#math-pedagogy

1 messages · Page 16 of 1

tight star
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What well-definedness is

wind ginkgo
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Well uhhh you have to already explain sets

tight star
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Only very informally

wind ginkgo
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And union intersection

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Huh hmmm

tight star
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Like

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I understand this is the rigorous def

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But I'm not gonna be rigorous just for the sake of it

wind ginkgo
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I mean, you are doing the same thing tbh when you explain it as input output

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Because you have to say a function is a machine that for each input has to output a single thing

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And then you can very logically reach this

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After you explain a relation as something that takes inputs and spits out outputs

tight star
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I mean...

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You can try this

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But I probably wouldn't go the relations route

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It's too much overhead

wind ginkgo
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I would have been much happier had someone done that. They tried and failed in our butchered foundations class.

tight star
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Tried and failed what?

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Also, I'm not really using this in a foundations class

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I mean using functions to explain other math concepts

wind ginkgo
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Ah

wind ginkgo
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But it was done poorly

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Imagine someone rushing through this while giving zero connection between each subject and no intuition.

tight star
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Mhm…

wind ginkgo
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By the end you were left wondering

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"Why did I learn all of this?" and you would then have to scramble to fill in holes left by this.

tight star
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Right, right

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I’m honestly not sure how I’d do a foundations class

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I’m a physicist..

wind ginkgo
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Hmm...

tight star
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How would you do it?

wind ginkgo
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Maybe with concrete examples?

tight star
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Like how would you make it cool

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But also not boring

wind ginkgo
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Probably by making it involved

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Having the class build it

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After basic logic

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And very barebones set theory

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Which would be the first class

tight star
wind ginkgo
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Well

tight star
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How do you teach logic even

wind ginkgo
# tight star How do you teach logic even

Well do some simple true, false, negation, and or, p,q =>, <=>,for all, for some. Recursion and the rest of the techniques can be built along the way and during homework

tight star
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Yeah but like

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That stuff feels quite boring to me

wind ginkgo
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Basically I would spend the first lecture giving the very basic tools

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Then

tight star
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Actually what’s the way you’d teach =>

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I’m curious

wind ginkgo
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I would start with boolean algebra (in spirit)

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So first explain truth valued and propositions

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Then do and and or

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Then implies by the classical example

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Negation and for all/ for some

tight star
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The classic example?

wind ginkgo
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It rained => The floor is wet

tight star
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I see

wind ginkgo
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But this first lecture would be the most boring

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For the rest I would do something guided discovery-esque

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So prepare questions that naturally lead the people solving them to build the set theory themselves

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And leave the unsolved stuff as homework

visual trout
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I like this philosophy

lelso.PNG
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the "coming to interesting results as soon as possible" part

tight star
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Hmm I might need to change my approach, but I’m not sure how

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I was trying to show someone that the converse of FTC part 1 fails

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But I couldn’t seem to…

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And now it seems I’ve made them upset

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I’m not sure what I did wrong :(

minor turtle
tight star
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I wonder if maybe conflicting variable names might’ve caused confusion

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This has definitely been like

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My toughest challenge yet as a person who wants to explain math

gritty tundra
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hey guys i built a network graph representing all the topics from pre-algebra to trigonometry that are needed for calculus

high_school_math.png
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what are your thoughts?

minor turtle
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lol

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arithmetic sequences aren’t required to talk about linear functions idt

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this image is very aesthetically pleasing btw

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these topics seem a bit more independent than what the diagram suggests, but i think one could in theory build a curriculum based on this

dapper flume
# gritty tundra hey guys i built a network graph representing all the topics from pre-algebra to...

If you are in the USA, I have found this site to be extremely useful. It creates a networked graph of every Common Core math standard from kindergarten through high school, giving great insight into what prerequisites a teacher should consider when teaching a standard.

Also, I like to build a progress map with my students throughout the year that looks like this. I get the sense that my students often feel like they are in the dark regarding their progress, and seeing it in the style of a "skill tree" seems to help

gritty tundra
dapper flume
gritty tundra
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oh i see

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i'm actually working on a bigger project too which is to map a bunch of undergrad math classes to that too

dapper flume
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Do you have a goal for doing this, or is it just something you were interested in mapping out?

gritty tundra
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and basically everything

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i built a react app to do this with

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this is what i uploaded to it

dapper flume
minor turtle
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especially since most fields are usually not very self-contained at all (unlike undergrad classes whose prerequisites and motivations for study are usually very clear)

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maybe an approximation is still useful but i doubt that such a thing could be mapped without people disagreeing what is even in the tree, let alone what’s dependent on another (it might end up as a much more tangled graph without nice properties if you’re trying to be pedagogically useful or even consider motivations)

tight star
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What struggles do people have with the idea of substitution? And what usually fixes those?

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For example, i have heard that sometimes people struggle with the idea of what it means to evaluate a function

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To me, substitution is really baked into the concept of a variable, so it’s challenging to understand what other people find difficult about it

minor turtle
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this is one of the primary struggles i find that people have with algebra

tight star
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I see

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And if I want to try a more functional approach to teaching

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I need to overcome this struggle

minor turtle
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yeah, it’s actually a really hard problem tbh. i still don’t have a set way that i deal with it if i’m teaching someone that stuff

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like it’s one of the things i’m the least opinionated about because i’m afraid to touch it

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:P

tight star
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gosh…

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it feels so fundamental to the idea of algebra

minor turtle
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like how do i get into the mind of someone who is trying to create a conception of what a variable is and how it relates to a function

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i want to either start with someone who is earlier or later because both types of people have “easy” to evaluate problems

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i find myself drilling fundamentals with them very often when i’m teaching stuff about variables, like i learned it. i want to be very hands-on, not worry about functions, try literally showing them unknown quantities (via stones in a bag) and letting them see what’s going on with equations with a physical scale

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and then students will learn what a variable/unknown is that way

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this approach could be extended to functions, though it feels a bit sloppy to do so

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maybe we don’t have to extend it to functions, i don’t hate the “machine that follows a rule” analogy, i was being prepped to hear that analogy pretty much my whole life anyways (albeit i was very lucky i think, since the assignments that we did during math class in elementary school were essentially implementations of functions but without the notation)

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but if one doesn’t have that upbringing, then that’s fine, we could just start with that analogy without any notation

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i’m unsure about all of this

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btw

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but i feel like this is an approach that i would feel comfortable using on an algebra student, provided that they can add numbers together, and do other basic operations (the four basic arithmetic operations are really all that’s required i think, obviously there is a bit more to elementary education than that but this topic specifically is quite narrow when you think about it for the first time)

minor turtle
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put two bags of marbles labeled into a box named y for example

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but substitution and functions at first are likely different enough concepts at this point that they’re learning how to do it

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i do want more perspectives tbh, i’m really taking some shots in the dark

tight star
cosmic ibex
# tight star You can see over in <#576508782637744130>

That was painful to read.
I think the underlying trouble (which never got resolved, but I can't blame you for that) is that they understand "necessary condition" wrong.
They started with the true facts that:

forall f( P(f) & Q(f) -> R(f) ) is true and provable
forall f( P(f) -> R(f) ) is false
and conclude "therefore the assumption that Q(f) is a necessary condition for R(f)".
That's actually not crazy based just on an everyday understanding of the word "necessary" -- it's just not how the word is used in mathematics, and they cut you off with "don't teach me logic" when you tried to engage with it.

tight star
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i see…

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what would you suggest?

minor turtle
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get a student who’s willing to learn

tight star
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i mean as a teacher

cosmic ibex
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I think at a certain point they had gotten so worked up that they were not going to concede no matter how well you explained anything.
You're talking to someone who's not even agreeing which way the student-teacher relationship between you goes, and salvaging that is not just a matter of pedagogy.

tight star
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hmm

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then what is it a matter of?

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or - how could i make sure we don’t get to that stage of being too worked up?

minor turtle
tight star
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hm, i thought i did that, but perhaps i could've done it better...

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like they said it was an iff multiple times

minor turtle
cosmic ibex
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I mean, sometimes you just have to cut your losses.
From their point of view it wasn't a matter of them learning from you -- it was "Pseudo is wrong and I'm very patiently explaining to them why they're wrong".
I think that kind of impedance mismatch is impossible to completely avoid in the social context where we're all just people talking on the internet without any hierarchy defined between us in advance.

tight star
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i see...

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this is difficult for me to accept, i think

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i've always been able to reach people with my explanations

minor turtle
# tight star i've always been able to reach people with my explanations

the first time i disagreed with someone in a similar way it happened to be someone with a little bit of status, and she didn’t understand that she was making a dichotomy where there wasn’t one, and no matter how much i explained it to her she wouldn’t listen to me, and it was really really frustrating.

