#math-pedagogy

1 messages · Page 31 of 1

slim path
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Depends on what the child infers. I could easily take that message and have the headcanon be that the other kids I am here with are dumb that's why the teacher asked me to shut up.

outer burrow
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the child may already think that

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and if so, i don't see how the message further reinforces it

slim path
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Which is why possibly reinforcing it is not a good idea

outer burrow
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yeah that's up to Will

torn apex
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How did you find this experience? I'm currently considering doing math tutoring myself but mostly as a fun hobby

burnt junco
arctic coral
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I did give a wrong solution, during my courses and figured something wrong with that solution later what kind of advice you would give me dealing with this situation i intend to have a discussion with the students about it, and take it as a learning opportunity

abstract grove
stark heron
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i will say it also personally forced me to fill in any knowledge gaps i mightve had, also gets you very good at communicating math which you may or may not be used to

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i guess, a lot of knowledge i took for granted i realized while tutoring 😭

lethal hornet
abstract grove
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I saw about eight or nine people either quit or get fired from mine. Then they fired me KEK

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they also fired a center director with a PhD in math. Along with three or four other center directors

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insane people who only care about their bottom line

solar sorrel
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I mean, if you seriously approach anything involving topos theory, higher category theory, or anything in theoretical computer science, then adopting a neutral constructivist approach seems the best way forward.

tight star
solar sorrel
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the object language is topoi is intuitionistic, so you can no longer use LEM.

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[you have to use continuations, which are .... not a pleasant programming construct]

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easiest way to solve that is to get someone more advanced than them in the same class to pop their bubble.

there are more advanced strategies you can use, but those require really skilled pedagogues which are effectively non-existent on a global scale.

plain valve
plain valve
zinc dove
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Hello, in your experience, do playing logic puzzle games helps learning/understanding math? If so, which game do you believe do this and why?

zinc dove
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Personally, I find games entertaining, specially video games, and trying to win the game by learning the patterns to solve the problems, is a nice way help learning math, as it's possible to translate (e.g. via analogies) the elements of the games to more abstract objects using symbols.
Looking at a post here https://redlib.perennialte.ch/r/math/comments/131qi17/puzzle_games_that_are_heavily_mathematical/, here's a list of games from that post (all of them available in Steam):

  • Baba is You
  • The Witness
  • Turing Complete
  • Opus Magnum
  • Taiji
  • Tactical Nexus
turbid zenith
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Honestly almost any game can help students learn mathematical thinking.

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I taught my liberal arts students about mathematical proof through having them play a few Tic-Tac-Toe variants from around the world this past semester and it went great!

tawny slate
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i think what is most useful about puzzles is methodological rather than conceptual

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many puzzles are designed in such a way that you can't really reason to the answer immediately, so you need to get your hands dirty and try a bunch of things, guess and check, brute force

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if there are too many possibilities, what logic or heuristics could you use to simplify the solution space?

and when you feel like youve exhausted everything but the answer still isnt clear, how do you systematically show that you've tried everything? when you find the gaps you missed, the solution lies in there

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there are some meta tips and tricks, such as knowing to try different orderings in addition to different things, to work backwards or locate logical constraints, but once youve got the big ones, the more nuanced and niche concepts are not really learnable generally other than by intuition, which is generally only built by practice and analysis

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to actually learn math specifically, that would depend on the specific puzzle i think. there are some phenomenal puzzle games here but i think not many of the ones listed are actually something you could cover in a math curriculum in some way, unless you got really creative

but i do agree that puzzle games are great nonetheless, because these skills are surprisingly lacking in a lot of students

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if you want some kind of game enrichment for math, i actually think board games are worth looking into as well

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a lot of board games you can actually apply to tangible math lessons

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a bit more advanced is taking any deck builder or engine builder (like Star Realms or Gizmos) and explaining the use of derivatives and acceleration in terms of efficiency

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or using Mascarade (which is really quite simple to learn and play) to model quantum superposition (which is intimidating)

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board games have the disadvantage that you need a physical copy and the people to play with, but the advantage is that it promotes a more hands on learning style and is much more social

torn apex
solar sorrel
solar sorrel
plain valve
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Ah ye I like this paper lol

quasi maple
quasi maple
tawny slate
# quasi maple yo calculus board game when lmk

just say that in, for instance star realms, the goal is for your damage per turn (1st deriv) to be higher than that of your opponent by the time N damage is accumulated

and so when you pick up econ cards, you are usually making your damage per turn worse, but it allows you to increase your damage per turn by more (2nd deriv) later

so these games would be solved and boring if you could calculate these precisely, but you can demonstrate how you can use these heuristics to help guide a general feel for why scrapping is so strong, why an aggressive deck can sometimes work and sometimes not work

tawny slate
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so pedagogically i dont prefer it in a math class, i think its more suited to english/philosophy/social studies actually

burnt junco
torn apex
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Thanks! I'm really happy with the Newton Raphson Slide, Taylor Synthesizer, & the Reimann Accordian. Those examples really bring to life the notion of integration

scarlet panther
# tawny slate if you want some kind of game enrichment for math, i actually think board games ...

Tabletop games definitely feel underrated to me when it comes to math learning. Simple space moving board games can obviously help kids practice counting, but card games like Yugioh involve plenty of arithmetic, as well as the ordinary logical thinking needed to wrap your head around the rules and strategize. There's so much mathematical thinking that goes into games generally speaking, whether its puzzle games, board games, or number heavy games that it saddens me when I see so-called educational game products that just slap pretty visuals onto a math worksheet. The Learning Company CD-ROM from the 90s-00s were great at that from what I can remember, if only because I remember them fondly looking back all these years later

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I plan to curate a steam library of phenomenal and perhaps educationally significant games for my kid one day so they have a collection to look back fondly on. The Witness is the #1 game going in there. In my mind, its a game you can return to over and over again that you can find interesting even as a beginner and then keep chipping away at as you get older and more capable

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Antechamber is of a similar vein

zinc dove
scarlet panther
haughty latch
torn apex
torn apex
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All I know is I played an ungodly amount of Number Munchers & Oregan trail as a kid

scarlet panther
turbid zenith
torn apex
turbid zenith
tawny slate
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i do have a spicy idea for a teaching assistant for combinatorics, but the full technical specification i haven't fully thought through yet, so if thats an interesting design problem anyone wants to try tackling feel free to DM me

scarlet panther
scarlet panther
tawny slate
# scarlet panther What do you mean by teaching assistant, like an AI?

not AI, a library for adhoc coding for presentations and visualizations

so imagine you're teaching combinatorics, doing an example problem, but students are having difficulty seeing why their solutions are wrong or why your solution is correct. a library where you interactively adhoc quick, easy to type and define functions that draw the bijections and casework explicitly i think helps give students a more concrete feel for combi, in the same way we use visualizations for geometry

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we have things like euclidea and desmos for algebra and geometry, and a lot of it is very easy and quick to deploy and use, but nothing of the sort for combinatorics

storm prairie
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As someone who does research in combinatorics I like that idea 🧐

tawny slate
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i just have too many projects on my plate to do this one, id love to collaborate with people on it to take some load off of me

scarlet panther
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So would the teaching assistant be a program like desmos in which you can input a combinatoric expression, and then it displays the operation as an animation or smth?

tawny slate
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yeah lemme type out an explicit simple example, but brb real quick getting food

scarlet panther
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Or would it be more like the user has to use your teaching assistant to create the animation visualization themselves? That's what I think of when I hear library

storm prairie
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I would guess like desmos or manim you do it yourself but there'd be ways to share/publish your work for others to use as well

scarlet panther
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The user would be the teacher, not the student in that sense

storm prairie
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I guess we'll hear what Cozmogrgdfschkipkhrshtensi has to say when they get back

storm prairie
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I've definitely coded visualizations for calc III in desmos

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And then just demonstrated on the board during class

scarlet panther
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Oh, speaking of things one idea I've wanted to see was a reverse graphing calculator, where you can take a graph form and manipulate it using simple controls within the graph, and it would update the equation that makes the graph. I'm pretty sure you can use desmos to do something like that, but I think it would work differently than I'm imagining

storm prairie
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I think I saw a video about making that idea into a game not too long ago

scarlet panther
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I recognize this youtuber lol

storm prairie
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It was the first (and only now that I think about it) video of his that I saw but the concept is something I had thought about before

storm prairie
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But they don't show the equation

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I could code it in Desmos for certain forms like polynomials and trig functions but they'd be limited in what you could change 🤔

scarlet panther
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Or I think I know what you mean

storm prairie
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MyOpenMath is like a math assignment software, you could search it and read the docs but idk if there's anything public you could look through
And since we're between semesters I can't open up a course to show it 😅

scarlet panther
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There are various graph editors

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like Khan academy has a similar kind of thing

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Where you can control a graph, and then check if its the answer or something, no?

storm prairie
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That... I'm not sure of 😅 I make my students use Khan Academy but haven't used it myself in a long time

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But it would make sense

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Like manipulate this graph of sin(x) to become 3sin(x)+4

scarlet panther
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Me neither tbh, when I used Khan academy, a single mistake could drop you from 97% mastery to 5%

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That was like 2014

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earlier

storm prairie
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Now they just have four levels instead of percentages

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Something, something, proficient, mastered

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idk what goes on in the backend though

scarlet panther
# storm prairie Like manipulate this graph of sin(x) to become 3sin(x)+4

But yeah, I feel like taking something like this and making it more accessible could have some educational merit. Like, if you wanted to explore function transformations in desmos, you would have to kind of know where to put the variables, and then manipulate each variable via a slider, which is good, but perhaps a bit of a learning curve for students that don't already know what they're doing? Not sure

storm prairie
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Well you'd do that on the backend, the student would just see the dots on the graph to drag

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Or at least I prepare a graph for the student if I want to do that type of thing

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I guess it's hard if the student wanted to do it themself 🤔

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Desmos teaching has a lot of premade calculators but I see the issue

tawny slate
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ok im back, im gonna lay out a rough picture of how this tool would work:

imagine im explaining the following problem: how many permutations of ABC are there?

obviously 3 x 2 x 1, but why?

to create the visualization, type some code that might look like:

set X = {A, B, C}
set Y = {a, b}
set Z = {c}
pull 1 from X
map Y to X
pull 1 from Y
map Z to Y
pull 1 from Z
visualize

and now it will animate the following:

draw A, B, C
then from each letter, branch into 2 paths, containing the not yet chosen letter, then draw 1 branch from each of those to write the remaining letter

then show how traversing this tree, one by one, creates the permutations like ABC, ACB, BAC, etc

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this could be one way to visualize, but you can give options to modularize it in several preset ways, or define something custom if you prepare in advance

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maybe have a built in routine for permutations entirely

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since its so ubiquitous

scarlet panther
storm prairie
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Yeah I would put the equation on the graph

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So they close the equations panel and only look at the graph panel

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Here I'll make a quick example

scarlet panther
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Sure lol

scarlet panther
tawny slate
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yeah

storm prairie
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Essentially this is all you see

Screenshot_2025-12-18_at_15.07.18.png
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Of course I could pretty it more

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I forgot to tell the thing to round to two decimal places on the variables

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So 💀 the equation will look crazy unless you use nice values

scarlet panther
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This might be a dumb question since I'm kinda ignorant about combinatorics, probability, and statistics, but could it extend to those wider areas, particularly probability? Because I think statistics for example is a particular pain point and this sounds like a tool that can help with that

storm prairie
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If it can do combinatorics visualizations I'm sure it could certainly do at least discrete probability

tawny slate
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probability i think of in several "categories" of concepts: in terms of counting, in terms of events, and geometric probability

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if you do like total successes / total outcomes, this is just counting, and the tool would work

scarlet panther
storm prairie
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😆

tawny slate
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it can be adapted to events, because it can do trees, so it probably could assign weights to those branches

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geometric probability is just geometry so desmos or whatever will just work

storm prairie
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Here I'll give you the one I use for cylindrical and spherical coordinates too

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Instructions are in the expressions menu

solar sorrel
torn apex
storm prairie
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That's really interesting

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fourier at its finest

torn apex
scarlet panther
scarlet panther
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You open/close the relevant folders depending on what form you wanna use

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Not sure if there's a way to automate something like that in desmos

storm prairie
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Take a look at my spherical/cylindrical one for automation within the graph

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i.e. buttons

storm prairie
scarlet panther
scarlet panther
storm prairie
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'easy' is an interesting way to put that

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XD

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I guess it is a simple solution though

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That was a technique I developed from doing animations in desmos after they introduced tickers and clickables within the graph

solar sorrel
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[I do seem to have some CSS issues, though.]

solar sorrel
tawny slate
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ok question for those with a bit more combi expertise

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is it a lies-to-children to say that counting is constructive?

