their attempt) and homework. What distinguishes the three is precisely the level of guidance. With exams obviously having the most, cause they should be solvable not only but the few best students (except for specific questions) and in a reasonable time

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The point being, of course, to be able to learn to solve problems somewhat autonomously

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But I agree that often you first need to get how a subject works, what the key ideas and methods are etc. before you can be autonomous

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Which is often not taken into account by the problem writers

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At least in an exercise session, we get the solution after a reasonable time of having thought about it

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So we can learn from it and try to apply it to the next problems

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Doesn't work for self-study though...

tawny slate
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i want to say firstly that i empathize, i have also sifted through lots of texts that explain something with a lot of difficulty or feel obtuse or dont have solutions

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that being said, having written some basic short texts, it is incredibly difficult to find that right balance between spoonfeeding answers and helping the students develop their skills and habits

there are certain things that are very difficult to teach explicitly, like abstract intuition, and certain things are flat out impossible to teach (muscle memory). you want to be clear and thorough but you also dont always want to handhold too much

prime gale
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like i think this motivation varies alllooottt on students many ppl like the more abstract side of math

prime gale
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like combinatroics is the most reachable thing u can do u give very cool problems without much theory[ if any]

tawny slate
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i usually turn to combinatorics and function growth as examples for why we are motivated to do math on a philosophical level

dim flicker
# vagrant meadow alright im about to expose myself for being a complete idiot. why do we make m...

https://academia.stackexchange.com/questions/56739/why-dont-graduate-math-texts-have-solutions-to-their-exercises

I agree with Paul Garrett here.

It seems that, yes, this is a cultural thing in math. If you look up posts on math stack exchange for why many textbooks don't include solutions, some of the most upvoted answers read like moralizing gatekeeping that advocates for everyone having to solve all their problems themselves or else they didn't struggle enough. I'm pretty sure the real reasons for not including solutions boil down to economics (solutions take a long time to write), yet there's a lot of gaslighting that books with no solutions have the most optimal pedagogy because you learn "life lessons".

Students don't learn efficiently by randomly blundering around trying to reinvent tools that took past mathematicians plenty of years to discover or decipher cryptic hints. And you can only really move forward in math by standing on the shoulders of others who already developed conceptual advances to deal with problems rather than being expected to continually reinvent the wheel.

Not including solutions in books may save time, and it may even be an effective way to source problems to use in assessments that are likely not on the internet, but I think it's disingenuous to frame it as a good learning tool.

Academia Stack Exchange
Why don't graduate math texts have solutions to their exercises?

I study mathematics and sometimes I have seen book that contains problems but not their solutions. Why are there no solutions available anywhere?

For example, the books I have read are:
Neukirch:

midnight scarab
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Honestly, I agree

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I also don't like the argument that if there are solutions then students are tempted to look at them without having tried enough

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Like solutions are really helpful

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Depriving students of those "for their own good" is a bit too paternalising especially for late ugrad/grad students who generally have enough maturity to know how they study best

#

And it's precisely at this level that textbooks have zero solutions and leave so many things for the reader to figure out

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Precisely cause writing things out would take them more time and effort than they were willing to invest

vagrant meadow
# tawny slate that being said, having written some basic short texts, it is incredibly difficu...

I'll say I generally agree with the idea it's not good to spoon feed. but to play devil's advocate, I think it would make math more accessible and approachable if the questions were easier. especially in today's world where it seems younger people are more impatient and less willing to bang their heads against a wall for "noble reasons".

imagine if a textbook had problems aimed for a level below the topic. would that actually do more harm than good? I feel like allowing intrigued and interested students who may not have a solid foundation in the topics usually necessary to take a rigorous course in the topic would benefit greatly.

#

this quarter I presented a lot of upper div linear algebra topics to my lower div students but in an accessible way. very few of them will have to take upper div linear (and the lower div class is super lackluster and has no proofs), and I got a lot of feedback that they really appreciated seeing how linear algebra is actually used without having to know how to prove anything.

quasi musk
# vagrant meadow alright im about to expose myself for being a complete idiot. why do we make m...

I used to feel this way as well when I first began taking graduate courses, especially towards the Stein & Shakarchi series. I think Folland and Evans are really well written, it's just that you need to have the appropriate background to be able to read them.

I believe that the books are intended for graduate students with caring professors that can fill in the gaps/details that are left in the material; I also believe that the exercises are more or less designed for you to have office hours if you get stuck.

I think there's a lot of value in working on problems that don't have a definitive answer in the back of the book, but it definitely makes self-studying or learning on your own harder. However, when you do solve a problem on your own, or you start to build up an intuition for how things are going to be done, then it feels very rewarding.

I spent 3 years out of graduate school, just solving Evans problems on my own. There were some where I had a lot of detail, and others where I missed somethings. During the qualifying prep last summer, the grad student running it was caught a little off-guard by how much of Evans I had solved on my own. Even people taking the PDE class this term were a little like "How are you supposed to think about this".