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it sucks to be the person on that end of it

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but it literally can’t be avoided

minor turtle
tight star
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that it's not an if and only if?

minor turtle
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no specifically dealing with the phenomenon that tropo said

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like coping with it

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or i felt like i had to cope with it for a while

cosmic ibex
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To be fair, "only if" is a super confusing phrasing if you try to understand it based on its individual words.
It simply needs to be learned as a piece of mathematicianese with a conventional meaning.

wind ginkgo
cosmic ibex
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My exact words. opencry

wind ginkgo
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Oh and @tight star mayhaps a good idea for that foundation-esque class you wanted to teach is adopting the teaching philosophy of Kenneth P. Bogart. The introduction to this book has always been very nice to read for me and a great inspiration. https://bogart.openmathbooks.org/

tawny slate
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society if everyone was rational and reasonable or something

image0.jpg
wind ginkgo
midnight scarab
vocal phoenix
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Living animals were inefficient so they were sacrificed on the altar of progress

tawny slate
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the resolution is so low

gritty tundra
minor turtle
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i see where you’re coming from, i think such an approximation might be useful and would definitely be very cool

sharp hawk
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I proved it in all details

But he came with the initial insight

signal estuary
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The fact that they teach a lot of very specific content and methods for solving certain problems

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I think it would be far better to decrease the number of specific content/methods that are taught and instead focus most of the time in math class on general problem solving techniques and just solving a lot of problems that are new enough so that you really have to think about how to solve it

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But that’s mostly because I learned math that way and it worked very well

tawny slate
# gritty tundra what are your thoughts?

so there was a guy who made a good point about how this is highly subjective from person to person, i also want to point out some things that imo were kinda weird about this

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it seems like you start with real numbers here, and then from that learn simplifying expressions

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dont think real numbers need to be introduced until irrationals are needed, and for some reason simplifying expressions is before add/sub both sides of an equation?

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i would imagine if those steps are that granular, "simplifying expressions" is much more difficult than you might be estimating

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and exponents are learned in parallel? i would only teach exponents after multiplication and addition properties are well understood, which isnt exactly mentioned anywhere

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i would imagine this tree is much more interconnected

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but otherwise good start

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i think these kinds if diagrams are less an objective thing and more of a reflection of your personal style of teaching and your gathered experiences

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results are more important i think, because at the end of the day, i think the reason we use this is to map out and target the students' weak points

gritty tundra
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i agree

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i made this more to show which concepts build on top of each other

tawny slate
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it is pretty cool to see everything laid out like that though

gritty tundra
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yea most people see this as a straight line from algebra 1 to the end of precalc

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also i put the laws of exponents down there because the concepts mainly center around expressions and not equations

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especially at the pre-algebra level

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but when you solve exponential equations, it's algebra 2

image.png
long pelican
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I’ve gone back and forth in my mind about whether I think people are rational, and at this stage I think people are best explained as rational, but mostly acting on incomplete information all the time

tawny slate
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i disagree

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i used to feel the same way, but i think after being exposed to some neuroscience, i think humans are social first, emotional second, rational last

long pelican
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Rational traditionally restricts itself to measurable things like money (as in economics) but I have a much broader view of aspects one can be rational in

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Social things like responding to pressure to fit in are included

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If you don’t allow that, then, by all means, humans are irrational, sure

tawny slate
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here is my interpretation of "rational"

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the reason why i put social in front of rational is because i think, given two test scenarios where the same person has to make the exact same decision, one clearly rational and one clearly irrational, humans will almost always choose the irrational decision given enough social pressure, and will almost always choose the rational decision absent that social pressure

tawny slate
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responding to social pressure is not necessarily rational, especially if they are pressuring you to do something you rationally oppose

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the question is when these forces come into conflict, which prevails?

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and in this sense, this is what makes humans incredibly fallible and easily exploited

long pelican
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I think you'll find it makes more sense to include social pressure in what is considered the decision making process, rather than apart from it

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Once you adopt this, the social pressure changes the calculus of what you want to get out of a decision -- it adds a factor you need to consider

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Doing something out of fear of getting ridiculed is very rational

midnight scarab
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Not if you're worse off in the end

long pelican
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Hmm but

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  1. Worse off on the thing without considering desire to avoid humiliation might total to better off overall
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  1. Even if you regret it later, doesn't mean you were irrational in the moment
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Rational decisions made with incomplete information are bound to sometimes be wrong

midnight scarab
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You usually decide based on what is (or what you feel is) more important for you at the moment, which is often an issue of feelings rather than rationality

long pelican
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You can view that as lack of information, namely the information about what you will eventually settle on as the proper weightings of the various factors in the far future

cosmic ibex
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Actions are not "rational" or not in isolation. It's always relative to what your goal is.

midnight scarab
midnight scarab
long pelican
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Obviously I'm challenging the modern-traditional idea that humans are fundamentally irrational

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Partly the definition of what is rational is way too narrow, and partly we don't give our subconscious brains enough credit

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e.g.

you just gave in to the pressure
under the hood if you examine what the brain is doing here, it's a lot more complicated than that

daring gulch
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Hmm I think you're operating under a few too many assumptions.

long pelican
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Would you like to say more?

daring gulch
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You never precisely defined what is "rational," for one. Rational in the sense of making the best choices for the individual? Under specific circumstances? Short term decisions vs. long-term decisions?

long pelican
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I did mention, though, that the current definition of what is rational is too narrow and it causes lots of problems

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making the best choices for the individual
can't be the definition, as this requires knowing information one doesn't have most of the time

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Short term decisions vs. long-term decisions
Both can be perfectly rational, even at the same time

ashen verge
midnight scarab
minor turtle
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does anyone here have experience writing expository notes? why did you do it, and are there any things that you would want to do differently if you could do it for the first time again?

tawny slate
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ok but at this point the conversation is devolving into semantics about what should be defined as "rational", but not about human nature and how we make decisions and think. thats why i was very technical when i defined what i meant

lets make this explicit with a hypothetical scenario:

you're not exactly poor but you certainly are looking to not waste money, as you need it in the short and long term. rationally, this means you should not be gambling at a casino, because you've studied probability and expected value. your friends invite you to come to the casino with them, and if you go you know they will strongly peer pressure you into gambling, and that you will probably gamble anyways and regret it

here, we have two forces coming into conflict, a "rational" choice that says to not go because not only is it not in your monetary interest, but if there is any social cost to not joining your friends, they should not be your friends in the first place as they dont respect you enough, so you should be assertive and decline, and a "social" choice, which is that the person wants a sense of belonging and acceptance within a group, even at the cost of short- or even long-term goals

you can argue all you want that it is in the person's self-interest to be sociable, as there is some abstract experience that makes them find the monetary cost to be worth it, but firstly, the example is to point out the difference between "rational" and "social", not to pick at a specific case, and secondly, by this logic every decision that every human being makes is rational, what does being irrational even mean? you're evaluating the decision by post-hoc and not prior to the decision being made. and thirdly, i could have chosen stronger, more explicit examples like addiction but that might be construed as cherry picking

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i dont deny that it helps to have a more broad definition of "rational" in the sense that because we are human, our feelings and abstract thoughts and intuition are sometimes critically important and we cannot separate ourselves from them, and i dont deny that in the real world you can never have perfect information and that you can make decisions that benefit you both short and long term, thats not the point

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the point is that once you separate decision-making factors into the categories of social, emotional, and logical, the human tendency is to vastly prioritize the factors in that order, even if the logical one is the most correct one and the person making the decision believes it. it is baked into our neuroscience from the moment we are born

this distinction is important to me as an educator because sometimes your students might hold strong emotional attachments to certain framing of certain problems or react to the classroom setting externally, and being aware of the way our neuroscience and psychology works helps reach the student better and guide them properly

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ive had students have viscerally emotional reactions to the fact that we say .9 repeating is equal to 1, despite being shown 3-4 different explanations for it, ive had one student that got extremely argumentative when i brought up a hypothetical problem where some rich guy was being taxed by a fraction of his wealth rather than a flat tax which prevented him from engaging with the math and was disruptive, and almost all of the students ive ever come across will at some point change their answers to a math problem if they find out they are in the small minority to match the answers of their peers, even when they did the scratch work and verified the correct answer themselves

these are all disruptive to the goal of actually learning math, which is why we show kids weird and interesting math to hone intuition, why we try to not bring up controversial topics if it is not the focus of the lesson, and why we explicitly guide the students to build confidence in their own logic and own work rather than following the masses

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the point is NOT that emotions or peer pressure is never "rational", and im certainly not going to define "rational" here as "following your heart"

tawny slate
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actually, just as a broader definition can sometimes help reframe an idea in a new light, sometimes a more specific definition can help us see important differences and nuance

long pelican
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If we are to separate the social, emotional, and rational spheres, all this suggests is that the "virtue" of being rational is simply the act of narrowing your decision making to a single goal, rather than multiple goals at the same time

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Is ignoring social aspects of a decision a virtue?