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the context here is that when you teach combi, part of what makes it so tricky is that you have to not only get to how youre counting something, but also check your answer

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and I find that while explaining why solutions are or aren't correct, you mostly just end up drawing out exactly how the counting works, step by step, as if you were doing programmatic steps and bijections

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in this sense, ive never really encountered any counting problems that arent "constructive" (in quotes because i dont know for sure if im not abusing this term in some technical way)

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im sure there are some esoteric examples of nonconstructive counting, but i wonder if any of that is practical or meaningful to students in a way they would ever encounter it or care

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so just outright telling them that counting is constructive as a mantra to instill better mindset and habits, does this constitute a lies-to-children?

midnight scarab
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I imagine there's also some application of the idea of translating counting into an expectation value of the indicator function

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Don't have a good example where it really buys you something nontrivial

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At least for exact counting

tawny slate
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i think incl-excl is still constructive, because you are making sets and then doing set operations, and i think derangements satisfy this too

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maybe expected value is a good example of being nonconstructive, but at that point i feel like you can sort of handwave and say that the probability or expected value itself is constructed in some sense, and so its not a lies-to-children necessarily

storm prairie
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That was one of the snags I actually had reading my first combinatorics book because I wasn't used to algorithmic proofs

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Is there even a nonconstructive way to count

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There's existence like Ramsey's but is that counting

tawny slate
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i don't consider that counting, no

storm prairie
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Right 🤔

storm prairie
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But counting itself if the axiomatic definition of countability is to algorithmically correspond each element to a successor

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At least when not dealing with bayesian I think probabilities boil down constructively as well from proportions

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Which then yields expected value

warm valley
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there are proofs in algebraic combinatorics that use "these two vector spaces corresponding to the things I count are isomorphic, therefore their dimensions are equal", which are typically nonexplicit in terms of writing down bases for both sides, and so i'd often consider them nonconstructive

storm prairie
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AH I didn't think about algebraic

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The union-closed sets conjecture is a great example if it's true

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The leading arguments are all nonconstructive

tawny slate
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uhhh could you guys eli5 that for me? not familiar with this

storm prairie
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What does eli5 mean 😅

tawny slate
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explain like im 5

storm prairie
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Yeah waes just told me 😅

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Well I can't explain the vector spaces but I can explain UCSC

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Although maybe not like you're 5 haha

quasi maple
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It doesn't have to be literal

storm prairie
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Essentially consider a union-closed family of sets
The conjecture states that there is some element x in at least half of the sets

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It's somewhat of a counting problem

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Ish

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Maybe I overhyped it now that I think about it

quasi maple
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https://en.wikipedia.org/wiki/Union-closed_sets_conjecture This thing?

Union-closed sets conjecture

The union-closed sets conjecture, also known as Frankl’s conjecture, is an open problem in combinatorics posed by Péter Frankl in 1979. A family of sets is said to be union-closed if the union of any two sets from the family belongs to the family. The conjecture states: For every finite union-closed family of sets, other than the empty famil...

storm prairie
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Regardless the leading method up till a while ago was by looking at the average frequency of an element and pushing it up near 50%

storm prairie
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Oh finite is important yes

quasi maple
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cf. closed operation, a family of sets is union-closed if for A and B in the family, the union A u B is also in the family

storm prairie
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And nonempty of course

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Yeah normally I'd say closed under union

quasi maple
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That'd be what I'd have said too, yh

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If you have such a family, the conjecture says that there's an element in at least half of the sets that are in that family

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In the image in the Wikipedia article above, E1 through E5 form such a family; v1 and v2 are satisfactory elements

storm prairie
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What Mr. Wires said

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It's funny there are 3 formulations in the wikipedia if we can get our paper published over here we'll add a fourth measure theory formulation

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Because weren't you wanting for more

quasi maple
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coolempirelore.pdf!?

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when's that coming out on arxiv

midnight scarab
storm prairie
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Hopefully next semester

storm prairie
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A student had such a problem earlier today actually

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"student" one of the question askers

tawny slate
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but i don't see how this leads to any kind of nonconstructive counting

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maybe it's esoteric enough that I can just say "counting is constructive" and just add a *footnote to mention the esoteric weirdness exists

storm prairie
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You have

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There must be 6 but you never counted 6 at all

tawny slate
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i get that abstract idea, just dont see how you get there from the vector space stuff

torn apex
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@scarlet panther @turbid zenith Here's your chance to bring your math game ideas to life (simple puzzles only!); so uh, I kinda made a JRPG mash-up called "Theorema: A Chrono Link". Gameplay is straight up "A Link To The Past".

There's gonna be 1 dungeon + 1 puzzle + 1 boss. Right now I tentatively have the MC as Emmy Noether and the puzzle is about arranging mirror crystals to hand-wave teach something about crystals and symmetry and light reflecting hand-wave

let's make math pedagogy real again ;P

image.png Screenshot_20251220_100525.png noether.png
stark heron
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Did you draw this? :)

torn apex
stark heron
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Oh, That's pretty lame

torn apex
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also, in my experience, it is infinitely easier to get kids/non-math normies/humans excited about making games when their games actually look good.

marsh compass
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there are already too many gamedev wannabes in this world 😂

stark heron
stark heron
solar sorrel
marsh compass
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i was talking about idea guys who do not like the frustrating and tedious part of game development

vocal phoenix
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I'm reminded of the line “Everybody does have a book in them, but in most cases that's where it should stay.”

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And I think several other incrementals based on ordinal arithmetic, since there's quite good synergy there

marsh compass
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the website link is wordsus though

vocal phoenix
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Galaxy is a general aggregator for incremental games, I don't think there's anything shady going on there

marsh compass
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i didn't know that, so i'll just let you ip grab me

vocal phoenix
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I don't need to grab your IP, I already have my own

scarlet panther
zinc dove
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I'll say it uses ai generated images

tawny slate
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i mean he already admitted to using AI images

marsh compass
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JRPG? More like JPEG

dire harbor
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You dont get it bro, bring your ideas to life, expand your market value share and touch base on next gen level learning integration technology

tawny slate
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the only thing any of those people should be touching is grass

torn apex
torn apex
torn apex
# scarlet panther By "made a JRPG math game" you dont mean put a prompt into an AI game generator ...

lol, do you think I honestly took time out of my saturday to generate AI slop and post it here? 😛

  • tldr: yes, the art is AI generated b/c I have 48 hours for this game jam?
  • long answer: by "ai generated", i spent half of my saturday morning coming up with a storyline, a style bible, some ideas on how to use hexagonal tiling as the grid as a shoutout to noether's ring theory/symmetry work, added some spells using RGB crystals to tie it to QCD
scarlet panther
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When you gonna link it lol

torn apex
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oh i'm uploading everything to github now just so i can cement my "wannabe gamedev" moniker 😂
gimme a few minutes

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also, the target audience is for my 8-10 year old nephews who are being gaslit into learning math through these gamejams...so let's set our expectations appropriately low😅

dire harbor
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Love the fireworks of someone obviously pretty hurt by some criticism and trying to cope

torn apex
dire harbor
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By love i mean hate, please do this in private i dont want to cringe at your sprialing

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And come back when youre ok

torn apex
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anyways, back to the main point the got lost, using gamedev as a vehicle to teach math is pretty fantastic, especially in the age of AI

  • I can turn each of my nephews into a character, weave them into stories and lessons
  • i can even let them use the character generator (llm-powered) to generate new stories
dire harbor
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Do whatever you want ig but this method of like "say thank you to the bullies and own it" is really sad

torn apex
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@turbid zenith @scarlet panther also check out the music, I made it for the tiktok generation 😂 :

Anyways, the main storyline is pretty much ripped from Chrono Trigger where the player goes back to pivotal moments in Noether/Grothendieck's life. I made Pierre Deligne the anti-hero who betrayes Grothendieck b/c someone has to be a villain. Feel free to give your thoughts or add any crazy suggestions. As long as I get them before Sunday afternoon, I'll probably be able to squeeze them in. These gamejams are extended family fun activities so go for it

I'm genuinely curious how broadly applicable this strategy for math education. My anecdata so far is pretty positive but that might be selection bias (I only tutor the nephews that reach out)

dire harbor
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Welp i tried to help but im not pushing further with somebody on the internet

dire harbor
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Oh man this music is something

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Its exactly as pathetic as youd expect lol

torn apex
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haha, i bring the unc/dad-joke energy. there's a hard-cut off for enjoyment for the kids which is right about the time they discover their friends are cooler than their family

dire harbor
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Ai is really good at replicating the style of nothing music

torn apex
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probably less AI and more lack of talent of the user. but also, i went on spotify and found the most popular tracklist for genalpha and based the melody and beats off of that

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🤷 my attitude is pick your battles; as long as they're having fun learning math, who cares

dire harbor
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Did you make this? Cause this sounds exactly like ai

torn apex
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i came up with the melody & bass/beat. lyrics are all AI i mean, it shows but also, it's hard writing lyrics especially about math😅

stark heron
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lord jesus

torn apex
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oh btw, this isn't supposed to be artisticly good at all! It's just a vehicle to get the kids inspired to learn about math

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  • artistically good for adults. i mean, it's top shelf dad joke cringe humor
dire harbor
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Oh god the "its not supposed to be good" os

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I wasnt really paying attention to the lyrics i was more focused on the background stuff and how ai generated it sounds i have a hard time believing this was made by a human

torn apex
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yeah, i'm not sure what you're point is anymore? are you trying to judge children's educational material by your adult taste?

dire harbor
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Yes actually, children dont deserve shitty low effort slop when its really not that hard to put in that effort

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I know that was supposed to be fallacious bs but its actually true lol i do actually hold childrens content to the same standard

torn apex
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you're right. I'm gonna go back to "not putting effort" with my nephews to flesh out the rest of the game. cheers!

spiral elbow
dire harbor
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Tried to help before, now im talking shit

scarlet panther
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Do you think it's ever worth it for a student to explore alternate number bases, whether its simply an overview of the concept or doing some arithmetic in different bases? If so, when or in what context? I think the only time I ever did such a thing in school was in a high school programming class when we were participating in a comp sci competition, and some of the questions were about binary and maybe hexadecimal or octal.

Part of me thinks learning a bit about number bases can help elucidate how place value works and break some of the underlying assumptions we have about numbers, but the obvious counter is that it can be really confusing, while not teaching any practical skills.

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I mean, tbh I think even people that know how the base system works have confusion about it. Like with the "dozenal" or "seximal" debates, where they create all these different words for numbers to refer to their exotic base systems when they could just, like, repurpose what already exists (primarily in the case of seximal)

stark heron
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i think it is valuable to understand our number system, that is it a convention, and that can be better illustrated by trying alternate bases
and imo, knowing what binary (at least) is should be pretty standard if we use technology daily, to me that sounds like knowing that we have cells in our body you know 🥲 maybe thats an exaggeration

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also helps with time though by the point they are prepared to deal in alternative bases they know time? i think?

scarlet panther
# stark heron i think it is valuable to understand our number system, that is it a convention,...