I believe that time off from graduate school where I really only had myself and this server to go based off of, really solidified a lot of my problem solving skills. Another issue is that these topics & problems can take a long time. Inevitably the academic terms compress what might take you 20 weeks to learn, into 10 weeks. Or vice versa, something you pick up in 2 weeks gets talked about for 10 weeks. It's a hard balancing act

daring gulch
prime gale
# rough agate how do you all explain complex numbers ?

the best way is prolly
starting with smth like

" Well we know about the positive integers have u ever wondered adding negative numbers help us solve problems with positive integers isnt that pretty damn cool ? . Well ok now say we want to repeat what more can we add well one crazy guy thought of adding x^2=-1 because it makes a lot of things in math more "nice" [ex:- now every non constant polynomial has roots]"

rough agate
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i mean the issue comes back down to how descartes called them imaginary

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which puts a lot of people off

prime gale
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like generally what ppl say is imaginary numbers help us do things with real nos

prime gale
# tawny slate my approach (adapted once again mostly from aops, but with some of my own added ...

that "next, mention that you can't just define anything arbitrarily, because it might lead to contradictions (feel free to use your favorite example, k/0 or whatever), so we need to first check to see that everything we expect to be able to do with the real numbers we know, we can also do with this new number" is very good like most high school explaination for some reason just skip this over

tidal shuttle
cosmic ibex
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Well, it does invalidate any reasoning based on order.
Even though that's often an acceptable trade-off, it would feel duplicitous to claim to students that it doesn't create any issues at all.

tight star
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yeah, order is exactly the thing you lose when going from R to C

tight star
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this kind of distinction between discovered vs. invented?

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i guess as a physicist i'm intrinsically biased to view math as a tool

tardy ember
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well no in the sense of like

prime gale
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ya thats true

tardy ember
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"we call this function sin and this function cos"

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is just a completely different type of thing from like, factorising quadratics

prime gale
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though the like the word "existence" in maths also dosent become clear in elemenary level its generally thought of as member of a set

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so like when u say compelx nos exist they obviously dont exist in R . [thats why the extending the things which we call "numbers" from positive->negative-> compelx works better and defining axioms for each extended set[so things stay constistent] is better coz that atleast comes closder to construction]

cosmic ibex
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Discovery-vs-invention is a philosophical distinction often not even agreed on by experts who know all the relevant math. So teaching children the "correct" classification of e.g. the complex numbers is probably too large of an ask.

modern trench
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We should teach the philosophy of mathematics

tawny slate
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im not sure there is much pragmatic value in the distinction between whether math is invented and discovered

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at best i think its just a matter of a specific context relative to how we as humans interpret it from like a psychology/pedagogy view

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i would rather students spend time on philosophy of science, art, morality, than philosophy of math

serene owl
tardy ember
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ok yeah i should definitely have phrased that better, what i meant had nothing to do with philosophy at all

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my intended point was that there's a distinction between stuff like "1 + 1 = 2", which does just outright follow from the definitions, "x^-1 = 1/x", which isn't deducible from just knowing how to take positive powers but which is the natural choice in the sense that it causes some nice properties to keep working, and "soh cah toa", which is entirely notation and doesn't really have any significance beyond being what most humans use

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and a fourth thing that very rarely happens but which they kind of act like it's what all of mathematics is, which is "just how the world is for no particular reason", like how physical laws are - impossible to deduce or guess from scratch, but also not decided by humans, just a fact that exists and that you have to know

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(i say "very rarely" and not "never" because arguably "here is the outcome of this computation-intensive procedure that you're not going to run yourself" is in this category)

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so with the complex numbers

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the fact that it's valid to construct them at all, is category 1 - you can just prove that if you take ordered pairs of real numbers and put these two operations on them then they form a field, or however you want to do it

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the fact that we're interested in them is more in the area of category 2 - they do turn out to be useful, but that's about it, it is logically valid to ignore them and just stick to R

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but then a lot of how i've seen them presented is in the region of not even noticing that the first three categories exist, and just saying "yeah we add a number i and declare that i^2 = -1", without addressing whether that's a valid thing to do or why we would be interested in it

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they're treating it as if whatever mechanism causes people to know things about mathematics that before that no humans on Earth knew about (i am using this rather awkward phrasing because if i say ||inventing|| mathematics or ||discovering|| mathematics then it will start another conversation about philosophy even though this is actually just a thing that happens), is some kind of inexplainable magic and not basically just thought

wispy slate
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What would you guys do if you have a bright student but have terrible grades cuz they didn't apply themselves? They said that they only learn math for the sake of learning math, and not for the sake of getting good grades. I get where theyre coming from, but im just really concerned bcz having terrible grades will definitely hurt em in the long run, and i don't wanna see theyre potential getting wasted.

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Sorry if this question is irrelevant

tawdry venture
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bright student but have terrible grades cuz they didn't apply themselves?
sounds an awful lot like this student has ADHD

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or some other mental shit going on

vagrant meadow
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to an extent i'd just say live and let live. there's value in failure and learning from your mistakes. if they understand the consequences, it's up to them to accept them. but grades aren't life and death. i had pretty bad grades in middle/high school, but over time i turned that around and was better for it. i just went to community college where i became more motivated, got really good grades, and then i got where i wanted to be. knowing what it's like to struggle also made me a better tutor, i think.

if this is a pre-uni student, i'd personally say don't harp on about it or nag them, that might hurt your rapport. if they're in uni, then they're probably an adult or close enough to make their own decisions. i'd just focus on teaching/helping them as best i can.

#

that's just my take, though.

wispy slate
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And i'm his older sibling

vagrant meadow
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ahhhhhhhhhh

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yeah i wouldn't worry about it tbh

wispy slate
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But he's going to uni soon

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And he really needs to improve his GPA

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To get into good uni

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He also said that he doesn't see any point in learning the math that they teach in school. He said that hs math is too computational and is basically just memorizing formulas, and plug and chug numbers. He thinks that that aspect of math will soon become obsolete bcz of AI. So he'd rather focus on getting bbetter at things that we as humans are naturally better at, which is the ability to think abstractly and critically, come up with out of the box ideas to a solution, and being creative

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So rn he practically dont follow the curriculum taught in school. Instead, he self studies through textbooks and the internet. He's currently on this book called calculus by spivak

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And he also has this weird perfectionist tendencies. From what i see, he has an insatiable thirst to learn everything there is to know about math. I told him that it's unrealistic, but he insist on still striving to do that cause otherwise he'd feel unsatisfied with himself. He studies math 24/7. He's like an addict at this point. Idk what to do

#

Would sending him to a adolescent psychologist would be a good idea?