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I like to abstract away "social" and "emotional" since there is nothing inherently special about these goals

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We might hold 10 or more goals in our brain at the same time

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Most of them very hard to communicate

tawny slate
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a human being in extreme emotional distress will make decisions and think things that normal human beings will not

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emotions can strongly strongly impact your thinking and decision making

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therefore, it makes sense to separate it from some other form of "logicalness" or "rationality"

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how many times in your life have you done something because you were angry or upset and then regretted it later?

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once again, to be perfectly clear, i am NOT saying that we should never take our emotions or social aspects out of our decision-making, its not a good thing nor is it possible, so it most certainly is not a virtue

long pelican
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What's the last emotional decision you know of (yourself or someone else) and how complicated do you think the decision making was?

tawny slate
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but there is a massive difference between "i think vaccines are generally safe based on the available scientific evidence" and "my friends all told me vaccines cause autism and will kill us all"

long pelican
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Oh that's a good example to discuss

tawny slate
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the last emotional decision i made was to decide to eat this bread

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i ate it because i wanted to eat it

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the decision making was not complicated in this case

long pelican
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Even in the bread case, if you were doing something at the time, then you had to decide whether you want to eat the bread now or finish part of what you're doing...

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Not consciously, mind you

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Even if emotional, it factors in

tawny slate
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once again, i am not disagreeing with you

long pelican
tawny slate
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not disagreeing, but there is a difference between teaching someone a math concept by providing them with a logical understanding and telling them it is because i said so or by emotionally pressuring them to

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you keep bringing up things that i dont disagree with but have no relevance to the point i made

long pelican
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That's not surprising since the point I'm making about rationality isn't directly a response to you

tawny slate
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ah

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ok then i have a different question

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can you provide me of the most concrete examples of a rational decision and an irrational decision

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for contrast and clarity

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and follow-up question: given your definition of rationality, does your point relate to math pedagogy in some way

long pelican
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Mmm, well, my view of irrational is that calling a decision irrational implies I don't understand the reasons behind it

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However, I'm open to using the narrow definition of "rational" for the purposes of this discussion

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In that case, a rational decision would be the decision to major in CS after hearing that jobs in CS are hot

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An irrational decision would be deciding on a restaurant and then changing last minute just because you like randomness (here, enjoyment of spontaneity would make it rational but not from the point of view of the goal of maximizing what you were originally maximizing)

tawny slate
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ok two issues

tawny slate
long pelican
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Oh I did miss necessary context

tawny slate
long pelican
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for this person, their ultimate goal is to make good money after graduation

tawny slate
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ok and i argue this is not necessarily rational

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i may have the goal and context but i do not see why that assertion/decision is rational

long pelican
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but "irrational" can be useful to communicate that you have no idea why they did what they did

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in an informal setting

tawny slate
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ok it sounds like the word "rational" has basically almost no meaning informally, it just amounts to "i understand" vs "i dont understand" (irrational)

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why bother using the term "rational" at all then?

long pelican
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I was getting to that!

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In the rationality community, there is a general goal of figuring out how to best arrive at true beliefs and then to teach that to the public. Traditionally (2010s-ish to now) the view is that rationality is the best way to approach having true beliefs, where rationality means using logic and reason over emotions and other things. I think it makes the fundamental mistake of explaining away those other factors as irrational, when in reality the brain is doing complex decision making involving all these other factors, not being irrational. Another complicating issue is that those other factors are often uncommunicable, so they cannot themselves explain what went into the decision because they don't have the practice of communicating that.

tawny slate
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feels like we are looping again

long pelican
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Maybe my usage of "irrational" was unclear above

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"stupid" might be better?

tawny slate
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again, thats just semantics, i dont know what you mean by either word and im trying to get clarification

long pelican
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it's the same way half the general public views vaccine denial as stupid

tawny slate
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ok but according to your definitions, it sounds like you would not label them as irrational

long pelican
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Right, vaccine denial at its core is about distrust of sources, not ignoring facts

tawny slate
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and thats the point

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by separating emotions from "rationality" or "logic"

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you make formal claims about these kinds of issues

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whereas if you lump them all together nothing has any meaning anymore

long pelican
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Distrust of sources isn't emotional

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It's inferential

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based on knowledge and past experience

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Prejudice is also inferential

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that's something a lot of people get wrong too

tawny slate
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are "emotional" and "inferential" not separate axis of categorization?

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can i not have inferential logical arguments?

long pelican
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Fine, but for distrust of sources, there's no reason to assume emotion is involved

tawny slate
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but the point is that by lumping them all together it is being reductive

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first of all, let me say that i am going to use the word "reasonable" for your definition of "rational" and "logical" for my definition of "rational" so they dont get mixed up

long pelican
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Sounds reasonable

tawny slate
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i can make an argument to say, for instance, that both of the statements:

  • there are an infinite number of primes
  • i like tomatoes
    are both reasonable
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but i would not call the second statement "logical"

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because it does not involve any [significant] form of formal arguments

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the first statement is reasonable regardless of who is interpreting it or how it is interpreted

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the second statement is necessarily contingent on my felt experience

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there is something massively qualitatively different here

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and that is the point of making this distinction

long pelican
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Neither one is making a inference of any kind... (well, the second one is inferring your messy brain configuration into "likes tomatoes" I guess)

tawny slate
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if you break "rationality" down into such tiny amorphous brain experiences, then yes all lived experiences are also made up of tiny inferences that you cant totally explain

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why do i think i like the taste? because i have felt this before and i would prefer it

long pelican
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Let's just go with "I like tomatoes" as irreducible lol

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Continue

tawny slate
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so basically the word is just as useless as your interpretation of "rational" if you take it that far

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its reasonable because i think its reasonable

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i cant explain it but i feel it so its reasonable

long pelican
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Only if we can reflect and communicate that far (and we can't)

tawny slate
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ok so now we are getting somewhere

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i would agree that one of the motivations of this distinction is whether or not we can explain it

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communicate it

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why do i like tomatoes? because i do

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sure, i can communicate it, but like, not in a way that will make others agree

long pelican
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In practice I'm completely fine with treating likes as atomic

tawny slate
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however, there are statements, like those in math, that once communicated can be completely understood

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those are fully communicable ideas

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where as my feelings towards tomatoes are only partially communicable

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there is no way i can transfer my qualia to someone else

long pelican
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Suppose we're treating likes as atomic, then a decision to eat tomatoes or not eat tomatoes when presented with this binary decision is nearly fully communicable too

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Of course there's the factor of "I'm full" that I thought of that can complicate that decision

tawny slate
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but the reasons for it are not fully communicable too

#

you are now arbitrarily deciding that that is an atomic statement

#

which, sure, i guess we do the same thing in math

long pelican
#

Well, I'm also claiming that likes being atomic is a good idea for now

tawny slate
#

the point is that each like statement is not only its own atomic statement

#

it is also a personal atomic statement

#

you cannot understand what i feel when i bite into a tomato

#

i cannot understand what you feel when you bite into a tomato

#

this is why this is a fundamentally emotional thing

#

im not saying its not reasonable, it absolutely is

#

but these are separate axis

#

each of these words have their own meaning and function

#

i think the word "rational" in the way you use it is not useless, its just highly highly generalized and abstracted and not applicable to many scenarios in practice

#

im saying there is a lot of value in the specificity to break down these different types of motivators, you seem to disagree

long pelican
#

Perhaps? But I noticed in many scenarios a stupid decision actually had reasons behind it and understanding those reasons is very powerful

tawny slate
#

sure, again, i totally agree

long pelican
#

But calling those reasons emotional is reductive

tawny slate
#

but that brings me to question 2, how does this relate to math pedagogy

#

all language is reductive

long pelican
#

The actual reasons, broken down into likes and whatnot

#

is better information than "You just acted on emotion"

tawny slate
#

there is no utterance that is not reductive

#

let alone a whole statement

long pelican
#

Um

tawny slate
#

i am not saying that emotions cant be more well understood either

long pelican
#

But calling those reasons emotional is reductive [and there is a better way]

tawny slate
#

well hold on

#

you said it is reductive but the reason you gave

tawny slate
#

nowhere in any of my claims did i ever imply that emotions cant be broken down or understood better

#

i made a point that generally speaking, humans are strongly wired to prioritize in a certain way

#

it is not an absolute statement and it most certainly is reductive

#

but so is the statement "labels are reductive"

#

because sometimes labels, and language in general, can help elucidate an idea

#

the only way to communicate is to start with something that has some kind of meaning, even if it is reductive, and build ideas with them