I can see the analogy of understanding something "practically" useless to most people, but fundamentally educational or important like, you know, 99% of stuff you learn in school, so like biology. Like baseline knowledge people should have so that they know how the world works around them, whether its how computers rely on computation in base-2 or how medicine can impact cells to fix a disease or smth. Clearly I'm more knowledgable about math than biology lol

scarlet panther
stark heron
stark heron
scarlet panther
# stark heron lol yeah its hard to justify any "impractical" knowledge to any bureaucracy thou...

I also think that high school math classes could benefit from more enrichment besides the traditional Algebra 1-Calculus track for less interested students, which is obviously a controversial topic. But like, recreational math is a thing, and there's a lot of interesting stuff out there that isn't necessarily rational functions and conic sections, like board lattices or magic squares and cellular automata and stuff. And it's not like those don't use a lot of math, it would just be a (albeit difficult) balancing act of interesting material, sufficient academic/educational rigor, and good teaching

#

I don't know what people here think about the whole "drop algebra 2/calculus" idea. I'm not super committal, ive heard evidence against it

abstract grove
#

there used to be a standardized test called the CHSPE, California high school proficiency exam, and it had different number bases as one of its topics. It surprised me because I don’t know of that topic being a standard part of any high school classes. But I taught some kids about it to help them prepare, and they usually got the idea in one lesson

scarlet panther
abstract grove
#

there would only be at most one or two questions about it, simple calculations like find 23_5 + 104_5

#

and it was multiple choice

scarlet panther
#

Would you do it by converting it through decimal, or just calculating in the base directly? Like, obiously that problem is easier to do directly

abstract grove
#

I seem to recall sometimes the answers would be in the given different base, and sometimes they’d be base 10

#

so it would depend

scarlet panther
#

I made some excel spreadsheets of times tables in different bases a little while back to look for patterns lol

halcyon glade
toxic talon
# dire harbor Its exactly as pathetic as youd expect lol

Everything you've said in this thread is unspeakably rude. Before you come back to interact with this server, take a long time to look inward and determine if this is really the kind of person you want to be. Does it make you happy? Do you get satisfaction out of tearing other people down like this? That is not behaviour that is encouraged, or indeed welcome, here.

toxic talon
strong hearth
#

Correlation is not causation, and causation need not be correlation
Trying to articulate this in a way that's palatable for students IltakCry I realised how tricky it is to unpack concepts this way

storm prairie
#

Just the basic arithmetic

#

Using boxes and grouping for division

#

etc.

#

Doing it in an unfamiliar base gets them to approach novelly (or so they say)

#

Part of me thinks learning a bit about number bases can help elucidate how place value works and break some of the underlying assumptions we have about numbers, but the obvious counter is that it can be really confusing, while not teaching any practical skills.
That part

#

On both counts

storm prairie
#

Unit conversion

#

I find for students who have trouble with ratios bringing out the clock is helpful

stark heron
storm prairie
#

Instead of saying I use one pound of sugar for every 3 liters of water I'll just move around the clock

stark heron
#

respectfully informing 👍

storm prairie
#

Sometimes it helps sometimes it doesn't

#

as usual

#

Seeing this varying jointly other than abstract concepts, but then again other bases brings it back to abstract concepts, idk

scarlet panther
storm prairie
#

Yeah

#

I like it for the same reason they chose 60

#

Nicely divisible for basic conversions

stark heron
#

have you seen that video about the .002 cents vs .002 dollars fiasco? i feel like that kind of "sense" with units falls into a similar category of knowledge

#

idk if im making sense here

storm prairie
#

Oh and of course for the actual hour and minute conversions - I can visually do miles per hour vs per minute on a clock, use it for sector area as well

#

But what point does the clock just become a marked circle

#

🤷‍♂️

storm prairie
stark heron
#

the fact he revised his teaching process as he moved up the ladder cracks me up

storm prairie
#

Oo 34 minutes I'll have to save it for morning

stark heron
#

haha ok

storm prairie
#

But I'll watch it

#

I think in general though mathematical reasoning 😅

stark heron
#

yes that vague thing

storm prairie
#

Maybe that's an argument to introduce more abstract concepts like different bases

abstract grove
#

that recording is very painful

scarlet panther
#

I'm having to pause at 3:37 and I'm not sure if I'll continue 😂

scarlet panther
storm prairie
#

How did they get a call with me and my relatives for the video

#

Down to providing an example, them affirming the correct answer, and then not connecting the dots

#

Anyway yes the rest for the morning

storm prairie
#

I wish I had one of the tests that I could take a picture of

#

After they learn a compendium of alternative arithmetic methods (for base 10) then they do base conversion, followed by the arithmetic in the given bases - all with some visual stimulus attached

toxic talon
storm prairie
#

i.e. grouping

scarlet panther
#

What kinds of alternative arithmetic methods?

storm prairie
#

Lattice multiplication, chinese method, box method for multiplication

#

Grouping/box method for division

scarlet panther
#

Hmm, what about subtraction?

storm prairie
#

For addition and subtraction using the number line, I can't remember if they do give and take

#

Because they don't do it for addition

#

Oh counting up

#

As well

#

Swapping and knowing when the answer should be positive and negative as well

scarlet panther
storm prairie
#

40-32

# 
33,34,35,36,37,38,39,40
1  2  3  4  5  6  7  8
#

The lower numbers counted on finger

#

The uppers counted aloud

scarlet panther
#

I'm only asking because I have a way of doing it that I started doing in other bases that I found helpful, and I wanna see if other people use it

#

What about the traditionally complicated subtractions? Something like 435-267, did you go that far

#

Oh right, this isn't in different bases though is it

storm prairie
#

Now that we have 1-digit subtradictions down

storm prairie
storm prairie
scarlet panther
#

Okay let me share you my "counting up" method of traditional subtraction

#

Once I figure out how lol

storm prairie
#

XD

#

I assume it's a mix of give-and-take and counting up?

#

Like

#

267 -> 270
3
270 -> 300
30
300 -> 400
100
400 -> 435
35

ans 35 + 100 + 30 + 3 = 168
Since the bigger number was the subtrahend our answer is positive

scarlet panther
#

More efficient

storm prairie
#

Using pure give and take you'd have

435 - 267
428 - 260
408 - 240
368 - 200
168

scarlet panther
storm prairie
#

Ah

#

That's actually the trap I fell into while trying to do the give and take 😂

#

5-7 => 2 wait

#

I assume it 's performed right to left still?

#

Yeah looks like it

scarlet panther
#

Yeah

#

Right to left, see how far up you have to go to get to the next digit. If you have to wrap around 10, carry the 1

storm prairie
#

Yeah this is nice

scarlet panther
#

It's helpful in different bases because of finding pairs that make 10 and such

#

And things like 12 - 5 = 5 in base_8 are unintuitive

storm prairie
#

Mmm yes 🤔 I guess the best benefit is not having to multiply carry

scarlet panther
#

Oh yeah that too, as you can see in the picture

#

Counting up is just easier than counting down too

#

The only friction would be in the wrap around concept

#

Dunno how intuitive that would be

#

But if anything I think its cute

storm prairie
#

Can anything be less intuitive than carrying XD

scarlet panther
storm prairie
#

That's what I meant

#

Wait what were you thinking of

scarlet panther
#

I mean I don't know how intuitive the concept of "count up and wrap around 10" is

storm prairie
#

I might have to do a small poll and see what people I know think

scarlet panther
#

Hehe, before you do let me warn you about one caveat

# 
 406
-297 <- If 1 carries over to 9, it makes 10, so it still carries over to the next place
----
 109
storm prairie
#

Ah

#

Yes that is true isn't it

#

Actually now that I actully perform it myself with this second example I think I've done this type of subtraction befro

#

I just considered it subtraction with negatives

#

6-7 = -1, flip the bit to get the value

#

Flipping the bit moves 1 to the next and so on

#

So you do end up with double carries but they get split up

#

You'll never have to multiple carry at the same time

#

I guess this is the same as borrowing the opposite direction yes

#

Very interesting

#

And you consider this more efficient than give and take

#

Interesting

#

I think personally I would be inclined to disagree but only on the premise that the traditional method is slower than that for me

#

And this feels equivalent to the traditional mthod

#

I'll think about it more tomororw

#

Hav eto be up in 6 hours 🙂

scarlet panther
#

@storm prairie Sorry got distracted. Which method was give and take? And in what ways does it compare to this method? Because this method is definitely quicker than traditional method for me, although traditional can be done a lot faster if you just skip the slashes, although that could cause more short term memory usage

scarlet panther
scarlet panther
#

So was just trying some subtraction in Hexadecimal (something I haven't tried before) and noticed a couple of interesting things:

1. Intuitively, I had no idea how to subtract a single digit from a two-digit number:
   1A  - My brain just froze looking at this. Trying to count down from 1A by D is just so foreign.
  - D    However! Counting up from D to get back to A made sense. D+3=10 (D->E,F,0) then +A, the total is A+3=D
    ?  = D

2. Subtraction by the radix digit (9 or F) while carrying a 1 is trivial:
    5BCA  - Contrary to what I said about carrying being unintuitive.
  - 3FFB    When you carry the 1 from B to F, it becomes 0 (with a carry), making C-0 trivial.
    ----    Similarly with the hundreds digit, it becomes B-0=B.
    1BCF    Contrast this from the trad. method, where the tens digit would process as 1B-F=C.

There were a couple of other nice properties I noticed, but not quite as clear to communicate.

#

The above might not look right on mobile

#

Image of the same thing, just in case

SPOILER_image.png
tawny slate
#

one of my major points is that society has a very strong implicit bias for base 10, in ways that we subconsciously often don't realize

#

like when you see the number 12345 or 4444 or 121121 our brains "light up" because they recognize some pattern in there, and we start thinking these numbers are somehow special

#

but if you can understand what base 10 actually means, that is it an arbitrary base, it can insulate you from this kind of bias, which can then insulate people from drawing weird supernatural superstitious connections, like thinking 7 is a lucky digit or numerology or something

#

I think these are things that don't have explicit "practical value", but really impact how you view the world in a deeper subconscious level, and these are important but underexamined and underappreciated for obvious reasons

scarlet panther
#

Thats to say once again that I think we can take time in math for example to enrich our knowledge about certain things, rather than stick to whats practical. And whats practical knowledge in math is a bit contentious at a certain level anyway😅

scarlet panther
scarlet panther
tawny slate
#

just a math video about funny number base things

#

but i wanted it to have some context instead of just being entirely silly fun

abstract grove
#

number theory is more or less the study of patterns in numbers that are independent of any base

scarlet panther
scarlet panther
#

I was doing a bit of "research" into different bases, I kinda enjoyed it

burnt junco
scarlet panther
#

Is this web based? Maybe just a default like system dark scheme would be best, most accessible anyway (i would hope)

burnt junco
#

its in quarto

#

so custom java

scarlet panther
#

Not sure then, maybe just reference other similar applications, or e-texts as thats what this looks like

solar sorrel
solar sorrel
solar sorrel
# torn apex <@209014240848117760> <@256252384731332612> also check out the music, I made it ...

I mean, SOTA advice on this is here:
https://justinmath.com/files/advice-on-upskilling.pdf
https://justinmath.com/files/the-math-academy-way.pdf

That's it. that is literally the best advice I have seen to learn non-proof-based mathematics.

For proof-based stuff, the most relevant would be Professor Po-Shen Loh.

[Lara Alcock is the only doing who has published research on this, but she focused on undergraduate students]

Everything else [like the standard reading recommendations from mathematics departments] is folklore and anecdata.

solar sorrel
# scarlet panther Do you think it's ever worth it for a student to explore alternate number bases,...