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Should he take meds?

modern trench
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Tell them they need good grades to get into a good math degree KEK

vagrant meadow
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there's all this pressure to get into a good uni right out of hs that just isn't warranted imo

dim musk
modern trench
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In high school

dim musk
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in my country there are courses that are listed as "philosophy of science"

dim musk
tawny slate
tawny slate
# wispy slate He also said that he doesn't see any point in learning the math that they teach ...

oooooh youve provided a lot of excellent details about what he is thinking and experiencing, this is good

ok so regarding psychologist/meds, my own personal opinion which you should absolutely take with a grain of salt is that a psycholgist might not hurt, if you can find the time and money, but really what you want out of the session is counseling and not meds, meds is always a last resort for anything debilitating that counseling cant fix

but if you want some arguments to help convince him, you could try some of the following:

  • in the same way that you cant do math without concepts, you also cant do math without computation. for example, you might come across a hard problem such as "prove that pi is irrational." and in the process you need to expand out very long and nasty power series (computation), in order to find some intuitive insights that will lead you to the solution (concepts). he should understand that practicing the computation side of things is not useless, as it builds muscle memory and insight in ways he doesn't expect yet
  • you can find a lot of deeper insight from HS math by doing more than simply "memorizing formulas". what i used to do was sit through a math class, and whenever anything was mentioned that was even remotely new to me, such as a formula ive never seen before, i dont just take it for granted that i have to memorize it. i try to derive it from what i already know, i try to imagine all of the ways it can be applied, until virtually none of the HS math i learned was memorized
  • in order to access more interesting and powerful math, not just through books, but through some of the best and brightest mathematical teachers and coaches, there is almost no better and more consistent method than to go to univ, but in order to do so, he needs good grades to demonstrate to the people at the univ that he is at least capable of the "plug and chug". if he cant even do the basic plug and chug, few math professionals will think of him as a serious candidate
#
  • those are some weirdly strong assertions about AI. we have had calculators (and very powerful ones) for a long time now, and they havent reduced our need to do computation without them on paper or mentally. more importantly, we will always have a need to do basic checkable computations, and AI does not even help much here. you need a good dose of mathematical maturity and conscientiousness to even put the correct inputs into the calculator (for instance, not realizing the calculator is set to degrees instead of radians or messing up order of operations). what kind of computational structure does AI offer that mitigates this problem? if anything, it only exacerbates it, as we have few ways of checking the AI's output other than just checking it by hand/traditional calculator regardless. AI's use is exceedingly limited in scope. he is also making a pretty strong claim about how AI is going to transform society, but why is he so confident?
  • (be careful about when you explain this one, given his issues) a thirst to learn about math is good, but not if it is to the point of being unhealthy. perhaps show him that it is completely impossible to learn about all math, and this can be proven mathematically. consider an uncomputable problem like finding the values of the busy beaver function. the fact that it is "uncomputable" also means that you can never find any kind of generalization that allows you to solve for its values. we will probably never know BB(6) in our lifetime and we will almost certainly never find BB(7). dont obsess over learning all of math because we can mathematically prove that its not possible, and so he will never be satisfied. he must learn to let go or else it jeapordizes what we can learn about math. we are fleshy meat humans, not robots, we have to take care of our health, physical or mental, in order to maintain the efficacy and efficiency of our abilities
#

you can also mention that a lot of this input comes from someone who is also obsessively passionate about math, who wants to learn everything, but is now teaching professionally and offering their advice and wisdom or something like that

#

maybe an appeal to authority may help some of that sink in

#

also helps if you get a bit stuck when/if he challenges you on anything, gives you an out

prime gale
prime gale
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like ok ig best thing one can say is to access the best quality of math education in colledge u have to prove ur uself to be capable for it

tight star
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How do you know if you’re handholding too much?

minor turtle
# wispy slate What would you guys do if you have a bright student but have terrible grades cuz...

yeah this is a relevant (and in fact much more interesting than it would seem at first glance) question, since i think it highlights two conflicting premises that students generally hold about their relationship with their math education: the first comes from what our society teaches our children, and it’s the premise that our grades reflect the ability to complete work that requires a higher level of thinking than just filling out paperwork or something, and also that grades reflect mastery of the current subject, and the ability to interface with the world using the material learned in class. Often, though, students will come to realize the second premise on their own, which is the idea that beyond a certification of basic competence (which can be acquired elsewhere, but grades are the easiest way to get that), grades are not a measure at all of academic ability, and one must learn a lot on their own in order to keep up with the pressure that academia seems to bring. it seems like your sibling holds this second principle and either doesn’t realize the ramifications of not having an actual certification that they can do the work that they’re supposed to do, or has a disorder that prevents them from starting the work (i have no doubt that people learning math on their own would be fine to do their homework if they were able to start), and needs help from either you or a professional, and in this respect it’s so important to communicate with them (you or your parents/guardians may have already been doing so, but i feel like communicating with them about the importance of grades and opening the possibility of seeing a professional about this specifically is important), because it might not be a matter of applying themselves, they seem to be just fine in working towards their goals, you just want to ensure that they’re set up to do well. this behavior absolutely will transfer to college if there is some psychological thing in the background