#

all language is reductive

long pelican
#

ok this discussion about reductiveness was not helpful, I didn't use that when you said abstracting away emotions is reductive

tawny slate
#

ok i see what youre saying

#

that does sound like im using a double standard here, let me first apologize

#

i guess im just saying that even though i understand and agree with you in the abstract, i disagree that there is no value in separating social, emotional, rational

#

i think sometimes you want to be more general, sometimes you want to be more specific

long pelican
#

Yes, in this part of the discussion I was saying being more specific is virtuous, while previously I was saying that "emotional" and "social" are not special labels

#

for that bit, what I meant is that "emotional" and "social" are just 2 of 10 possible categories

#

7 unnamed

#

(or more)

tawny slate
#

sure

long pelican
#

So decision making will first be approximately: how much do you value social, how much do you value emotional, how much do you value money, how much do you value physical health, etc, etc... then after those weightings you calculate (not consciously) the course of action

#

Rationality folk would have you believe that if you don't pick a single goal and effectively consciously optimize it, you aren't being rational

tawny slate
#

i think the bulk of the community does give that vibe/impression, and i do feel you

#

i do agree with you that full robot-mode rationalizing every decision and choice does not necessarily give you the best outcomes, because humans are fallible

#

i agree that not going full robot doesnt mean youre irrational

long pelican
#

I like how robot is being used as a synonym of having a single goal 😛 I was about to discuss how that idea has infected AI research

tawny slate
#

and this is coming from someone who obssessively rationalizes everything by second nature

#

like, this is a cool abstract thing i took away from model theory

#

given some basic axioms it seems like you can only do so much

#

but if you ponder that some nonstandard models exist, you realize you get a lot of weirdness

#

there are all kinds of existing frameworks built into your brain

#

and when you try to be hyper-rational, the danger is not realizing when you're using that framework that isnt rational, but you think it is

#

things you take for granted on a deep subconscious level

#

im with you here

#

but look, im just trying to explain how best to teach math here

long pelican
#

I did have some thoughts about the relationship between rationality and what you said about teaching math earlier but I will have to scroll up to recall it...

#

brb

#

By the way, new thought: you all are undoubtedly familiar with the experience of deciding whether you should ask a question now, or wait until the end of class, or just figure it out on your own. Complex decision making process that is, interesting to ponder how it works

#

I'm already seeing quite a few layers of inference being used

#

(also very practical for math-pedagogy since maximizing the chances your students will decide to ask the question now is one of the best things you can try to do)

tawny slate
#

oh yes, in that case this is actually a good example of my original point

#

students are worried about the embarrassment or shame from asking questions during class rather than asking immediately to get the most out of the class

#

hence social/emotional > rational

#

i want the students to ask questions and ask as soon as possible

long pelican
tawny slate
#

the students act differently in the class setting than they would in private

long pelican
#

social/emotional > goal of understanding math

tawny slate
#

in this case sure, but i claim something stronger than that

#

for instance, even if one knows they are right and believes it by logic and reasoning, can be strayed away by peer pressure or public ridicule

#

you can unconvince someone of the answer they know is right by bullying them

long pelican
#

Asch experiment right?

#

Unlike the popular conclusion from that experiment, my conclusion is that doubting yourself when you're doing against literally everyone else in your social group can actually be pretty rational

#

for the goal of having a true belief about that thing

#

being new increases the chances of self-doubt, being senior decreases it

#

I think that's a universal experience 😛

tawny slate
#

and thats basically part of what i was getting at, yep

tawny slate
minor turtle
# tawny slate sorry for burying this, but is there a more specific context for why youre askin...

i’m just writing for myself really, i’m organizing my thoughts and want them to be comprehensible enough for a reader to be able to follow along with it, but also i don’t just want to copy the book (which so often happens when i’m doing pencil/paper notes, and in some poorly typed papers on the internet) , but honestly i do want to know a bit about what your writing process is like even if it’s not the exact same format

#

like how do you talk to a reader

#

or viewer in your case

#

is it any different knowing that readers can’t ask questions in real time

tawny slate
#

its a little different

#

when im writing for viewers of the video, there is a much higher standard i think

#

because it isnt solely the exposition and the facts being presented

#

youre essentially also telling a kind of story

#

you need a hook, a motivation, you need to connect with the viewer about why they should care, and then you have to follow up on that promise

#

as for the writing process, i have to make sure that all of that is planned out and strucutred before i draft up the script

#

and the script has to be in presentation form, not just some notes i can pull out at any time

#

once i go through several revisions, i test the script with some people close to me to see if it does hit the goals i want

#

like a checklist

#

it needs to not only be easy to understand and clear, it needs to be motivated, it needs to be interesting, it has to inspire them and they feel like they took something of value, and ideally it is memorable and leaves an impression

#

and then the cherry on top is the balancing act with the clickbait thumbnail and title, like i want people to see it, but i also dont want to make it so ridiculous and dishonest that i feel gross and lowers my integrity

#

feel free to ask if you want any more details or if you want to see my channel

#

i can show you videos that worked well and videos that bombed

tardy ember
#

or at least this doesn't accurately describe the position of rationalists i've seen

#

in particular, "a single goal" is a technical term here, it means a mapping of world states to real numbers

#

which definitely can include what looks from a human perspective like a very complicated system of several different things that you want

long pelican
#

Ya, I have seen weighted averages of "what you want"

#

I think pretending you have a single goal is quite common though

#

for organized groups like a company even more so

tardy ember
long pelican
#

(for a company it'll be, for example, profit, for a charity based on EA, it'll be something like QALY per dollar)

long pelican
#

Which can be inaccurate

tardy ember
#

"irrationality" looks like deciding that buying an apple for £5 is worth it, and selling it for £3 is also worth it, (and also assigning £ positive value), which means someone can make money off of you by just repeatedly buying and selling you an apple, because your preferences are inconsistent

long pelican
#

It's exactly in these thought experiments that I noticed that rationalists work with single goals

tardy ember
# long pelican Which can be inaccurate

oh yeah, deciding that you've found the entirety of your utility function when in fact you're missing huge amounts of it is a common mistake by people who don't get it i think

tardy ember
#

the same ideas still generalise: even if "what you want" is actually extremely complicated, it's still pretty obviously stupid to have preferences that aren't transitive

#

but you aren't going to find anything in the maths that says you can't "want multiple things" as long as you have some defined rate of what you do when there's a tradeoff

#

you can want a lot of things, just don't do something where you can rearrange to get more of all of it simultaneously, because that's irrationality

#

a utility function cannot be irrational, that's a type error

#

(although thinking you have a certain utility function can be, in the sense that it's a belief and beliefs can be wrong)

long pelican
#

Why is nontransitive preferences stupid?

#

I know abstractly it is illogical

#

but why in a human is it stupid

#

Or maybe my real question is

#

What if they just appear nontransitive

tardy ember
#

...how would they "appear nontransitive" without actually being nontransitive

#

but ok if you have states A B C, and you expend effort to get from A to B, and to get from B to C, and then to get from C to A, because your preferences are A < B < C < A rather than some more complicated reason like "because you like variety" ...then what are you doing

#

to make it a better example let's say what you're actually doing is changing which of these events will occur at some fixed future time

#

so by changing A to B to C to A you have accomplished literally nothing

long pelican
#

Classic treadmill

#

You spend effort to get to the same place

tardy ember
#

ok yes fine, let's say you don't benefit from doing this except in the sense that you changed the outcome

long pelican
#

In the workplace the appearance of doing something has positive utility, in the gym the thought of you getting healthier by exercise has positive utility

tardy ember
long pelican
#

Another scenario is you are exploring

#

and you end up at B then C then A due to influx of new information

tardy ember
long pelican
#

Actually the constant dynamic of new information means nothing can be nontransitive tbh

tardy ember
long pelican
#

The value from changing it can just outweigh nontransitivity

tardy ember
#

it can outweigh your preferences between A B C

#

there's no nontransitivity happening here

#

but yes, that's also possible

long pelican
#

Well if changing has value then you'll always climbing while being at the same place

#

talking about transitivity won't be very useful anymore

tardy ember
long pelican
#

Why the singular goal thinking lol

#

You don't need to do that

#

It still works

#

Another way to think about it, you can never truly return to your original state

tardy ember
#

but at some point you would want to stop moving in order to sleep otherwise you'll become less effective at changing things in the long-term

tardy ember
# long pelican Why the singular goal thinking lol

well because otherwise you would end up doing something else that you value more that also achieves that things are changing, like talking to your friends - which changes the state of the channel you're talking to them in by adding messages to it, and is also valuable because social whatever

long pelican
#

regardless it allows A B C A with no rational contradictions

#

and no assumption of singular goal necessary

tardy ember
#

it allows A B C A'