I recall that the "new math" people tried doing this in the 60s and 70s, but it does not seem to work that well for the median person [at least, using the sort of scaffolding from back then]: https://en.wikipedia.org/wiki/New_Math

New Math

New Mathematics or New Math was a dramatic but temporary change in the way mathematics was taught in grade schools which started in France and spread to many other countries between 1950s and 1970s.

solar sorrel
scarlet panther
scarlet panther
stark heron
tight star
stark heron
#

(also, this channel should probably remain discussing actual pedagogy)

stark heron
#

i know a lot of teachers do that on a small scale for "discovery days"

turbid zenith
turbid zenith
turbid zenith
#

But I think the best take on teaching alternative number bases in an accessible way was James Tanton's "Exploding Dots"

torn apex
# solar sorrel The instrumental backing was hand-made, IIRC: https://discord.com/channels/26888...

thanks for the pointers and advice! I'll try to incorporate them in the next kids game jam. I'm still trying to wrap this one before the weekend gets away

also, technically the music is 100% AI generated: i hummed the melody into Fruity Loops plugins to midi-fi it and the same with the bass. Not a musician so auto-tuned the heck out it and crushed the note timing to whatever was nearest 1/8 (iirc). I then fed the auto-gen lyrics + auto-tunified tracks into Suno to generate the voice + replace with actual music instruments.

I'm sure to a musician it sounds robotic as heck but to my + my nephews untrained ears, it's great!

especially because this took me like 10 minutes which means I can do this live with the kids

torn apex
# turbid zenith I have to know why you tagged me in particular XD But these are cute

hah! sorry, I thought you taught elementary/highschool but I might've hallucinated that 😛

I keep trying to explain "The Big Idea" is using AI/LLMs as a teaching aid and one vector I'm exploring is using the vehicle of game development. I 100% believe normies+kids need to learn that there is an unavoidable "grind" aspect to learning math but I also 100% believe you have to incentivize the learning with fun in the beginning, especially for a human who is not innately in-love with math at birth.

From my anecdata with kids, co-creating games with them has given me the best results ("Hey kid, do you like fortnite? do you want to learn how to make fortnite?" is my line); co-creating a game with a kid where the game itself teaches them something has been terrific in terms of memory retention

The necessary condition for all of this is the Generative AI; there's absolutely no way that even I could devote spare time to create even programmer art, much less something that captivates a normie or a kid.

But now, GenAI massively reduces the cost for creating "filler" or "placeholder" quality art by anyone meaning from a general math educator pov, this is attainable by normal junior high/highschool teachers

#

maybe a better metaphor is that it's like the invention of the calculator. yes, ofcourse kids should not be allowed to use a calculator before they learn their multiplication tables. but that also doesn't mean teachers can't use it for calculating the class grades or something.
with AI, it's the fact that it can significantly lower barriers to learning. For example, even in gamedev/vfx, it takes an entire spectrum of disciplines to make a game (music, art, tech art, programming, engine programming, game design, audio). With GenAI, an aspiring student game developer can use GenAI to fill in the other parts for their game. For an artist, they can use GenAI to "program" their prototypes.

#

Of course this is all based on anecdata with selection bias and is highly speculative at this point, which I thought that was the point of this channel-to discuss the pedagogy of normie/k-12 Math Education?

quiet jackal
#

you don’t need to saturate a child’s dopamine receptors with AI generated dogshit to get them interested in math

#

it sounds like youre proposing that schools have students use AI to make scratch projects

torn apex
# quiet jackal you don’t need to saturate a child’s dopamine receptors with AI generated dogshi...

I have trouble taking you seriously with a comment like this b/c this sounds like someone who's never worked with kids.
fwiw, I never proposed anything other than literally sharing what has worked over the last couple of months with my nephews, one of which is non-verbal autistic.
I'm also keen to hear other people's attempts and approaches (less so armchair quarterbacking; us stem people love to think we can deduce our way to anything without actually, you know, trying stuff in the real world)

turbid zenith
#

And I teach college, but I do work with high school students over the summers at a gifted program

torn apex
turbid zenith
#

Hmm, okay. The names of all the tools were what tipped me off.

#

How old are the students you made those two songs for?

torn apex
torn apex
# turbid zenith And I teach college, but I do work with high school students over the summers at...

super cool! I have a prototype math DSL working that's targeted towards gifted math students to generate those types of interactive math visualizations

My hypothesis is that there's an order of magnitude of improvement if we create DSLs oriented around human comprehension (i.e. the mathematician) instead of around machine comprehension.
For example, there's no programmer love or UX given to theorem provers like Lean which makes it difficult for adoption

turbid zenith
#

They're pretty, but as with anything AI related, do make sure they all work before you post them. The cross product one doesn't work very well at all.

torn apex
turbid zenith
torn apex
#

here are some of the previous character generations of the kids from previous "gamejams" (i.e. me gas lighting them into learning about math)

content.png content.png
turbid zenith
#

That you could paste here?

quiet jackal
#

what math are they learning

turbid zenith
#

I have some questions, but seeing the lyrics would help me better formulate them.

torn apex
# turbid zenith I have some questions, but seeing the lyrics would help me better formulate them...

ok, but brace yourself for a very bad attempt at doing a GenAlpha version of School House Rock

[Acoustic Banjo Intro]
(Fast finger-picking banjo riff)
(Stomp... Stomp... Stomp...)
(Whistle)
Woo!
This one’s for the scholars in the back!
(Laughs)
We goin' back to Göttingen!
Come on!

[Verse 1]
(Rhythmic, sassy delivery)
Walked into the boys' club, chalk in my hand
Ready to rewrite the laws of the land
They said "No girls allowed," tried to shut the door
But Hilbert said, "Gentlemen, this ain't a bathhouse, it's a war!"
I got the rings, I got the fields, I got the abstract style
Making Einstein rewrite his notes and crack a smile
Variables changing, but the logic stays the same
Put some respect on Emmy Noether's name!

[Pre-Chorus]
(Building energy - Snare roll)
Can't destroy it, can't create it!
Only transform, calculate it!
(Hey!)
The universe is balanced, baby, can't you see?
It's all locked in perfect harmony!

[Chorus]
(Explosive Beat Drop - Heavy Kick + Fiddle)
It’s a Symmetry! (Woo!)
A Conservation!
We got a theorem sweeping through the whole nation!
If you shift it in time, you save the Energy!
If you spin it around, that’s Angular Velocity!
Don't be a zero, get with the hero
It’s the Noether boogie, baby, here we go!
(Banjo breakdown)
turbid zenith
#

Okay, yeah most of this I remember now.

#

Here's my question ... what do you want 8 and 10 year olds to get out of this song?

torn apex
# quiet jackal what math are they learning

It depends on whatever they're having trouble with. Multiplication/negative numbers. If they don't have anything, then I indulge them in fictionalized math history. I basically just pretend that math equations/discoveries are magic in an alternate world so it makes coming up with stuff easy

also...they're 8 & 10 and it's only meant for them🙃

turbid zenith
#

Because what I'm seeing from this is a lot of name-dropping of words having to do with what Noether studied (and the "this ain't a bathhouse" line did make me smile since I know that's a thing Hilbert famously said, give or take some linguistic liberties — wondering if you came up with that or if the AI did), but there's no context from which they could learn what any given one of those things means.

torn apex
turbid zenith
#

Contrast this with the plethora of songs from a TV show I grew up with and is very near and dear to my heart: Square One TV.

torn apex
#

And ofcourse, down the road, it'd be interesting to actually try to put more effort into the songs given how accessible it is to generate them on the fly with the kids

turbid zenith
#

What seems to be the effect of all this on the kids?

torn apex
#

btw, I literally started doing this a month ago so there's lots of experimentation to be had. It's fun to just get them excited about it but I've been really taken aback by the potential

#

heh, i'm also just realizing that all my anecdata is heavily heavily biased by the fact that Z (nephew) is autistic so I don't know if his exceptionally positive reaction will translate

turbid zenith
#

Okay, so he reacts positively in the moment to it, but what does it do for his mathematical learning?

torn apex
#

and also I don't really know anything about math education so i'm just winging it (and why i joined this channel 😅 )

turbid zenith
#

Okay. Improved engagement is a good start, but like some others I worry about the possibility that it's just a dopamine hit and may not help the students learn the mathematics.

torn apex
turbid zenith
#

It's an easy trap to fall into — kids like music and games, so if we add music and games, they'll learn!

torn apex
#

also, this isn't in lieu of actual homework

turbid zenith
#

There's a lot more that goes into it than that.

torn apex
#

this is distinctly part of their "fun & hobby" time

turbid zenith
#

What I'd be also interested in is why they hate math

torn apex
#

All the usual answers and probably all the usual underlying exigent circumstances not directly related to the subject matter

#

Have you had much success in math education for people who don't like it or have a negative conditioning to it?

turbid zenith
#

I do think this is a use case that, in principle, I'm okay with for GenAI — these kinds of small-scale fun generations. My caveats are (1) you need to be careful and always keeping in mind that the goal should be for them to learn, not just entertained, and (2) I would heavily advise against making something like this a large scale or especially commercial without involving artists, musicians, etc.

turbid zenith
torn apex
turbid zenith
#

Okay. That's useful to know.

#

I did something kind of similar a few summers ago when I had a whole back-and-forth with ChatGPT to create a play based on the story of the cubic in Italy.

#

And the students at the summer program I taught did a "cold reading" I think it's called? Where they basically had roles and did their best to get into character. But without moving around, just at their desks.

#

The students had fun with it, and the students who did theatre had some good feedback on what it did well and where it was lacking.

#

What's your day job by the way?

torn apex
turbid zenith
#

I don't really have resources but I'm happy to answer any questions.

#

But my main point I'll say as you're on your way out is that I encourage you to look seriously into, how do you move from entertaining them to teaching them?

#

That should be your next goal.

torn apex
turbid zenith
#

Games are a powerful motivator, but they don't have to be video games. You can just as easily hook them with board games and puzzles and the like.

torn apex
torn apex
turbid zenith
#

Especially if they're intrinsic games.

#

(Where the math you're trying to teach them is actually essential to the game's mechanics.)

scarlet panther
tawny slate
turbid zenith
#

Probably in response to calling it "dog shit"

tawny slate
#

i think if you have any good ideas beyond a game jam, you should take those seriously and actually do a full development on them

#

that doesn't mean it has to be big, but like treat it as a complete project to be distributed

turbid zenith
#

I agree, thought at that point you should look into getting people to generate assets rather than relying on AI.

tawny slate
#

because id rather see each person complete 1 or 2 fantastic projects useful to a general public than have one person do 100 small projects that are developed so quickly we have no clue not only how effective they are but also sacrificing their quality

#

can you just focus and work on like a single thing at a time

turbid zenith
#

Because at that point if you're using AI then you're taking work away from someone who at the very least could probably use that project for their portfolio.

tawny slate
#

also sure, people are going to have negative responses to genAI usage, its inevitable, its a highly politically charged topic, but if you are just immediately defensive and don't know how to separate the constructive from the insulting then you're only hurting yourself

#

what matters is not how pretty the final product looks, not that looks dont matter, but that's not how success is done

#

engagement in this channel and interest is great but so far its mostly just been you excitedly sharing the AI generated assets of your game jam wip project more than actually asking about actual pedagogy, so idk if its just naivete or if you're not actually interested in pedagogy

#

actually why am i even typing i forgot he blocked me catshrug

scarlet panther
turbid zenith
#

The "just pick up a pencil bro" argument kind of misses that point

#

(And the entire rabbit hole of the commodification of hobbies and talents that it leads down)

turbid zenith
scarlet panther
#

By "good graphics" I just mean something innately appealing, not any particular level of design or detail

turbid zenith
#

I do think attention to visual design can help, and lack of attention to visual design can hurt. But only so much.

scarlet panther
#

Ooh, let me introduce a concrete example

turbid zenith
#

Like if it feels "thrown together" that can hurt

stark heron
scarlet panther
#

This was a proof-of-concept thing I posted here a couple days ago of a "reverse" graphing calculator: https://www.desmos.com/calculator/969xoaztxk.