#

i’m not a psychologist and don’t want to diagnose them with anything, but if these rationalizations do happen to be an underlying psychological thing (like ADHD, which doesn’t mean they’re bad, or whatever other stereotype, plenty of incredible mathematicians that i know have ADHD), then it really is important to see a psychologist about it. i have had similar issues in high school, i saw a psychologist and it really helped me.

minor turtle
minor turtle
upper solar
#

Even if he finds solving things easy, he is still making an trade-off: saving on memorisation by sacrificing speed.
if he wants to learn all of maths he's going to need that speed.
memorisation gives the benefits of 1) speed from more time/brainpower freed up 2) pattern recognition, clever new uses from a different area of math won't appear in your head if you don't have results memorised (most of maths is invented by pattern recognising collections of computations anyway)
personally I found going chapter by chapter in school tedious, as the questions were largely the same with different numbers. If this is the case, for a self motivated person the answer is 1) variety 2) speed practice 3) above and beyond techniques for both theory and computation
Variety can be found by just taking 1/2 question from all different chapters of the book working that way.
Speed practice can be quite relaxing honestly, especially if you practice doing more complicated stuff in your head.
Above and beyond techniques incorporate theory results to shortcut a lot of the computation, e.g. nomial expansions through pascals triangle (and even pascals simplex)

#

@wispy slate

wise onyx
vagrant meadow
#

i'd like to get opinions on something. i'm trying to write something on jordan forms, and i feel the reason they're so confusing and weird is because students aren't really told that it's all a result on finitely generated modules over a PID. i.e. without the abstract algebra context, it's all just strange and things sort of come out of nowhere.

so without actually going through the proof for the structure theorem, what do you all think of at least explaining that the structure theorem exists and how it guarantees the existence of the jordan blocks/invariant subspaces? how non-simple parts of the sum imply a sense of nilpotency and how that corresponds to not being diagonalizable etc. then i can just focus on a single eigenspace/jordan block and use change of basis to understand how to actually find the jordan basis.

i figure that like if a student is in linear algebra and doesn't know ring theory, then it's probably better to know that there's something behind the curtain and accept that it's simply beyond their current understanding than it is to just memorize a bunch of theorems about the existence of the jordan form and a generalized eigenbasis without understanding the underlying logic.

cosmic ibex
#

My immediate feeling is that if the students don't know enough ring theory to already understand what it means to say that a vector space with a linear operator is the same as a k[X]-module, then they'd just be confused by bringing it up.
Even more if they're not going to learn a proof of either of the theorems. Then you're just replacing one opaque theorem that at least speaks in terms of something they're familiar with (bases and matrices) with another opaque theorem that speaks about something abstract that they're not familiar with.

austere delta
# vagrant meadow i'd like to get opinions on something. i'm trying to write something on jordan f...

I think it would be better to focus on (generalized) eigenvalues.

Like what is a Jordan block saying? Well, as they should probably know if a linear transformation T: V -> V has enough eigenvector you can diagonalize it. And you're guaranteed to have at least one eigenvector, great! So say E is your eigenspace. If E is all of V you're done. If not we have to get clever... But hey! T maps E to E, so induces a map on V/E -> V/E. And this map will again have an eigenvector, great so we found something that's like an eigenvector except instead of being T(v) = sv it's T(v) = sv + e for some e in E.

Package this information into a matrix and you basically have Jordan blocks

#

It's also helpful to think geometrically of how any linear automorphism of the plane is some combination of rotation, stretching and shearing, with shearing being the nontrivial jordan blocks

daring gulch
# wispy slate He's 16

My opinion is that someone at that age simply doesn't have the luxury of saying "hs math is too computational for my tastes." Unless they're Feynman and have already completed integral and differential calculus at age 15, so what if it's just memorizing formulas and plug and chug? You'll probably need them later anyways. Also, AI isn't going to change shit in professional mathematics, or any higher math beyond calculus, really.

vagrant meadow
# cosmic ibex My immediate feeling is that if the students don't know enough ring theory to al...

my thought was more like "it's okay that this doesn't make sense because it's actually more advanced than you think". bc I'd rather they don't think "this is taught in linear algebra so it must necessarily be at my level. and if I don't understand what's going on that means I'm stupid and not smart enough to understand these proofs and theorems that seem to come out of nowhere."
but I see your point

midnight scarab
#

Idk the extent of your students' knowledge, but Dunford decomposition is a significant step towards Jordan and doesn't need all the cyclicity nonsense that can be done without k[X]-modules, though perhaps not in a very enlightening manner (I'm looking at you pointwise minimal polynomial)

#

Plus Dunford is good enough if you want to take powers or exponentiate matrices

vagrant meadow
tight star
#

Yeah I’d never actually seen this description before either

#

Honestly like
 quotient spaces got introduced in linalg but we never really worked with them

austere delta
#

Well, if there's one more advanced topic worth focusing on in linalg, I think quotient spaces is a good contender

tight star
#

Hmm i think

#

We didn’t cover anything like the universal property of the quotient

#

So, the quotient space kinda felt hard to work with?