#

but yes it is true that essentially any behaviour is the rational way to achieve some goal

long pelican
long pelican
tardy ember
tardy ember
long pelican
#

Instead of using your brain's subconscious processing on hundreds of internal goals, estimate them with a couple of quantifiable goals and use conscious processing instead

tardy ember
#

like if you do genuinely reflect on it and conclude that you value the opinions of your friends a billion times more than anything else, then sure go for it

long pelican
#

Don't think the brain will weigh friends' opinions at 1 billion times anything else for length of time more than 0.5 seconds

tardy ember
#

if you decide that's not how your goals work, then giving in to peer pressure when you know that isn't in your interests is irrational and you should do it less

tardy ember
#

"rationality" is whatever works best, you don't get to look at a strategy that would perform really well and declare that instead you're going to use the "rational" strategy that works less well

long pelican
#

Isn't the contention of rationality that what "works best" is measured by those estimated quantifiable goals

#

what else can one mean by "works best" in that context

tardy ember
#

your subconscious mind is fairly stupid but it is also an extremely useful tool and you should find the best way to use it to achieve your goals

tardy ember
#

replacing your goals with some other set of goals that's easier to achieve, does not achieve your goals, so don't do that

long pelican
#

You're at a local maximum for whatever your goals are almost tautologically

#

getting out of that requires doing something against your goals for a bit

tardy ember
#

no i'm not

#

or i guess local maximum maybe?? but like

long pelican
#

yep, local is important

tardy ember
#

i would prefer if i was immortal, and i'm not

long pelican
#

That's a singular goal and not what you have

#

and all paths to being immortal while not compromising on your other goals are not accessible to you at the moment

tardy ember
#

that... is true...?

#

i don't see why it's relevant at all?

#

why do we care about local maximums

long pelican
#

Challenging the assumption that the subconscious brain is stupid

#

And that what you should do is achieve your goals better

#

What you're actually doing when applying rationality is either pretending your goals are different or changing your goals or going against your goals (temporarily)

tardy ember
#

i would absolutely take something that would be a huge inconvenience for a month and then make me immortal*, so it doesn't really look like i care about local maximums?

long pelican
#

But not if you didn't know about the outcome

tardy ember
#

*(i do have more conditions on what would make this desirable than literally just "unable to die", but still)

long pelican
#

The anticipation of the outcome is definitely increasing your utility

tardy ember
#

?????

#

no it isn't

#

i do not intrinsically value thinking that i'm going to be immortal

long pelican
#

Then why are you doing it? 😎

#

You're doing singular goal thinking again

#

Utility is not just being immortal

#

Happiness counts too

tardy ember
#

oh the fact that we're talking about one particular outcome that i think is good means i'm doing "singular goal thinking", ok

long pelican
#

I mean you wouldn't climb that hill if you had no idea you'd be immortal at the end (even if it was the case you'd be immortal at the end)

tardy ember
tardy ember
long pelican
#

Knowledge of being immortal at the end makes your brain shape the hill so it's not a hill anymore

tardy ember
#

also i actually think that "incorrectly thinking i'm going to be immortal" is by itself net-negative

tardy ember
#

you're acting like i'm a local optimiser when i'm just not?

long pelican
#

Sounds like there's the map that your brain encodes internally and there's the map that you are aware of and/or talk about and they are two different things

tardy ember
#

???

#

we're not arguing about the world or about how i perceive the world, we're arguing about what my utility function is

long pelican
#

I'm not talking map of the world

tardy ember
#

map of what then

long pelican
#

I'm talking map of the utility over the space of actions

#

err

#

space of states

tardy ember
#

ok so you're saying that my intuition doesn't perfectly match what i'd say my utility function is?

#

yeah obviously

long pelican
#

Yes. and I've been discussing the "true" utility function encoded by your brain this whole time

tardy ember
#

well that's not the "true" one

long pelican
#

hypothetically true in the thought experiment

#

of course

tardy ember
#

ok then this hypothetical is about someone else who isn't me and behaves nothing like me and i don't see what your point is

long pelican
#

You think that doing a month's work in anticipation of immortality is an example of leaving a local maximum in utility

tardy ember
#

yes

long pelican
#

I'm saying, actually, your utility is increasing during that process

tardy ember
#

well "utility over all futures" does, in the sense that a world where i'm closer to immortality is "better" because i want to be immortal

#

but utility of that particular timeslice... no??

long pelican
#

Yes, but we already know the brain's utility function isn't just about the given instant in time at each time

#

err

#

Like

#

I'll phrase it differently

#

Anticipation of something really good in the future affects utility now

tardy ember
#

yes in the sense of consequentialism, no in the sense that i think you mean it

long pelican
#

Not consequentialism!

tardy ember
#

expecting something good to happen and being wrong can actually be net-negative imo

long pelican
#

I'm talking about the brain's utility function, not some "ground truth" utility function

#

To the brain, not knowing it's wrong, there cannot be an effect of being wrong on its utility function

#

until that knowledge enters

tardy ember
#

ok wait so why are we talking about this "brain's utility function" thing

long pelican
#

I've been explaining that we are at local maxima for our utility function

#

or rather settle into one

tardy ember
#

well this "brain's utility function" isn't my utility function

long pelican
#

That's exactly the crux I think

#

Rationality says, use a different utility function than your brain's utility function

#

That's all, I think

#

Well, that + try to make it understandable and measurable and consciously optimize it

#

avoid fallacies, etc

tardy ember
#

like what this actually means is "think about what you want instead of just going with whatever sounds best in the moment"

tardy ember
long pelican
#

yes so for example if money is just one of the things you want, but now you've decided you'll focus on money as a singular goal (!) (ok, can be money along with a couple other goals) that's an example of the first step to applying rationality

long pelican
tardy ember
tardy ember
long pelican
#

This "what you want," is this not already encoded by the brain's utility function?

#

I assume no

#

Then it's either a subset of it or something different from it

#

That's changing right?

tardy ember
#

it's "encoded" in the sense that it's in there somewhere

#

but like

#

my "brain's utility function" ...isn't a utility function

#

it's a mess

long pelican
#

It's behaviors but a utility function is the main component of it (in my opinion)

#

something like happiness, satisfaction, peace

#

and then things feed into that

tardy ember
# tardy ember it's a *mess*

it's the pile of puzzle pieces that you assemble together to get more information on what your values actually are

long pelican
#

Ya so my viewpoint is getting your values into something communicable in language is losing information, not gaining information

tardy ember
long pelican
#

That is part of what feeds into the part labeled "satisfaction"

tardy ember
tardy ember
tardy ember
long pelican
#

Yeah contradictory values happen a lot, usually when you learn them from different social groups (parents and then peers, for example)

#

but those just feed into utility function indirectly, I think?

tardy ember
#

because that's what rationality is, achieving whatever your values already are

#

and i don't think my intuitions are the full story there but they are also in a sense the only information i can possibly get about what really matters, so it would be stupid to ignore them

long pelican
#

I do know the Sequences phrase "finding what your values actually are" in terms of discovery

#

But I increasingly cannot avoid realizing that it's modification or pretense

#

example, if you had contradictory values the rational thing is to modify them

tardy ember
#

if i had contradictory values, then the rational thing is to go "wait what" and introspect further

long pelican
#

in the end, something will change though?

#

Ok to be clear I was never talking about "arbitrary" modification

tardy ember
#

well it will "change" in the sense that gaining knowledge always changes the state of your brain

#

my moral intuitions respond to arguments in the same way that my causal intuitions about the world do

#

i might introspect further and realise that actually the "contradiction" is a complete illusion, because the two intuitions apply to subtly different scenarios or give outputs that are in fact compatible or etc.

#

or i might realise that one or both of them really should have some extra condition because the reasoning behind it breaks down in some particular weird case

#

or i might realise that one or both of them is just plain wrong

#

because while some of these intuitions do just report themselves as primitives requiring no justification, like "suffering is bad", a lot of them, when poked, will unfold into an argument of some kind

#

because they know that they are simple approximations, and, not always but sometimes, they will look at a logical argument and say "ok yeah you're right, i was wrong"

long pelican
#

If you show some stats on how safe vaccines are and after that, they say "well, nah I still think vaccines are unsafe", are they illogical and/or irrational?