The idea is you control the graph directly and it displays to you the equation for that graph as you change it. It's very cumbersome as is; to switch between the forms of the line graph, you have to manually select/deselect the relevant folders, and while the controls and labels work decently, the presentation of the equations has a lot to be desired.

So when I play around with this thing, I don't exactly get the feeling of exploration/discovery I think the tool should yield, and I'm not sure if it's because the concept is simply not good, or if the execution is poor.

#

btw, for fun, here's one for quadratics too, but it doesn't actually display the equations, so it's pretty much useless for its purpose 😂

#

@turbid zenith Just atting you since we were talking about it

turbid zenith
#

Hmmmm

#

I just tried it on my iPad, and I do think sometimes the graphics got in the way — the numbers overlapping each other, that kind of thing

#

But as a proof of concept I think it's great

#

I think what may be missing more than anything is an accompanying set of lines to use as a goal

scarlet panther
scarlet panther
#

Although this does go back to my main point about how much precise execution matters in making pedagogical experiences valuable. Something like this I think needs a high level of good execution to work even a little bit, whereas tried and true things can work quite well with minimal execution (like flashcards)

turbid zenith
#

Speaking of Square One TV

#

Requesting a Mr. Glitch emoji for the server please

dapper flume
# scarlet panther Pedagogically, how important do you think good presentation (e.g. appealing visu...

I think "appeal" is not that important, but I have a strong personal belief (albeit based more on my cross-disciplinary experience than real research) that intentional representation of visual information is really important. I teach at the middle school level, and in my experience, a good visual shows evidence of being much more accessible to all kinds of learners, especially those whose language skills are developing.

Intentional representation of information means displaying information faithfully and intuitively while highlighting the intended connections between bits of information. For example, color coding variables that correspond to words in a problem help English language learners draw those connections much more thoroughly, and even students who are totally fluent in English benefit from this visual tool as a way to help them build abstract thinking skills. Students also simply like it more, because they have much more bandwidth to look and listen at the same time when there aren't so many words to read.

My slides are not always very good looking from an objective sense. I took graphic design in college and high school, and I bring back the concepts I learned in those courses to make all my visuals more accessible and logically organized. Beauty is usually a side effect rather than an intention.

turbid zenith
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And not show us your slides and such :V

dapper flume
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I'd be happy to! ...once winter break starts 😌

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Need to sort a LOT of files because I forget where my best ones are lowkey

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Named everything after the curriculum lesson number

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Maybe I'll pull some good recent ones in a little bit

turbid zenith
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I would LIKE to think my visual design is decent but I never formally learned anything ;w;

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And I'm 100% with you on color

scarlet panther
scarlet panther
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Do you think something like color-coding word problems generally (if it makes sense) is something that could be beneficial? Or could it be like "cheating" lol

solar sorrel
solar sorrel
# stark heron really dont understand why you would encourage a lack of creativity here. its di...

There is a difference between ‘mastery of the medium’ and ‘the virtue of creativity’ that are being conflated here.

yes, most people who are using AI tools are probably, "on average", not very creative.

[this is actually relevant for mathematics pedagogy since mastery of the mathematics medium is quite malleable, but ‘virtue of creativity’ as a personality trait is a lot less malleable]

stark heron
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to be honest im struggling to understand your point
(but i will say i disagree that creativity is not malleable. assuming you mean it is not able to be learned? i truly think its a skill to be practiced like any other. now that is a pedagogy debate there..)

torn apex
# scarlet panther I think you're right, and color-coding things like that sounds super smart and s...

I also would love to see examples! I agree a lot with this sentiment. Also tangentially related, in my experience, there's a lot of semantic impedence mismatch from what people label as "beautiful"/"appealing" vs. what they mean...especially on here where not everyone is a native speaker of english. In my experience, designers vs. artists vs. programmers subconsciously mean very different things when they talk about visual "appeal" or "beauty".

For me personally, I would have used the label "appealing" to describe "visuals more accessible and logically organized"/"autonymically color coding variables"

scarlet panther
# torn apex I also would love to see examples! I agree a lot with this sentiment. Also tange...

Yeah, I think the way I meant to use appeal is compatible with that, but there are lots of ways to take it. For example, for me a simple worksheet with arithmetic problems is more appealing to me than a similar worksheet that wastes space with cute pictures of cartoon characters that have little to do with the math. But idk, maybe a child finds it appealing and thats all that matters 😂

outer burrow
# scarlet panther This was a proof-of-concept thing I posted here a couple days ago of a "reverse"...

execution good, concept bad. a lot of additional complexity compared to the obvious approach with 2 sliders, yet i don't see what the purpose is. also, a restriction of yours is that m can take on only a value in {-3, -2, ..., 3} and b a value in {-4, -3, ..., 4}. bounded by small values and only integers, and the code is complex enough that if i wanted to change that, i can't easily figure out what to change

i guess you gain the ability to touch the line to manipulate it and see what that does to m. so yes, it's a "reverse graph". but, changing m and seeing how that affects the line teaches all the same concepts. i don't see the point of going to all this effort when the obvious way (2 sliders) is so much easier and works just fine

for whatever reason, i like your quadratic one a lot more.

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i'm trying to understand from the POV of someone who doesn't understand y=mx+b. i can see that the red dot for b makes it clear that b is the y-intercept. (we could also have this dot in the normal 2-slider approach)

the visual presentation is clear. however, it would be more fun/interactive for a student to be given the 2 sliders (no dots, no colors) and try to figure out what the sliders do without further instruction. hopefully, they'd play around for 1-2 min and figure it out. if not, we discuss, and i give the answer after they give it a shot. i believe this would result in a better understanding in the end because the student would've derived the answer themselves (or at least tried to).

whether most 6th graders would actually put effort into the activity, idk. but i think they'd put more effort into it than listening to a lecture, at least.

overall, the colors obviously make the presentation more clear. but they may not actually enhance student understanding.

scarlet panther
# outer burrow execution good, concept bad. a lot of additional complexity compared to the [obv...

I liked the feel of the quadratic one too more, possibly because (1) it's not discrete (the reason the line one was discrete was mostly because of the equation formatting) and (2) There's no equation complicating the view. Thinking about it a little more, I think you're right that having sliders detached from the graph (maybe still in the view, but just not connected to the function) would make it easier to see the direct relationship between the variable and the equation. Maybe I'll try to compare the approaches with a more complicated function (for fun lol)

outer burrow
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definitely being "real"-valued feels a lot nicer.

one other factor is that you simplified the parabola representation. in the vertex-form equation, we have (v_x, v_y) (detached from each other, it's (x-v_x) and +v_y on the outside), whereas your representation groups them into the vertex as (v_x, v_y). then you have a

so, three variables vs. two (kinda). i like the simplicity. though, you eventually do have to get to the equation

scarlet panther
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You think a cubic would be something worth trying?

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Not sure what function form to try next, I'm about to check tho

outer burrow
scarlet panther
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Are there even cubic "forms" that aren't just the standard form?

outer burrow
# outer burrow nah, this requires a lot of effort and cubics aren't worth it

by the time the student starts seeing cubics, they would have already learned linear and quadratic (they should have, but schools kinda just pass people who really should retake, so this isn't always true). at that point, they should be able to derive properties of the graph from the equation themselves (i.e., make the desmos graph yourself).

scarlet panther
sharp canopy
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I'm teaching my students what parallelograms are and whenever I explain it their brains have no idea what I said what's another way of explaining

abstract grove
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at the risk of asking the obvious, have you tried showing pictures of actual parallelograms and going “see these two sides here? we call them opposite sides, and look.. they’re parallel”?

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and right away observing that rectangles and squares also count as parallelograms?

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if they’re having trouble with it, maybe they need a review of parallel lines

tawny slate
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basically a translation until the x^2 term becomes 0

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and the depressed cubic has quite a few applications, including making deriving the cubic formula much more reasonable

outer burrow
slim path
cold trench
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my gf's little sister is in high school and doesn't understand that a symbol doesn't always mean the same thing in every scenario

e.g. an SAT practice problem had the vertex form of a parabola (commonly written as y = a(x-h)² + k) but had a constant k that didn't represent the y-value of the vertex, so she got confused

that's a more complicated example, but even easy stuff where the most commonly written symbol in an equation is replaced with a different symbol. she can't grasp that it doesn't matter what it is, and I'm having a tough time getting this point across. she's really bad at abstracting things and finding connections and patterns in math, which I've tried to help her with but is so difficult when her foundation is shaky

does someone have resources or explanations that can help her out?

normal aurora
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the letter is like a role (werewolf, mafia boss, mayor etc)

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am I cooking

midnight scarab
cold trench
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I don't think so, she plays roblox a lot lol

tight star
cold trench
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fair, but words have defined meanings (even if multiple) while symbols in math (like k for example) can be assigned to pretty much any meaning

dapper flume
# cold trench my gf's little sister is in high school and doesn't understand that a symbol doe...

This is a math literacy gap that I imagine can be patched with some literacy training.

An Idea i have is to teach a line of inquiry that helps her to understand math better. As math-savvy people, we understand the structure and purpose of mathematical definitions for concepts, variables, and symbols. This is not how definitions work in other contexts and is not intuitive to everyone. We can teach an explicit, flowchart-like model to help a developing math reader to get it right more often.

For example, for that SAT question, a useful line of questions might have been:

  • What do these symbols normally mean? Is this what they mean here?
  • Which symbols in this question have been defined by the problem? Which symbols must be inferred from context?
  • What is the goal of the problem? What should the structure of my answer look like?
  • How can I use the information they gave me to answer the question?
cold trench
dapper flume
# cold trench I like these questions, especially the second one. do you know of any videos or ...

None come to mind, as I learned this kind of thing mostly in college and in professional practice.

If you want to help her, I would model how you would answer these questions, check if she understands, and then have her just try to parse a complex expression. Take for example:

"Josh is analyzing how the height of a mass on a spring, given by h, in inches, changes over time, t, in seconds."

  1. Would it make sense to say *h=6 * seconds in this context?
  2. Would it make sense to say t=6 seconds in this context?
  3. Josh wrote down the equation $h(t) = a * \cos(kt)* to describe the motion. What do you think a and k might represent in this context?

.
.
.
This example may not work, but it could help her highlight what exactly she should be paying attention to.

cold trench
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thank you so much for your help!! if anyone else has additional input, please lmk - but this helps a lot

tight star
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I guess we also don’t tend to be completely arbitrary in what symbols we use for things either

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You could have a function x : R -> R given by x(f) = f^2

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but I feel like people would be a little weirded out by that

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compared to f(x) = x^2

turbid zenith
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Maybe you could do what I do with the Pythagorean theorem lol

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I ask my students to tell me what the Pythagorean theorem says

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And invariably they all say "A squared plus B squared equals C squared"

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So I show them an acute triangle and ask "does a² + b² = c²?"
"No, it has to be a right triangle!"

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Then I show them a right triangle with a as the longest side and ask "does a² + b² = c²?"
"No, c has to be the longest side"

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"Okay, so the Pythagorean Theorem isn't really about a² + b² = c² after all! Rather it says, in a right triangle, leg² + leg² = hypotenuse². The symbols a, b, and c on their own have no intrinsic meaning."

cold trench
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this is a great idea! it reminds me of the "teach me how to make a peanut butter sandwich" and similar videos lol

turbid zenith
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Yeah that was part of the inspiration!

cold trench
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this would break her brain but should show her what's going on

turbid zenith
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And it puts the students right at the point where it creates the needed cognitive dissonance

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With something very visual and visceral

cold trench
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for sure, and very applicable to many kinds of problems and situations too

turbid zenith
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You could also as part of it, to drive it home, show her a 3-4-5 right triangle and ask
"What's c?"