#

Like, i knew the definition in terms of equivalence classes

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But I only ever really used it in rank-nullity

hollow musk
#

I do wish they were a standard topic in linear algebra courses too.

vagrant meadow
#

i feel like in terms of being a resource for linear algebra students, i might just focus on the basis/coordinate vector approach and offhandedly mention that the existence is a deeper result using the structure theorem. then maybe have a second part on quotient spaces and how that relates to the structure theorem,

#

i only really understood what a quotient anything was in my ring theory course and it made perfect sense there (funnily enough, i needed to learn quotient rings to understand quotient groups). it's really just a super useful topic. i need to deep dive into quotient spaces because the only application i'm currently aware of is a slick proof for rank-nullity.

tight star
vagrant meadow
#

what do you mean?

tight star
#

Have you heard of universal properties?

vagrant meadow
#

so i guess not?

tight star
#

Ok ok

#

They’re a concept from a field of maths called category theory

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The categorical perspective is to focus on what something “does”, how it relates to other things, as opposed to what something “is”

#

Can i give an example with sets first?

vagrant meadow
#

sure

tight star
#

So, take a set X

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You have some equivalence relation R on it

#

And from this you can form the quotient set X/R

#

And were given the definition of what X/R is - its elements are equivalence classes of elements of X

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Other quotients have similar constructions

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The categorical perspective focuses on what X/R does, how to use it, how it relates to other sets

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And here's how - functions out of X/R naturally correspond to functions out of X which respect R

image.png
#

And by "respecting R", I mean that whenever xRy, $f(x) = f(y)$

burnt vesselBOT
#

Pseudonium

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tight star
#

Does that make sense?

vagrant meadow
tight star
#

Mhm mhm

#

Note, by the way, that nothing about the categorical definition actually requires R to be an equivalence relation

#

You could let R be an arbitrary relation

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And you could say "hmm, I wonder whether there exists a set X/R with the following property"

#

"functions out of X/R naturally correspond to functions out of X respecting R"

#

And it turns out there does exist such a set! You quotient X by the smallest equivalence relation generated by R

#

But I think the categorical definition here is a little cleaner - you don't really care what the particular construction of X/R "is", you just care what it "does", how to use it, how it relates to other sets

#

Is that fine?

vagrant meadow
# tight star Is that fine?

yeah that generally makes sense to me. i don't know much category theory, but i think you explained it pretty well.

tight star
#

Cool

vagrant meadow
#

i actually really appreciate it because it makes the idea a lot more general than the way i was originally thinking about it

tight star
#

Moreover, using something called the Yoneda Lemma, you can actually deduce what this "correspondence" has to look like

#

Here $q : X \to X / R$ is the quotient map that sends an element $x$ to its equivalence class $q(x) = [x]$

burnt vesselBOT
#

Pseudonium

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tight star
#

Given any function $\tilde f : X / R \to Z$, you can just compose with $q$ to get a function $f = \tilde f \circ q : X \to Z$

burnt vesselBOT
#

Pseudonium

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tight star
#

And this function will respect the relation R

#

But, importantly, you can also go backwards! If you have a function $f : X \to Z$ that respects R, you can "divide by q" to get a unique $\tilde f : X / R \to Z$ such that $f = \tilde f \circ q$

burnt vesselBOT
#

Pseudonium

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tight star
#

You could almost say $\tilde f = f / q$

burnt vesselBOT
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Pseudonium

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tight star
#

Hope that makes sense?

tight star
vagrant meadow
# tight star But, importantly, you can also go backwards! If you have a function $f : X \to Z...

mind = blown!
okay so like this is how i sort of naively understood it originally

  • ideals are defined in like the way such that we can quotient by them in a way that preserves the "structure" which "happens" to be preserved by ring homomorphisms
  • repeat this logic for quotient groups
    but, no, it looks like my reasoning was backwards i think? i guess it's kind of a chicken and egg situation. but basically the equivalence relation defined by an ideal/normal subgroup is one that preserves the ring/group operations (which are also preserved by homomorphisms). so since homomorphisms respect the ideal structure, we can have a well defined quotient structure that is preserved well by the homomorphisms.

i always thought the FHT was like... kind of obvious or easy to prove. but i'm seeing that it seems to be based on a much more general principle.

hopefully i didn't just butcher your carefully given and thought out explanation lol

tight star
#

Yes, that's what I was going onto!

tight star
#

Given a group G, and a subset $N \subseteq G$, you can define a relation $x \sim y \iff x^{-1} y \in N$

burnt vesselBOT
#

Pseudonium

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tight star
#

It turns out that when $N$ is a normal subgroup, this is an equivalence relation that respects the group operation

burnt vesselBOT
#

Pseudonium

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tight star
#

So if $x \sim y$ and $z \sim w$ then $xz \sim yw$

burnt vesselBOT
#

Pseudonium

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tight star
#

You can recover $N$ as the equivalence class of the identity

burnt vesselBOT
#

Pseudonium

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tight star
#

And again, there's this duality between what $G/N$ 'is' - its elements are left cosets $gN$, with group multiplication $gN \star hN = gh N$

burnt vesselBOT
#

Pseudonium

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tight star
#

And what $G/N$ "does" from the categorical perspective

burnt vesselBOT
#

Pseudonium

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tight star
#

Which is actually the same as what $X/R$ does, in a sense - group homs out of $G/N$ are group homs out of $G$ which respect the equivalence relation, which turns out to be the same as a group hom out of $G$ where $N$ is contained in the kernel

burnt vesselBOT
#

Pseudonium

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tight star
#

It's the same story for rings

#

Vector spaces

#

Modules

#

Topological spaces

#

The idea of what a quotient "is" will change from place to place

#

But it turns out they all agree on what a quotient "does"

#

You consider maps out of some object $X$ that satisfy some condition - in this case, the condition is having to respect a relation

burnt vesselBOT
#

Pseudonium

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tight star
#

And you want to identify these with just maps out of some quotient $X/R$ that don't have to obey any conditions

burnt vesselBOT
#

Pseudonium

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tight star
#

I've not told you how to make X/R, or even that such a quotient exists

#

I haven't told you what X/R "is"