#

let's say studies, not stats

tardy ember
#

well, depends on their priors

#

and also their goals if they have some reason other than "because they think it's true" to say that vaccines are unsafe

long pelican
#

I would go as far as to say that being shown some studies or stats on something should not move your needle very much

#

*sociological studies

tardy ember
#

well, again, depends on your priors

long pelican
#

Well not even that

tardy ember
#

if someone thinks science is generally trustworthy and just somehow managed to not find out that the scientific consensus is that vaccines are safe, then seeing a load of studies saying vaccines are safe probably should update them

long pelican
#

Just that 80% of people who show you a study or stat on something have an agenda and have had the ability to select studies/stats

tardy ember
#

...ok yes fair that is true

#

...although if you have the (incorrect) prior that all studies always give exactly the objectively correct result, then seeing studies that show vaccines are safe would update you more than they would if you had a more accurate picture of how accurate studies are

long pelican
#

Of course

regal beacon
#

Hello how are you

minor turtle
coral copperBOT
regal beacon
#

Hello how are you

tawdry venture
tawny slate
#

that dude banned

rapid tusk
#

ignores redirects
refuses to elaborate
leaves

wind ginkgo
#

Anyways question: Would a guided discovery format work for topology at least to some extent?

tight star
#

What are you suggesting specifically?

tawny slate
#

i think it can absolutely work, depending on how its executed

#

so yeah, what did you have in mind

wind ginkgo
#

Something like students naturally coming to generalize the notion of open and closed sets from analysis, maybe through first generalizing the idea of a distance.

tight star
#

How does one generalise the idea of a distance?

wind ginkgo
#

Well I meant the usual distance on R

tight star
#

I’m interested because I feel like I haven’t satisfactorily answered for myself how one gets from metric to topological spaces

wind ginkgo
#

I am still thinking about that. Possibly it has to go through the idea of sequences.

#

Or being close to a point rather.

#

Approaching a point with some regularity.

#

This is key.

#

If you get to the idea of open balls and convergence you can naturally do away with the open balls at that point.

#

So something similar to the local basis idea. And then you can build the axioms from there like Hausdorff did.

cosmic ibex
#

It feels like a rather big expectation to have students invent the point-set definition of topology from scratch just by generalizing from R^n. There's a lot of different generalization one could equally well end up with -- for example, should topological spaces be required to be Hausdorff? Or first countable? Learning how those choices are settled is not as much a question of figuring out what works (they all do, for different purposes), but just of learning which one, mostly by historical accident, got to have the name of "topological space".

#

And that's even before we consider how to discover for oneself that the "right" formalization is by axiomatizing open sets, rather than, say, the concept of "interior point", or neighborhood systems, or closure operators, or even closed sets.

tight star
#

You’ve also gotta emphasise what makes thinking topologically actually useful, I think

#

Cause its still a while before they meet any non-metrizable topologies (aside from the obvious ones like indiscrete)

daring gulch
tight star
#

I don’t see how this is a generalisation..

daring gulch
#

stick

#

||/s||

cosmic ibex
remote jacinth
#

thank you

cloud zealot
# wind ginkgo Anyways question: Would a guided discovery format work for topology at least to ...

I suspect a guided discovery format would not work unless you taught topology very slowly. Since point-set topology is supposed to cover a broad range of topics for a broad audience with varying backgrounds and interests, teaching with this format might preclude you with from assigning exercises that might extend or illustrate facets of core theorems in the interest of maintaining reasonable workloads if you decide to mainly focus on "reinventing" theorems. One thing you should keep in mind is that using theorems to solve problems can lead to insight into theorems aside from reproving them.

minor turtle
# wind ginkgo Something like students naturally coming to generalize the notion of open and cl...

they should already know about metric spaces if they’re taking a topology course. there’s enough time for the eager topology student to do plenty of exploration if they want to, the subject is just too deep to not allow for it, and at that point the students should have enough mathematical maturity that the discovery part will come whether you decide to actually teach them or make them do it themselves

modern trench
tight star
vocal phoenix
#

That last one is only needed if your definition of "neighborhood" isn't limited to "open neigbhorhood"

#

Opinion seems divided over which definition of "neighborhood" is superior; I've always preferred the open one.

#

Although without open you'll probably still need something like the last axion; just saying if nbd(x,S), then for every y in S we have nbd(y,S)

tight star
#

yeah idk im not sure if ive ever found a topology definition that felt satisfactory

vocal phoenix
#

The metric one 😄

tight star
#

i mean for general topology

vocal phoenix
#

I know I know

modern trench
vocal phoenix
#

True

#

Never mind then

modern trench
#

This axiomitization is how I justify open sets are important anyway

#

I think there's a closure axiomatization too

vocal phoenix
#

There is and it's even worse.

tight star
#

im not sure I’ll ever find a nice axiomatization of topology

vocal phoenix
#

I don't mind the one with open sets

tight star
#

it’s probably still my favourite

#

but it’s not great

modern trench
#

Open sets are easiest to work with

#

But usually one would need to get motivation from studying the R^n or metric space cases

#

Whereas I feel closure and neighbourhood system are motivated on their own

vocal phoenix
#

Yes, which is why I'm a great proponent of introducing topology in that order.

#

First in R/R^n, then metric, then general

#

Which is an interesting contract with measure theory, where I feel starting with the context of R and the Lebesgue measure specifically, is counterproductive

#

You should start with the general language, but of course repeatedly bring up the specific example of R and Lebesgue

wispy ridge
#

What program is used to make figures like these?

image.png
#

This is from DoCarmo differential geometry

#

I know he probably just drew it on paper, but what tools people use today to male those images?

cloud zealot
tawdry venture
#

TikZ

noble hare
#

Interested in peoples opinions on color coding equations/expressions. Motivated by seeing the image below and finding it somewhat harder to parse than if there were no colour. (But this could also be due to the fact that I hadn't read the previous context, along with various other factors like number of and choice of colours). I imagine there is a good middle ground where its very effective.

244390699431100416.png
vocal phoenix
#

In general it's probably useful, and I sometimes do that in lectures, but I don't think this particular example is great.

noble hare
#

Yeah I wanna stress that the OP was using it for their own use rather than to explain to someone else, so it clearly worked for them. But made me wonder about effective use to convey things to someone else

vocal phoenix
#

I do sometimes color-code things in more complex expressions, based on the role of individual parts or relationships between them.

#

Although I do that in lectures, my own notes are monochromatic.

cosmic ibex
#

I went through a phase in high school where I used different pen colors in my homework to encode typefaces -- red for italic letters, blue for upright.
My poor teachers must have cursed me ...

tawny slate
#

basically i think color-coding is one particular way to "highlight" something, putting the reader's attention on it

#

cognitive science has shown that directed attention is a required component to learning, so in order to help the reader see the parts that are important or relevant, it can help to highlight them

#

in this particular case, the different colors add so much information that it doesnt focus the attention but detracts attention from where it should go, which is why it doesn't work as well here

#

ofc, this is also pretty contextless

#

you could also use color-coding as some kind of key, making the diagram or expression some kind of reference rather than some part of exposition

#

but in many instances of this kind of use case, i would rather a different approach be used if possible, as it can be difficult for colorblind people and there might be more illuminating ways to direct attention to the main ideas

#

i can only think of niche situations where this is a good idea imo

daring gulch
# noble hare Interested in peoples opinions on color coding equations/expressions. Motivated ...

Colors are definitely helpful but use them very sparingly imo, only where the most emphasis is needed. Otherwise, you risk distracting both you (from obsessing over coloring) and your audience ("The COLORS, what do they MEAN?!").
Your image in particular uses very high contrast (bright?) colors, which may be why it's harder to read. The light red 1 and other more muted colors would be better.

drowsy otter
#

Sorry to distract from the subject, but I have a question about teaching.

I'm planning on teaching functional skills maths to adults who may or may not have English as their first language.

I've read the learning objectives of Entry Level 1, and the ones pictured in the screenshot below seem to be basic-level skills that I believe everyone should know. Too basic, if you ask me. However, I don't want to assume anything about these students' prior knowledge, or neglect to teach them this if they do not know.

Keep in mind these students will be adults who aren't working, so I don't want to insult them by teaching them content they would already know, but I don't want to assume that they do either. After all, these adults may or may not have English as their first language, so may need to understand numbers before going on to more complicated stuff. However, it's debatable (at least to me) whether or not I should skip this very basic part of the module.

Thank you guys for your help. It may sound silly, but I want to do what's best for my audience, especially since they're not grade-school kids, but adults who do not quite yet have access to GCSE's. Therefore, I don't want to treat them like kids.

image.png image.png
long pelican
#

This is typically uncommon advice but I would suggest finding more about what they know and what they don't know, as much as you can, and then use that knowledge to help craft a syllabus. This avoids the problem of having to guess what's level appropriate for them based on little to no information

modern trench
dapper flume
# noble hare Interested in peoples opinions on color coding equations/expressions. Motivated ...