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And if she says "5"...
"Nope — c is a letter. There's nothing in the triangle marked c."

cold trench
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that's really good, and reminds me of another problem she faces

turbid zenith
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You could then add the markings yourself but point out you could have easily used p,q,r or x,y,z ... but conventions are useful

cold trench
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kinds of problems where she has a function f(x)=... and has something happen at point c, so find c

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she struggles with the distinction between x and c

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I try to explain this to her, and it helps but I don't think it fully clicks

turbid zenith
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Oh like ... suppose f(x) = 3x + 1, and suppose f(c) = 10, find c?

cold trench
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it's usually in calculus problems with like intermediate value theorem or smth iirc

turbid zenith
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That's honestly a hard distinction to describe ... I still haven't found a pithy explanation myself

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The distinction between a variable, a literal constant, and a parameter

cold trench
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yeahhhh, I've explained it as "c is one of the x values that satisfies the condition" sort of thing and she understands in the moment but quickly forgets

turbid zenith
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understands in the moment but quickly forgets
Story of my life when it comes to sooooo many students

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Classic time when that happens is overapplication of "the A of a B is the B of the A"

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(a + b)² = a² + b² etc

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Or using the = sign to mean "and the next step is"

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e.g. x³ = 3x²

cold trench
shadow flower
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are you sure she’s not working in characteristic 2

cold trench
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ah man mb mb got me there

outer burrow
# turbid zenith > understands in the moment but quickly forgets Story of my life when it comes t...

"understands in the moment but quickly forgets" doesn't make sense to me

if a student forgets that fast, it generally means they didn't actually understand (and i'd say they didn't even "understand in the moment") in the first place.

for example, i have a really hard time seeing how someone can work through (2+3)^2 = 25 != 13 = 2^2 + 3^2 yet still make the mistake (a+b)^2 = a^2 + b^2. if it's in a large class and some student doesn't actually pay attention to the example, then sure. or maybe they're just starting to break their bad habit, also fine. but with a little practice, if they understood the example, they should stop making the general mistake, and they shouldn't forget quickly.

maybe i have a stricter definition of "understand". i would never assume someone understands something after just listening to a lecture about it. i would only say someone "understands" x after they've shown that they can apply x on HW or assessments, wouldn't forget x quickly, etc.

i like feynman's quotes -

  • "If you cannot explain something in simple terms, you don't understand it"
  • "What I cannot create, I do not understand"
  • "If you want to master something, teach it"
cosmic ibex
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Depends on what it is, I think. It feels very plausible that someone can "understand in the moment" how to carry out a procedure but forget that later.

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Forgetting a conceptual understanding is more iffy, but still feels like it would be possible if there are enough moving parts.

little cave
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How do I make the symbols look pretty

cosmic ibex
little cave
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Nah like in discord my bad

#

Will $$\sum$$ work?

burnt vesselBOT
#

xxxVsjuicewrld

667877312792100865.png
little cave
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Nope

cosmic ibex
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When someone fails to internalize (a+b)² != a² + b², my general assumption is that it must be because they have somehow not grasped that the fundamental meaning of an algebraic expression is to describe calculations. If they have internalized algebra as just a collection of disembodied rewriting rules, where "plug in a number and evaluate" is just one random operation among many they may be asked to do, it may not be obvious to them that a numerical counterexample is the gold standard for showing that a rewriting is invalid -- because they haven't grokked that preserving calculational meaning is the purpose of the rewriting.

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(Consider that there seems to be a fair amount of people who will happily describe a function (say, squaring) by writing down something like

1 = 1
2 = 4
3 = 9
4 = 16
If one has missed the meaning of = badly enough to put that on paper, one similarly has no chance of really understanding why it's wrong to write (2+3)² = 2²+3², merely because the two sides evaluate to different numbers!)

outer burrow
little cave
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Normalize the counterexample

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Also yeah they don't know what equals means

cosmic ibex
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A subtle thing here is that when we move onward to abstract algebra with axiomatically defined structures, then some of the rewriting rules are just procedures that work because we say so! And then it becomes a much more involved thing to refuse a rewriting with a concrete counterexample.

outer burrow
little cave
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I would say it's pretty prevalent from my experience

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Not if they've been given a function as much, but if asked to make up a function which accomplishes some task? It gets ugly

cosmic ibex
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There are enough who do it that I don't think we can just brush it off as the individual people being "kinda shit with math". They must have been failed by the teaching they've received, somehow.

little cave
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One thing I hope to try if I get to teach college algebra is to at least have => for "implies"

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Instead of just = for "equals"

outer burrow
# cosmic ibex There are enough who do it that I don't think we can just brush it off as the in...

i don't like this mindset. generally, the teaching didn't fail them. they (and their parents) failed themselves. this feels mean to say, but i believe it's true.

public school is free for everyone in the US. for this basic stuff, the teachers are good enough. i myself grew up in a pretty shitty city with low educational outcomes. the education was sufficient to learn this. more advanced stuff, mostly okay but room for improvement (and i was lucky enough to move out anyway). but this, yes.

cosmic ibex
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It's not difficult to imagine how the misconception arises. If kids learn to write 54+13=67 in early grades just because "that's how you write it", it's easy for them to internalize = as meaning something like "and then we do the appropriate procedure, and the result of the appropriate procedure is ..."
Then fast-forward several years until a different teacher attempts to introduce them to algebra and assumes they already know the = sign, so it doesn't need to be explained explicitly -- or if it is explained, the explanation doesn't need to be stressed.
It feels believably possible to come through school without ever being told outright that the verb "to equal" means "to be the same as".

outer burrow
#

one google/AI search away from getting over any such misconception. if you have a computer and internet, there's really nothing to blame nowadays except yourself imo

cosmic ibex
#

Google searches are only useful if you know you have a misconception.

turbid zenith
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Plenty of studies have been done about equals sign misconceptions. They're pervasive.

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The same students will say that "7 = 2 + 5" is wrong because "you have to write it as 2 + 5 = 7"

outer burrow
# cosmic ibex Google searches are only useful _if you know you have a misconception_.

that's why we have AI where you can paste your work. it's really good at catching mistakes. then you'll figure out you have a misconception. it'll even point out your misconception. i've been using it myself to check over some linear algebra exercises i've been doing (never took proof-based lin alg in school)

back in the day, we had to have a teacher or classmate do this. now, there's no excuse

turbid zenith
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Or that "9 = 9" is nonsense because no operation has been specified

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And AI is also good at making mistakes with the user none the wiser.

outer burrow
#

teachers make mistakes too lmao

turbid zenith
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Yes, and when teachers make mistakes, that's bad too.

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It doesn't excuse the AI when it makes a mistake.

little cave
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My favorite is "a/a+b = 1/b"
To me this is the same problem. You can show this doesn't work by plugging in a few pairs of numbers.

turbid zenith
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I'm too tired to look up the logical fallacy there.

outer burrow
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let me tell a story, my 4th grade teacher gave us a standardized test question in lecture


A. the gold
B. the feathers
C. they weigh the same```

she said A. i pointed out the mistake. she insisted A. no classmate backed me up. i'm still pissed
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AI doesn't have to be perfectly accurate. more accurate than dumbahh teachers is already good. and i'd say it's way more accurate than that

turbid zenith
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Just sent this to ChatGPT

image.png
outer burrow
turbid zenith
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🙂

outer burrow
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see? humans are very fallible

turbid zenith
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I very deliberately said "5 feathers" to see if it would overcorrect and assume it was the old classic riddle

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And it did

outer burrow
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well yeah it's been trained on the old classic riddle data. but why purposely give it a bad input... garbage in garbage out, what did you expect?

anyway, it did say "5 pounds" vs "5 pounds". this clearly implies that it's talking about a different problem than what you gave...

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(i can't believe this is how i'm spending christmas xd)

lethal hornet
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how is a learner supposed to know what the garbage is?

turbid zenith
#

That was quite a whiplash of a backpedal

little cave
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Another place where this type of thing happens is chain rule

outer burrow
turbid zenith
lethal hornet
turbid zenith
little cave
outer burrow
turbid zenith
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They'll blindly copy stuff from it without even taking a moment to think about it.

outer burrow
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and they'll get bad grades on tests where you can't use AI. although i do agree it's unfortunate

little cave
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Ehhh they're being passed on the tests

lethal hornet
little cave
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At least in America

turbid zenith
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I remember it got this wrong, and lots and lots of students just went ahead and copied it

image.png
outer burrow
turbid zenith
#

Are you an educator?

lethal hornet
vocal phoenix
outer burrow
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yes, i was an educator, detailed explanation [redacted]

turbid zenith
vocal phoenix
turbid zenith
#

Poe's law

#
  • can't tell sarcasm without emoji these days
outer burrow
vocal phoenix
outer burrow
turbid zenith
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But anyway, I think we agree that students need immediate feedback from a trusted source to not internalize misconceptions

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I think we disagree on whether AI is a trusted source

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And I say this as someone who uses ChatGPT a lot.

lethal hornet
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same. anything i use from it (i use it to debug code sometimes), i verify by searching for it (usually on stack overflow or some other source of documentation) or ask it to show its sources

outer burrow
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all i'm saying, it's very good for the undergrad level math questions i ask it (linear algebra, real analysis, and even some CS research stuff)

i obviously don't use it for middle school/high school math, but if it can handle what i ask it, i am confident it works very well for that level as well

little cave
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Right but you have to understand that it is most useful to you in terms of a check because you already have a lot of mathematical maturity

lethal hornet
outer burrow
turbid zenith
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Which is why AI could eventually be a useful resource. It's not there yet enough for my satisfaction.

outer burrow
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if i'm learning myself, i pester GPT. if there's something wrong, hopefully i catch the mistake and then i'll go for a trusted source (i.e., google, stackoverflow, wikipedia, teacher). if not, then i assume the misconception will manifest in some later homework or exam. that's another chance for me to catch the misconception myself (then goto trusted source) or for the grader to catch it.

lethal hornet
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why is that the mindset, AI first?

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i go to AI usually as a last resort

outer burrow
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rn i'm reading from a textbook and doing exercises. i'm not asking AI for the initial explanation because the textbook is written very well. but if something is tricky, yes i go to GPT first. if there is a problem (i don't like it or it's wrong), i go to google. i also ask GPT to check my work for exercises. same thing, if something goes wrong i look up the exercise on google (i.e., trusted source)

lethal hornet
#

im not saying that wikipedia, stack overflow, etc (i don't consider google a source) are infallible, but i have a higher degree of confidence in these sources than an AI

turbid zenith
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The students we're discussing don't.

lethal hornet
turbid zenith
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It's likely easier, all in one place

outer burrow
#

and more "interactive"

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i can't just give google all this, but i can and did give it to GPT to check

    Recall Definition 8.6.2. A subset $A$ of the rational numbers is called
    a \textit{cut} if it possesses the following three properties:

    \begin{enumerate}
        \item [(c1)] $A \neq \emptyset$ and $A \neq \mathbb{Q}$
        \item [(c2)] If $r \in A$, then $A$ also contains every rational $q < r$.
        \item [(c3)] $A$ does not have a maximum; that is, if $r \in A$, then there exists $s \in A$ with $r < s$.
    \end{enumerate}

    \enum{
        \item 
        \begin{enumerate}
            \item [(c1)] $r-1 \in C_r$ ($\mathbb{Q}$ is closed under addition).
        Hence $C_r \neq \emptyset$.
        $r \in \mathbb{Q}$ is not in $C_r$.
...