#

Instead we've been a bit more pragmatic and just decided what X/R should do, how we want to use it

#

That's the idea behind category theory

vagrant meadow
tight star
#

I mean I'm not really an algebraist

#

I'm a physicist

#

And I love category theory cause it feels like physics

vagrant meadow
#

i mean that instead of spending two quarters of the year learning PDEs, i should have been focusing more on algebra and topics like category theory. but whatever we'll see.

as much as i loved and got value from your explanation using category theory, i don't know if i'm gonna use it in my resource aimed at linear algebra students 😆
but we'll see haha

tight star
#

Mhm mhm

#

I think I have seen some people use commutative diagrams when teaching linear algebra

#

It can help for things like change of basis

#

One thing category theory does is make you an expert at manipulating functions

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And this is just a generally good math skill

#

Also, for some reason category theory is usually only taught at the masters or graduate level

#

So I'm not sure you'd have seen it anyway

vagrant meadow
# tight star So I'm not sure you'd have seen it anyway

yeah never have :/
my UG was not... amazing. certainly not the kind of school to have a cutting edge UG category theory course. even topology is only available at the grad level there lol (and complex analysis / grad algebra and other classes have a high chance of getting cancelled and just not being offered at all during the year)

tight star
#

Favourite course this year though

#

Seems like you’re doing well for yourself now though!

vagrant meadow
#

im trying haha

tight star
#

Which is basically all you can do anyway

hollow musk
tight star
#

Because, as it turns out, functions are a good way to explain things

#

And category theory gives you tools to work with functions

#

So far it's been pretty successful

hollow musk
#

interesting!

#

I feel as though category theory is one of those things that is seen as not generally teachable (beyond very surface level stuff) to undergraduate students, so this is quite intruiging to hear

tight star
#

And - did it seem accessible?

hollow musk
#

I'm nowhere near the level to comprehend what you had written, haha

tight star
#

Ah

hollow musk
#

perhaps in a couple of years!

tight star
#

Perhaps

hollow musk
#

(I will still be an UG by then)

tight star
#

Universal properties just tend to crop up in lots and lots of places

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Cause as it turns out, when mathematicians define things

#

They want it to “do” something useful mathematically

#

And that’s exactly the categorical perspective

minor turtle
tight star
minor turtle
#

dude i fucking hate this eeveethink emoji

hollow musk
hollow musk
minor turtle
#

i want to say something like “category theory isn’t synonymous with mathematics” in a way that emphasizes that while category theory is useful for many things, if you’re a student it’s often more useful to study things like, say, manifolds without first having to abstract it away with category theory

#

not so much to do with nature

#

sorry i misspoke it’s late at night for me

#

and i still don’t know if i’m getting the point across exactly as i want it

tight star
#

Not even most maths

#

It’s just a useful perspective to have, i think

#

Sometimes it’s better to focus on what something is, and sometimes it’s better to focus on what something does

#

The lesson from category theory isn’t that one perspective is the right one

#

It’s that what matters is having lots of perspectives and the ability to translate between them

midnight scarab
#

I think Smay is referring to a phenomenon I have indeed observed of people being obsessed with the categorical viewpoint and, for example, being unwilling to get their hands dirty and do the explicit computations when necessary (or, for Smay' point, missing what is special about this particular category that warrants an entire course on it)

tight star
#

I'm a physicist, after all

midnight scarab
#

Me too, and I'm being killed by these multi-parameter integral asymptotics I'm trying to do rn, and unfortunately cat theory cannot help me 😔

tight star
#

mhm mhm

wind ginkgo
#

I personally enjoy dirty computation and diagram chasing

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And their intersection

wind ginkgo
#

If I can do dirty computation for cheaper with diagrams I will

tight star
#

Ok this is probably a silly question but - why are functions so seemingly underused in math pedagogy? Why doesn't everyone teach things in terms of functions?

tawdry venture
#

what kind of "teaching in terms of functions" are you thinking of specifically

#

also it might depend on the region

#

bc i know that where i am the syllabi involve functions a fair bit

tight star
#

A student was confused about limits

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Something like $\lim_{n \to \infty} f(n)$

burnt vesselBOT
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Pseudonium

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tight star
#

The confusion was whether $n$ was infinitely large or not

burnt vesselBOT
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Pseudonium

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tight star
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I said it was a natural number, so it's finite

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But they asked "if it's finite, how big is it?"

#

And it was a little difficult to convey the concept of variable in a way that made sense to them

tawdry venture
#

questionable!

tight star
#

It also came up in calculus where you have $\lim_{ \Delta x \to 0}$

burnt vesselBOT
#

Pseudonium

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tawdry venture
#

yeah idk like this student has a conceptual gap there i think

tight star
#

They had the idea that

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$\Delta x$ was "infinitely small"

burnt vesselBOT
#

Pseudonium

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tight star
#

I said no, it's always finite

#

And again their question was "if it's finite, how small is it?"