Color is a clarifying tool which can be used against that fact. I took classes in design before I got into math and education, and we did a lot related to color psychology and how to emphasize a subject. When it comes to equations,

  • Color only what needs to be colored. I think generally 3 distinct colors is a soft upper limit unless you know what you are doing.
  • Keep colors consistent as the equation evolves
  • Saturated colors only really work on light backgrounds most of the time. Consider pastel for dark backgrounds for readability.
dapper flume
midnight scarab
#

The "up to 20" is killing me. Especially for adults

#

And like after 20 it's all smooth sailing, it's just twenty-one ...

dapper flume
# midnight scarab And like after 20 it's all smooth sailing, it's just twenty-one ...

When it comes to comparing, ordering, and adding/subtracting, I think keeping it below 20 at first is generally a good idea for tangibility. For example, it's not too cumbersome to move a group of 8 objects and a group of 5 objects together to count 13 objects (8+5=13), but doing the same thing with something like 48+92 is just intangible to a beginner. For adults it might not take long to get to this point, but remember that for the most foundational learners, even large numbers are too abstract to tackle right away.

tawny slate
#

fun somewhat related fact: the idea that any two consecutive integers differ by 1 is actually not something humans innately know

#

it is explicitly a concept that has to be taught

#

there are neuroscience studies that explain how these kinds of ideas are mapped out and learned in the brain and there are indigenous tribes of people who dont know this

tight star
#

is it related to how we naturally think multiplicatively rather than additively?

tawny slate
#

probably, in the sense that:

  • we do know that we tend to think larger numbers are closer together (99 and 100 are closer together than 9 and 10)
  • i dont know if we can technically say this is a "multiplicative" mode of thinking or if it, for instance, has more to do with spatial or symbolic thinking
  • not sure if this idea is explicitly related to the differ by 1 idea, but seems like it is
cosmic ibex
#

Well 99 and 100 are closer together than 9 and 10, relatively speaking. I've seen this being characterized as a misperception, but to me it feels completely rational and desired to have intuition about relative differences rather than strictly about absolute subtraction results. The former seem to matter a lot more in real-life situations.

midnight scarab
tawny slate
#

which is why i do believe it is logarithmic but i come just short of asserting that it is (maybe it is, but i am personally not well-versed enough to declare it)

cosmic ibex
#

Yes, what I was reacting to was mostly the notion (I don't recall if it was here I saw it, tbh) that this cognitive phenomenon is somehow a shortcoming of our intiution about numbers.

daring gulch
cosmic ibex
#

They did.

dapper flume
cursive schooner
pastel ledge
lethal leaf
tight star
#

So something interesting i came across, to do with issues people have with variables

#

The yoneda lemma came up in a number theory problem

#

Specifically we were trying to prove that, for integers m, n, $m | n \iff \forall d \in \mathbb{Z}, d | m \implies d | n$

burnt vesselBOT
#

Pseudonium

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tight star
#

There was no issue with the $\implies$ direction which I thought would be the harder direction

burnt vesselBOT
#

Pseudonium

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tight star
#

It follows from transitivity of divisibility

#

But the $\impliedby$ direction actually caused a fair bit of issues

burnt vesselBOT
#

Pseudonium

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tight star
#

We eventually got to it but i had to really walk them through it

#

It turns out that they’d thought of the step “substitute d = m” pretty quickly, but thought they weren’t allowed to do it

#

They didn’t know they were allowed to change variables like that

#

Though they weren’t able to articulate why they thought this step was disallowed

#

I wonder how one could fix this misconception…

tawdry venture
#

what has yoneda got to do with this

#

you can phrase it as
"we know that for any integer d which divides m, d also divides n. in particular this is true when d is taken to be m itself."

#

thats how i have always thought of it.

tight star
#

oh this is just the yoneda lemma applied to an appropriate category

#

the category theory isn’t the main part here

tight star
tight star
tawdry venture
#

maybe try showing some problems where the key step is to instantiate a forall'd variable with a specific number like 5

tawdry venture
tight star
#

yeah I did that afterwards

tight star
tawdry venture
#

what pedagogical value does it bring to the table?

#

how??

tight star
#

cause we were using universal properties

tawdry venture
#

show me the lemma itself and exactly how you are applying it and why this is useful pedagogically in a NT context

tight star
#

and then yoneda naturally comes up

tight star
#

so, here’s the first problem we used it for

#

let $a, b \in \mathbb{Z}$ such that $\text{gcd}(a, b) = 1$

burnt vesselBOT
#

Pseudonium

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tight star
#

Then show that $\text{gcd}(a - b, a + b)$ is either 1 or 2

burnt vesselBOT
#

Pseudonium

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tight star
#

We used universal properties to help with this

#

like you can use the universal property of gcd as part of solving this

tawdry venture
#

i get your argument with divisibility as a universal property, phrased entirely in terms of divisibility itself.

tight star
#

mhm

#

yeah like I didn’t explicitly introduce a category or anything

tawdry venture
#

what i'm objecting to is: why dress it in dozens of previously unknown words of category theory jargon?

#

what good does that do?

tight star
#

I mean it worked?

#

it wasn’t really dozens of words..

tawdry venture
#

how did you state the yoneda lemma

tight star
#

well

tight star
#

Though that was actually after the problem

#

We used it in a more specific way for the problem itself

#

I can show you how we solved the problem?

#

the only new words were really “Yoneda lemma” and “universal property”

#

everything else was just standard NT

tawdry venture
tight star
#

cause it worked?

tawdry venture
#

if you are not planning to tie them in with anything at all in the future then why introduce them at all

#

feels like an overcomplication to the extreme

tight star
#

it really wasn’t that complicated..

tight star
#

I just cared about finding a nice way to solve the problem

#

and I happened to find a nice way using universal properties and yoneda

#

that’s all?

tight star
#

you can disagree with my use of category theory, that’s fine

halcyon glade
#

I usually try to come up with solutions that the helpee would be able to get on their own, without introducing any sort of jargon

#

And then the jargon can be introduced as a "bonus" at the end if you really want

tight star
#

hm, right

#

i mean i guess I could’ve avoided using yoneda lemma

#

like those words

#

but - how else would you call the universal property of gcd?

twin lichen
tight star
#

I don’t think they would’ve known what meet was..

#

considering I didn’t either

tardy ember
#

do you think they would've known what a universal property was?

tight star
#

no

#

but im asking about what I would’ve called it without new jargon

tardy ember
#

...if you wanted to avoid "new jargon" then why did you call it a "universal property"

tight star
#

(and again, the category theory really isn’t the important part here)

tight star
#

but I was told I should

#

so im asking what I should’ve done instead

#

but anyway this was just a question about teaching variables

#

I guess everyone latches on to the category theory instead of engaging with that

vocal phoenix
#

If you brought in Galois theory we'd have latched onto that as well

twin lichen
#

The language of category theory is unnecessarily general to treat this, in my opinion. Posets deal with relations - is a <= b? Categories deal with hom-sets: there is a set of morphisms from a to b.
Relations and propositions can be recast as sets, by making a proposition into a set that has either 0 or 1 elements, but this is occasionally confusing and makes it harder to parse statements.

For example, saying that
a <= gcd(x,y) if and only if a <= x and a <= y

is equivalent to saying that
Hom(a, gcd(x,y)) \cong Hom(a, x) \times Hom(a, y)
but then one has to decipher that

  • bijection of sets is being used as a stand-in for logical equivalence of propositions
  • the Cartesian product of sets is being used as a stand-in for logical conjunction

In both cases i think this is unnecessarily complicated if we are talking about propositions. Posets are only degenerate categories and the language of categories is poorly adapted to discussing the theory of posets.

tight star
#

Like

#

All I used was

#

$\forall d \in \mathbb{Z}, d | m \wedge d | n \iff d | \text{gcd}(m, n)$

burnt vesselBOT
#

Pseudonium

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tight star
#

what am I supposed to call this?

#

as not new jargon

vocal phoenix
#

I'm not sure it needs a specific term in the context of this particular problem, although throwing in "we call this kind of thing universal property" isn't a problem in my opinion.

tight star
#

well

#

it’s a mess of symbols, so it’d be convenient to have a name for it

#

I used the name “universal property of gcd”

#

if you have an alternative suggestion, im open to hearing it

tight star
#

I did not mention categories or morphisms or anything like that

twin lichen
#

it is the meet.

tight star
#

so I should’ve said “gcd is the meet of m and n”?