[pretend there's a lot more stuff here]
\end{mysol}```
#

i make plenty of mistakes, and it does catch them.

turbid zenith
#

At this point I don't think I'm going to get through on why "students have no excuse" is a fundamental misunderstanding of how things work for learners who aren't at a point where they can judge whether something is correct

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So I'm out, have a good night

outer burrow
#

that's okay, agree to disagree. merry christmas

little cave
#

Could you agree that one who is not at fault may still hold further responsibility?

outer burrow
little cave
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No no

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It's almost always the student's fault

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I'm interested in seeing students be more self sufficient with these types of things

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I just happen to believe that further clarity can sometimes be an attack vector towards this goal

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For example, ai probably won't viciously roast you for writing mathematical nonsense on your paper

#

Vicious being a subjective term

outer burrow
# little cave For example, ai probably won't viciously roast you for writing mathematical nons...

yeah it's said some bullshit for my research stuff (no or little training data). still somewhat useful but definitely less. it's said some bullshit for typst too, tries to insert backslashes \ like in latex because it's been trained on latex, not typst.

but the context is we're talking about < high school math here. and the training data for that is great. and i think it performs excellent in this context.

little cave
quasi maple
outer burrow
outer burrow
quasi maple
little cave
quasi maple
quasi maple
#

Inferential skills are a massive reason to prefer an actual teacher over an AI chatbot, on that ground

outer burrow
quasi maple
#

I mean, this was only last month

image.png
little cave
quasi maple
#

But if you go through the algebra you should notice something glaringly wrong

quasi maple
outer burrow
# quasi maple I mean, this was only **last month**

i agree for detailed computations like this, it is bad. i would never trust a computation from it without verifying. that's another reason not to ask it to solve your homework for you (along with the fact that it's cheating and you only cheat yourself, etc.)

i think it's good at explaining concepts and pointing out flaws in my own work

quasi maple
#

But don't you see that this is not a complicated question here?

#

At best, a 16-yo should be able to answer this question correctly

little cave
#

Right, and this is the type of thing on your high school math homework

outer burrow
#

maybe doesn't matter too much, but GPT gives the correct answer for me

image.png
#

what i would do is take x=1 and try it. very easy to see if it's wrong. if it doesn't immediately seem wrong, then i go and try to understand the steps

quasi maple
#

And it did for me too today - but think from the perspective of a school-grade student who is having to resort to this to check

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More often then not, they're not going to have the wherewithal to know how to spot faults in the first place

outer burrow
#

great opportunity to learn this crucial skill then

quasi maple
#

I note, a not-insignificant number of my tutees stumble on questions that ask them to go through a student's working out and explain whether it's wrong

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And to clarify - I teach people doing their A Levels, i.e. the examinations just before university

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It is then not unreasonable to assume that a larger proportion of younger (to use US terminology, e.g. middle school) students will not have the understanding to know how to check a GPT'd response

little cave
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Which is what I said when this discussion arose

outer burrow
# quasi maple It is then not unreasonable to assume that a larger proportion of younger (to us...

again, i just see it as an opportunity to learn this skill then. we can even ask gpt how we can verify its work.

the fact that your students don't know how to do it means they haven't had enough experience with verifying. so maybe they could've benefitted from this.

and whether they know that they can't immediately trust it, they should. everyone says gpt can be inaccurate, so i think they'd know

quasi maple
#

Again, consider if you don't have a teacher; who would be asking you to verify the AI's response?

outer burrow
outer burrow
quasi maple
#

Being aware of AI BS is the problem I was trying to get at

outer burrow
#

before gpt, who was there to make me verify my own work when the teacher's not available? still me

quasi maple
#

Okay, but there are kids growing up in this case without a "before GPT" time to reminisce about

outer burrow
quasi maple
#

Even if people tell person A that something can be wrong, that doesn't inherently mean person A will check, though, for the record

#

For evidence, see most general elections, internationally /hj

little cave
#

Right but we are in the predicament of working with students whose self verification function ultra scuffed

#

And we're trying to fix that

quasi maple
#

Humans may be born "rational", but that doesn't equate to them being very "logical" - this is a skill that needs to be learnt

little cave
#

And although it may seem counter intuitive, we don't believe that having ai is actually helping said students, even though in theory it should, and it's not our fault that it's not

quasi maple
#

But an LLM is trained to mimic human speech (written or otherwise), not to be logical

outer burrow
quasi maple
#

Example of the logical argument that I was hoping you'd fall into lmao

little cave
#

I'd be curious to see if maybe a course in debate or on court cases would help these people with the bad self verification function

quasi maple
#

"may be... but that doesn't mean" exists as a conditional in English that means "Whether or not [condition], it is not the case that [outcome]"

quasi maple
outer burrow
#

that's what i perceived it as

quasi maple
#

No

#

It often does; but the issue is that the language is ambiguous

#

And as an LLM is mimicking language, not mathematics, it too is ambiguous

#

that was my point

outer burrow
#

i'm gonna hop off. good discussion.

midnight scarab
chrome tendon
tawny slate
# turbid zenith And if she says "5"... "Nope — c is a letter. There's nothing in the triangle ma...

i think we need to be a little bit careful here, because while technically yes this makes the point and helps remind ourselves of the abstract concept here, in the context where this question is asked, the answer is in fact almost always 5 (because no one ever asks this question solely for the purpose to trick someone with a nonsense question), and if the student latches too hard to this idea, then now you have the problem in the other direction, where they will be unable to grasp the contextual language that is present everywhere

turbid zenith
#

Agreed. Worth it to point out the purpose yet arbitrariness of conventions.

turbid zenith
#

Does anyone have access to this article?
https://pubs.nctm.org/view/journals/tcm/3/3/article-p128.xml?rskey=FWuSN5&result=3&print

NCTM Publications
Multicultural Mathematics and Alternative Algorithms

Up until recently I wasn't even aware that other people in the world did things [arithmetic algorithms] differently. I thought God sent these. That's the way of the world. The first day you [to another teacher] were talking about some way you did things differently in Ireland. It never occurred to me. I thought there was a world standard.

torn apex
turbid zenith
#

Is this the format it was given in?

#

There seem to be diagrams missing

lethal leaf
#

Couldn't find a way to get the original scan

turbid zenith
#

Ahh well thank you!

#

I can probably figure it out from here anyway

pure light
dapper jewel
turbid zenith
#

RIP my institution doesn’t give me access

little cave
#

didn't wrath of math make a video on this

#

https://www.youtube.com/watch?v=GQE3Vq2twPM
9:43 he goes over a different occurrence of a similar phenomenon from a different source

YouTube
Wrath of Math
so this is 2nd grade subtraction

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Join Wrath of Math to get exclusive videos, lecture notes, and more:
https://www.youtube.com/channel/UCyEKvaxi8mt9FMc62MHcliw/join

More Math Chats: https://www.youtube.com/playlist?list=PLztBpqftvzxXQDmPmSOwXSU9vOHgty1RO
Business Inquiries: wrathofmathlessons@gmail.com...

▶ Play video
#

and the comments are turned off...wonder what the store is there

quiet jackal
turbid zenith
#

Oh I remember this story

pure light
#

even without institutional access it says you get a monthly free article quota if you sign up for an account

turbid zenith
#

Oh nice, I'll try that

wild solstice
turbid zenith
#

I actually meant what cloud said 😛

wild solstice
pure light
#

going to have to remove the piracy website (discord tos)

torn apex
#

@turbid zenith @scarlet panther @dapper flume 's an example of what I call "apealing" + "beautiful" in a way that really adds to understanding the material:
https://ikrima.github.io/topos.noether/con-math/constructive-ch2-integration.html

I think this is where AI really shines. Now I could code something like that up but there's no way I could make the design aspect of it look right.

context: these are my own notes for myself on Constructive Analysis so the point isn't that I made something beautiful...instead, it's that you can copy-paste the prompt template that took me 2 hours to generate and apply that to your own notes

dapper flume
turbid zenith
#

How DID you choose the colors?

torn apex
# turbid zenith How DID you choose the colors?

no no no, credit is all due to the AI from my prompt. I just tell it in a lot of detail on what to do as if I'm a designer myself. (ofcourse, I can't do any of that but I can tell it to "Focus on the human-computer interaction UX affordances. Be sure to take into account a color neutral, yadda yadda yadda"

You can probably copy paste your own ( @dapper flume 's) detailed breakdown, reworded as either questions or instructions, and have Google Gemini 3.0 "fix" your own slides.

So I myself didn't choose the colors, the AI did. But I told it to choose colors to fit X,Y,Z conditions and to satisfy those conditions as if they were a person with A,B,C credentials

#

maybe an example would help? If you have some notes or slides that you want spruced up, I can run it through the same prompt template. I guess I can also share the prompt template but the only unique thing about it is that it's just really long (~I spend about ~5 - 10 minutes on each)

#

ah, I also forgot to add the source material is Constructive Analysis by Bishop. As I'm reading the book, I'm basically making these notes for myself. Normally I'd just tell myself that I'd write down notes and never end up dong anything 😅

dapper flume
#

I feel that i am satisfied with how my visual materials look when I make them. I intentionally pour a diverse concoction of my experience and personality into my work, so I find AI has little to no place in my workflow.

torn apex
#

makes sense. this really only helps those of us who don't have developed design or art sensibilities

dapper flume
#

Depending on the purpose of the widgets you displayed, I would have very different types of feedback. They seem to be ideal for supplementing a rigorous course, best for contrasting verbose knowledge with visual intuition.

torn apex
#

happy to hear as much as you wanna provide

dapper flume
#

If I used AI I feel like I'd lowkey take a lot longer than just doing it myself. If I got that website as a result of my prompt, I would be initially satisfied and impressed, but there's a lot of details I'm noticing I'd want to change.

For example, on the Riemann Sum graph, I see the labels "0" and "2" for the x-axis, and no scale for the y-axis. Sometimes it is better to remove details for clarity, but in this case, I would need to see the x-axis to understand why the rectangles stop at the x-axis. Otherwise, it seems like an arbitrary decision to stop the rectangles there.

I also pay a lot of attention to how my visual design cues the viewer to draw connections between representations. If I am lucky enough to be making something interactive or animated, I want different representations of a concept to respond to each other. When I watch 3b1b "essence of calculus," I am drawn to the balance Grant makes between visualizing the big picture and visualizing the details. Being able to see all the rectangles converge to the area under the curve, and also seeing where on that diagram Δx is, is super helpful for new learners to draw that connection. On the website you sent, I'd be interested in seeing some sort of interplay between the Riemann Sum visual and the error plot. Maybe a vertical line that shows where on the plot we happen to be for a given Riemann Sum.

I could go on in an increasingly nitpicky fashion, and a lot of my feedback will boil down to my preferences, but this is what I mean by intentionality. I believe visual design is pedagogical, but if and only if the designer considers it that way and plans accordingly.

torn apex
dapper flume
#

That's right. A person should channel their energy through their own practice and talent. An instructor who is fantastic at live demos should not over-prioritize their powerpoints when designing lessons. They should teach according to what they know well. If you're a speaker, speak. If you're a facilitator, facilitate. If you're a thespian, act. And if you're a designer, design.

Since I believe it is best for professional educators to be exposed to the basics of all of these modes of instruction, I hope to increase visual literacy for US public math educators at some point in the medium future in professional development sessions.

Having the tools to understand how to be intentional is great even for those using an AI-assisted workflow.

#

.
.
I fear I talk like I'm some professional designer with the most mind-blowing powerpoints. The truth is, they're pretty silly. They're full of my personality, which is humorous, a bit unserious, and very very student- centered. I believe my basics, however, are very strong. That goes a long way to supporting kids with autism, ADHD, dyslexia, and visual impairments as they try to use my resources to help them learn.

torn apex
#

Could not agree more

#

That goes a long way to supporting kids with autism, ADHD, dyslexia, and visual impairments as they try to use my resources to help them learn.

This is purely my own anecdata but honestly, the things that help the "special needs" children is almost exactly the same thing that helps struggling students become mediocre students and mediocre students become exceptional kids

#

It reminds me of that episode of some UK show way back in the day where they swapped public school kids with private ritzy kids for some amount of time

dapper flume
torn apex
#

there was one kid who was a bit of a delinquent who turned 180 degrees because you could see for the first time in his life, he saw that it was possible to succeed.

it was so heart breaking at the end when they had to go back. I think he even asked on camera if he could stay behind.