tight star
tawdry venture
#

i would not even say that delta x or n have any fixed value

#

cause the point, conceptually, is that they approach 0 or infty respectively

tight star
#

Right, yeah

#

I found a way to fix their intuition using functions

#

So, $f$ is a function

burnt vesselBOT
#

Pseudonium

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tight star
#

It takes in a natural number $\mathbb{N}$ and spits out a real number $\mathbb{R}$

burnt vesselBOT
#

Pseudonium

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tight star
#

So $f : \mathbb{N} \to \mathbb{R}$

burnt vesselBOT
#

Pseudonium

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tight star
#

And then the way I explained limit as

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"lim" is a function

#

It takes in a function $\mathbb{N} \to \mathbb{R}$

burnt vesselBOT
#

Pseudonium

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tight star
#

And spits out either a number, or infinity, or -infinity, or undefined

#

So it's a function $\text{Limit} : (\mathbb{N} \to \mathbb{R}) \to \mathbb{R} \cup {\infty, - \infty, \text{ undefined}}$

burnt vesselBOT
#

Pseudonium

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tight star
#

Defined by $\text{Limit}(f) = \lim_{n \to \infty} f(n)$

burnt vesselBOT
#

Pseudonium

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tight star
#

And this worked surprisingly well

rain hawk
#

fascinating, I have tried something like this and it just made it more confusing for them I think. Maybe I did it poorly

tight star
#

I mean, it's entirely possible I got lucky, but

#

I have been testing out ideas like this recently

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Expressing things as functions wherever I can

#

And I've been surprised at how well it's worked

#

For example, one can think of algebraic expressions as functions

rain hawk
#

wdym

tight star
burnt vesselBOT
#

Pseudonium

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tight star
#

You plug in a pair of numbers $(a, b)$

burnt vesselBOT
#

Pseudonium

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tight star
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And you get back a number

rain hawk
#

oh yeah, tbh isn't this the usual way of describing such expressions?

tight star
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Well, as a practical example

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They did an equality $4 a^2 b = 4 a^2 4 b^2$

burnt vesselBOT
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Pseudonium

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tight star
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And I showed them why this wasn't true by showing the functions were different on the input $(2, 1)$

burnt vesselBOT
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Pseudonium

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tight star
rain hawk
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perhaps. perhaps

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I actually think that the opposite perspective, that there is a difference between e.g. polynomial and polynomial functions, is something that I wish was taught a little better

tight star
#

I do recognise that, but

#

Polynomials on their own come way later

#

Compared to polynomial functions

rain hawk
#

this is true

tight star
#

And hopefully by that point you have enough mathematical maturity to help distinguish them

#

I also used currying to demonstrate what the partial derivative was

rain hawk
#

the distinction becomes important as soon as we start talking about polynomial division (eg 4x/x = 4 as a polynomial but if f(x) = 4x and g(x) = x and h(x) = 4 then f(x)/g(x) != h(x) because the right is defined at 0 and the leftist isn't)

#

this sort of thing is particularly why I think that distinguishing them should be done earlier than is actually done

tight star
#

Wdym by polynomial division

#

That isn't really a thing?

rain hawk
#

https://en.wikipedia.org/wiki/Polynomial_long_division
yes it is?

Polynomial long division

In algebra, polynomial long division is an algorithm for dividing a polynomial by another polynomial of the same or lower degree, a generalized version of the familiar arithmetic technique called long division. It can be done easily by hand, because it separates an otherwise complex division problem into smaller ones. Sometimes using a shorthand...

tight star
#

You wouldn't notate that as a fraction though

#

Long division takes in a pair of polynomials

#

And outputs a quotient and a remainder

#

It's a function $\mathbb{R}[X] \times \mathbb{R}[X] \to \mathbb{R}[X] \times \mathbb{R}[X]$

burnt vesselBOT
#

Pseudonium

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rain hawk
#

idk, maybe this is a regional thing because I definitely see it notated as a fraction

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like, when I teach highschoolers and whatnot

tight star
#

Hmm

#

I wouldn't say 4x/x is a polynomial

#

Or that it equals 4 as a polynomial

rain hawk
#

https://www.youtube.com/watch?v=3eowXt_LNbg maybe this is an american thing, idk, but it definitely feels like the usual way to speak

YouTube
Khan Academy
Polynomial division introduction | Algebra 2 | Khan Academy

Keep going! Check out the next lesson and practice what you’re learning:
https://www.khanacademy.org/math/algebra2/x2ec2f6f830c9fb89:poly-div/x2ec2f6f830c9fb89:poly-div-by-x/v/dividing-polynomials-by-x

When we divide the polynomial p(x) by q(x) we are basically asking "what should we multiply by q(x) to get p(x)?" If this sounds familiar, it's ...

▶ Play video
tight star
#

Which seems to me more like thinking of them as functions

#

Rather than as their own objects

rain hawk
#

I can try to find a specific source, but resources like this are very inconsistent about that

#

they usually treat them as functions, but occasionally do not

rapid tusk
#

someone is being difficult in one of the pre uni channels again - i feel like im being a bit too harsh but how does one even deal with someone so unable/unwilling to help themselves 😭

tardy ember
#

at this point i don't think "unable/unwilling to help themselves" is the relevant thing to analyse

rapid tusk
#

they have serious attitude issues

tardy ember
#

they're not engaging with the conversation as having semantic meaning, they're just treating it as some kind of game

rapid tusk
#

such a flippant approach just wastes everyone’s time 😭

#

unfortunately they do not seem to have enough sense to realize that

tardy ember
#

well it's not a waste of their time, according to them, because they enjoy it

rapid tusk
#

just blocked them, no more will they waste my time

tardy ember
#

obviously it doesn't help anyone with learning maths but again i don't think that's their goal at that point

rapid tusk
#

their stated goal was to “be the smartest in their class”

#


 you can probably guess what issues that’ll bring

#

they appear to be a middle schooler or smth

tardy ember
#

...yeah well like, does that actually mean knowing more maths/whatever, or does it mean being able to win the conversational game of looking smart (as judged by them)