#

im not exactly sure how this is better or worse than what I already did

#

this is someone who was trying to learn some proofs through NT

#

I doubt they would’ve met meet before (hehe)

#

especially since, uh, I didn’t see that word until my category theory course actually

#

i don’t remember directly working with posets

#

anyway, lesson learned, next time I won’t mention that what im talking about is in any way related to category theory

#

because for some reason people will just tunnel-vision on that

#

I’d appreciate anyone who has something to say about the actual Q I wanted answered

tight star
vocal phoenix
#

I probably would have talked generally about what we're actually doing when we use letters and quantifiers.

tight star
#

right

#

so I should’ve explained “forall” more, then?

vocal phoenix
#

And in particular that the letters aren't important as such, and the "m" represents "any integer number you want"

tight star
#

mhm, right

vocal phoenix
#

It's a bit hard to say what I would do since I don't have the other side of how that conversation would have gone, but that would have been the gist of it.

tight star
#

sure that’s fair

#

i will try that out next time then

vocal phoenix
#

I'd have focused on how the letters are "placeholders" and in particular since you can put any number in the place of "d", you can also put "m" there, because that is a number

tight star
#

i see i see

#

i think i might’ve tried to phrase this using functions then

#

but yeah - if im going to be using meets/universal properties, i should make sure the student understands the quantifiers properly

#

that makes sense

vocal phoenix
#

I think I wouldn't bring in any "outside" concepts such as functions, posets, categories, general algebra, what have you.

tight star
vocal phoenix
#

If someone struggles with the concept of "variable", I think it's best to stick to specifics

tight star
#

If you’re curious, that was the convo

tight star
#

Though - we do need to talk variables if we’re using gcd(a, b) right

vocal phoenix
#

I sort of get why you love category theory, because it's so general

tight star
#

im not sure i could avoid them..

tight star
vocal phoenix
#

But in my experience as a teacher, getting too general too early is often not as great as talking in terms of very concrete objects

tight star
#

but yes, it’s quite general

vocal phoenix
#

(one notable exception being the Lebesgue integral, because it makes no sense to start with the Lebesgue measure and integral on its own instead of going full measure theory immediately)

tight star
#

oh yeah totally

#

(has never done measure theory)

vocal phoenix
#

Even if on a more advanced level that would be the best approach

tight star
#

i mean im curious how you’d approach the problem

#

To me the way via using gcd as a meet felt the most natural

vocal phoenix
#

Number theory is a bit outside of my experience but as I say, I'd try to stick to the "domain of knowledge" I can expect the student to have.

#

For the same reason I wouldn't bring in general measure theory if someone asked me why a function that's undefined at a or b has an integral on [a,b] anyway.

#

I'd stick to the framework of Riemann integration even though I would hate it

tight star
#

sure

vocal phoenix
#

Although I'd definitely mention that there's a language and toolset that makes this sort of thing much easier to handle

tight star
#

i would say that I don’t think I really brought in much category theory

#

I didn’t define categories or functors or natural transformations or Hom functors or anything

#

I just used yoneda as a name for this result about divisibility

twin lichen
#

but yoneda is a more advanced concept than categories, functors, natural transformations and hom functors so somebody who tries to actually understand what you meant would have to understand all those concepts before they can understand the Yoneda lemma

tight star
#

I mean

#

I can just give the result of yoneda

#

and that’s understandable

#

I don’t have to give all the prerequisites

#

also I gave a talk about the yoneda lemma in the category of matrices

#

I did not define categories, functors, natural transformations, Hom functors

#

but people understood it just fine

#

like, people can understand specific instantiations of a theorem without having to understand the full theorem

#

is the point im making, I think

#

and indeed, usually when I use category theory in my explanations here, im using specific instantiations of general results

tawny slate
#

ok so as someone who had once planned to learn category theory but has not yet found the time

#

can you either eli5 or roleplay this instruction with me

#

i tried to follow the help channel but i got lost the moment yoneda was mentioned

tight star
tawny slate
#

prove that gcd(2a+b, a+2b)= 1 or 3

#

i can see that whenever a and b are congruent mod 3 we get the gcd is 3

tight star
#

Have you met the universal property of gcd

tawny slate
#

im not sure how to show that this is the only way to get a gcd other than 3

tawny slate
tight star
#

It’s this:

#

Let $m, n \in \mathbb{Z}{\geq 0}$. Then $\text{gcd}(m, n) \in \mathbb{Z}{\geq 0}$ is the unique integer satisfying $\forall d \in \mathbb{Z}_{\geq 0}, d | m \wedge d | n \iff d | \text{gcd}(m, n)$

burnt vesselBOT
#

Pseudonium

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tight star
#

Does that make sense?

#

It lets you interconvert between two divisibility statements and one

#

So for example, gcd(6, 9) = 3

#

This means that d | 6 and d | 9 if and only if d | 3

tawny slate
#

ok, i see that

tight star
#

Do you want to spend more time on this, or should we get to the problem?

tawny slate
#

i am not totally clear about why its called a "universal property", i imagine it has to do with the fact that this property, no matter how you define divisibility, as long as it satisfies this property, will get you basically the standard gcd definition

tight star
tawny slate
#

but i suppose its not relevant to this problem specifically

tight star
#

We haven’t specified what the gcd “is” directly

#

Instead

#

We’ve specified what it “does”

#

How to use it

#

How it relates to every other integer in the “universe”

#

What it lets us do is interconvert between two divisibility statements and one

#

That’s the idea behind a universal property - instead of directly specifying what something is (through a construction), you specify what you want it to do

#

In particular, I haven’t actually shown that such a gcd exists

#

It turns out it does though - the usual def of gcd satisfies this property

#

Is that ok?

tawny slate
tight star
#

So like

#

The gcd of 6 and 9

#

It lets us convert 2 divisibility statements

#

d | 6 and d | 9

tawny slate
#

the rest seems fine, although not rigorously demonstrated but i get the idea

tight star
#

Into a single statement

#

d | 3

#

And we can also convert back - given d | 3, we can get d | 6 and d | 9

#

That’s what I mean?

tawny slate
#

okayyyyy

#

sure, go ahead

tight star
#

Ok cool

tawny slate
#

now whats the next step

tight star
#

So we want to show $\text{gcd}(2a + b, a + 2b)$ is either 1 or 3

burnt vesselBOT
#

Pseudonium

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tight star
#

Let’s start with the universal property

#

So, $\forall d \in \mathbb{Z}_{\geq 0}, d | \text{gcd}(2a + b, a + 2b) \iff d | 2 a + b \wedge d | a + 2b$

burnt vesselBOT
#

Pseudonium

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tight star
#

Is that fine?

tawny slate
#

yep

tight star
#

Then, if d | 2a + b and d | a + 2b

#

Then d | 2(2a + b) - (a + 2b)

#

So d | 3a

#

Is that fine?

tawny slate
#

ok yeah i follow

tight star
#

So, we can show that $d | 2a + b \wedge d | a + 2b \implies d | 3a \wedge d | 3b$

burnt vesselBOT
#

Pseudonium

547515245258735616.png
tight star
#

Then, we can use the universal property again

#

So $d | 3a \wedge d | 3b \iff d | \text{gcd}(3a, 3b)$

burnt vesselBOT
#

Pseudonium

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tight star
#

Is that fine?

tawny slate
#

huuuuuh

#

ok sick

tight star
#

Wait is that good or

tawny slate
#

yes very good

tight star
#

Ok cool

#

Now

#

We had as an assumption that gcd(a, b) = 1, right?

#

It turns out this implies gcd(3a, 3b) = 3, you can argue using Bezout’s lemma

#

Is that ok, or would you like to see how that’s done?

tawny slate
#

so the rest of it is i think very intuitive, i could probably prove from first principles without bezouts

#

bezouts, like, i dont even remember offhand

#

ok so this works, i now understand this problem

tight star
#

Ok sure

tawny slate
#

so now the question is

#

i could have gotten here with the hint:
if d|a and d|b, then d| (any linear combination of a and b)

tight star
#

Sure?

#

That was something id worked through earlier (in a different help channel)

tawny slate
#

assuming the basics there, like ignoring degen cases involving zero and non-integers

#

sure

#

but my next question is

#

what was the point of bringing in the language of "universal property"

#

i feel like, from a student's perspective, i didnt need that

tight star
#

Well, I haven’t quite finished the problem..

tawny slate
#

oh?

#

ok please continue

tight star
#

So

#

If we chain together our equivalences

#

We’ve shown that $\forall d \in \mathbb{Z}, d | \text{gcd}(a + 2b, 2a + b) \implies d | 3$

#

Right?

burnt vesselBOT
#

Pseudonium

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tawny slate
#

oh waaaaait

#

ohhhh i see, yes

#

ok yeah

tight star
#

This holds for any nonnegative integer

#

So in particular

#

It holds for $d = \text{gcd}(a + 2b, 2a + b)$

burnt vesselBOT
#

Pseudonium

547515245258735616.png
tight star
#

Which obviously divides itself

#

So, we get that $\text{gcd}(a + 2b, 2a + b) | 3$

burnt vesselBOT
#

Pseudonium

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tight star
#

Meaning the gcd is either 1 or 3 - those are the only factors of 3

#

And that finishes it!

#

Does that make sense?

tawny slate
#

yeah

tight star
#

This step is essentially where we apply the yoneda lemma

tawny slate
#

ok whoa