#

he ended up getting a bunch of donations to transfer and the town made a scholarship just for him. A ~decade later, he was the graduating valedictorian.

It turned into a rare super wholesome moment

#

The darker metapoint being: is this anec-data proving that pedagogy is the defining difference between excellent for the delinquent vs. valedectorian version of that kid? 🤷

dapper flume
# torn apex The darker metapoint being: is this anec-data proving that pedagogy is the defin...

It is relevant, yeah. Access is probably more important than pedagogy in this regard. I have at least one student i know i could not teach math to, even with infinite educational resources, simply because she does not get enough food and sleep at home to function at a normal level. Right now, she needs food and sleep. Then, she can learn all the math she wants.

What I do with intentional visual design is to break down as many of these barriers as I can at the pedagogical level. I cannot fix all problems of access, but I can support struggling readers and those who struggle with abstract thinking.

dapper flume
#

Here's a sample of the kinds of critiques I make and fix related to worksheets/assessments for Algebra I (this is a mockup, not a real curriculum problem)

Screenshot_20251228_125706_Samsung_Notes.jpg Screenshot_20251228_125736_Samsung_Notes.jpg
turbid zenith
dapper flume
tawny slate
# torn apex I agree with all of that. But it also took you X amount of hours/years to develo...

we all start somewhere. like many others in this channel, i am far from an expert if you solely look at my qualifications. in fact, i have almost no qualifications in almost anything i actually work on professionally or hobby-wise

if pedagogy is soemthing you care about, then rather than mope about how you lack the skills that are so central and core to pedagogy and use AI to supplant them, you should spend that time actually developing and honing those skills. and i genuinely think this kind of fixed mindset is one of the biggest dangers of generative AI. studies show that people who use generative AI use their brains much less compared to those who don't use any reference material or even a classic search engine. you end up hamstringing yourself in the long run

dont make excuses for yourself, just do it

tawny slate
# torn apex The darker metapoint being: is this anec-data proving that pedagogy is the defin...

probably actually, depending on context. more modern neuroscience seems to indicate that while there are some aspects of intelligence that are likely immutable, the development of skills and overall brain health is extremely sensitive to certain environmental factors, such as healthy socialization in early age

but research in this area is still fairly new and there are lots of confounding factors, so take that with a grain of salt

if nothing else, assuming strong environmental influence is probably "safer" because as i just mentioned, growth mindset is demonstrably very healthy to have, human brains are still very plastic, and an encouraging and positive education for all is a moral thing to strive for

little cave
#

has the discussion been had about whether introducing rigor-like ideas is a reasonable strategy in helping a student understand a concept

#

Like, if I'm a TA and a calc 1 student is struggling with say just the mechanics of an integral, you can get the point across without referencing adding up little rectangles, but are you just kicking the can down the road?

compact isle
#

I'd be careful with formalism with students that aren't pure math majors, in my experience its often more confusing than helpful. That said, it is useful to give people an idea of what is going on

stark heron
#

ive had students that really appreciate the rigor and those that cry if they see a symbol. probably what blake said, depends on the major and background.

tawny slate
#

yeah for most engineers, "close enough is good enough"

#

they do need a rigorous understanding of error analysis, because that is actually important, but that's not rigor in the same way

compact isle
#

It somewhat depends on the subject its a bit hard to say

#

I think with more applied math its better to focus on giving good explanations and heuristics rather than anything rigorous

little cave
#

I understand that the people pleasing option is to do your best to balance the broom on your nose, and the students will be happier with you in the short term, and honestly likely in the long term. But I'm also asking if this approach creates a compounding problem which results in net loss which is hard to measure.

compact isle
#

sounds quite hard to measure so I don't really know if there's a concrete example

#

but I will mention that its very easy to overestimate the ability for people to follow and digest mathematical rigor if they haven't been immersed in it for quite a while

little cave
#

I understand this completely, it took me years and years and I'm still just alright. But there is also the point that you would have to start somewhere if that is what you need.

midnight scarab
compact isle
#

I would say it's good for a calc student to know that an integral is a limit of Riemann sums, but teaching them how to make it precise is imo not helpful

little cave
midnight scarab
#

And while they may not need to know about them to mechanically compute an antiderivative, they certainly tell them why they should care, and what it means

compact isle
midnight scarab
little cave
# compact isle Sure but if that's what they needed they would be in a pure math program

here would be my main argument against that. Most of what non-math majors "do" with math is more or less rote, algorithmic, etc, especially before high school. As they progress, we start to sprinkle in more conceptual checks, usually as a word problem, where the student is asked to build the problem to apply the algorithm to. I don't know where you can train this ability, if you don't have it, outside of a more rigorous environment, because it requires more familiarity with the way that the tools operate

#

so I guess what I am saying to be more concrete, is that I can get most people up to speed on how to apply the "integrate this thing" algorithm, but I have much less success in helping people reliably do problems where say the throughline is 1. find a sum with finite terms (edited, said finite sum), 2. transform to integral, 3. apply the "integrate this thing" algorithm

tawny slate
#

i find that the pedagogical role of rigor includes things like:

  • helping the student understand where there might be flaws in their logic
  • helping convince the student that the claims and concepts and ideas are sound and reasonable
  • elucidating intuition
    etc
#

if rigor isnt helping to explicitly resolve at least some of these goals, then while im not saying it shouldn't be taught, i just dont think it's the highest priority

#

like explaining to a student that equations are logical statements that can be true or false is elucidating because that means if they solve 2(x+2) = 2x+2 and get to 4=2 they dont think "oh crap, what did i do wrong?" and instead "hmm I wonder if the equation is just false"

#

i think limits are at least helpful when you get to things like integrals of 1/x, where you have weird behavior at asymptotes and poles and other infinite behavior, but otherwise i dont think it adds too much, at least if the goal is strictly calc mastery

turbid zenith
little cave
#

A larger dictionary provides more avenue for explanation

#

I'm really not even saying to prove anything, maybe just get across the idea more concretely

#

I see people not know that any point on a circle satisfies the equation of said circle, and the easiest thing I can think would help them is some work on how we describe a set

turbid zenith
rapid tusk
#

just because you could describe it that way doesn't necessarily mean you should

little cave
#

I think you can argue that the idea of function is not getting fully translated

#

Things like like function vs function evaluated at given input (function evaluated at a point is confusing)

turbid zenith
rapid tusk
turbid zenith
#

Gotcha

turbid zenith
#

I remember being in high school and I didn't understand the point of saying (f + g)(x) when I could just do f(x) + g(x)

#

I didn't realize that the entire point was we were adding more abstract things together

#

Instead of adding numbers we're adding functions, and defining what that would even mean

rapid tusk
# rapid tusk ^

at least at lower levels it might confuse more than it helps from what ive seen

turbid zenith
#

I'll agree that set formalism often confuses lower level students

#

But you can sneak in set ideas without having to hit them with set-builder notation

#

Like when you try to make the graph of y = 2x + 1, you're trying to find all the points that make that true. You can test out a whole bunch of points and you end up with a line, and you get the idea that all the points on that line satisfy that equation.

little cave
#

I fail to see how set builder notation is the hard part

turbid zenith
#

Students see symbols and shut off.

#

I've seen it time and time again.

little cave
#

Stop doing that

rapid tusk
#

how do you motivate that formalism in a setting where students aren't receptive

little cave
#

It's not motivated, it's just a way to say it

#

It's notation lol

turbid zenith
#

I think you underestimate just how terrified many students are of notation.

little cave
#

I mean

turbid zenith
#

Again, magical incantations.

little cave
#

I have a tasteful distaste for fancy notation

#

But set builder is just english

turbid zenith
#

At least for many of the students I've worked with.

little cave
#

But yeah I get what you mean

#

Especially if it isn't the answer to a homework problem

halcyon glade
stark heron
# little cave Stop doing that

a sin of education is forgetting what it is like to not understand, which includes proofs and rigorous definitions in general lol

#

cant think of a better way to express that but whatever

quiet jackal
#

who dont deeply understand the material and consequently cant articulate intuition

tight star
#

3b1b has talked about this too

little cave
#

What if I didn't use the word rigor and just said solid less half baked notation

#

Would that change minds

#

Solid notation is a large part of rigor

turbid zenith
little cave
# turbid zenith Such as what?

F: X -> Y for any function and S = {(x,y) such that e1(x,y) = e2(x,y)} for any set where e1 and e2 are expressions using x and y are what I would start with if I was in charge. So basically what I've already said.

stark heron
#

good luck with that

turbid zenith
#

That will lose 99% of my students.

little cave
#

But do you think it's possible that if students DID have it, that it could help some students?

#

I think you'd lose tho if you were to say try to use epsilon delta with students

#

I understand there's a limit, no pun intended

turbid zenith
#

I don't think I've met a student where starting with the formalism would help.

quiet jackal
#

the same thing is true about math papers and math talks

quiet jackal
#

i think i had to learn it in precalculus

little cave
#

A half liner way of me saying "yes, a circle is all the points satisfying this condition"

rapid tusk
#

being comprehensible should take precedence over being totally formally correct at the lower levels

#

symbol vomit is more likely to just scare people off

little cave
# rapid tusk how is this supposed to help 😭😭

Maybe I'm not being clear. You wouldn't literally write e1 = e2, you would write the relevant description. For example, x^2 + y^2 = 1 for the circle centered at the origin with radius one. Then you know that if (x0,y0) is on the circle, then we must have x0^2 + y0^2 = 1. This is less symbol vomit than english.

#

I'm not saying tell them to write e1 and e2, just write the condition for the graph

rapid tusk
#

oh ic

compact isle
#

I mean set notation is a pretty standard part of most calculus classes is it not?

#

at least for multivariable calculus the classes I've TA'd for have had plenty of "express this shape as a set" questions

#

(students often screw them up)

turbid zenith
#

I think if you wrote it as "The set of all points where x² + y² = 1" you could get the idea across just fine

turbid zenith
#

And yeah I had to learn set notation in precalculus but a lot of students either don't seem to anymore or have forgotten it

#

Not to mention students who haven't taken precalculus.

stark heron
#

my high school never did any of that unfortunately it was a stupid online class for precalc

#

i learned set notation in university ;_;

turbid zenith
#

I do remember that in university I thought in order to count as math there had to be lots and lots of symbols 😛

lethal leaf
#

especially when they could not read their own work

tall bolt
lethal leaf
#

Yea I tell them the same thing

#

Use words

tawny slate
tawny slate
#

and i mostly teach middle/high school students

#

so my point is that yes there is a place for rigor, but that rigor needs to be motivated

#

in this case its "yo youve been using this thing for years, do you even know what this thing even is or why we care?"

#

i still contend that rigor is fine as long as it is to help the student and not to massage your own ego that you have adequately proved the problem to yourself

slim path
# tawny slate and i mostly teach middle/high school students

I presently teach high school students. I beg to differ here. Not everyone, everywhere takes math because they want to and like it. If we were dealing with these students alone, then yes, rigor is helpful. When you have students who couldn't care less but pass the exams and be done with it, rigor has the opposite effect.

tawny slate
#

yeah no disagreement there

#

maybe i missed that context in this convo sorry

tight star
#

I’ve seen it in cat theory with the idea that “more abstract = more good”

solar sorrel
#

[category theory becomes slightly less inscrutable when you have a networked medium like the nLab wiki to help ground category theoretic constructions in ‘concrete’ examples]

solar sorrel
# lethal leaf One of my biggest struggles as a discrete math TA was getting students to stop t...

You might be interested in reading this: https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/

What's new
Terence Tao
There’s more to mathematics than rigour and proofs

The history of every major galactic civilization tends to pass through three distinct and recognizable phases, those of Survival, Inquiry and Sophistication, otherwise known as the How, Why, and Wh…

tight star
#

It’s not really about being as abstract as possible, but finding the right level of abstraction

solar sorrel