#

also tbf they did get an answer to the question they asked

rapid tusk
rapid tusk
#

perhaps i should be less readily handing out answers like that anyway

tight star
rapid tusk
#

they got mad that i called them out for not wanting to put in any work themselves

tight star
#

I mean they just asked a Q

rapid tusk
#

googling “proof of sum of divisors formula” likely could’ve answered their question immediately

tight star
#

Sometimes people prefer to talk to a human

rapid tusk
#

ig

tight star
#

Like there’s plenty of times where I’ve asked a Q here

#

I know I can google it

rapid tusk
tight star
#

But there’s a kind of feedback that’s helpful when talking with a person

#

That you just don’t get with Google

rapid tusk
#

one could read the proof online and then come with specific questions about it, maybe they need clarification on a step or two

#

neglecting to even do that is like “if you’re not willing to put in the work, why should we?”

tight star
#

Or they could come and ask here first

#

I think that’s ok

rain hawk
#

someone asked what 5/2 is recently

tight star
#

Mhm

rapid tusk
#

i think that’s a little more understandable if it’s on more elementary topics

#

if the student themselves is at a more elementary level

rain hawk
#

they asked "what is 5/2" and someone responded "2.5" and they said "oh thanks" and closed the thread

tight star
#

Lol

rapid tusk
#

bleak well that’s certainly one way to do it

#

lmao

tight star
#

That’s hilarious

rapid tusk
#

(no calculator)

tight star
rapid tusk
#

(in which I probably crossed a few lines too many 😭)

tight star
#

I guess there’s some history here im unfamiliar with

rapid tusk
#

they are in over their head

tight star
#

Cause it did honestly seem like you blew up at them out of nowhere

rapid tusk
#

they want to be the smartest in their class but are stumbling over relatively basic things

tight star
#

Sure

rapid tusk
#

and then when we try to see what’s going on we just get completely incomprehensible responses

tight star
#

Like?

rapid tusk
#

some really deep fundamental confusions over taking square roots

#

it felt like viper and i were talking in circles trying to explain things to him

tight star
#

Hmm, I’d be interested to see this

rapid tusk
#

just search his messages in #prealg-and-algebra (again, pardon my getting rather heated as the conversation progresses 😭)

#

you should be able to find it pretty quickly

tight star
#

You do seem to uh

#

Resort to capital letters quickly..

rapid tusk
#

yea i gotta cut back on that ik

tight star
#

I guess this is a confusion between preimage and square root

#

I remember learning category theory and learning that you can make sqrt(9) = +-3

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And have it make sense

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Which I thought was cool

rapid tusk
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I don’t think he’d understand if we tried to cast it in those terns bleak

tight star
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I’m not so sure!

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Like, I’ve found thinking in terms of functions to be useful

rapid tusk
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ye

tight star
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The issue is essentially

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There’s two common meanings of “square root”

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There’s the function $\sqrt \space : [0, \infty) \to \mathbb{R}$

rapid tusk
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yeah

burnt vesselBOT
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Pseudonium

547515245258735616.png
tight star
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Ah whatever

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But there’s also another common square root

rapid tusk
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empty brace maybe?

tight star
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$\sqrt{} : \mathbb{R} \to P(\mathbb{R})$

burnt vesselBOT
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Pseudonium

547515245258735616.png
tight star
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Both of these are perfectly well-defined functions

rapid tusk
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P being power set ?

tight star
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Mhm

rapid tusk
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aight

tight star
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And these are usually confused

rapid tusk
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not enough distinction is made between the two in my experience

tight star
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Mhm

rapid tusk
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certainly when I was in middle and high school the difference was completely brushed under the rug

tight star
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Mhm mhm

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I think when I was in school

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It wasn’t clear to me that the latter was even a function

rapid tusk
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once I realized that sqrt(x^2)=abs(x) it became clearer why things were the way they were

tight star
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I think maybe there’s too much of a focus on just

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Functions which take in a number and spit out a number

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And I’ve been experimenting with having more general functions in my explanations

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People seem to grok the concept easier than I’d expect

rapid tusk
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:o interesting

tight star
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I was even able to explain how to do a partial derivative with an ordinary derivative m

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Using currying

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Because someone was confused about why we had partial derivatives

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And not just ordinary ones

rapid tusk
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I think i saw part of that lol

tight star
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And like - they actually seemed to understand a fair amount of it

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I think maybe students have this feeling that there’s something “more” beyond functions which take in a number and spit out a number

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Like with square roots

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Or with partial derivatives

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So what I was experimenting with was

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Rather than saying “ew that’s not allowed to be a function”

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Say “it is a function, just not one you’re used to”

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And so far, this has seemed to work well

rapid tusk
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yea

tight star
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I also did this with algebraic expressions

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And also with propositions

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You can view a proposition with a free variable as a function

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And this viewpoint helped someone in deciding whether a proposition had a free variable or not

rapid tusk
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damn

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I took a discrete class where they absolutely butchered propositional and first order logic LMAO

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should probably learn more in my own time

wind ginkgo
wind ginkgo
tight star
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huh...

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why??

wind ginkgo
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So √ makes more sense

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And a bunch of other reasons

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Mainly Burbaki

wind ginkgo
wind ginkgo
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Also personally

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IMO the best way to learn functions is through relations and in an explicit fashion

tight star
wind ginkgo
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A function f is just a special relation on D(f)UcoD(f)

tight star
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hmmm

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I mean

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I do like input-output machine

wind ginkgo
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this is

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Exactly that

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in a sense

tight star
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Right, but then you have to get into the weeds of set theory and all...

wind ginkgo
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Well

tight star
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If I took this approach, I'd have to explain what sets and unions etc are

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And what relations are